This project was funded and sponsored by Wharton Customer Analytics.

Description

This package implements the BG/BB, BG/NBD and Pareto/NBD models, which capture/project customer purchase patterns in a typical non-contractual setting.

Details

While these models are developed on a customer-by-customer basis, they do not necessarily require data at such a granular level. The Pareto/NBD requires a "customer-by-sufficient-statistic" matrix (CBS), which consists of each customer's frequency, recency (the time of their last transactions) and total time observed - but the timing of each and every transaction (other than the last) is not needed by the model. If, however, you do have the granular data in the form of an event log (which contains at least columns for customer identification and the time of each transaction, and potentially more columns such as transaction amount), this package provides functions to convert it to a CBS. You can use dc.ReadLines to get your event log from a comma-delimited file to an event log usable by this package; it is possible to use read.table or read.csv, but formatting will be required afterwards. You can then convert the event log directly to a CBS (for both the calibration and holdout periods) using dc.ElogToCbsCbt. As the name suggests, this function also produces a customer-by-time matrix (CBT). This matrix consists of a row for every customer and a column for every date, and is populated by a statistic of your choice (reach, frequency, or spend). It is not necessary for any of the models presented in this package, but is used as a building block to produce the CBS.

The BG/NBD model requires all the same inputs as the Pareto/NBD model.

The BG/BB model requires the same information as the Pareto/NBD model, but as it models discrete transaction opportunities, this information can be condensed into a recency-frequency matrix. A recency-frequency matrix contains a row for every recency/frequency combination in the given time period, and each row contains the number of customers with that recency/frequency combination. Since frequency will always be less than or equal to recency, this matrix will contain (n)(n-1)/2 + 1 rows at most, with n as the number of transaction opportunities (of course, the maximum number of rows for pooled data - for customers with varying numbers of transaction opportunities - will be the sum of the above equation for each unique number of transaction opportunities). You can convert a CBS to recency-frequency matrices using dc.MakeRFmatrixCal and dc.MakeRFmatrixHoldout.

If you want to test the data contained in the package, or have data formatted as a customer-by-sufficient-statistic or recency-frequency matrix, a good starting place would be pnbd.EstimateParameters, bgnbd.EstimateParameters, or bgbb.EstimateParameters.

Following that, pnbd.PlotFrequencyInCalibration, bgnbd.PlotFrequencyInCalibration and bgbb.PlotFrequencyInCalibration will give a check that the model fits the data in-sample. Further plotting functions, comparing actual and expected results, are labelled "pnbd.Plot...", "bgnbd.Plot..." and "bgbb.Plot...". The building blocks of these functions are also provided: pnbd.LL, bgnbd.LL bgbb.LL, pnbd.pmf, bgnbd.pmf, bgbb.pmf, pnbd.Expectation, bgnbd.Expectation, bgbb.Expectation, pnbd.ConditionalExpectedTransactions, bgnbd.ConditionalExpectedTransactions, and bgbb.ConditionalExpectedTransactions may be of particular interest.

This package uses the following conventions:

The time period used to estimate the model parameters is called the calibration period. Users may be accustomed to this being called the estimation period, or simply being referred to as "in-sample". Function parameter names generally follow this convention: for example, "n.cal" is used to refer to the number of transaction opportunities in the calibration period.

The time period used to validate model performance is called the holdout period. Users may be accustomed to this being called the validation period, or simply being referred to as "out-of-sample". Function parameters relating to this time period are generally appended with ".star". For example, n.star is used to refer to the number of transaction opportunities in the holdout period.

As described in the papers referenced below, the BG/BB, BG/NBD and Pareto/NBD models are generally concerned with repeat transactions, not total transactions. This means that a customer's first transaction in the calibration period is usually not part of the data being modeled - this is due to the fact that a new customer generally does not show up "on the company's radar" until after their first purchase has taken place. This means that the modal number of repeat purchases tends to be zero. If your data does not have a relatively large number of customers with zero transactions, but does have a relatively large number of customers with one transaction, and the estimation functions are struggling, the problem is most likely that you are including customers' very first transactions. Some of the data-conversion functions have examples illustrating how to work with data that includes this very first transaction. Note that this does not apply to the holdout period; in the holdout period, we already know about the customer and take all of their previous transactions into account.

Author(s)

Maintainer: Gabi Huiber ghuiber@gmail.com [contributor]

Authors:

• Lukasz Dziurzynski

• Edward Wadsworth

• Daniel McCarthy [contributor]

Other contributors:

• Peter Fader [contributor]

• Elea McDonnell Feit [contributor]

• Bruce Hardie [contributor]

• Arun Gopalakrishnan [contributor]

• Eric Schwartz [contributor]

• Yao Zhang [contributor]

References

See https://www.brucehardie.com for papers, notes, and datasets relating to applied probability models in marketing.

Fader, Peter S., and Bruce G.S. Hardie. “A Note on Deriving the Pareto/NBD Model and Related Expressions.” November. 2005. Web. http://www.brucehardie.com/notes/008/

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.” Journal of Marketing Research Vol.42, pp.415-430. November. 2005. http://www.brucehardie.com/papers.html

Fader, Peter S., and Bruce G.S. Hardie. “Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model.” September. 2006. Web. http://www.brucehardie.com/notes/012/

Fader, Peter S., and Bruce G.S. Hardie. “Creating an RFM summary using Excel.” December. 2008. Web. http://www.brucehardie.com/notes/022/

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. “Customer-Base Analysis in a Discrete-Time Noncontractual Setting.” Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. http://www.brucehardie.com/papers/020/

Jerath, Kinshuk, Peter S. Fader, and Bruce G.S. Hardie. “Customer-Base Analysis on a 'Data Diet': Model Inference Using Repeated Cross-Sectional Summary (RCSS) Data.” June. 2011. Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1708562 or doi: 10.2139/ssrn.1708562

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. ““Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model.” Marketing Science Vol.24, pp.275-284. Spring. 2005. http://www.brucehardie.com/papers.html

Fader, Peter S., Hardie, Bruce G.S., and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web. http://www.brucehardie.com/notes/021/palive_for_BGNBD.pdf

See Also

Useful links:

• Report bugs at https://github.com/ghuiber/BTYD/issues

Add Logs

Description

Given log(a) and log(b), returns log(a + b)

Usage

addLogs(loga, logb)

Arguments

BG/BB Conditional Expected Transactions

Description

Calculates the number of expected transactions in the holdout period, conditional on a customer's behavior in the calibration period.

Usage

bgbb.ConditionalExpectedTransactions(params, n.cal, n.star, x, t.x)

Arguments

Details

E(X(n, n+n*) | alpha, beta, gamma, delta, x, t.x, n). This function requires the holdout period to immediately follow the calibration period.

n.cal, n.star, x, and t.x may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

The number of transactions a customer is expected to make in the n.star transaction opportunities following the calibration period, conditional on their behavior during the calibration period.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

 
params <- c(1.20, 0.75, 0.66, 2.78)
# the number of transactions a customer is expected
# to make in the 10 transaction opportunities
# following the calibration period, which consisted
# of 6 transaction opportunities (during which they
# made 3 transactions, the last of which occurred
# in the 4th opportunity)
bgbb.ConditionalExpectedTransactions(params, n.cal=6, n.star=10, x=3, t.x=4)

# We can also use vectors as input:
bgbb.ConditionalExpectedTransactions(params, n.cal=6, n.star=1:10, x=3, t.x=4)
bgbb.ConditionalExpectedTransactions(params, n.cal=6, n.star=10, x=1:4, t.x=4)

BG/BB Discounted Expected Residual Transactions

Description

Computes the number of discounted expected residual transactions by a customer, conditional on their behavior in the calibration period.

Usage

bgbb.DERT(params, x, t.x, n.cal, d)

Arguments

Details

DERT(d | alpha, beta, gamma, delta, x, t.x, n). This is the present value of the expected future transaction stream for a customer with x transactions and a recency of t.x in n.cal transaction opportunities, discounted by a rate d.

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The present value of the expected future transaction stream for a particular customer.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web. See equation 14.

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# Compute DERT for a customer who made 3 transactions
# in the calibration period(consisting of 6 transaction
# opportunities), with the last transaction occurring
# during the 4th transaction opportunity, discounted at
# 10%.
bgbb.DERT(params, x=3, t.x=4, n.cal=6, d=0.1)

# We can also compare DERT for several customers:
bgbb.DERT(params, x=1:6, t.x=6, n.cal=6, d=0.1)

BG/BB Parameter estimation

Description

Estimates parameters for the BG/BB model.

Usage

bgbb.EstimateParameters(
  rf.matrix,
  par.start = c(1, 1, 1, 1),
  max.param.value = 1000
)

Arguments

Details

The best-fitting parameters are determined using the bgbb.rf.matrix.LL function. The sum of the log-likelihood for each customer (for a set of parameters) is maximized in order to estimate paramaters.

A set of starting parameters must be provided for this method. If no parameters are provided, (1,1,1,1) is used as a default. It may be useful to use starting values for parameters that represent your best guess of the heterogeneity in the transaction and dropout rates of customers. It may be necessary to run the estimation from multiple starting points to ensure that it converges. To compare the log-likelihoods of different parameters, use bgbb.rf.matrix.LL.

The lower bound on the parameters to be estimated is always zero, since BG/BB parameters cannot be negative. The upper bound can be set with the max.param.value parameter.

Value

Vector of estimated paramaters.

See Also

bgbb.rf.matrix.LL

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)
# log-likelihood of estimated parameters
bgbb.rf.matrix.LL(est.params, rf.matrix)

BG/BB Expectation

Description

Returns the number of transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in the first n transaction opportunities.

Usage

bgbb.Expectation(params, n)

Arguments

Details

E(X(n) | alpha, beta, gamma, delta)

Value

Mean of the BG/BB probability mass function.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# Expected number of transactions that a randomly chosen customer
# will make in the first 10 transaction opportunities.
bgbb.Expectation(params, n=10)

# We can also compare expected transactions over time:
bgbb.Expectation(params, n=1:10)

BG/BB Heatmap of Holdout Period Expected Transactions

Description

Plots a heatmap based on the conditional expected holdout period frequency for each recency-frequency combination in the calibration period.

Usage

bgbb.HeatmapHoldoutExpectedTrans(
  params,
  n.cal,
  n.star,
  xlab = "Recency",
  ylab = "Frequency",
  xticklab = NULL,
  title = "Heatmap of Conditional Expected Transactions"
)

Arguments

Details

E(X(n, n+n*) | alpha, beta, gamma, delta, x, t.x, n). This function requires the holdout period to immediately follow the calibration period.

Value

A matrix containing the conditional expected transactions in the holdout period for each recency-frequency combination in the calibration period. The rows represent calibration period frequencies, and the columns represent calibration period recencies.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.ConditionalExpectedTransactions

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# Plot a heatmap of conditional expected transactions in
# a holdout period of 5 transaction opportunities, given
# that the calibration period consisted of 6 transaction
# opportunities.
bgbb.HeatmapHoldoutExpectedTrans(est.params, n.cal=6, n.star=5)

BG/BB Log-Likelihood

Description

Calculates the log-likelihood of the BG/BB model.

Usage

bgbb.LL(params, x, t.x, n.cal)

Arguments

Details

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

A vector of log-likelihoods as long as the longest input vector (x, t.x, or n.cal).

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

params <- c(1.20, 0.75, 0.66, 2.78)

# Returns the log likelihood of the parameters for a customer who
# made 3 transactions in a calibration period with 6 transaction opportunities,
# with the last transaction occurring during the 4th transaction opportunity.
bgbb.LL(params, x=3, t.x=4, n.cal=6)

# We can also give vectors as function parameters:
set.seed(7)
x <- sample(1:3, 10, replace = TRUE)
t.x <- sample(3:5, 10, replace = TRUE)
n.cal <- rep(5, 10)
bgbb.LL(params, x, t.x, n.cal)  

BG/BB P(Alive)

Description

Uses BG/BB model parameters and a customer's past transaction behavior to return the probability that they will be alive in the transaction opportunity following the calibration period.

Usage

bgbb.PAlive(params, x, t.x, n.cal)

Arguments

Details

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

P(alive at n+1 | alpha, beta, gamma, delta, x, t.x, n)

Value

Probability that the customer is alive at the (n+1)th transaction opportunity. If x, t.x, and/or n.cal are of length greater than one, then this will be a vector of probabilities (containing one element matching each element of the longest input vector).

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

params <- c(1.20, 0.75, 0.66, 2.78)

# The probability that a customer who made 3 transactions in
# the calibration period (which consisted of 6 transaction
# opportunities), with the last transaction occurring at the
# 4th transaction opportunity, is alive at the 7th transaction
# opportunity
bgbb.PAlive(params, x=3, t.x=4, n.cal=6)

# The input parameters may also be vectors:
bgbb.PAlive(params, x=1, t.x=1:6, n.cal=6)

BG/BB Plot Dropout Rate Heterogeneity

Description

Plots and returns the estimated beta distribution of Theta (customers' propensities to drop out).

Usage

bgbb.PlotDropoutRateHeterogeneity(params)

Arguments

Details

This returns the distribution of each customer's geometric parameter that determines their lifetime (using the BG/BB assumption that a customer's lifetime can be modeled with a geometric distribution).

Value

Distribution of customers' propensities to drop out.

Examples

params <- c(1.2, 0.75, 0.66, 2.78)
bgbb.PlotDropoutRateHeterogeneity(params)
params <- c(0.2, 1.5, 3.2, 6)
bgbb.PlotDropoutRateHeterogeneity(params)

BG/BB Plot Frequency vs Conditional Expected Frequency

Description

Plots the actual and conditional expected number of transactions made by customers in the holdout period, binned according to calibration period frequencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

bgbb.PlotFreqVsConditionalExpectedFrequency(
  params,
  n.star,
  rf.matrix,
  x.star,
  trunc = NULL,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

Arguments

Value

Holdout period transaction frequency comparison matrix (actual vs. expected), binned by calibration period frequency.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# get the holdout period transactions
x.star <- donationsSummary$x.star

# number of transaction opportunities in the holdout period
n.star <- 5

# Plot holdout period transactions
bgbb.PlotFreqVsConditionalExpectedFrequency(est.params, n.star, rf.matrix, x.star, trunc=6)

BG/BB Plot Frequency in Calibration Period

Description

Plots the actual and expected number of customers who made a certain number of repeat transactions in the calibration period. Also returns a matrix with this comparison.

Usage

bgbb.PlotFrequencyInCalibration(
  params,
  rf.matrix,
  censor = NULL,
  plotZero = TRUE,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

Arguments

Value

Calibration period repeat transaction frequency comparison matrix, actual vs. expected.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# plot actual vs. expected frequencies in the calibration period
bgbb.PlotFrequencyInCalibration(est.params, rf.matrix)

BG/BB Plot Frequency in Holdout

Description

Plots the actual and expected number of customers who made a certain number of transactions in the holdout period, binned according to holdout period frequencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

bgbb.PlotFrequencyInHoldout(
  params,
  n.cal,
  rf.matrix.holdout,
  censor = NULL,
  plotZero = TRUE,
  title = "Frequency of Repeat Transactions",
  xlab = "Holdout period transactions",
  ylab = "Customers"
)

Arguments

Value

Holdout period repeat transaction frequency comparison matrix (actual vs. expected).

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
rf.matrix.holdout <- donationsSummary$rf.matrix.holdout
# donationsSummary$rf.matrix and donationsSummary$rf.matrix.holdout already
# have appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# number of periods in the calibration period
n.cal = max(rf.matrix[,"n.cal"])

bgbb.PlotFrequencyInHoldout(est.params, n.cal, rf.matrix.holdout)

BG/BB Plot Recency vs Conditional Expected Frequency

Description

Plots the actual and conditional expected number of transactions made by customers in the holdout period, binned according to calibration period recencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

bgbb.PlotRecVsConditionalExpectedFrequency(
  params,
  n.star,
  rf.matrix,
  x.star,
  trunc = NULL,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expected Transactions by Recency"
)

Arguments

Value

Holdout period transaction frequency comparison matrix (actual vs. expected), binned by calibration period recency.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# get the holdout period transactions
x.star <- donationsSummary$x.star

# number of transaction opportunities in the holdout period
n.star <- 5

# Compare holdout period transactions.
bgbb.PlotRecVsConditionalExpectedFrequency(est.params, n.star, rf.matrix, x.star, trunc=6)

BG/BB Tracking Cumulative Transactions Plot

Description

Plots the actual and expected cumulative total repeat transactions by all customers for the calibration and holdout periods. Also returns a matrix with this comparison.

Usage

bgbb.PlotTrackingCum(
  params,
  rf.matrix,
  actual.cum.repeat.transactions,
  xlab = "Time",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions"
)

Arguments

Details

The holdout period should immediately follow the calibration period. This function assumes that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(n.cal) - n.cal rather than assuming that they are all 0).

Value

Matrix containing actual and expected cumulative repeat transactions.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)
# donationsSummary$rf.matrix already has appropriate column names
rf.matrix <- donationsSummary$rf.matrix

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# get the annual repeat transactions, and transform them into
# a cumulative form
actual.inc.repeat.transactions <- donationsSummary$annual.trans
actual.cum.repeat.transactions <- cumsum(actual.inc.repeat.transactions)

# set appropriate x-axis
x.tickmarks <- c( "'96","'97","'98","'99","'00","'01","'02","'03","'04","'05","'06" )

# plot actual vs. expected transactions. The calibration period was 6 periods long.
bgbb.PlotTrackingCum(est.params, rf.matrix, actual.cum.repeat.transactions, xticklab=x.tickmarks)

BG/BB Tracking Incremental Transactions Plot

Description

Plots the actual and expected incremental total repeat transactions by all customers for the calibration and holdout periods. Also returns a matrix of this comparison.

Usage

bgbb.PlotTrackingInc(
  params,
  rf.matrix,
  actual.inc.repeat.transactions,
  xlab = "Time",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Incremental Transactions"
)

Arguments

Details

The holdout period should immediately follow the calibration period. This function assumes that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(n.cal) - n.cal rather than assuming that they are all 0).

Value

Matrix containing actual and expected incremental repeat transactions.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

Examples

data(donationsSummary)
# donationsSummary$rf.matrix already has appropriate column names
rf.matrix <- donationsSummary$rf.matrix

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# get the annual repeat transactions
actual.inc.repeat.transactions <- donationsSummary$annual.trans

# Set appropriate x-axis
x.tickmarks <- c( "'96","'97","'98","'99","'00","'01","'02","'03","'04","'05","'06" )

# Plot actual vs. expected transactions. The calibration period was 6 periods long.
bgbb.PlotTrackingInc(est.params, rf.matrix, actual.inc.repeat.transactions, xticklab=x.tickmarks)

BG/BB Plot Transaction Rate Heterogeneity

Description

Plots and returns the estimated beta distribution of P (customers' propensities to purchase).

Usage

bgbb.PlotTransactionRateHeterogeneity(params)

Arguments

Details

This returns the distribution of each customer's Bernoulli parameter, which determines the level of their purchasing (using the BG/BB assumption that purchasing on the individual level can be modeled with a Bernoulli distribution).

Value

Distribution of customers' propensities to purchase.

Examples

params <- c(1.2, 0.75, 0.66, 2.78)
bgbb.PlotTransactionRateHeterogeneity(params)
params <- c(0.2, 1.5, 3.2, 6)
bgbb.PlotTransactionRateHeterogeneity(params)

BG/BB Posterior Mean Dropout Rate

Description

Computes the mean value of the marginal posterior value of Theta, the geometric dropout process parameter.

Usage

bgbb.PosteriorMeanDropoutRate(params, x, t.x, n.cal)

Arguments

Details

E(Theta | alpha, beta, gamma, delta, x, t.x, n). This is calculated by setting l = 0 and m = 1 in bgbb.PosteriorMeanLmProductMoment.

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The posterior mean dropout rate.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.rf.matrix.PosteriorMeanDropoutRate

Examples

 
data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# return the posterior mean dropout rate vector
bgbb.rf.matrix.PosteriorMeanDropoutRate(est.params, rf.matrix)

BG/BB Posterior Mean (l,m)th Product Moment

Description

Computes the ⁠(l,m)⁠th product moment of the joint posterior distribution of P (the Bernoulli transaction process parameter) and Theta (the geometric dropout process parameter).

Usage

bgbb.PosteriorMeanLmProductMoment(params, l, m, x, t.x, n.cal)

Arguments

Details

E((P)^l(Theta)^m | alpha, beta, gamma, delta, x, t.x, n)

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The expected posterior ⁠(l,m)⁠th product moment.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See equation 17.

BG/BB Posterior Mean Transaction Rate

Description

Computes the mean value of the marginal posterior value of P, the Bernoulli transaction process parameter.

Usage

bgbb.PosteriorMeanTransactionRate(params, x, t.x, n.cal)

Arguments

Details

E(P | alpha, beta, gamma, delta, x, t.x, n). This is calculated by setting l = 1 and m = 0 in bgbb.PosteriorMeanLmProductMoment.

x, t.x, and n.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The posterior mean transaction rate.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.rf.matrix.PosteriorMeanTransactionRate

Examples

 
data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# return the posterior mean transaction rate vector
bgbb.rf.matrix.PosteriorMeanTransactionRate(est.params, rf.matrix)

BG/BB Probability Mass Function

Description

Probability mass function for the BG/BB.

Usage

bgbb.pmf(params, n, x)

Arguments

Details

P(X(n)=x | alpha, beta, gamma, delta). Returns the probability that a customer makes x transactions in the first n transaction opportunities.

Parameters n and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of X(n)=x, conditional on model parameters.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.pmf.General

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# The probability that a customer made 3 transactions in the first
# 6 transaction opportunities.
bgbb.pmf(params, n=6, x=3)

# Vectors may also be used as arguments:
bgbb.pmf(params, n=6, x=0:6)

BG/BB General Probability Mass Function

Description

Calculates the probability that a customer will make x.star transactions in the first n.star transaction opportunities following the calibration period.

Usage

bgbb.pmf.General(params, n.cal, n.star, x.star)

Arguments

Details

P(X(n, n + n*) = x* | alpha, beta, gamma, delta). This is a more generalized version of the bgbb.pmf. Setting n.cal to 0 reduces this function to the probability mass function in its usual format - the probability that a user will make x.star transactions in the first n.star transaction opportunities.

It is impossible for a customer to make a negative number of transactions, or to make more transactions than there are transaction opportunities. This function will throw an error if such inputs are provided.

n.cal, n.star, and x.star may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of X(n, n + n*) = x*, given BG/BB model parameters.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.pmf

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# Probability that a customer will make 3 transactions in the 10
# transaction opportunities following the 6 transaction opportunities
# in the calibration period, given BG/BB parameters.
bgbb.pmf.General(params, n.cal=6, n.star=10, x.star=3)

# Vectors may also be provided as input:
# Comparison between different frequencies:
bgbb.pmf.General(params, n.cal=6, n.star=10, x.star=1:10)
# Comparison between different holdout transaction opportunities:
bgbb.pmf.General(params, n.cal=6, n.star=5:15, x.star=3)

BG/BB Discounted Expected Residual Transactions using a recency-frequency matrix

Description

Computes the number of discounted expected residual transactions by a customer, conditional on their behavior in the calibration period.

Usage

bgbb.rf.matrix.DERT(params, rf.matrix, d)

Arguments

Value

The present value of the expected future transaction stream for a particular customer.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web. See equation 14.

See Also

bgbb.DERT

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# compute DERT for a customer from every row in rf.matrix,
# discounted at 10%.
bgbb.rf.matrix.DERT(est.params, rf.matrix, d = 0.1)

BG/BB Log-Likelihood using a recency-frequency matrix

Description

Calculates the log-likelihood of the BG/BB model.

Usage

bgbb.rf.matrix.LL(params, rf.matrix)

Arguments

Value

The total log-likelihood of the provided data in rf.matrix.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.LL

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

params <- c(1.20, 0.75, 0.66, 2.78)
bgbb.rf.matrix.LL(params, rf.matrix)

BG/BB Posterior Mean Dropout Rate using a recency-frequency matrix

Description

Computes the mean value of the marginal posterior value of Theta, the geometric dropout process parameter.

Usage

bgbb.rf.matrix.PosteriorMeanDropoutRate(params, rf.matrix)

Arguments

Details

E(Theta | alpha, beta, gamma, delta, x, t.x, n). This is calculated by setting l = 0 and m = 1 in bgbb.PosteriorMeanLmProductMoment.

rf.matrix has columns x⁠, ⁠t.x⁠, and ⁠n.cal'.

Value

The posterior mean dropout rate.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.PosteriorMeanDropoutRate

Examples

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# return the posterior mean dropout rate vector
bgbb.rf.matrix.PosteriorMeanDropoutRate(est.params, rf.matrix)

BG/BB Posterior Mean Transaction Rate using a recency-frequency matrix

Description

Computes the mean value of the marginal posterior value of P, the Bernoulli transaction process parameter.

Usage

bgbb.rf.matrix.PosteriorMeanTransactionRate(params, rf.matrix)

Arguments

Details

rf.matrix has columns x⁠, ⁠t.x⁠, and ⁠n.cal'.

Value

The posterior mean transaction rate.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.PosteriorMeanTransactionRate

Examples

 
data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# return the posterior mean transaction rate vector
bgbb.rf.matrix.PosteriorMeanTransactionRate(est.params, rf.matrix)

BG/NBD Conditional Expected Transactions

Description

E[X(T.cal, T.cal + T.star) | x, t.x, r, alpha, a, b]

Usage

bgnbd.ConditionalExpectedTransactions(
  params,
  T.star,
  x,
  t.x,
  T.cal,
  hardie = TRUE
)

Arguments

Details

T.star, x, t.x and T.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Number of transactions a customer is expected to make in a time period of length t, conditional on their past behavior. If any of the input parameters has a length greater than 1, this will be a vector of expected number of transactions.

References

Fader, Peter S.; Hardie, Bruce G.S.and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web. http://www.brucehardie.com/notes/021/palive_for_BGNBD.pdf

See Also

bgnbd.Expectation

Examples

params <- c(0.243, 4.414, 0.793, 2.426)
# Number of transactions a customer is expected to make in 2 time
# intervals, given that they made 10 repeat transactions in a time period
# of 39 intervals, with the 10th repeat transaction occurring in the 35th
# interval.
bgnbd.ConditionalExpectedTransactions(params, T.star=2, x=10, t.x=35, T.cal=39)

# We can also compare expected transactions across different
# calibration period behaviors:
bgnbd.ConditionalExpectedTransactions(params, T.star=2, x=5:20, t.x=25, T.cal=39)

BG/NBD Parameter Estimation

Description

Estimates parameters for the BG/NBD model.

Usage

bgnbd.EstimateParameters(
  cal.cbs,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  method = "L-BFGS-B",
  hessian = FALSE
)

Arguments

Details

The best-fitting parameters are determined using the bgnbd.cbs.LL function. The sum of the log-likelihood for each customer (for a set of parameters) is maximized in order to estimate parameters.

A set of starting parameters must be provided for this method. If no parameters are provided, (1,3,1,3) is used as a default. These values are used because they provide good convergence across data sets. It may be useful to use starting values for r and alpha that represent your best guess of the heterogeneity in the buy and die rate of customers. It may be necessary to run the estimation from multiple starting points to ensure that it converges. To compare the log-likelihoods of different parameters, use bgnbd.cbs.LL.

The lower bound on the parameters to be estimated is always zero, since BG/NBD parameters cannot be negative. The upper bound can be set with the max.param.value parameter.

This function may take some time to run.

Value

Vector of estimated parameters.

References

Fader, Peter S.; Hardie, and Bruce G.S.. "Overcoming the BG/NBD Model's #NUM! Error Problem." December. 2013. Web. http://brucehardie.com/notes/027/bgnbd_num_error.pdf

See Also

bgnbd.cbs.LL

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# starting-point parameters
startingparams <- c(1.0, 3, 1.0, 3)

# estimated parameters
est.params <- bgnbd.EstimateParameters(cal.cbs = cal.cbs,
                                       par.start = startingparams)

# complete object returned by \code{\link[optimx]{optimx}}
optimx.set <- bgnbd.EstimateParameters(cal.cbs = cal.cbs,
                                       par.start = startingparams,
                                       hessian = TRUE)

# log-likelihood of estimated parameters
bgnbd.cbs.LL(est.params, cal.cbs)

BG/NBD Expectation

Description

Returns the number of repeat transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in a given time period.

Usage

bgnbd.Expectation(params, t, hardie = TRUE)

Arguments

Details

E(X(t) | r, alpha, a, b)

Value

Number of repeat transactions a customer is expected to make in a time period of length t.

References

Fader, Peter S.; Hardie, Bruce G.S.and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web. http://www.brucehardie.com/notes/021/palive_for_BGNBD.pdf

See Also

bgnbd.ConditionalExpectedTransactions

Examples

params <- c(0.243, 4.414, 0.793, 2.426)

# Number of repeat transactions a customer is expected to make in 2 time intervals.
bgnbd.Expectation(params, t=2, hardie = FALSE)

# We can also compare expected transactions over time:
bgnbd.Expectation(params, t=1:10)

BG/NBD Expected Cumulative Transactions

Description

Calculates the expected cumulative total repeat transactions by all customers for the calibration and holdout periods.

Usage

bgnbd.ExpectedCumulativeTransactions(
  params,
  T.cal,
  T.tot,
  n.periods.final,
  hardie = TRUE
)

Arguments

Details

The function automatically divides the total period up into n.periods.final time intervals. n.periods.final does not have to be in the same unit of time as the T.cal data. For example: - if your T.cal data is in weeks, and you want cumulative transactions per week, n.periods.final would equal T.star. - if your T.cal data is in weeks, and you want cumulative transactions per day, n.periods.final would equal T.star * 7.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Vector of expected cumulative total repeat transactions by all customers.

See Also

bgnbd.Expectation

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

params <- c(0.243, 4.414, 0.793, 2.426)

# Returns a vector containing cumulative repeat transactions for 273 days.
# All parameters are in weeks; the calibration period lasted 39 weeks.
bgnbd.ExpectedCumulativeTransactions(params,
                                     T.cal = cal.cbs[,"T.cal"],
                                     T.tot = 39,
                                     n.periods.final = 273,
                                     hardie = TRUE)

BG/NBD Log-Likelihood

Description

Calculates the log-likelihood of the BG/NBD model.

Usage

bgnbd.LL(params, x, t.x, T.cal)

Arguments

Details

x, t.x and T.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

A vector of log-likelihoods as long as the longest input vector (x, t.x, or T.cal).

See Also

bgnbd.EstimateParameters

bgnbd.cbs.LL

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# random assignment of parameters
params <- c(0.5, 6, 1.2, 3.3)
# returns the log-likelihood of the given parameters
bgnbd.cbs.LL (params, cal.cbs)

# compare the speed and results to the following:
cal.cbs.compressed <- dc.compress.cbs(cal.cbs)
bgnbd.cbs.LL(params, cal.cbs.compressed)

# Returns the log likelihood of the parameters for a customer who
# made 3 transactions in a calibration period that ended at t=6,
# with the last transaction occurring at t=4.
bgnbd.LL(params, x=3, t.x=4, T.cal=6)

# We can also give vectors as function parameters:
set.seed(7)
x <- sample(1:4, 10, replace = TRUE)
t.x <- sample(1:4, 10, replace = TRUE)
T.cal <- rep(4, 10)
bgnbd.LL(params, x, t.x, T.cal)

BG/NBD P(Alive)

Description

Uses BG/NBD model parameters and a customer's past transaction behavior to return the probability that they are still alive at the end of the calibration period.

Usage

bgnbd.PAlive(params, x, t.x, T.cal)

Arguments

Details

P(Alive | X=x, t.x, T.cal, r, alpha, a, b)

x, t.x, and T.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability that the customer is still alive at the end of the calibration period. If x, t.x, and/or T.cal has a length greater than one, then this will be a vector of probabilities (containing one element matching each element of the longest input vector).

References

Fader, Peter S.; Hardie, Bruce G.S.and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web.

Examples

params <- c(0.243, 4.414, 0.793, 2.426)

bgnbd.PAlive(params, x=23, t.x=39, T.cal=39)
# P(Alive) of a customer who has the same recency and total
# time observed.

bgnbd.PAlive(params, x=5:20, t.x=30, T.cal=39)
# Note the "increasing frequency paradox".

# To visualize the distribution of P(Alive) across customers:

data(cdnowSummary)
cbs <- cdnowSummary$cbs
params <- bgnbd.EstimateParameters(cbs, par.start = c(0.243, 4.414, 0.793, 2.426))
p.alives <- bgnbd.PAlive(params, cbs[,"x"], cbs[,"t.x"], cbs[,"T.cal"])
plot(density(p.alives))

BG/NBD Plot Dropout Probability Heterogeneity

Description

Plots and returns the estimated gamma distribution of p (customers' probability of dropping out immediately after a transaction).

Usage

bgnbd.PlotDropoutRateHeterogeneity(params, lim = NULL)

Arguments

Value

Distribution of customers' probabilities of dropping out.

Examples

params <- c(0.243, 4.414, 0.793, 2.426)
bgnbd.PlotDropoutRateHeterogeneity(params)
params <- c(0.243, 4.414, 1.33, 2.426)
bgnbd.PlotDropoutRateHeterogeneity(params)

BG/NBD Plot Frequency vs. Conditional Expected Frequency

Description

Plots the actual and conditional expected number transactions made by customers in the holdout period, binned according to calibration period frequencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

bgnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

Arguments

Details

This function requires a censor number, which cannot be higher than the highest frequency in the calibration period CBS. The output matrix will have (censor + 1) bins, starting at frequencies of 0 transactions and ending at a bin representing calibration period frequencies at or greater than the censor number.

Value

Holdout period transaction frequency comparison matrix (actual vs. expected).

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# number of transactions by each customer in the 39 weeks
# following the calibration period
x.star <- cal.cbs[,"x.star"]

# parameters estimated using bgnbd.EstimateParameters
est.params <- c(0.243, 4.414, 0.793, 2.426)
# the maximum censor number that can be used
max(cal.cbs[,"x"])

# plot conditional expected holdout period frequencies,
# binned according to calibration period frequencies
bgnbd.PlotFreqVsConditionalExpectedFrequency(est.params, 
                                             T.star = 39, 
                                             cal.cbs, 
                                             x.star, 
                                             censor = 7)

BG/NBD Plot Frequency in Calibration Period

Description

Plots a histogram and returns a matrix comparing the actual and expected number of customers who made a certain number of repeat transactions in the calibration period, binned according to calibration period frequencies.

Usage

bgnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor,
  plotZero = TRUE,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

Arguments

Details

This function requires a censor number, which cannot be higher than the highest frequency in the calibration period CBS. The output matrix will have (censor + 1) bins, starting at frequencies of 0 transactions and ending at a bin representing calibration period frequencies at or greater than the censor number. The plot may or may not include a bin for zero frequencies, depending on the plotZero parameter.

Value

Calibration period repeat transaction frequency comparison matrix (actual vs. expected).

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs 
# cal.cbs already has column names required by method

# parameters estimated using bgnbd.EstimateParameters 
est.params <- c(0.243, 4.414, 0.793, 2.426) 
# the maximum censor number that can be used
max(cal.cbs[,"x"])

bgnbd.PlotFrequencyInCalibration(est.params, cal.cbs, censor=7)

BG/NBD Plot Actual vs. Conditional Expected Frequency by Recency

Description

Plots the actual and conditional expected number of transactions made by customers in the holdout period, binned according to calibration period recencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

bgnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)

Arguments

Details

This function does bin customers exactly according to recency; it bins customers according to integer units of the time period of cal.cbs. Therefore, if you are using weeks in your data, customers will be binned as follows: customers with recencies between the start of the calibration period (inclusive) and the end of week one (exclusive); customers with recencies between the end of week one (inclusive) and the end of week two (exclusive); etc.

The matrix and plot will contain the actual number of transactions made by each bin in the holdout period, as well as the expected number of transactions made by that bin in the holdout period, conditional on that bin's behavior during the calibration period.

Value

Matrix comparing actual and conditional expected transactions in the holdout period.

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs 
# cal.cbs already has column names required by method

# number of transactions by each customer in the 39 weeks following 
# the calibration period 
x.star <- cal.cbs[,"x.star"]

# parameters estimated using bgnbd.EstimateParameters 
est.params <- c(0.243, 4.414, 0.793, 2.426)

# plot conditional expected holdout period transactions, 
# binned according to calibration period recencies
bgnbd.PlotRecVsConditionalExpectedFrequency(est.params, 
                                            cal.cbs, 
                                            T.star = 39,
                                            x.star)

BG/NBD Tracking Cumulative Transactions Plot

Description

Plots the actual and expected cumulative total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

bgnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  n.periods.final = NA,
  hardie = TRUE,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions"
)

Arguments

Details

actual.cu.tracking.data does not have to be in the same unit of time as the T.cal data. T.tot will automatically be divided into periods to match the length of actual.cu.tracking.data. See bgnbd.ExpectedCumulativeTransactions.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Matrix containing actual and expected cumulative repeat transactions.

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# Cumulative repeat transactions made by all customers across calibration
# and holdout periods
cu.tracking <- cdnowSummary$cu.tracking

# parameters estimated using bgnbd.EstimateParameters
est.params <- c(0.243, 4.414, 0.793, 2.426)

# All parameters are in weeks; the calibration period lasted 39
# weeks and the holdout period another 39.
bgnbd.PlotTrackingCum(est.params, 
                      T.cal = cal.cbs[,"T.cal"], 
                      T.tot = 78, 
                      actual.cu.tracking.data = cu.tracking, 
                      hardie = TRUE)

BG/NBD Tracking Incremental Transactions Comparison

Description

Plots the actual and expected incremental total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

bgnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  n.periods.final = NA,
  hardie = TRUE,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions"
)

Arguments

Details

actual.inc.tracking.data does not have to be in the same unit of time as the T.cal data. T.tot will automatically be divided into periods to match the length of actual.inc.tracking.data. See bgnbd.ExpectedCumulativeTransactions.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Matrix containing actual and expected incremental repeat transactions.

Examples

data(cdnowSummary)
cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# Cumulative repeat transactions made by all customers across calibration
# and holdout periods
cu.tracking <- cdnowSummary$cu.tracking
# make the tracking data incremental
inc.tracking <- dc.CumulativeToIncremental(cu.tracking)

# parameters estimated using bgnbd.EstimateParameters
est.params <- c(0.243, 4.414, 0.793, 2.426)

# All parameters are in weeks; the calibration period lasted 39
# weeks and the holdout period another 39.
bgnbd.PlotTrackingInc(est.params, ,
                      T.cal = cal.cbs[,"T.cal"],
                      T.tot = 78,
                      actual.inc.tracking.data = inc.tracking, 
                      hardie = TRUE)

BG/NBD Plot Transaction Rate Heterogeneity

Description

Plots and returns the estimated gamma distribution of lambda (customers' propensities to purchase).

Usage

bgnbd.PlotTransactionRateHeterogeneity(params, lim = NULL)

Arguments

Details

This returns the distribution of each customer's Poisson parameter, which determines the rate at which each customer buys.

Value

Distribution of customers' propensities to purchase.

Examples

params <- c(0.243, 4.414, 0.793, 2.426)
bgnbd.PlotTransactionRateHeterogeneity(params)
params <- c(0.53, 4.414, 0.793, 2.426)
bgnbd.PlotTransactionRateHeterogeneity(params)

BG/NBD Log-Likelihood Wrapper

Description

Calculates the log-likelihood sum of the BG/NBD model.

Usage

bgnbd.cbs.LL(params, cal.cbs)

Arguments

Details

Note: do not use a compressed cal.cbs matrix. It makes quicker work for Pareto/NBD estimation as implemented in this package, but the opposite is true for BG/NBD. For proof, compare the definition of the bgnbd.cbs.LL to that of pnbd.cbs.LL.

Value

The total log-likelihood of the provided data.

See Also

bgnbd.EstimateParameters

bgnbd.LL

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# random assignment of parameters
params <- c(0.5, 6, 1.2, 3.3)
# returns the log-likelihood of the given parameters
bgnbd.cbs.LL(params, cal.cbs)

# compare the speed and results to the following:
cal.cbs.compressed <- dc.compress.cbs(cal.cbs)
bgnbd.cbs.LL (params, cal.cbs.compressed)

# Returns the log likelihood of the parameters for a customer who
# made 3 transactions in a calibration period that ended at t=6,
# with the last transaction occurring at t=4.
bgnbd.LL(params, x=3, t.x=4, T.cal=6)

# We can also give vectors as function parameters:
set.seed(7)
x <- sample(1:4, 10, replace = TRUE)
t.x <- sample(1:4, 10, replace = TRUE)
T.cal <- rep(4, 10)
bgnbd.LL(params, x, t.x, T.cal)

Define general parameters

Description

This is to ensure consistency across all functions that require common bits and bobs.

Usage

bgnbd.generalParams(params, func, x, t.x, T.cal, T.star = NULL, hardie = NULL)

Arguments

Value

a list with things you need for bgnbd.LL, bgnbd.PAlive and bgnbd.ConditionalExpectedTransactions

See Also

bgnbd.LL

bgnbd.PAlive

bgnbd.ConditionalExpectedTransactions

BG/NBD Probability Mass Function

Description

Probability mass function for the BG/NBD.

Usage

bgnbd.pmf(params, t, x)

Arguments

Details

P(X(t)=x | r, alpha, a, b). Returns the probability that a customer makes x repeat transactions in the time interval (0, t].

Parameters t and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of X(t)=x conditional on model parameters. If t and/or x has a length greater than one, a vector of probabilities will be returned.

References

Fader, Peter S.; Hardie, Bruce G.S.and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web.

Examples

params <- c(0.243, 4.414, 0.793, 2.426)
# probability that a customer will make 10 repeat transactions in the
# time interval (0,2]
bgnbd.pmf(params, t=2, x=10)
# probability that a customer will make no repeat transactions in the
# time interval (0,39]
bgnbd.pmf(params, t=39, x=0)

# Vectors may also be used as arguments:
bgnbd.pmf(params, t=30, x=11:20)

Generalized BG/NBD Probability Mass Function

Description

Generalized probability mass function for the BG/NBD.

Usage

bgnbd.pmf.General(params, t.start, t.end, x)

Arguments

Details

P(X(t.start, t.end)=x | r, alpha, a, b). Returns the probability that a customer makes x repeat transactions in the time interval (t.start, t.end].

It is impossible for a customer to make a negative number of repeat transactions. This function will return an error if it is given negative times or a negative number of repeat transactions. This function will also return an error if t.end is less than t.start.

t.start, t.end, and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of x transaction occuring between t.start and t.end conditional on model parameters. If t.start, t.end, and/or x has a length greater than one, a vector of probabilities will be returned.

References

Fader, Peter S.; Hardie, Bruce G.S.and Lee, Ka Lok. “Computing P(alive) Using the BG/NBD Model.” December. 2008. Web.

Examples

params <- c(0.243, 4.414, 0.793, 2.426)
# probability that a customer will make 10 repeat transactions in the
# time interval (1,2]
bgnbd.pmf.General(params, t.start=1, t.end=2, x=10)
# probability that a customer will make no repeat transactions in the
# time interval (39,78]
bgnbd.pmf.General(params, t.start=39, t.end=78, x=0)

CDNOW event log data

Description

Data representing the purchasing behavior of 2,357 CDNOW customers between January 1997 and June 1998, in event log format.

Format

A comma-delimited file representing an event log with 6,919 entries. It has 5 columns: The customer's ID in the master dataset, the customer's ID in this dataset (which represents 1/10th of the master dataset), the date of the transaction in the format "%Y%m%d" (e.g. 19970225), the number of CDs purchased, and the dollar value of the transaction.

Details

The customers in this data represent 1/10th of the cohort of customers who made their first transactions with CDNOW in the first quarter of 1997. CDNOW was an online retailer, selling music and related products on the web since 1994.

Source

Can be found online.

CDNOW repeat transaction data summary

Description

Data representing the purchasing behavior of 2,357 CDNOW customers between January 1997 and June 1998, summarized as a customer-by-time matrix and a vector of cumulative weekly transactions.

Usage

data(cdnowSummary)

Format

A named list of four elements:

• cbs A customer-by-time matrix with four columns: frequency ("x"), recency ("t.x"), length of observation in the calibration period ("T.cal"), and number of transactions in the holdout period ("x.star"). Each row represents a customer.

• cu.tracking A vector containing cumulative transactions for every week in both the calibration and estimating periods (78 weeks total). This vector contains the sum of transactions across all customers.

• est.params A vector containing estimated values for the four Pareto/NBD parameters: r, alpha, s, and beta, in that order. This estimation was made using pnbd.EstimateParameters, and is included here to avoid having to run the relatively time-consuming parameter estimation function in examples.

• m.x A vector containing the average value of each customer's repeat transactions. Used in examples for spend functions.

Details

The customers in this data represent 1/10th of the cohort of customers who made their first transactions with CDNOW in the first quarter of 1997. CDNOW was an online retailer, selling music and related products on the web since 1994.

Source

The data was put together using data conversion functions included in this package. The original event log is included (see cdnowElog).

Build CBS matrix from CBT matrix

Description

Given a customer-by-time matrix, yields the resulting customer-by-sufficient-statistic matrix.

Usage

dc.BuildCBSFromCBTAndDates(cbt, dates, per, cbt.is.during.cal.period = TRUE)

Arguments

Details

The customer-by-sufficient statistic matrix will contain the sum of the statistic included in the customer-by-time matrix (see the cbt parameter), the customer's last transaction date, and the total time period for which the customer was observed.

Value

Customer-by-sufficient-statistic matrix, with three columns: frequency("x"), recency("t.x") and total time observed("T.cal"). See details. Frequency is total transactions, not repeat transactions.

Examples

elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

# Transaction-flow models are about interpurchase times. Since we
# only know purchase times to the day, we merge all transaction on
# the same day. This example uses dc.MergeTransactionsOnSameDate
# to illustrate this; however, we could have simply used dc.CreateReachCBT
# instead of dc.CreateFreqCBT to obtain the same result.
merged.elog <- dc.MergeTransactionsOnSameDate(elog)
cutoff.date <- as.Date("1997-09-30")
freq.cbt <- dc.CreateFreqCBT(merged.elog)
cal.freq.cbt <- freq.cbt[,as.Date(colnames(freq.cbt)) <= cutoff.date]
holdout.freq.cbt <- freq.cbt[,as.Date(colnames(freq.cbt)) > cutoff.date]

cal.start.dates.indices <- dc.GetFirstPurchasePeriodsFromCBT(cal.freq.cbt)
cal.start.dates <- as.Date(colnames(cal.freq.cbt)[cal.start.dates.indices])
cal.end.dates.indices <- dc.GetLastPurchasePeriodsFromCBT(cal.freq.cbt)
cal.end.dates <- as.Date(colnames(cal.freq.cbt)[cal.end.dates.indices])
T.cal.total <- rep(cutoff.date, nrow(cal.freq.cbt))
cal.dates <- data.frame(cal.start.dates, 
                        cal.end.dates, 
                        T.cal.total)

# Create calibration period customer-by-sufficient-statistic data frame,
# using weeks as the unit of time.
cal.cbs <- dc.BuildCBSFromCBTAndDates(cal.freq.cbt, 
                                      cal.dates,
                                      per="week", 
                                      cbt.is.during.cal.period=TRUE)
# Force the calibration period customer-by-sufficient-statistic to only contain
# repeat transactions (required by BG/BB and Pareto/NBD models)
cal.cbs[,"x"] <- cal.cbs[,"x"] - 1

holdout.start <- cutoff.date+1
holdout.end <- as.Date(colnames(holdout.freq.cbt)[ncol(holdout.freq.cbt)])
holdout.dates <- c(holdout.start, holdout.end)

# Create holdout period customer-by-sufficient-statistic data frame, using weeks
# as the unit of time.
holdout.cbs <- dc.BuildCBSFromCBTAndDates(holdout.freq.cbt, 
                                          holdout.dates,
                                          per="week", 
                                          cbt.is.during.cal.period=FALSE)                        

Build Customer-by-Time Matrix from Event Log

Description

Creates a customer-by-time matrix from an event log.

Usage

dc.BuildCBTFromElog(elog, statistic = "freq")

Arguments

Value

Customer-by-time matrix.

Create Frequency Customer-by-Time Matrix

Description

Creates a customer-by-time matrix with total number of transactions per time period.

Usage

dc.CreateFreqCBT(elog)

Arguments

Value

Frequency customer-by-time matrix.

Examples

# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)

# Given that the dates are in the order year-month-day,
# it is not strictly necessary to convert the date column
# to date formats. However, it is good practice:
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

freq.cbt <- dc.CreateFreqCBT(elog)

Create Reach Customer-by-Time Matrix

Description

Creates a customer-by-time matrix with 1's in periods that a customer made a transaction and 0's otherwise.

Usage

dc.CreateReachCBT(elog)

Arguments

Value

Reach customer-by-time matrix.

Examples

# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)

# Given that the dates are in the order year-month-day,
# it is not strictly necessary to convert the date column
# to date formats. However, it is good practice:
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

reach.cbt <- dc.CreateReachCBT(elog)

Create Spend Customer-by-Time Matrix

Description

Creates a customer-by-time matrix with spend per time period.

Usage

dc.CreateSpendCBT(elog, is.avg.spend = FALSE)

Arguments

Value

Spend customer-by-time matrix.

Examples

# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5);

# Given that the dates are in the order year-month-day,
# it is not strictly necessary to convert the date column
# to date formats. However, it is good practice:
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

spend.cbt <- dc.CreateSpendCBT(elog)

Cumulative to Incremental

Description

Converts a vector of cumulative transactions to a vector of incremental transactions.

Usage

dc.CumulativeToIncremental(cu)

Arguments

Value

Vector of incremental transactions.

Dissipate Event Log

Description

Filters an event log, keeping a fraction of the original event log.

Usage

dc.DissipateElog(elog, dissipate.factor)

Arguments

Value

Reduced event log.

Convert Event Log to CBS and CBT Matrices

Description

Uses an event log to return calibration period CBT and CBS, holdout period CBT and CBS, and summary data for each customer (including times of first and last transactions).

Usage

dc.ElogToCbsCbt(
  elog,
  per = "week",
  T.cal = max(elog$date),
  T.tot = max(elog$date),
  merge.same.date = TRUE,
  cohort.birth.per = T.cal,
  dissipate.factor = 1,
  statistic = "freq"
)

Arguments

Details

This function automatically removes customers' first transactions, meaning that the output matrices will only contain repeat transaction information.

Value

A list of items: - ⁠$cal⁠ list with CBS and CBT from the calibration period - ⁠$holdout⁠ list with CBS and CBT from holdout period - ⁠$cust.data⁠ data frame with each customer's first and last transaction details

Examples

 
# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)

elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

data <- dc.ElogToCbsCbt(elog, per="week", T.cal=as.Date("1997-09-30"))

Filter Customer by Birth

Description

Filters an event log, keeping all transactions made by customers who made their first transactions in the given time interval.

Usage

dc.FilterCustByBirth(elog, cohort.birth.per)

Arguments

Value

event log with only rows from customers who made their first transaction within the birth period.

Examples

# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)

# converting the date column to Date objects is
# necessary for this function.
elog$date <- as.Date(elog$date, "%Y%m%d")

# starting date. Note that it must be a Date object.
start.date <- as.Date("1997-01-01")
# ending date. Note that it must be a Date object.
end.date <- as.Date("1997-01-31")

# Filter the elog to include only customers who made their
# first transaction in January 1997
filtered.elog <- dc.FilterCustByBirth(elog, c(start.date, end.date))

Get First Purchase Periods from Customer-by-Time Matrix

Description

Uses a customer-by-time matrix to return a vector containing the periods in which customers made their first purchase.

Usage

dc.GetFirstPurchasePeriodsFromCBT(cbt)

Arguments

Value

a vector containing the indices of periods in which customers made their first transactions. To convert to actual dates (if your customer-by-time matrix has dates as column names), use colnames(cbt)[RESULT]

Get Last Purchase Periods from Customer-by-Time Matrix

Description

Uses a customer-by-time matrix to return a vector containing the periods in which customers made their last purchase.

Usage

dc.GetLastPurchasePeriodsFromCBT(cbt)

Arguments

Value

a vector containing the indices of periods in which customers made their last transactions. To convert to actual dates (if your customer-by-time matrix has dates as column names), use colnames(cbt)[RESULT]

Check the inputs to functions that use this common pattern

Description

A bunch of functions whose names start with pnbd take a set of four parameters as their first argument, and then a set of vectors or scalars such as x or T.cal as their subsequent arguments. This function started out as pnbd.InputCheck() and it was meant to run input checks for any number of such subsequent vector arguments, as long as they all met the same requirements as x, t.x and T.cal in pnbd.LL: meaning, the length of the longest of these vectors is a multiple of the lengths of all others, and all vectors are numeric and positive.

Usage

dc.InputCheck(params, func, printnames = c("r", "alpha", "s", "beta"), ...)

Arguments

Details

With an extra argument, printnames, pnbd.InputCheck() could also accommodate input checks for functions whose names start with bgbb, bgnbd, and spend so it was basically useful everywhere. That's when it became dc.InputCheck(). params can have any length as long as that length is the same as the length of printnames, so dc.InputCheck() can probably handle mixtures of distributions for modeling BTYD behavior that are not yet implemented.

By other arguments ... here we mean a bunch of named vectors that are used by functions that call dc.InputCheck, such as x, t.x, T.cal, etc. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. Vector recycling is a good way to get into trouble. Keep vectors to the same length and use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

If all is well, a data frame with everything you need in it, with nrow() equal to the length of the longest vector in ...

See Also

pnbd.LL pnbd.ConditionalExpectedTransactions

Make Calibration Period Recency-Frequency Matrix

Description

Make a calibration period recency-frequency matrix.

Usage

dc.MakeRFmatrixCal(
  frequencies,
  periods.of.final.purchases,
  num.of.purchase.periods,
  holdout.frequencies = NULL
)

Arguments

Value

A matrix with all possible frequency-recency combinations, and the number of customers with each combination. It contains columns for frequency ("x"), recency ("t.x"), number of transaction opportunities in the calibration period ("n.cal"), number of customers with this combination of recency, frequency, and number of periods observed ("custs"), and optionally, number of transactions in the holdout period ("x.star").

Examples

elog <- dc.ReadLines(system.file("data/discreteSimElog.csv", package="BTYD"),1,2)
elog[,"date"] <- as.Date(elog[,"date"])

cutoff.date <- as.Date("1977-01-01")
cbt <- dc.CreateReachCBT(elog)
cal.cbt <- cbt[,as.Date(colnames(cbt)) <= cutoff.date]
holdout.cbt <- cbt[,as.Date(colnames(cbt)) > cutoff.date]

cal.start.dates.indices <- dc.GetFirstPurchasePeriodsFromCBT(cal.cbt)
cal.start.dates <- as.Date(colnames(cal.cbt)[cal.start.dates.indices])
cal.end.dates.indices <- dc.GetLastPurchasePeriodsFromCBT(cal.cbt)
cal.end.dates <- as.Date(colnames(cal.cbt)[cal.end.dates.indices])
T.cal.total <- rep(cutoff.date, nrow(cal.cbt))
cal.dates <- data.frame(cal.start.dates, cal.end.dates, T.cal.total)

# Create calibration period customer-by-sufficient-statistic data frame,
# using years as the unit of time.
cal.cbs <- dc.BuildCBSFromCBTAndDates(cal.cbt,
                                      cal.dates,
                                      per="year",
                                      cbt.is.during.cal.period=TRUE)

holdout.start <- as.Date(colnames(holdout.cbt)[1])
holdout.end <- as.Date(tail(colnames(holdout.cbt),n=1))
# The (-1) below is to remove the effect of the birth period - we are only
# interested in repeat transactions in the calibration period.
frequencies <- (cal.cbs[,"x"] - 1)
periods.of.final.purchases <- cal.cbs[,"t.x"]
num.of.purchase.periods <- ncol(cal.cbt) - 1

# Create a calibration period recency-frequency matrix
cal.rf.matrix <- dc.MakeRFmatrixCal(frequencies,
                                    periods.of.final.purchases,
                                    num.of.purchase.periods)

Make Holdout Period Recency-Frequency Matrix

Description

Creates a recency-frequency matrix for the holdout period.

Usage

dc.MakeRFmatrixHoldout(holdout.cbt)

Arguments

Value

recency-frequency matrix for the holdout period, with three columns: frequency ("x.star"), recency ("t.x.star"), number of transaction opportunities in the holdout period ("n.star"), and the number of customers with each frequency-recency combination ("custs").

Make Recency-Frequency Matrix Skeleton

Description

Creates a matrix with all possible recency and frequency combinations.

Usage

dc.MakeRFmatrixSkeleton(n.periods)

Arguments

Details

Makes the structure in which to input data for recency-frequency matrices.

Value

Matrix with two columns: frequency ("x") and recency ("t.x"). All possible recency-frequency combinations in the calibration period are represented.

Merge Customers

Description

Takes two CBT or CBS matrices and ensures that the second one has the same row names as the first.

Usage

dc.MergeCustomers(data.correct, data.to.correct)

Arguments

Details

Care should be taken in using this function. It inserts zero values in all rows that were not in the original holdout period data. This behavior does not cause a problem if using CBT matrices, but will cause a problem if using CBS matrices (for example, the output will report all customers with a holdout period length of zero). However, this particular issue is easily fixed (see examples) and should not cause problems.

A work-around to avoid using this function is presented in the example for dc.BuildCBSFromCBTAndDates - build the full CBT and only use the columns applying to each particular time period to construct separate CBTs, and from them, CBSs. That is a much cleaner and less error-prone method; however, on occasion the data will not be available in event log format and you may not be able to construct a CBT for both time periods together.

Value

Updated holdout period CBT or CBS.

Examples

elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")
cutoff.date <- as.Date("1997-09-30")
cal.elog <- elog[which(elog[,"date"] <= cutoff.date),]
holdout.elog <- elog[which(elog[,"date"] > cutoff.date),]

# Create calibration period CBT from cal.elog
cal.reach.cbt <- dc.CreateReachCBT(cal.elog)
# Create holdout period CBT from holdout.elog
holdout.reach.cbt <- dc.CreateReachCBT(holdout.elog)

# Note the difference:
nrow(cal.reach.cbt)            # 2357 customers
nrow(holdout.reach.cbt)        # 684 customers

# Create a "fixed" holdout period CBT, with the same number
# of customers in the same order as the calibration period CBT
fixed.holdout.reach.cbt <- dc.MergeCustomers(cal.reach.cbt, holdout.reach.cbt)
nrow(fixed.holdout.reach.cbt)  # 2357 customers

# You can verify that the above is correct by turning these into a CBS
# (see \code{\link{dc.BuildCBSFromCBTAndDates}} and using
# \code{\link{pnbd.PlotFreqVsConditionalExpectedFrequency}}, for example

# Alternatively, we can fix the CBS, instead of the CBS:

cal.start.dates.indices <- dc.GetFirstPurchasePeriodsFromCBT(cal.reach.cbt)
cal.start.dates <- as.Date(colnames(cal.reach.cbt)[cal.start.dates.indices])
cal.end.dates.indices <- dc.GetLastPurchasePeriodsFromCBT(cal.reach.cbt)
cal.end.dates <- as.Date(colnames(cal.reach.cbt)[cal.end.dates.indices])
T.cal.total <- rep(cutoff.date, nrow(cal.reach.cbt))
cal.dates <- data.frame(cal.start.dates, cal.end.dates, T.cal.total)

# Create calibration period customer-by-sufficient-statistic data frame,
# using weeks as the unit of time.
cal.cbs <- dc.BuildCBSFromCBTAndDates(cal.reach.cbt,
                                      cal.dates,
                                      per="week",
                                      cbt.is.during.cal.period=TRUE)

# Force the calibration period customer-by-sufficient-statistic to only
# 	contain repeat transactions (required by BG/BB and Pareto/NBD models)
cal.cbs[,"x"] <- cal.cbs[,"x"] - 1

holdout.start <- cutoff.date+1
holdout.end <- as.Date(colnames(fixed.holdout.reach.cbt)[ncol(fixed.holdout.reach.cbt)])
holdout.dates <- c(holdout.start, holdout.end)

# Create holdout period customer-by-sufficient-statistic data frame,
# using weeks as the unit of time.
holdout.cbs <- dc.BuildCBSFromCBTAndDates(holdout.reach.cbt,
                                          holdout.dates,
                                          per="week",
                                          cbt.is.during.cal.period=FALSE)

# Note the difference:
nrow(cal.cbs)            # 2357 customers
nrow(holdout.cbs)        # 684 customers

# Create a "fixed" holdout period CBS, with the same number
# of customers in the same order as the calibration period CBS
fixed.holdout.cbs <- dc.MergeCustomers(cal.cbs, holdout.cbs)
nrow(fixed.holdout.cbs)  # 2357 customers

# Furthermore, this function will assign a zero value to all fields
# that were not in the original holdout period CBS. Since T.star is the
# same for all customers in the holdout period, we should fix that:
fixed.holdout.cbs[,"T.star"] <- rep(max(fixed.holdout.cbs[,"T.star"]),nrow(fixed.holdout.cbs))

Merge Transactions on Same Day

Description

Updates an event log; any transactions made by the same customer on the same day are combined into one transaction.

Usage

dc.MergeTransactionsOnSameDate(elog)

Arguments

Value

Event log with transactions made by the same customer on the same day merged into one transaction.

Plot Log-Likelihood Contour

Description

Makes a contour plot of a loglikelihood function that varies over two designated parameters, centered around a set of previously estimated parameters.

Usage

dc.PlotLogLikelihoodContour(
  loglikelihood.fcn,
  vary.or.fix.param,
  predicted.params,
  ...,
  n.divs = 3,
  new.dev = FALSE,
  num.contour.lines = 10,
  zoom.percent = 0.9,
  allow.neg.params = FALSE,
  param.names = c("param 1", "param 2", "param 3", "param 4")
)

Arguments

Details

The contour plot will have the first parameter labelled "vary" on the x-axis, and the second parameter labelled "vary" on the y-axis. It will extend out by (n.divs * zoom.percent) in both directions and both dimensions from the estimated parameter values. The exception is if allow.neg.params is FALSE. In this case, the contour plot will end at zero if it would have extended into negative parameter values.

The estimated parameter values will be indicated by the intersection of two red lines.

See Also

dc.PlotLogLikelihoodContours

Examples

# **Examples for BG/BB model:
data(donationsSummary)
rf.matrix <- donationsSummary$rf.matrix

# starting-point parameters
bgbb.startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
bgbb.est.params <- bgbb.EstimateParameters(rf.matrix, bgbb.startingparams)

# set up parameter names for a more descriptive result
bgbb.param.names <- c("alpha", "beta", "gamma", "delta")

# plot a log-likelihood contour of alpha and beta, the unobserved
# parameters for the beta-Bernoulli transaction process of the BG/BB.
# Note that allow.neg.params has been set to false as BG/BB parameters
# cannot be negative:
dc.PlotLogLikelihoodContour(bgbb.rf.matrix.LL, 
                            c("vary", "vary", "fix", "fix"), 
                            bgbb.est.params, 
                            rf.matrix = rf.matrix, 
                            n.divs = 15, 
                            num.contour.lines = 15, 
                            zoom.percent = 0.2, 
                            allow.neg.params = FALSE, 
                            param.names = bgbb.param.names)

# plot a log-likelihood contour of gamma and delta, the unobserved
# parameters for the beta-geometric dropout process of the BG/BB.
# Note that allow.neg.params has been set to false as BG/BB parameters
# cannot be negative:
dc.PlotLogLikelihoodContour(bgbb.rf.matrix.LL, 
                            c("fix", "fix", "vary", "vary"), 
                            bgbb.est.params, 
                            rf.matrix = rf.matrix, 
                            n.divs = 15, 
                            num.contour.lines = 15, 
                            zoom.percent = 0.2, 
                            allow.neg.params = FALSE, 
                            param.names = bgbb.param.names)

# **Example for Pareto/NBD model:
data(cdnowSummary)
cbs <- cdnowSummary$cbs

# Speed up calculations:
cbs <- dc.compress.cbs(cbs)

# parameters estimated using pnbd.EstimateParameters
pnbd.est.params <- cdnowSummary$est.params

# set up parameter names for a more descriptive result
pnbd.param.names <- c("r", "alpha", "s", "beta")

# plot a log-likelihood contour of r and s, the shape parameters
# of the transaction and dropout process models (respectively).
# Note that allow.neg.params has been set to false as Pareto/NBD
# parameters cannot be negative:
dc.PlotLogLikelihoodContour(pnbd.cbs.LL, 
                            c("vary", "fix", "vary", "fix"), 
                            pnbd.est.params, 
                            cal.cbs = cbs, 
                            hardie = TRUE,
                            n.divs = 20, 
                            num.contour.lines = 20, 
                            zoom.percent = 0.1, 
                            allow.neg.params = FALSE, 
                            param.names = pnbd.param.names)

# **Example for BG/NBD model:
data(cdnowSummary)
cbs <- cdnowSummary$cbs

# parameters estimated using bgnbd.EstimateParameters
bgnbd.est.params <- cdnowSummary$est.params

# set up parameter names for a more descriptive result
bgnbd.param.names <- c("r", "alpha", "s", "beta")

# plot a log-likelihood contour of r and s, the shape parameters
# of the transaction and dropout process models (respectively).
# Note that allow.neg.params has been set to false as BG/NBD
# parameters cannot be negative:
dc.PlotLogLikelihoodContour(bgnbd.cbs.LL, 
                            c("vary", "fix", "vary", "fix"), 
                            bgnbd.est.params, 
                            cal.cbs = cbs, 
                            n.divs = 20, 
                            num.contour.lines = 20, 
                            zoom.percent = 0.1, 
                            allow.neg.params = FALSE, 
                            param.names = bgnbd.param.names)

Plot Log-Likelihood Contours

Description

Creates a set of contour plots, such that there is a contour plot for every pair of parameters varying.

Usage

dc.PlotLogLikelihoodContours(
  loglikelihood.fcn,
  predicted.params,
  ...,
  n.divs = 2,
  multiple.screens = FALSE,
  num.contour.lines = 10,
  zoom.percent = 0.9,
  allow.neg.params = FALSE,
  param.names = c("param 1", "param 2", "param 3", "param 4")
)

Arguments

Details

For each contour plot, the non-varying parameters are kept constant at the predicted values.

The contour will extend out by (n.divs * zoom.percent) in both directions and both dimensions from the estimated parameter values. The exception is if allow.neg.params is FALSE. In this case, the contour plot will end at zero if it would have extended into negative parameter values.

The estimated parameter values will be indicated by the intersection of two red lines.

See Also

dc.PlotLogLikelihoodContour

Examples

# **Example for BG/BB model:
data(donationsSummary)
rf.matrix <- donationsSummary$rf.matrix

# starting-point parameters
bgbb.startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
bgbb.est.params <- bgbb.EstimateParameters(rf.matrix, bgbb.startingparams)

# set up parameter names for a more descriptive result
bgbb.param.names <- c("alpha", "beta", "gamma", "delta")

# plot-log likelihood contours:
dc.PlotLogLikelihoodContours(bgbb.rf.matrix.LL,
                             bgbb.est.params, 
                             rf.matrix = rf.matrix, 
                             n.divs = 5, 
                             num.contour.lines = 8, 
                             zoom.percent = 0.3, 
                             allow.neg.params = FALSE, 
                             param.names = bgbb.param.names)

# **Example for Pareto/NBD model:
data(cdnowSummary)
cbs <- cdnowSummary$cbs

# Speed up calculations:
cbs <- dc.compress.cbs(cbs)

# parameters estimated using pnbd.EstimateParameters
pnbd.est.params <- cdnowSummary$est.params

# set up parameter names for a more descriptive result
pnbd.param.names <- c("r", "alpha", "s", "beta")

# plot log-likelihood contours:
dc.PlotLogLikelihoodContours(pnbd.cbs.LL, 
                             pnbd.est.params, 
                             cal.cbs = cbs, 
                             hardie = TRUE,
                             n.divs = 5, 
                             num.contour.lines = 15, 
                             zoom.percent = 0.3, 
                             allow.neg.params = FALSE, 
                             param.names = pnbd.param.names)

# **Example for BG/NBD model:
data(cdnowSummary)
cbs <- cdnowSummary$cbs

# parameters estimated using bgnbd.EstimateParameters
bgnbd.est.params <- cdnowSummary$est.params

# set up parameter names for a more descriptive result
bgnbd.param.names <- c("r", "alpha", "s", "beta")

# plot log-likelihood contours:
dc.PlotLogLikelihoodContours(bgnbd.cbs.LL, 
                             bgnbd.est.params, 
                             cal.cbs = cbs, 
                             n.divs = 5, 
                             num.contour.lines = 15, 
                             zoom.percent = 0.3, 
                             allow.neg.params = FALSE, 
                             param.names = bgnbd.param.names)

Read Lines

Description

Given a .csv file that throws errors when read in by the usual read.csv and read.table methods, loops through the file line-by-line and picks out the customer, date, and sales (optional) transaction data to return an event log.

Usage

dc.ReadLines(csv.filename, cust.idx, date.idx, sales.idx = -1)

Arguments

Details

Once this function has been run, you may need to convert the date column to Date objects for the event log to work with other functions in this library. See the as.Date function in the base R package for more details.

Examples

# Create event log from file "cdnowElog.csv", which has
# customer IDs in the second column, dates in the third column, and
# sales numbers in the fifth column.
elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)

# convert date column to date objects, as required by some other functions
elog$date <- as.Date(elog$date, "$Y%m%d")

Remove Time Between

Description

This function creates a new event log, with time in the middle removed. Used, for example, in sports with off-seasons.

Usage

dc.RemoveTimeBetween(elog, day1, day2, day3, day4)

Arguments

Details

The four date parameters must be in ascending order.

Value

list - elog1 the event log with all elog$date entries between day1 and day2 - elog2 the event with all elog$date entries between day3 and day4 - elog3 elog1 combined with elog2, with all dates from elog2 reduced by the time removed between elog1 and elog2

Examples

elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD"),2,3,5)
elog[,"date"] <- as.Date(elog[,"date"], "%Y%m%d")

# Use the cdnow data to return a 6 month event log for January, February,
# March, October, November, December.
period.one.start <- as.Date("1997-01-01")
period.one.end <- as.Date("1997-03-31")
period.two.start <- as.Date("1997-10-01")
period.two.end <- as.Date("1997-12-31")
reduced.elog <- dc.RemoveTimeBetween(elog, period.one.start, period.one.end,
                                     period.two.start, period.two.end)

# Note that the new elog will go up to June 30 at a maximum, since we
# are only using 6 months of data starting on January 1
max(reduced.elog$elog3$date)  # "1997-06-30"

Split Up Event Log for Repeat Transactions

Description

Turns an event log into a repeat transaction event log, removing customers' first transactions. Also returns a data frame with information about customers' first and last transactions.

Usage

dc.SplitUpElogForRepeatTrans(elog)

Arguments

Value

A named list: - repeat.trans.elog an event log containing only repeat transactions - cust.data data frame containing the first and last transaction information for each customer

Write Line

Description

Writes any number of arguments to the console.

Usage

dc.WriteLine(...)

Arguments

Details

The code is literally: cat(..., fill=TRUE); flush.console();

Check model params

Description

Check model parameters for correctness.

Usage

dc.check.model.params(printnames, params, func)

Arguments

Value

Stops the program if there is something wrong with the parameters.

Compress Customer-by-Sufficient-Statistic (CBS) Matrix

Description

Combines all customers with the same combination of recency, frequency and length of calibration period in the customer-by-sufficient-statistic matrix, and adds a fourth column labelled "custs" (with the number of customers belonging in each row).

Usage

dc.compress.cbs(cbs, rounding = 3)

Arguments

Details

This function is meant to be used to speed up log-likelihood and parameter estimation functions in the Pareto/NBD (pnbd) set of functions. How much faster those function run depends on how similar customers are. You can used compressed CBS matrices in BG/NBD estimation too, but there will be no speed gains there over using un-compressed CBS data.

This function only takes columns "x", "t.x", and "T.cal" into account. All other columns will be added together - for example, if you have a spend column, the output's spend column will contain the total amount spent by all customers with an identical recency, frequency, and time observed.

Value

A customer-by-sufficient-statistic matrix with an additional column "custs", which contains the number of customers with each combination of recency, frequency and length of calibration period.

Examples

# Create a sample customer-by-sufficient-statistic matrix:
set.seed(7)
x <- sample(1:4, 10, replace = TRUE)
t.x <- sample(1:4, 10, replace = TRUE)
T.cal <- rep(4, 10)
ave.spend <- sample(10:20, 10, replace = TRUE)
cbs <- cbind(x, t.x, T.cal, ave.spend)
cbs

# If cbs is printed, you would note that the following
# sets of rows have the same x, t.x and T.cal:
# (1, 6, 8); (3, 9)

dc.compress.cbs(cbs, 0)   # No rounding necessary

# Note that all additional columns (in this case, ave.spend)
# are aggregated by sum.

Discrete simulated annual event log data

Description

Data simulated using BG/BB model assumptions. Contains annual transaction behavior for a period of 14 years, for a cohort of 10,000 customers who made their first transactions in 1970.

Format

A comma-delimited file representing an event log with 52,432 entries. It has 2 columns: The customer's ID and the date of the transaction in standard R date format.

Details

This dataset was simulated in order to illustrate certain data-conversion functions (see dc.MakeRFmatrixCal).

Discrete donation data summary

Description

This dataset contains a recency-frequency matrix capturing the discrete transaction behavior of 11,104 customers over 6 transaction opportunities, summarized as a recency-frequency matrix and a vector of annual transactions.

Usage

data(donationsSummary)

Format

A named list:

$rf.matrix

A matrix with 22 rows (for each possible recency-frequency combination in 6 calibration period transaction opportunities) and 4 columns: number of transactions during the calibration period ("x"), recency in the calibration period ("t.x"), number of transaction opportunities in the calibration period ("n.cal"), and number of customers with this recency-frequency combination in the calibration period ("custs").

$rf.matrix.holdout

A matrix with 15 rows (for each possible recency-frequency combination in 5 holdout period transaction opportunities) and 4 columns: number of transactions during the holdout period ("x.star"), recency in the holdout period ("t.x.star"), number of transaction opportunities in the holdout period ("n.star"), and number of customers with the recency-frequency combination in the holdout period ("custs").

$x.star

A vector with 22 elements, containing the number of transactions made by each calibration period recency-frequency bin in the holdout period. It is in the same order as $rf.matrix.

$annual.sales

A vector with 11 elements, containing the number of transactions made by all customers in each time period in both the calibration and holdout periods.

Details

Data from "a major nonprofit organization located in the midwestern United States that is funded in large part by donations from individuals. In 1995 the organization "acquired" 11,104 first-time supporters; in each of the following six years, these individuals either did or did not support the organization."

This dataset contains, for each possible in-sample recency/frequency combination in the 1995 cohort, the number of customers and the number of transactions they made during the validation period.

Source

Data can be found online at http://www.brucehardie.com/notes/010/ (Associated Excel spreadsheet)

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. “Customer-Base Analysis in a Discrete-Time Noncontractual Setting.” Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. http://www.brucehardie.com/notes/020/

Use Bruce Hardie's Gaussian hypergeometric implementation

Description

In benchmarking pnbd.LL runs more quickly and it returns the same results if it uses this helper instead of hypergeo, which is the default. But h2f1 is such a barebones function that in some edge cases it will keep going until you get a segfault, where hypergeo would have failed with a proper error message.

Usage

h2f1(a, b, c, z)

Arguments

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

See Also

hypergeo

Pareto/NBD Conditional Expected Transactions

Description

Uses Pareto/NBD model parameters and a customer's past transaction behavior to return the number of transactions they are expected to make in a given time period.

Usage

pnbd.ConditionalExpectedTransactions(
  params,
  T.star,
  x,
  t.x,
  T.cal,
  hardie = TRUE
)

Arguments

Details

E[X(T.cal, T.cal + T.star) | x, t.x, r, alpha, s, beta]

T.star, x, t.x, and T.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Number of transactions a customer is expected to make in a time period of length t, conditional on their past behavior. If any of the input parameters has a length greater than 1, this will be a vector of expected number of transactions.

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

See Also

pnbd.Expectation

Examples

params <- c(0.55, 10.56, 0.61, 11.64)
# Number of transactions a customer is expected to make in 2 time
# intervals, given that they made 10 repeat transactions in a time period
# of 39 intervals, with the 10th repeat transaction occurring in the 35th
# interval.
pnbd.ConditionalExpectedTransactions(params, 
                                     T.star = 2, 
                                     x = 10, 
                                     t.x = 35, 
                                     T.cal = 39, 
                                     hardie = TRUE)

# We can also compare expected transactions across different
# calibration period behaviors:
pnbd.ConditionalExpectedTransactions(params, 
                                     T.star = 2, 
                                     x = 5:20, 
                                     t.x = 25, 
                                     T.cal = 39, 
                                     hardie = TRUE)

Pareto/NBD Discounted Expected Residual Transactions

Description

Calculates the discounted expected residual transactions of a customer, given their behavior during the calibration period.

Usage

pnbd.DERT(params, x, t.x, T.cal, d, hardie = TRUE)

Arguments

Details

DERT(d | r, alpha, s, beta, X = x, t.x, T.cal)

x, t.x, T.cal may be vectors. The standard rules for vector operations apply

• if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

Value

The number of discounted expected residual transactions for a customer with a particular purchase pattern during the calibration period.

References

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. "RFM and CLV: Using Iso-Value Curves for Customer Base Analysis." Journal of Marketing Research Vol.42, pp.415-430. November. 2005. http://www.brucehardie.com/papers.html

See equation 2.

Note that this paper refers to what this package is calling discounted expected residual transactions (DERT) simply as discounted expected transactions (DET).

Examples

# elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD2"),2,3)
# elog[, 'date'] <- as.Date(elog[, 'date'], format = '%Y%m%d')
# cal.cbs <- dc.ElogToCbsCbt(elog)$cal$cbs
# params <- pnbd.EstimateParameters(cal.cbs, hardie = TRUE)
params <- c(0.5629966, 12.5590370, 0.4081095, 10.5148048)

# 15% compounded annually has been converted to 0.0027 compounded continuously,
# as we are dealing with weekly data and not annual data.
d <- 0.0027

# calculate the discounted expected residual transactions of a customer
# who made 7 transactions in a calibration period that was 77.86
# weeks long, with the last transaction occurring at the end of
# the 35th week.
pnbd.DERT(params, 
          x = 7, 
          t.x = 35, 
          T.cal = 77.86, 
          d, 
          hardie = TRUE)

# We can also use vectors to compute DERT for several customers:
pnbd.DERT(params, 
          x = 1:10, 
          t.x = 30, 
          T.cal = 77.86, 
          d, 
          hardie = TRUE)

Pareto/NBD Parameter Estimation

Description

The best-fitting parameters are determined using the pnbd.cbs.LL function. The sum of the log-likelihood for each customer (for a set of parameters) is maximized in order to estimate parameters.

Usage

pnbd.EstimateParameters(
  cal.cbs,
  par.start = c(1, 1, 1, 1),
  max.param.value = 10000,
  method = "L-BFGS-B",
  hardie = TRUE,
  hessian = FALSE
)

Arguments

Details

A set of starting parameters must be provided for this method. If no parameters are provided, (1,1,1,1) is used as a default. It may be useful to use starting values for r and s that represent your best guess of the heterogeneity in the buy and die rate of customers. It may be necessary to run the estimation from multiple starting points to ensure that it converges. To compare the log-likelihoods of different parameters, use pnbd.cbs.LL.

The lower bound on the parameters to be estimated is always zero, since Pareto/NBD parameters cannot be negative. The upper bound can be set with the max.param.value parameter.

This function may take some time to run. It uses optimx for maximum likelihood estimation, not optim.

Value

Unnamed vector of estimated parameters by default, optimx object with everything if hessian is TRUE.

References

Fader, Peter S.; Hardie, and Bruce G.S.. "Overcoming the BG/NBD Model's #NUM! Error Problem." December. 2013. Web. http://brucehardie.com/notes/027/bgnbd_num_error.pdf

See Also

pnbd.cbs.LL

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# starting-point parameters
startingparams <- c(0.5, 6, 0.9, 8)

# estimated parameters
est.params <- pnbd.EstimateParameters(cal.cbs = cal.cbs, 
                                      par.start = startingparams, 
                                      method = 'L-BFGS-B',
                                      hardie = TRUE)
                                      
# complete object returned by \code{\link[optimx]{optimx}}
optimx.set <- pnbd.EstimateParameters(cal.cbs = cal.cbs, 
                                      par.start = startingparams, 
                                      hardie = TRUE, 
                                      hessian = TRUE)

# log-likelihood of estimated parameters
pnbd.cbs.LL(est.params, cal.cbs, TRUE)

Pareto/NBD Expectation

Description

Returns the number of repeat transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in a given time period.

Usage

pnbd.Expectation(params, t)

Arguments

Details

E(X(t) | r, alpha, s, beta)

Value

Number of repeat transactions a customer is expected to make in a time period of length t.

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

See Also

pnbd.ConditionalExpectedTransactions

Examples

params <- c(0.55, 10.56, 0.61, 11.64)

# Number of repeat transactions a customer is expected to make in 2 time intervals.
pnbd.Expectation(params = params, 
                 t = 2)

# We can also compare expected transactions over time:
pnbd.Expectation(params = params, 
                 t = 1:10)

Pareto/NBD Expected Cumulative Transactions

Description

Calculates the expected cumulative total repeat transactions by all customers for the calibration and holdout periods.

Usage

pnbd.ExpectedCumulativeTransactions(params, T.cal, T.tot, n.periods.final)

Arguments

Details

The function automatically divides the total period up into n.periods.final time intervals. n.periods.final does not have to be in the same unit of time as the T.cal data. For example: - if your T.cal data is in weeks, and you want cumulative transactions per week, n.periods.final would equal T.star. - if your T.cal data is in weeks, and you want cumulative transactions per day, n.periods.final would equal T.star * 7.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Vector of expected cumulative total repeat transactions by all customers.

See Also

pnbd.Expectation

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

params <- c(0.55, 10.56, 0.61, 11.64)

# Returns a vector containing cumulative repeat transactions for 546 days.
# All parameters are in weeks; the calibration period lasted 39 weeks
# and the holdout period another 39.
pnbd.ExpectedCumulativeTransactions(params = params, 
                                    T.cal = cal.cbs[,"T.cal"], 
                                    T.tot = 78, 
                                    n.periods.final = 546)

Pareto/NBD Log-Likelihood

Description

Calculates the log-likelihood of the Pareto/NBD model.

Usage

pnbd.LL(params, x, t.x, T.cal, hardie = TRUE)

Arguments

Value

A vector of log-likelihoods as long as the longest input vector (x, t.x, or T.cal).

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

See Also

pnbd.EstimateParameters

Examples

# Returns the log likelihood of the parameters for a customer who
# made 3 transactions in a calibration period that ended at t=6,
# with the last transaction occurring at t=4.
pnbd.LL(params, x=3, t.x=4, T.cal=6, hardie = TRUE)

# We can also give vectors as function parameters:
set.seed(7)
x <- sample(1:4, 10, replace = TRUE)
t.x <- sample(1:4, 10, replace = TRUE)
T.cal <- rep(4, 10)
pnbd.LL(params, x, t.x, T.cal, hardie = TRUE)

Pareto/NBD P(Alive)

Description

Uses Pareto/NBD model parameters and a customer's past transaction behavior to return the probability that they are still alive at the end of the calibration period.

Usage

pnbd.PAlive(params, x, t.x, T.cal, hardie = TRUE)

Arguments

Details

P(Alive | X=x, t.x, T.cal, r, alpha, s, beta)

x, t.x, and T.cal may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability that the customer is still alive at the end of the calibration period. If x, t.x, and/or T.cal has a length greater than one, then this will be a vector of probabilities (containing one element matching each element of the longest input vector).

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

Examples

data(cdnowSummary)
cbs <- cdnowSummary$cbs
params <- pnbd.EstimateParameters(cbs, hardie = TRUE)

pnbd.PAlive(params, x=0, t.x=0, T.cal=39, TRUE)
# 0.2941633; P(Alive) of a customer who made no repeat transactions.

pnbd.PAlive(params, x=23, t.x=39, T.cal=39, TRUE)
# 1; P(Alive) of a customer who has the same recency and total
# time observed.

pnbd.PAlive(params, x=5:20, t.x=30, T.cal=39, TRUE)
# Note the "increasing frequency paradox".

# To visualize the distribution of P(Alive) across customers:
p.alives <- pnbd.PAlive(params, cbs[,"x"], cbs[,"t.x"], cbs[,"T.cal"], TRUE)
plot(density(p.alives))

Pareto/NBD Plot Discounted Expected Residual Transactions

Description

Plots discounted expected residual transactions for different combinations of calibration period frequency and recency.

Usage

pnbd.Plot.DERT(params, x, t.x, T.cal, d, hardie = TRUE, type = "persp")

Arguments

Details

The length of the calibration period T.cal must be a single value, not a vector.

Value

A matrix with discounted expected residual transaction values for every combination of calibration period frequency x and calibration period recency t.x.

References

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. "RFM and CLV: Using Iso-Value Curves for Customer Base Analysis." Journal of Marketing Research Vol.42, pp.415-430. November. 2005. http://www.brucehardie.com/papers.html

Note that this paper refers to what this package is calling discounted expected residual transactions (DERT) simply as discounted expected transactions (DET).

Examples

# The RFM and CLV paper uses all 78 weeks of the cdnow data to
# estimate parameters. These parameters can be estimated as follows:
# elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD2"),2,3)
# elog[, 'date'] <- as.Date(elog[, 'date'], format = '%Y%m%d')
# cal.cbs <- dc.ElogToCbsCbt(elog)$cal$cbs
# pnbd.EstimateParameters(cal.cbs, hardie = TRUE)

# (The final function was run several times with its own output as
# input for starting parameters, to ensure that the result converged).

params <- c(0.5629966, 12.5590370, 0.4081095, 10.5148048)

# 15% compounded annually has been converted to 0.0027 compounded continously,
# as we are dealing with weekly data and not annual data.
d <- 0.0027

pnbd.Plot.DERT(params = params, 
               x = 0:14, 
               t.x = 0:77, 
               T.cal = 77.86, 
               d = d, 
               hardie = TRUE, 
               type = "persp")
pnbd.Plot.DERT(params = params, 
               x = 0:14, 
               t.x = 0:77, 
               T.cal = 77.86, 
               d = d, 
               hardie = TRUE, 
               type="contour")

Pareto/NBD Plot Dropout Rate Heterogeneity

Description

Plots and returns the estimated gamma distribution of mu (customers' propensities to drop out).

Usage

pnbd.PlotDropoutRateHeterogeneity(params, lim = NULL)

Arguments

Details

This returns the distribution of each customer's exponential parameter that determines their lifetime (using the Pareto/NBD assumption that a customer's lifetime can be modeled with an exponential distribution).

Value

Distribution of customers' propensities to drop out.

Examples

params <- c(0.55, 10.56, 0.61, 11.64)
pnbd.PlotDropoutRateHeterogeneity(params)
params <- c(0.55, 10.56, 3, 11.64)
pnbd.PlotDropoutRateHeterogeneity(params)

Pareto/NBD Plot Frequency vs. Conditional Expected Frequency

Description

Plots the actual and conditional expected number transactions made by customers in the holdout period, binned according to calibration period frequencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

pnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  hardie = TRUE,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

Arguments

Details

This function requires a censor number, which cannot be higher than the highest frequency in the calibration period CBS. The output matrix will have (censor + 1) bins, starting at frequencies of 0 transactions and ending at a bin representing calibration period frequencies at or greater than the censor number.

Value

Holdout period transaction frequency comparison matrix (actual vs. expected).

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# number of transactions by each customer in the 39 weeks
# following the calibration period
x.star <- cal.cbs[,"x.star"]

# parameters estimated using pnbd.EstimateParameters
est.params <- cdnowSummary$est.params
# the maximum censor number that can be used
max(cal.cbs[,"x"])

# plot conditional expected holdout period frequencies,
# binned according to calibration period frequencies
pnbd.PlotFreqVsConditionalExpectedFrequency(params = est.params, 
                                            T.star = 39, 
                                            cal.cbs = cal.cbs, 
                                            x.star = x.star, 
                                            censor = 7, 
                                            hardie = TRUE)

Pareto/NBD Plot Frequency in Calibration Period

Description

Plots a histogram and returns a matrix comparing the actual and expected number of customers who made a certain number of repeat transactions in the calibration period, binned according to calibration period frequencies.

Usage

pnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor,
  hardie = TRUE,
  plotZero = TRUE,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

Arguments

Details

This function requires a censor number, which cannot be higher than the highest frequency in the calibration period CBS. The output matrix will have (censor + 1) bins, starting at frequencies of 0 transactions and ending at a bin representing calibration period frequencies at or greater than the censor number. The plot may or may not include a bin for zero frequencies, depending on the plotZero parameter.

Value

Calibration period repeat transaction frequency comparison matrix (actual vs. expected).

Examples

data(cdnowSummary)
cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# parameters estimated using pnbd.EstimateParameters
est.params <- cdnowSummary$est.params
# the maximum censor number that can be used
max(cal.cbs[,"x"])

pnbd.PlotFrequencyInCalibration(params = est.params, 
                                cal.cbs = cal.cbs, 
                                censor = 7, 
                                hardie = TRUE)

Plot Pareto/NBD Rate Heterogeneity

Description

A helper for plotting either the estimated gamma distribution of mu (customers' propensities to drop out), or the estimated gamma distribution of lambda (customers' propensities to purchase).

Usage

pnbd.PlotRateHeterogeneity(params, func, lim = NULL)

Arguments

Value

Depending on the value of func, either the distribution of customers' propensities to purchase or the distribution of customers' propensities to drop out.

See Also

pnbd.PlotDropoutRateHeterogeneity

pnbd.PlotTransactionRateHeterogeneity

Pareto/NBD Plot Actual vs. Conditional Expected Frequency by Recency

Description

Plots the actual and conditional expected number of transactions made by customers in the holdout period, binned according to calibration period recencies. Also returns a matrix with this comparison and the number of customers in each bin.

Usage

pnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  hardie = TRUE,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)

Arguments

Details

This function does bin customers exactly according to recency; it bins customers according to integer units of the time period of cal.cbs. Therefore, if you are using weeks in your data, customers will be binned as follows: customers with recencies between the start of the calibration period (inclusive) and the end of week one (exclusive); customers with recencies between the end of week one (inclusive) and the end of week two (exlusive); etc.

The matrix and plot will contain the actual number of transactions made by each bin in the holdout period, as well as the expected number of transactions made by that bin in the holdout period, conditional on that bin's behavior during the calibration period.

Value

Matrix comparing actual and conditional expected transactions in the holdout period.

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# number of transactions by each customer in the 39 weeks following
# the calibration period
x.star <- cal.cbs[,"x.star"]

# parameters estimated using pnbd.EstimateParameters
est.params <- cdnowSummary$est.params

# plot conditional expected holdout period transactions, binned according to
# calibration period recencies
pnbd.PlotRecVsConditionalExpectedFrequency(params = est.params, 
                                           cal.cbs = cal.cbs, 
                                           T.star = 39, 
                                           x.star = x.star, 
                                           hardie = TRUE)

Pareto/NBD Tracking Cumulative Transactions Plot

Description

Plots the actual and expected cumulative total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

pnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  n.periods.final = NA,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions"
)

Arguments

Details

actual.cu.tracking.data does not have to be in the same unit of time as the T.cal data. T.tot will automatically be divided into periods to match the length of actual.cu.tracking.data. See pnbd.ExpectedCumulativeTransactions.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Matrix containing actual and expected cumulative repeat transactions.

Examples

data(cdnowSummary)

cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# Cumulative repeat transactions made by all customers across calibration
# and holdout periods
cu.tracking <- cdnowSummary$cu.tracking

# parameters estimated using pnbd.EstimateParameters
est.params <- cdnowSummary$est.params

# All parameters are in weeks; the calibration period lasted 39
# weeks and the holdout period another 39.
pnbd.PlotTrackingCum(params = est.params, 
                     T.cal = cal.cbs[,"T.cal"], 
                     T.tot = 78, 
                     actual.cu.tracking.data = cu.tracking)

Pareto/NBD Tracking Incremental Transactions Comparison

Description

Plots the actual and expected incremental total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

pnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  n.periods.final = NA,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions"
)

Arguments

Details

actual.inc.tracking.data does not have to be in the same unit of time as the T.cal data. T.tot will automatically be divided into periods to match the length of actual.inc.tracking.data. See pnbd.ExpectedCumulativeTransactions.

The holdout period should immediately follow the calibration period. This function assume that all customers' calibration periods end on the same date, rather than starting on the same date (thus customers' birth periods are determined using max(T.cal) - T.cal rather than assuming that it is 0).

Value

Matrix containing actual and expected incremental repeat transactions.

Examples

data(cdnowSummary)
cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# Cumulative repeat transactions made by all customers across calibration
# and holdout periods
cu.tracking <- cdnowSummary$cu.tracking
# make the tracking data incremental
inc.tracking <- dc.CumulativeToIncremental(cu.tracking)

# parameters estimated using pnbd.EstimateParameters
est.params <- cdnowSummary$est.params

# All parameters are in weeks; the calibration period lasted 39
# weeks and the holdout period another 39.
pnbd.PlotTrackingInc(params = est.params, 
                     T.cal = cal.cbs[,"T.cal"], 
                     T.tot = 78, 
                     actual.inc.tracking.data = inc.tracking)

Pareto/NBD Plot Transaction Rate Heterogeneity

Description

Plots and returns the estimated gamma distribution of lambda (customers' propensities to purchase).

Usage

pnbd.PlotTransactionRateHeterogeneity(params, lim = NULL)

Arguments

Details

This returns the distribution of each customer's Poisson parameter, which determines the level of their purchasing (using the Pareto/NBD assumption that purchasing on the individual level can be modeled with a Poisson distribution).

Value

Distribution of customers' propensities to purchase.

Examples

params <- c(0.55, 10.56, 0.61, 11.64)
pnbd.PlotTransactionRateHeterogeneity(params)
params <- c(3, 10.56, 0.61, 11.64)
pnbd.PlotTransactionRateHeterogeneity(params)

Pareto/NBD Log-Likelihood

Description

Calculates the log-likelihood of the Pareto/NBD model.

Usage

pnbd.cbs.LL(params, cal.cbs, hardie = TRUE)

Arguments

Value

The log-likelihood of the provided data.

References

Fader, Peter S., and Bruce G.S. Hardie. "A Note on Deriving the Pareto/NBD Model and Related Expressions." November. 2005. Web. http://www.brucehardie.com/notes/008/

See Also

pnbd.EstimateParameters

pnbd.LL

Examples

data(cdnowSummary)
cal.cbs <- cdnowSummary$cbs
# cal.cbs already has column names required by method

# random assignment of parameters
params <- c(0.5, 8, 0.7, 10)
# returns the log-likelihood of the given parameters
pnbd.cbs.LL (params, cal.cbs, TRUE)

# compare the speed and results to the following:
cal.cbs.compressed <- dc.compress.cbs(cal.cbs)
pnbd.cbs.LL (params, cal.cbs.compressed, TRUE)

Define general parameters

Description

This is to ensure consistency across all functions that require the likelihood function, or the log of it, and to make sure that the same implementation of the hypergeometric function is used everywhere for building A0.

Usage

pnbd.generalParams(params, x, t.x, T.cal, func, hardie = TRUE)

Arguments

Details

This function is only ever called by either pnbd.LL or pnbd.PAlive so it returns directly the output that is expected from those calling functions: either the log likelihood for a set of customers, or the probability that a set of customers with characteristics given by x, t.x and T.cal, having estimated a set of params, is still alive. Either set of customers can be of size 1.

Value

A vector of log likelihood values if func is pnbd.LL, or a vector of probabilities that a customer is still alive if func is pnbd.PAlive.

See Also

pnbd.LL

pnbd.PAlive

pnbd.DERT

Pareto/NBD Probability Mass Function

Description

Probability mass function for the Pareto/NBD.

Usage

pnbd.pmf(params, t, x, hardie = TRUE)

Arguments

Details

P(X(t)=x | r, alpha, s, beta). Returns the probability that a customer makes x repeat transactions in the time interval (0, t].

Parameters t and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of X(t)=x conditional on model parameters. If t and/or x has a length greater than one, a vector of probabilities will be returned.

References

Fader, Peter S., and Bruce G.S. Hardie. “Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model.” Sept. 2006. Web. http://www.brucehardie.com/notes/012/

Examples

params <- c(0.55, 10.56, 0.61, 11.64)
# probability that a customer will make 10 repeat transactions in the
# time interval (0,2]
pnbd.pmf(params, t=2, x=10, hardie = TRUE)
# probability that a customer will make no repeat transactions in the
# time interval (0,39]
pnbd.pmf(params, t=39, x=0, hardie = TRUE)

# Vectors may also be used as arguments:
pnbd.pmf(params = params, 
         t = 30, 
         x = 11:20, 
         hardie = TRUE)

Generalized Pareto/NBD Probability Mass Function

Description

Generalized probability mass function for the Pareto/NBD.

Usage

pnbd.pmf.General(params, t.start, t.end, x, hardie = TRUE)

Arguments

Details

P(X(t.start, t.end)=x | r, alpha, s, beta). Returns the probability that a customer makes x repeat transactions in the time interval (t.start, t.end].

It is impossible for a customer to make a negative number of repeat transactions. This function will return an error if it is given negative times or a negative number of repeat transactions. This function will also return an error if t.end is less than t.start.

t.start, t.end, and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of x transaction occuring between t.start and t.end conditional on model parameters. If t.start, t.end, and/or x has a length greater than one, a vector of probabilities will be returned.

References

Fader, Peter S., and Bruce G.S. Hardie. "Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model." Sept. 2006. Web. http://www.brucehardie.com/notes/012/

Fader, Peter S., Bruce G.S. Hardie, and Kinshuk Jerath. "Deriving an Expression for P (X(t, t + tau) = x) Under the Pareto/NBD Model." Sept. 2006. Web. http://www.brucehardie.com/notes/013/

Examples

# probability that a customer will make 10 repeat transactions in the
# time interval (1,2]
data("cdnowSummary")
cal.cbs <- cdnowSummary$cbs
params <- pnbd.EstimateParameters(cal.cbs = cal.cbs, 
                                  method = "L-BFGS-B",
                                  hardie = TRUE)
pnbd.pmf.General(params, t.start=1, t.end=2, x=10, hardie = TRUE)
# probability that a customer will make no repeat transactions in the
# time interval (39,78]
pnbd.pmf.General(params, 
                 t.start = 39, 
                 t.end = 78, 
                 x = 0, 
                 hardie = TRUE)

Spend Parameter Estimation

Description

Estimates parameters for the gamma-gamma spend model.

Usage

spend.EstimateParameters(
  m.x.vector,
  x.vector,
  par.start = c(1, 1, 1),
  max.param.value = 10000
)

Arguments

Details

The best-fitting parameters are determined using the spend.LL function. The sum of the log-likelihood for each customer (for a set of parameters) is maximized in order to estimate parameters.

A set of starting parameters must be provided for this method. If no parameters are provided, (1,1,1,1) is used as a default. It may be necessary to run the estimation from multiple starting points to ensure that it converges. To compare the log-likelihoods of different parameters, use spend.LL.

The lower bound on the parameters to be estimated is always zero, since gamma-gamma parameters cannot be negative. The upper bound can be set with the max.param.value parameter.

Value

Vector of estimated parameters.

Examples

## Not run: 
data(cdnowSummary)
ave.spend <- cdnowSummary$m.x
tot.trans <- cdnowSummary$cbs[,"x"]

# There will be many warnings due to the zeroes that are
# included in the data above. To avoid them, use the following:
# (see example for spend.LL)

ave.spend <- ave.spend[which(tot.trans > 0)]
tot.trans <- tot.trans[which(tot.trans > 0)]

# We will let the spend function use default starting parameters
spend.EstimateParameters(ave.spend, tot.trans)

## End(Not run)

Spend Log-Likelihood

Description

Calculates the log-likelihood of the gamma-gamma model for customer spending.

Usage

spend.LL(params, m.x, x)

Arguments

Details

m.x and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of log-likelihoods.

Value

The log-likelihood of the gamma-gamma model. If m.x or x has a length greater than 1, this is a vector of log-likelihoods.

References

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.” Journal of Marketing Research Vol.42, pp.415-430. November. 2005. Web.

Examples

## Not run: 
data(cdnowSummary)
ave.spend <- cdnowSummary$m.x
tot.trans <- cdnowSummary$cbs[,"x"]
# params <- c(6.25, 3.74, 15.44) # in original documentation. check below:
params <- spend.EstimateParameters(m.x.vector = ave.spend, x.vector = tot.trans)
# get the total log-likelihood of the data and parameters
# above. There will be many warnings due to the zeroes that are
# included in the data. If you wish to avoid these warnings, use:

# ave.spend <- ave.spend[which(tot.trans > 0)]
# tot.trans <- tot.trans[which(tot.trans > 0)]

# Note that we used tot.trans to remove the zeroes from ave.spend.
# This is because we need the vectors to be the same length, and it
# is possible that your data include customers who made transactions
# worth zero dollars (in which case the vector lengths would differ
# if we used ave.spend to remove the zeroes from ave.spend).

sum(spend.LL(params, ave.spend, tot.trans))

# This log-likelihood may be different than mentioned in the
# referenced paper; in the paper, a slightly different function
# which relies on total spend (not average spend) is used.

## End(Not run)

Conditional expected transaction value

Description

Calculates the expected transaction value for a customer, conditional on the number of transaction and average transaction value during the calibration period.

Usage

spend.expected.value(params, m.x, x)

Arguments

Details

E(M | p, q, gamma, m.x, x).

m.x and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

The expected transaction value for a customer conditional on their transaction behavior during the calibration period. If m.x or x has a length greater than one, then a vector of expected transaction values will be returned.

References

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.” Journal of Marketing Research Vol.42, pp.415-430. November. 2005. Web.

Examples

## Not run: 
data(cdnowSummary)
ave.spend <- cdnowSummary$m.x
tot.trans <- cdnowSummary$cbs[,"x"]
# params <- c(6, 4, 16); # in original documentation. rounded values of:
params <- spend.EstimateParameters(m.x.vector = ave.spend, x.vector = tot.trans);
# calculate the expected transaction value of a customer
# who spent an average of $35 over 3 transactions.
spend.expected.value(params, m.x=35, x=3)

# m.x and x may be vectors:
spend.expected.value(params, m.x=30:40, x=3)
spend.expected.value(params, m.x=35, x=1:10)
spend.expected.value(params, m.x=30:40, x=1:11)

## End(Not run)

Define general parameters

Description

This is to ensure consistency across all spend functions.

Usage

spend.generalParams(params, func, m.x, x)

Arguments

Details

This function is only ever called by functions defined in the original BTYD package, such as spend.LL, spend.marginal.likelihood or spend.expected.value so it returns directly the output that is expected from those calling functions.

Value

That depends on func: 1. If func is spend.marginal.likelihood, the marginal distribution of a customer's average transaction value (if m.x or x has a length greater than 1, a vector of marginal likelihoods will be returned). 2. If func is spend.LL, the log-likelihood of the gamma-gamma model; if m.x or x has a length greater than 1, this is a vector of log-likelihoods. 3. If func is spend.expected.value, the expected transaction value for a customer conditional on their transaction behavior during the calibration period. If m.x or x has a length greater than one, then a vector of expected transaction values will be returned.

See Also

spend.LL

spend.marginal.likelihood

Gamma-gamma marginal likelihood

Description

Calculates the marginal likelihood of a customer's average transaction value.

Usage

spend.marginal.likelihood(params, m.x, x)

Arguments

Details

m.x and x may be vectors. The standard rules for vector operations apply - if they are not of the same length, the shorter vector will be recycled (start over at the first element) until it is as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

This function will issue a warning if any of m.x or x is 0, and will return a marginal likelihood of 0 for those values.

f(m.x | p, q, gamma, x).

Value

The marginal distribution of a customer's average transaction value. If m.x or x has a length greater than 1, a vector of marginal likelihoods will be returned.

References

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.” Journal of Marketing Research Vol.42, pp.415-430. November. 2005. Web.

See equation 3.

Examples

params <- c(6, 4, 16)

# calculate the marginal distribution of the average transaction value
# of a customer who spent an average of $35 over 3 transactions.
spend.marginal.likelihood(params, m.x=35, x=3)

# Several values can also be computed at once:
spend.marginal.likelihood(params, m.x=30:40, x=3)
spend.marginal.likelihood(params, m.x=35, x=1:10)
spend.marginal.likelihood(params, m.x=30:40, x=1:11)

Plot Actual vs. Expected Average Transaction Value

Description

Plots the actual and expected densities of average transaction values, and returns a vector with each customer's average transaction value probability.

Usage

spend.plot.average.transaction.value(
  params,
  m.x.vector,
  x.vector,
  xlab = "Average Transaction Value",
  ylab = "Marginal Distribution of Average Transaction Value",
  title = "Actual vs. Expected Average Transaction Value Across Customers"
)

Arguments

Value

a vector with the probability of each customer's average transaction value.

See Also

spend.marginal.likelihood

Examples

## Not run: 
data(cdnowSummary)
ave.spend <- cdnowSummary$m.x
tot.trans <- cdnowSummary$cbs[,"x"]
# params <- c(6.25, 3.74, 15.44) # in original documentation. check below:
params <- spend.EstimateParameters(m.x.vector = ave.spend, x.vector = tot.trans)

# Plot the actual and expected average transaction value across customers.
f.m.x <- spend.plot.average.transaction.value(params, ave.spend, tot.trans)

## End(Not run)

Subtract Logs

Description

Given log(a) and log(b), returns log(a - b)

Usage

subLogs(loga, logb)

Arguments