Data Generation

Description

Generating data from COST DGP, assuming Markov process of order one

Usage

Data.COST(n,n.total,seed1,coord,par.t)

Arguments

Value

Author(s)

Yanlin Tang, Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Examples

library(COST)
n = 500
n.total = 1001
seed1 = 22222
coord = cbind(rep(c(1,3,5)/6,each=3),rep(c(1,3,5)/6,3))
par.t = c(0,1,1,0.5,1.5,100)
dat = Data.COST(n,n.total,seed1,coord,par.t)
#it returns a data set with dimension 501*9

one-step ahead forecast by separate time series analysis

Description

one-step ahead forecast, analyzing the time series at each location separately with a t copula, including: (i) point forecast, either conditional median or mean; (ii) 95% forecast intervals, which can also be adjusted by the users; (iii) m (m=500 by default) random draws from the conditional distribution for each location, can be used for multivariate rank after combining all the locations together

Usage

Forecasts.CF(par,Y,seed1,m)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

one-step ahead forecast by Gaussian copula

Description

one-step ahead forecast by Gaussian copula, including: (i) point forecast, either conditional median or mean; (ii) 95% forecast intervals, which can also be adjusted by the users; (iii) m (m=500 by default) random draws from the conditional distribution, can be used for multivariate rank

Usage

Forecasts.COST.G(par,Y,s.ob,seed1,m,isotropic)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

one-step ahead forecast by t copula

Description

one-step ahead forecast by t copula, including: (i) point forecast, either conditional median or mean; (ii) 95% forecast intervals, which can also be adjusted by the users; (iii) m (m=500 by default) random draws from the conditional distribution, can be used for multivariate rank

Usage

Forecasts.COST.t(par,Y,s.ob,seed1,m,isotropic)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

one-step ahead forecast by Gaussian process fitting

Description

one-step ahead forecast by Gaussian process fitting, including: (i) point forecast, either conditional mean; (ii) 95% forecast intervals, which can also be adjusted by the users; (iii) m (m=500 by default) random draws from the conditional distribution, can be used for multivariate rank

Usage

Forecasts.GP(par,Y,s.ob,seed1,m,isotropic)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

new location prediction by Gaussian copula

Description

new location prediction by Gaussian copula, where the copula dimension is extended, and the marginal CDF of the new location is estimated by neighboring information; it gives 0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n, conditional on observed locations at time n-1 and n; both point and interval predictions are provided

Usage

Predictions.COST.G(par,Y,s.ob,s.new,isotropic)

Arguments

Value

0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

new location prediction by t copula

Description

new location prediction by t copula, where the copula dimension is extended, and the marginal CDF of the new location is estimated by neighboring information; it gives 0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n, conditional on observed locations at time n-1 and n; both point and interval predictions are provided

Usage

Predictions.COST.t(par,Y,s.ob,s.new,isotropic)

Arguments

Value

0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

new location prediction by Gaussian process method

Description

new location prediction by Gaussian process method, and the marginal mean and variance of the new location is estimated by neighboring information; it gives 0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n, conditional on observed locations at time n-1 and n; both point and interval predictions are provided

Usage

Predictions.GP(par,Y,s.ob,s.new,isotropic)

Arguments

Value

0.025-, 0.975- and 0.5-th conditional quantiles of the conditional distribution for each new location, at time n

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Wind speed data from 10 sites

Description

The data set is a subset of the data we used in the paper, with 10 sites and 6-month long time series.

Usage

data(Wind6month)

Format

A 4320*10 matrix from 10 locations, date ranges from Sep 22, 2014 to Dec 20, 2014, 180 days

BiddleButte

wind speed from site BiddleButte

ForestGrove

wind speed from site ForestGrove

HoodRiver

wind speed from site HoodRiver

HorseHeaven

wind speed from site HorseHeaven

Megler

wind speed from site Megler

NaselleRidge

wind speed from site NaselleRidge

Roosevelt

wind speed from site Roosevelt

Shaniko

wind speed from site Shaniko

Sunnyside

wind speed from site Sunnyside

Tillamook

wind speed from site Tillamook

Source

https://transmission.bpa.gov/business/operations/wind/MetData/default.aspx

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Examples

data(Wind6month)
Y.ob = Wind6month[,-3]
Y.newloc = Wind6month[,3]
dim(Y.ob) #4320*9, data at 9 locations, with length 4320 (hours)
length(Y.newloc) #4320, time series at the new location

example for one-step ahead forecast

Description

Example for one-step ahead forecast for Gaussian Process and our COST method with Gaussian and t copulas, where the data are generated from COST DGP, where the parameters are assumed to be known; the parameters can be obtained by the “optim" function. Assuming that data are observed at d=9 locations, and n+1 time points, where the last time point is for validation.

Usage

example.forecast(n,n.total,seed1)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Examples

library(COST)
#settings
seed1 = 2222222
n.total = 101 #number of total time points, including the burning sequence
n = 50 #number of time points we observed
example.forecast(n,n.total,seed1)
#OUTPUTS

# $COST.t.fore.ECP #whether the forecast interval includes the true value at n+1
# [1] 1 1 1 1 1 1 1 1 1
#
# $COST.t.fore.ML #length of the forecast interval
# [1] 0.7036 4.1318 4.8749 2.7615 3.7398 5.8186 4.4532 4.9251 6.3757
#
# $COST.t.fore.rank #multivariate rank
# [1] 162
#
#
# $COST.G.fore.ECP #whether the forecast interval includes the true value at n+1
# [1] 1 1 1 1 1 1 1 1 1
#
# $COST.G.fore.ML #length of the forecast interval
# [1]  0.7035 4.1316 4.8656 2.7611 3.7388 5.7913 4.4458 4.9036 6.3727
#
# $COST.G.fore.rank #multivariate rank
# [1] 186
#

# $GP.fore.ECP #whether the forecast interval includes the true value at n+1
# [1] 1 0 0 1 1 1 1 1 1
#
# $GP.fore.ML #length of the forecast interval
# [1] 0.4879 2.0449 3.4436 2.2107 2.9170 4.4537 4.2169 5.5789 7.3689
#
# $GP.fore.rank #multivariate rank
# [1] 17

example for new location prediction

Description

Example for new location prediction, Gaussian process method, and our COST method with Gaussian and t copulas, where the parameters are assumed to be known; the parameters can be obtained by the “optim" function. Data are generated at 13 locations and n time points, and assume that 9 locations are observed, and 4 new locations need prediction at time n, conditional on 9 locations at time points n-1 and n.

Usage

example.prediction(n,n.total,seed1)

Arguments

Value

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Examples

library(COST)
#settings
n.total = 101 #number of total time points, including the burning sequence
n = 50 #number of time points we observed
seed1 = 22222
example.prediction(n,n.total,seed1)

#OUTPUTS

# $COST.t.pre.ECP #whether the prediction interval includes the true value, time point n
# [1] 1 1 1 1
#
# $COST.t.pre.ML #length of the prediction interval
# [1] 1.445576 2.146452 2.260688 2.706681
#
# $COST.t.pre.med.error #point prediction error, using conditional median
# [1]  0.01127162 -0.03222058 -0.22081051  0.57831480
#
# $COST.G.pre.ECP #whether the prediction interval includes the true value, time point n
# [1] 1 1 1 1
#
# $COST.G.pre.ML #length of the prediction interval
# [1] 1.445576 2.432646 2.260688 2.914887
#
# $COST.G.pre.med.error #point prediction error, using conditional median
# [1] 0.01127162 -0.03222058 -0.22081051  0.57831480
#
# $GP.pre.ECP #whether the prediction interval includes the true value, time point n
# [1] 1 1 1 1
#
# $GP.pre.ML #length of the prediction interval
# [1] 0.8345359 1.4096642 1.5948724 2.3419428
#
# $GP.pre.med.error #point prediction error, using conditional median
# [1] 0.09447685 -0.05889409 -0.08923935  0.58494684

Locations of 10 sites

Description

Locations of 10 sites.

Usage

data(location)

Format

Locations of 10 sites, 10*2 matrix in Cartesian coordinate system

Source

https://transmission.bpa.gov/business/operations/wind/MetData/default.aspx

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

Examples

s.ob = location[-3,2:3]
s.new = location[3,2:3]

negtive log-likelihood for separate time series analysis

Description

negtive log-likelihood for separate time series analysis, copula-based semiparametric method from Chen and Fan (2006), assuming t copula for each time series and Markov process of order one, with marginal distribution estimated by espirical CDF, and it is for correlation parameter estimation

Usage

logL.CF(par,Yk,dfs)

Arguments

Value

the negative log-likelihood

Author(s)

Yanlin Tang and Huixia Judy Wang

References

1.Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. Journal of Econometrics 130, 307–335.\ 2.Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

negtive log-likelihood for Gaussian copula

Description

gives the negtive log-likelihood of the Gaussian copula, with empirical CDF plugin, and it is for parameter estimation in the correlation matrix

Usage

logL.COST.G(par,Y,s.ob)

Arguments

Value

the negative log-likelihood

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

negtive log-likelihood for t copula

Description

gives the negtive log-likelihood of the t copula, with empirical CDF plugin, and it is for parameter estimation in the correlation matrix

Usage

logL.COST.t(par,Y,s.ob)

Arguments

Value

the negative log-likelihood

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

negtive log-likelihood of Gaussian process

Description

negtive log-likelihood of Gaussian process, with mean vector and variance vector obtained by the empirical version, and it is for parameter estimation in the correlation matrix

Usage

logL.GP(par,Y,s.ob)

Arguments

Value

the negative log-likelihood

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.

multivariate rank of a vector

Description

calculating the multivariate rank of a vector among a set of vectors, used to evaluate the performance of conditional distribution, and the rank would be uniform when the conditional distribution is estimated well

Usage

rank.multivariate(y.test,y.random,seed1)

Arguments

Value

the multivariate rank of the observed (verifying) vector at time n+1

Author(s)

Yanlin Tang and Huixia Judy Wang

References

Yanlin Tang, Huixia Judy Wang, Ying Sun, Amanda Hering. Copula-based semiparametric models for spatio-temporal data.