| Type: | Package |
| Title: | Tests of Randomness and Tests of Independence |
| Version: | 1.2.0 |
| Description: | Functions for testing randomness for a univariate time series with arbitrary distribution (discrete, continuous, mixture of both types) and for testing independence between random variables with arbitrary distributions. The test statistics are based on the multilinear empirical copula and multipliers are used to compute P-values. The test of independence between random variables appeared in Genest, Nešlehová, Rémillard & Murphy (2019) and the test of randomness appeared in Nasri (2022). |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| LazyData: | true |
| Depends: | R (≥ 3.5.0), doParallel, parallel, foreach, stats, copula |
| Imports: | ggplot2,survey |
| RoxygenNote: | 7.2.3 |
| NeedsCompilation: | yes |
| Packaged: | 2024-02-13 12:43:42 UTC; Utilisateur |
| Author: | Bouchra R. Nasri [aut, cre, cph], Bruno N Remillard [aut], Johanna G Neslehova [aut], Christian Genest [aut] |
| Maintainer: | Bouchra R. Nasri <bouchra.nasri@umontreal.ca> |
| Repository: | CRAN |
| Date/Publication: | 2024-02-14 00:04:09 UTC |
Dependogram for Kendall's tau and Spearman's rho
Description
This function, used in EstDepSerial, draws the P-values of Kendall's tau and Spearman's rho for a given number of lags.
Usage
AutoDep(out)
Arguments
out |
List of the output of EstDepSerial (P-values, subsets) |
References
B.R Nasri (2021). Tests of serial dependence for arbitrary distributions
Examples
out <-EstDepSerial(SimAR1Poisson(c(5,0.4),100),10)
AutoDep(out)
Dependogram for Cramer-von Mises statistics
Description
This function, used in EstDep, TestIndCopula and TestIndSerCopula, draws the P-values of the Moebius Cramer-von Mises statistics from the multilinear copula and their combination for a tests of randomness for k consectives values X(1), ..., X(k) or for a test of independence between random variables.
Usage
Dependogram(out, stat = "CVM")
Arguments
out |
List of the output from EstDep, EstDepSerial, TestIndCopula or TestIndSerCopula (P-values, subsets) |
stat |
Name of statistics to be used (default is "CVM") |
References
Genest, Neslehova, Remillard & Murphy (2019). Testing for independence in arbitrary distributions
Examples
x <- matrix(rnorm(250),ncol=5)
out <-TestIndCopula(x)
Dependogram(out)
Dependogram for Moebius correlations
Description
This function, used in EstDepMoebius and EstDepSerialMoebius plot the graphs of the correlation statistics of Spearman, van der Waerden and Savage as a function of the subsets for tests of randomness or test of independence between random variables. Under the null hypothesis, the statistics should be independent N(0,1).
Usage
DependogramZ(out, n)
Arguments
out |
List of the output from EstDep, EstDepSerial, TestIndCopula or TestIndSerCopula (P-values, subsets) |
n |
Number of observations |
References
Nasri & Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances
Examples
x <- matrix(rnorm(250),ncol=5)
out <-EstDepMoebius(x)
DependogramZ(out,50)
Kendall's tau and Spearman's rho statistics for testing independence between random variables
Description
This function computes the matrix of pairs of Kendall's tau and Spearman's rho statistics between random variables with arbitrary distributions.
Usage
EstDep(x, graph = FALSE)
Arguments
x |
Data matrix |
graph |
Set to TRUE for a dependogram for all pairs of Kendall's taus and Spearman's rhos. |
Value
stat |
List of Kendall's tau and Spearman's rho statistics from multilinear copula, and test combinations LB |
pvalue |
P-values for the tests statistics |
References
Genest, Neslehova, Remillard & Murphy (2018). Testing for independence in arbitrary distributions
Examples
x <- matrix(rnorm(500),ncol=10)
out <-EstDep(x)
Dependence measures and statistics for test of independence between random variables
Description
This function computes copula-based dependence measures for Moebius versions of Spearmans's rho, van der Waerden's coefficient, and Savage's coefficient, as well as their combination for tests of independence between random variables.
Usage
EstDepMoebius(x, trunc.level = 2, graph = FALSE)
Arguments
x |
Data matrix |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
graph |
Set to TRUE if one wants the dependogram of P-values for the Moebius statistics |
Value
stat |
List of statistics (spearman, vdw, savage) and test combinations Ln and Ln2 (only pairs) |
pvalue |
P-values for the tests |
cardA |
Cardinaly of the subsets for the Moebius statistics |
subsets |
Subsets for the Moebius statistics |
References
B.R Nasri & B.N. Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances
Examples
x <- matrix(rnorm(250),ncol=5)
out <-EstDepMoebius(x,3)
Kendall's tau and Spearman's rho statistics for testing randomness in a univariate time series
Description
This function computes Kendall's tau and Spearman's rho statistics for tests of randomness in a time series with arbitrary distribution for pairs (X[i],X[i+k]), k=1:lags
Usage
EstDepSerial(x, lag, graph = FALSE)
Arguments
x |
Time series |
lag |
Number of lags |
graph |
Set to TRUE for a dependogram for Kendall's tau and Spearman;s rho |
Value
stat |
List of Kendall's tau and Spearman's rho statistics from multilinear copula, and test combinations LB |
pvalue |
P-values for the tests statistics |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
out <-EstDepSerial(SimAR1Poisson(c(5,0.4),100),10)
Dependence measures and statistics for test of randomness for a univariate time series
Description
This function computes copula-based dependence measures for Moebius versions of Spearmans's rho, van der Waerden's coefficient, and Savage's coefficient, as well as their combination for tests of randomness for p consecutive values Y(1), ..., Y(p).
Usage
EstDepSerialMoebius(y, p, trunc.level = 2, graph = FALSE)
Arguments
y |
Time series |
p |
Number of consecutive observations |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
graph |
Set to TRUE if one wants the dependogram of P-values for the Moebius statistics |
Value
stat |
List of statistics (spearman, vdw, savage) and test combinations Ln and Ln2 (only pairs) |
pvalue |
P-values for the tests |
card |
Cardinaly of the subsets for the Moebius statistics |
subsets |
Subsets for the Moebius statistics |
References
B.R Nasri & B.N. Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances
Examples
y<- SimAR1Poisson(c(5,0.2),100)
out <- EstDepSerialMoebius(y,4,4)
Quantile function of margins
Description
This function computes the quantile of seven cdf used in the simulatuons of Nasri (2022).
Usage
Finv(u, k)
Arguments
u |
Vector of probabilities |
k |
Marginal distribution: [1] Bernoulli(0.8), [2] Poisson(6), [3] Negative binomial with r = 1.5, p = 0.2, [4] Zero-inflated Poisson (10) with w = 0.1 and P(6.67) otherwise, [5] Zero-inflated Gaussian, [6] Discretized Gaussian, [7] Discrete Pareto(1) |
Value
x |
Vector of quantiles |
Author(s)
Bouchra R. Nasri January 2021
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
x = Finv(runif(100),2)
Simulation of a AR(1) Poisson process
Description
Conditionally on the past, X[t] is Poisson with lambda[t] = a+bX[t-1]
Usage
SimAR1Poisson(param, n)
Arguments
param |
Param[1] = a>0, param[2] = b, 0<=b <1 (for stationarity) |
n |
Length of the series. |
Value
X |
Simulated series |
Examples
data <- SimAR1Poisson(c(5,0.4),500)
Simulation of a copula-based time series
Description
This function simulates a Markovian time series (p-Markov for the Farlie-Gumbel-Morgenstern copula) with uniform margins using a copula family for the joint distribution of U[t], U[t-1].
Usage
SimCopulaSeries(family, n, tau = 0, param = NULL)
Arguments
family |
"ind", "tent", "gaussian", "t" , "clayton", "fgm", "frank", "gumbel", joe" , "plackett" |
n |
length of the time series |
tau |
Kendall's tau of the copula family |
param |
extra copula parameter: for "fgm", param is the dimension of the copula; for "t", param = nu |
Value
U |
Simulated time series |
Author(s)
Bouchra R. Nasri January 2021
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
U = SimCopulaSeries("fgm",100,0.2, 3) # for the FGM, |tau|<= 2/9
Computes the Moebius Cramer-von Mises statistics for the test of independence between random variables
Description
This function computes Moebius Cramer-von Mises statistics for a tests of independence for matrix of data x.
Usage
Sn_A(x, trunc.level)
Arguments
x |
Matrix of observations. |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
Value
stats |
Cramer-von Mises Moebius statistics |
cardA |
Cardinality of subsets |
M |
Matrix for multitpliers bootstrap for stats |
Asets |
Vector of (0,1) for Moebius subsets |
Sn |
Cramer-von Mises Sn statistic |
J |
Matrix for multipliers bootstrap for Sn |
References
Genest, Neslehova, Remillard & Murphy (2019). Testing for independence in arbitrary distributions
#'@examples X <- rnorm(100,5) out <- Sn_A(X,3)
Computes the Moebius Cramer-von Mises statistics for the test of randomness
Description
This function computes Moebius Cramer-von Mises statistics for a tests of randomness for observations X(1), ..., X(p).
Usage
Sn_Aserial(x, p, trunc.level)
Arguments
x |
Time series. |
p |
Number of consecutive observations for the test. |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
Value
stats |
Cramer-von Mises Moebius statistics |
cardA |
Cardinality of subsets |
M |
Matrix for multitpliers bootstrap for stats |
Asets |
Vector of (0,1) for Moebius subsets |
Sn |
Cramer-von Mises Sn statistic |
J |
Matrix for multipliers bootstrap for Sn |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
#'@examples X <- SimAR1Poisson(c(5,0.2),100) out <- Sn_Aserial(X,5,3)
Computes the Moebius Cramer-von Mises statistics for the test of randomness
Description
This function he Moebius Cramer-von Mises statistics for a tests of randomness for random vectors Y(1), ...l, X(p).
Usage
Sn_AserialVec(Y, p, trunc.level = 2)
Arguments
Y |
Time series. |
p |
Number of consecutive observations for the test. |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
Value
stats |
Cramer-von Mises Moebius statistics |
cardA |
Cardinality of subsets |
M |
Matrix for multitpliers bootstrap for stats |
Asets |
vector of (0,1) for Moebius subsets |
Sn |
Cramer-von Mises Sn statistic |
J |
Matrix for multipliers bootstrap for Sn |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
#'@examples Y <- data(Y) out <- Sn_Aserial(Y,5,2)
Computes the Cramer-von Mises statistic Sn for the test of randomness
Description
This function computes the Cramer-von Mises statistic for a tests of randomness for p consecutives values X(1), ..., X(p). Useful for traditional bootstrapping.
Usage
Sn_serial(x, p)
Arguments
x |
Time series |
p |
Number of consecutive observations |
Value
stats |
Cramer-von Mises statistics Sn |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
#'@examples X <- SimAR1Poisson(c(5,0.2),100) out <- Sn_serial(X,5,3)
Statistics and P-values for a test of independence between random variables
Description
This function computes Cramer-von Mises statistics and their combination for a tests of independence between random variables with arbitrary distributions. The P-values are computed using Gaussian multipliers.
Usage
TestIndCopula(
x,
trunc.level = 2,
B = 1000,
par = FALSE,
ncores = 2,
graph = FALSE
)
Arguments
x |
Data matrix |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
B |
Number of multipliers samples (default = 1000) |
par |
Set to TRUE if one prefers paraller computing (slower) |
ncores |
Number of cores for parallel computing (default is 2) |
graph |
Set to TRUE if one wants the dependogram of P-values for the Moebius statistics |
Value
stat |
List of Cramer-von Mises statistics cvm, Sn from the multilinear copula, and test combinations Tn and Tn2 (only pairs) |
pvalue |
Approximated P-values for the tests using Gaussian multipliers |
card |
Cardinaly of the subsets for the Moebius statistics |
subsets |
Subsets for the Moebius statistics |
References
Genest, Neslehova, Remillard & Murphy (2019). Testing for independence in arbitrary distributions
Examples
x <- matrix(rnorm(250),ncol=5)
out <-TestIndCopula(x)
Statistics and P-values for a test of randomness for a univariate time series
Description
This function computes Cramer-von Mises statistics from the multilinear copula and their combination for tests of randomness of p consecutives values X(1), ..., X(p). The p-values are computed using Gaussian multipliers.
Usage
TestIndSerCopula(
x,
p,
trunc.level = 2,
B = 1000,
par = FALSE,
ncores = 2,
graph = FALSE
)
Arguments
x |
Time series |
p |
Number of consecutive observations |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
B |
Number of multipliers samples (default = 1000) |
par |
Set to TRUE if one prefers paraller computing (slower) |
ncores |
Number of cores for parallel computing (default = 2) |
graph |
Set to TRUE if one wants the dependogram of P-values for the Moebius statistics |
Value
stat |
List of Cramer-von Mises statistics cvm, Sn, and test combinations Tn and Tn2 (only pairs) |
pvalue |
Approximated P-values for the tests using Gaussian multipliers |
card |
Cardinaly of the subsets for the Moebius statistics |
subsets |
Subsets for the Moebius statistics |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
X <- SimAR1Poisson(c(5,0.2),100)
out <- TestIndSerCopula(X,5,3)
Statistics and P-values for a test of randomness for a multivariate time series
Description
This function computes Cramer-von Mises statistics from the multilinear copula and their combination for a tests of randomness for p consecutives values of random vectors X(1), ..., X(p). The p-values are computed using Gaussian multipliers.
Usage
TestIndSerCopulaMulti(x, p, trunc.level = 2, B = 1000, graph = FALSE)
Arguments
x |
Time series matrix |
p |
Number of consecutive vectors |
trunc.level |
Only subsets of cardinality <= trunc.level (default=2) are considered for the Moebius statistics. |
B |
Number of multipliers samples (default = 1000) |
graph |
Set to TRUE if one wants the dependogram of P-values for the Moebius statistics |
Value
stat |
List of Cramer-von Mises statistics cvm, tilde Sn, and test combinations tilde Tn and tilde Tn2 (only pairs), as defined in Nasri(2022). |
pvalue |
Approximated P-values for the tests using Gaussian multipliers |
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
data(Y)
out <- TestIndSerCopulaMulti(Y,5,5)
AR(1) Poisson with parameters
Description
Simulated AR(1) Poisson sequence of length n=100 with parameters c(5,0.4).
Usage
data(X)Format
Count data.
Examples
data(X)
acf(X)
Bernoulli sequence
Description
Simulated Bernoulli sequence.
Usage
data(Xbin)Format
Count data.
Examples
data(Xbin)
plot(Xbin)
VAR(1) Poisson with parameters
Description
Simulated VAR(1) Poisson sequence of length n=100.
Usage
data(Y)Format
Count data.
Examples
data(Y)
acf(Y)
Function to perform multiplier bootstrap for tests of randomness or independence
Description
This function simulates Cramer-von Mises statistics using Gaussian multipliers.
Usage
bootstrap(M, J, n)
Arguments
M |
n x n x m vector with MM[i,j] = 1(Xi <= Xj) and C=mean(M[,j]); |
J |
n x n vector for bootstrapping Sn. |
n |
length of the series. |
Value
cvm_sim |
Simulated value of the Cramer-von Mises statistics |
sn_sim |
simulated value of the Cramer-von Mises statistic Sn |
References
B.R Nasri (2021). Tests of serial dependence for arbitrary distributions
Horseshoecrabs dataset
Description
Horseshoe Crab Data from Table 3.2 of Agresti(2007). This data set consists of five variables, three of which are categorical, measured on 173 female crabs, each having a male attached in her nest.
Usage
data(horseshoecrabs)Format
Data frame with 173 rows and 5 variables:
X1: Color of the female (1: light medium, 2: medium, 3: dark medium, 4: dark)
X2: Spine condition (1: both good. 2: one worn or broken, 3: both worn or broken)
X3: Carapace width (cm)
X4: Number of satellites, i.e., other males around the female
X5: Weight (kg)
References
Agresti, A. (2007). An Introduction to Categorical data analysis, John Wiley & Sons, Wiley Series in Probability and Statistics, 2nd edition.
Examples
data(horseshoecrabs)
x =data.matrix(horseshoecrabs)
out = TestIndCopula(x,trunc.level=5,graph=TRUE)
Fetal lamb dataset
Description
240 body movement measurements of a fetal lamb at consecutive 5 second intervals.
Usage
data(lamb)Format
Count data.
References
Leroux B, Putterman M (1992). Maximum Penalized Likelihood estimation for independent and Markov-dependent Mixture models.Biometrics, 48, 545–558.
Examples
data(lamb)
plot(lamb)
Computes unique values, cdf and pdf
Description
This function computes the unique values, cdf and pdf for a series of data.
Usage
preparedata(x)
Arguments
x |
Vector |
Value
values |
Unique (sorted) values |
m |
Number of unique values |
Fn |
Empirical cdf of the unique values |
fn |
Empirical pdf of the unique values |
References
B.R. Nasri (2022). Tests of serial dependence for arbitrary distributions
C. Genest, J.G. Neslehova, B.N. Remillard and O. Murphy (2019). Testing for independence in arbitrary distributions.
#'@examples x = c(0,0,0,2,3,1,3,1,2,0) out <- prepare_data(x)
Data-driven selection of p for the test of randomness
Description
This function uses a AIC/BIC type criterion to select p based on the data.
Usage
select_p(X, p0 = 2, d = 5, q = 2.4, lambda = 0.25)
Arguments
X |
Time series |
p0 |
Minimum value of p (default is 2) |
d |
Maximum value of p (default is 5) |
q |
Constant for selecting between AIC and BIC type penalty (default is 2.4) |
lambda |
Penalty term (default is 0.25); small values lead to p=d, large value lead to p=p0 |
Value
p |
Selected value of p |
References
B.R Nasri (2021). Tests of serial dependence for arbitrary distributions
Examples
X <- SimAR1Poisson(c(5,0.2),100)
out <- select_p(X)
Computes the Kendall's taus and Spearman's rho for tests of randomness
Description
This function Computes the Kendall's taus and Spearman's rho for tests of independence used in EstDep
Usage
stat_dep(x, y)
Arguments
x |
Vector of length n |
y |
Vector of length n |
Value
tau |
Kendall's taus for lags 1:lag |
rho |
Spearman's rhos for lags 1:lag |
s2 |
Estimated variance of Spearman's rho |
References
Genest, Neslehova, & Remillard (2017). Asymptotic behavior of the empirical multilinear copula process under broad conditions
Examples
x <- SimAR1Poisson(c(5,0.2),100)
y <- SimAR1Poisson(c(5,0.2),100)
out <- stat_dep(x,y)
Computes the Kendall's taus and Spearman's rho for tests of randomness
Description
This function Computes the Kendall's taus and Spearman's rho for tests of randomness.
Usage
stat_dep_ser(x, lag)
Arguments
x |
Time series |
lag |
Number of lags. |
Value
tau |
Kendall's taus for lags 1:lag |
rho |
Spearman's rhos for lags 1:lag |
References
B.R Nasri (2021). Tests of serial dependence for arbitrary distributions
Examples
X <- SimAR1Poisson(c(5,0.2),100)
out <- stat_dep_ser(X,10)