| Title: | Statistical Tests for Functional Time Series |
| Version: | 0.1.0 |
| Description: | A collection of statistical hypothesis tests of functional time series. While it will include more tests when the related literature are enriched, this package contains the following key tests: functional stationarity test, functional trend stationarity test, functional unit root test, to name a few. |
| License: | GPL-2 |
| Encoding: | UTF-8 |
| Language: | en-US |
| RoxygenNote: | 7.1.1 |
| Depends: | R (≥ 2.10) |
| Imports: | e1071 |
| Suggests: | testthat (≥ 3.0.0) |
| Config/testthat/edition: | 3 |
| NeedsCompilation: | no |
| Packaged: | 2021-08-18 05:10:54 UTC; cspun |
| Author: | Yichao Chen [aut], Chi Seng Pun [cre, aut] |
| Maintainer: | Chi Seng Pun <cspun@ntu.edu.sg> |
| Repository: | CRAN |
| Date/Publication: | 2021-08-19 08:10:05 UTC |
Test Statistic in Functional KPSS Test
Description
Calculate test statistic R_N for functional KPSS test, which Was constructed in Kokoszka and Young (2016) and Chen and Pun (2019).
Usage
FKPSS_teststat(X)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
Value
The value of test statistic R_N for functional KPSS test.
References
Chen, Y., & Pun, C. S. (2019). A bootstrap-based KPSS test for functional time series. Journal of Multivariate Analysis, 174, 104535.
Kokoszka, P., & Young, G. (2016). KPSS test for functional time series. Statistics, 50(5), 957-973.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
FST_teststat_TN(EX)
Test Statistic M_N in Functional Stationarity Test
Description
Calculate test statistic M_N for functional stationarity test, which was constructed in Horvath et al. (2014).
Usage
FST_teststat_MN(X)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
Value
The value of test statistic M_N calculated for functional stationarity test.
References
Horvath, L., Kokoszka, P., & Rice, G. (2014). Testing stationarity of functional time series. Journal of Econometrics, 179(1), 66-82.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
FST_teststat_MN(EX)
Test Statistic T_N in Functional Stationarity Test
Description
Calculate test statistic T_N for functional stationarity test, which was constructed in Horvath et al. (2014).
Usage
FST_teststat_TN(X)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
Value
The value of test statistic T_N calculated for functional stationarity test.
References
Horvath, L., Kokoszka, P., & Rice, G. (2014). Testing stationarity of functional time series. Journal of Econometrics, 179(1), 66-82.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
FST_teststat_TN(EX)
Integrals of Squared Brownian Bridge
Description
Generate a dataset of independent simulated values of \int_0^1{B^2(t)}dt, where B is a standard Brownian bridge on [0,1].
Generate a dataset of independent simulated values of \int_0^1{B^2(t)}dt, where B is a standard Brownian bridge on [0,1].
Usage
dataset_bb(NUM)
dataset_bb(NUM)
Arguments
NUM |
Number of simulated values generated in the dataset. |
Value
A vector with length equals to NUM.
A vector with length equals to NUM.
Examples
dataset_bb(10)
dataset_bb(10)
Integrals of Squared Brownian Motion
Description
Generate a dataset of independent simulated values of \int_0^1{W^2(t)}dt, where W is a standard Brownian motion on [0,1].
Usage
dataset_bm(NUM)
Arguments
NUM |
Number of simulated values generated in the dataset. |
Value
A vector with length equals to NUM.
Examples
dataset_bm(10)
Integrals of Squared De-meaned Brownian Bridge
Description
Generate a dataset of independent simulated values of \int_0^1{[B(t)-\int_0^1{B(x)dx}]^2}dt, where B is a standard Brownian bridge on [0,1].
Usage
dataset_dbb(NUM)
Arguments
NUM |
Number of simulated values generated in the dataset. |
Value
A vector with length equals to NUM.
Examples
dataset_dbb(10)
Integrals of Squared Second-level Brownian Bridge
Description
Generate a dataset of independent simulated values of \int_0^1{[W(t)+(2x-3x^2)W(1)+(-6x+6x^2)\int_0^1{W(x)dx}]^2}dt, where W is a standard Brownian motion on [0,1].
Usage
dataset_sbb(NUM)
Arguments
NUM |
Number of simulated values generated in the dataset. |
Value
A vector with length equals to NUM.
Examples
dataset_sbb(10)
Default Kernel Function
Description
Default kernel function used in calculating the estimated long-run covariance function.
Usage
default_kernel(x)
Arguments
x |
The variable. |
Value
The kernel value.
Examples
default_kernel(1)
default_kernel(0)
Eigenvalues of An Estimated Long-run Covariance Function
Description
Calculate the eigenvalues of an estimated long-run covariance function.
Usage
est_eigenvalue(este, K, h_power, estev)
Arguments
este |
Estimated errors in the long-run covariance function, inputted in a matrix form with each row representing each observation of the functional error values on equidistant points of any prespecified interval. |
K |
Kernel function in the estimation of the long-run covariance function. |
h_power |
Power of sample size 'N' (valued in (0,1)) for the smoothing bandwidth in the kernel function. |
estev |
Number of the largest eigenvalues chosen in the output. |
Value
A vector of first 'estev' largest eigenvalues in descending order.
Examples
N<-100
EE<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EE[i,1]<-temp[1]
for (j in 2:100) {
EE[i,j]<-EE[i,j-1]+temp[j]
}
}
est_eigenvalue(este=EE,K=default_kernel,h_power=1/2,estev=10)
Functional KPSS Test
Description
KPSS test for functional time series with different methods on determining the critical values of the test statistic. The Monte Carlo method (for a large sample size) was constructed in Kokoszka and Young (2016) and the bootstrap-based methods (both the simple bootstrap method and moving block bootstrap method, for a small/moderate sample size) were constructed in Chen and Pun (2019).
Usage
fkpsst(
X,
ALPHA = 0.05,
METHOD = "MC",
K = default_kernel,
h_power = 2/5,
est_ev = nrow(X),
MCNsim = 10000,
sbb_set = NULL,
M = 1000,
b = ceiling((2 * nrow(X))^(1/3))
)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
ALPHA |
Significance level of the test. The default value is 5%. |
METHOD |
Method to determine the critical value of the test statistic. The default value 'METHOD="MC"' represents the Monte Carlo method. 'METHOD="SB"' represents the simple bootstrap method and 'METHOD="MBB"' represents the moving block bootstrap method. |
K |
Kernel function in the estimation of the long-run covariance function, which is only effective in the Monte Carlo method. The default function is 'default_kernel' function in this package. |
h_power |
Power of sample size 'N' (valued in (0,1)) for the smoothing bandwidth, which is only effective in the Monte Carlo method. The default value is 2/5. |
est_ev |
Number of the largest eigenvalues chosen to estimate the limiting distribution, which is only effective in the Monte Carlo method. The default value is the sample size 'N'. |
MCNsim |
Number of Monte Carlo datasets generated in the Monte Carlo method, which is only effective in Monte Carlo method. The default value is 10000. |
sbb_set |
A vector of independent simulated data generated from the function 'dataset_sbb', which is only effective and essential in Monte Carlo method. |
M |
Number of bootstrap datasets generated in the bootstrap method, which is only effective in bootstrap methods. The default value is 1000. |
b |
Block length used in the moving block bootstrap method, which is only effective in the moving block bootstrap method. The default value is ceiling((2N)^(1/3)), where 'N' is the sample size. |
Value
The result of the test is presented with the value of test statistic and its p-value under the null hypothesis of trend stationarity.
References
Chen, Y., & Pun, C. S. (2019). A bootstrap-based KPSS test for functional time series. Journal of Multivariate Analysis, 174, 104535.
Kokoszka, P., & Young, G. (2016). KPSS test for functional time series. Statistics, 50(5), 957-973.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
EXI<-rep(0,100)
set.seed(1)
EXI[1]<-rnorm(1,0,1)
for (j in 2:100) {EXI[j]<-EXI[j-1]+rnorm(1,0,1)}
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
fkpsst(X=EX,METHOD="SB")
Functional Stationarity Test
Description
Stationarity test for functional time series with different methods on determining the critical values of the test statistic. The Monte Carlo method was constructed in Horvath et al. (2014), while the bootstrap-based methods have not been validated in the literature (although such an option is provided, please use them at your own risk).
Usage
fst(
X,
STAT = 1,
ALPHA = 0.05,
METHOD = "MC",
K = default_kernel,
h_power = 2/5,
est_ev = nrow(X),
MCNsim = 10000,
bb_set = NULL,
dbb_set = NULL,
M = 1000,
b = ceiling((2 * nrow(X))^(1/3))
)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
STAT |
Test statistic used in the stationarity test. The default value 'STAT=1' represents 'T_N' and 'STAT=2' represents 'M_N' in original paper. |
ALPHA |
Significance level of the test. The default value is 5%. |
METHOD |
Method to determine the critical value of the test statistic. The default value 'METHOD="MC"' represents the Monte Carlo method. 'METHOD="SB"' represents the simple bootstrap method and 'METHOD="MBB"' represents the moving block bootstrap method. |
K |
Kernel function in the estimation of the long-run covariance function, which is only effective in the Monte Carlo method. The default function is 'default_kernel' function in this package. |
h_power |
Power of sample size 'N' (valued in (0,1)) for the smoothing bandwidth, which is only effective in the Monte Carlo method. The default value is 2/5. |
est_ev |
Number of the largest eigenvalues chosen to estimate the limiting distribution, which is only effective in the Monte Carlo method. The default value is the sample size 'N'. |
MCNsim |
Number of Monte Carlo datasets generated in the Monte Carlo method, which is only effective in Monte Carlo method. The default value is 10000. |
bb_set |
A vector of independent simulated data generated from the function 'dataset_bb', which is only effective and essential in Monte Carlo method with 'STAT=1'. |
dbb_set |
A vector of independent simulated data generated from the function 'dataset_dbb', which is only effective and essential in Monte Carlo method with 'STAT=2'. |
M |
Number of bootstrap datasets generated in the bootstrap method, which is only effective in bootstrap methods. The default value is 1000. |
b |
Block length used in the moving block bootstrap method, which is only effective in the moving block bootstrap method. The default value is ceiling((2N)^(1/3)), where 'N' is the sample size. |
Value
The result of the test is presented with the value of test statistic and its p-value under the null hypothesis of local stationarity.
References
Horvath, L., Kokoszka, P., & Rice, G. (2014). Testing stationarity of functional time series. Journal of Econometrics, 179(1), 66-82.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
fst(X=EX,METHOD="SB")
Functional Unit Root Test
Description
Unit root test for functional time series with different methods on determining the critical values of the test statistic. The Monte Carlo method was constructed in Chen and Pun (2021), while the bootstrap-based methods have not been validated in the literature (although such an option is provided, please use them at your own risk).
Usage
furt(
X,
ALPHA = 0.05,
TYPE = "R",
METHOD = "MC",
K = default_kernel,
h_power = 2/5,
est_ev = nrow(X) - 1,
MCNsim = 10000,
bm_set = NULL,
M = 1000,
b = ceiling((2 * (nrow(X) - 1))^(1/3))
)
Arguments
X |
The functional time series being tested, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
ALPHA |
Significance level of the test. The default value is 5%. |
TYPE |
Type of hypothesis test. The default value 'TYPE="R"' represents the right-tailed test, which is used when the alternative hypothesis is trend stationarity model. 'TYPE="L"' represents the left-tailed test, which is used when the alternative hypothesis is simple stationarity model or AR(1) model. |
METHOD |
Method to determine the critical value of the test statistic. The default value 'METHOD="MC"' represents the Monte Carlo method. 'METHOD="SB"' represents the simple bootstrap method and 'METHOD="MBB"' represents the moving block bootstrap method. |
K |
Kernel function in the estimation of the long-run covariance function, which is only effective in the Monte Carlo method. The default function is 'default_kernel' function in this package. |
h_power |
Power of sample size 'N' (valued in (0,1)) for the smoothing bandwidth, which is only effective in the Monte Carlo method. The default value is 2/5. |
est_ev |
Number of the largest eigenvalues chosen to estimate the limiting distribution, which is only effective in the Monte Carlo method. The default value is the sample size 'N'. |
MCNsim |
Number of Monte Carlo datasets generated in the Monte Carlo method, which is only effective in Monte Carlo method. The default value is 10000. |
bm_set |
A vector of independent simulated data generated from the function 'dataset_bm', which is only effective and essential in Monte Carlo method. |
M |
Number of bootstrap datasets generated in the bootstrap method, which is only effective in bootstrap methods. The default value is 1000. |
b |
Block length used in the moving block bootstrap method, which is only effective in the moving block bootstrap method. The default value is ceiling((2N)^(1/3)), where 'N' is the sample size. |
Value
The result of the test is presented with the value of test statistic and its p-value under the null hypothesis of functional random walk.
References
Chen, Y., & Pun, C. S. (2021). Functional Unit Root Test. Available at SSRN.
Examples
N<-100
EE<-matrix(rep(0,N*100),ncol=100)
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EE[i,1]<-temp[1]
for (j in 2:100) {
EE[i,j]<-EE[i,j-1]+temp[j]
}
}
EX[1,]<-EE[1,]
for (i in 2:N) {EX[i,]<-EX[i-1]+EE[i,]}
furt(X=EX,METHOD="SB")
Partial Sum
Description
Calculate the partial sum matrix of a functional time series.
Usage
partial_sum(X)
Arguments
X |
The functional time series being calculated, inputted in a matrix form with each row representing each observation of the functional data values on equidistant points of any prespecified interval. |
Value
A square matrix of partial sums of the functional time series. The column (row) dimension of the output matrix is equal to the column dimension of the input matrix.
Examples
N<-100
EX<-matrix(rep(0,N*100),ncol=100)
set.seed(1)
for (i in 1:N) {
temp<-rnorm(100,0,1)
EX[i,1]<-temp[1]
for (j in 2:100) {
EX[i,j]<-EX[i,j-1]+temp[j]
}
}
partial_sum(EX)