Multivariate Dependence Trend Test

Description

Computes the test statistic for the multivariate dependence trend (MDT) test and the p-value.

Usage

MDT_test(data, width, alpha = 0.05, Nbs = 1000)

Arguments

Details

The multivariate dependence trend (MDT) test allows to check for trend in the data series dependence structure. The moving window technique has been employed in order to take into account the dependence evolution, represented by Kendall's \tau, according to time.

The test proposes a null hypothesis of no trend, against an alternative hypothesis of a monotonic trend in the dependence structure. Goutali and Chebana (2024) propose the following test statistic:

Let X = (x_1, x_2, ..., x_n) and Y = (y_1, y_2, ..., y_n) be two variables of a data series, the test statistic of the MDT test is given by:

T_{MDT}=\tau_n(\tau_{nw}(X,Y),T')

where \tau_n is the empirical version of bivariate Kendall's \tau. The series of the Kendall's \tau obtained through moving window with width w is \tau_{nw}(X, Y), this series has length q = n - w +1. T' = (1, 2, ..., q) is the time order of the rolling window series.

The choice of width is a trade-off. A small w increases the number of rolling windows for reliable analysis, while a large w is necessary to have sufficient values to identify the dependence structure. The p-value is computed using a bootstrap procedure.

Value

Named list containing:

• statistic, the MDT test statistic.

• p.value, computed p-value computed using bootstrap.

• result, the result of the test. If TRUE the trend is considered significant.

References

Goutali, D., and Chebana, F. (2024). Multivariate overall and dependence trend tests, applied to hydrology, Environmental Modelling & Software, 179, doi:10.1016/j.envsoft.2024.106090

See Also

• kendall.tau : Function from the package VGAM used for computing the bivariate Kendall's \tau.

• rollapply : Function from the package zoo used to apply the moving window technique.

• samp.bootstrap: Function from the resample package, used to generate the samples necessary to perform bootstrapping.

Examples

# CASE 1: Only trend in the dependence structure
# Sample data:
DependenceStructure <- generate_data("dependenceStructure")

width <- 10

# Perform the mdt test:
mdt <- MDT_test(DependenceStructure, width, alpha = 0.05, Nbs = 1000)
print(mdt)


# CASE 2: Only trend in the marginal distributions
# Sample data:
MarginalTrend <- generate_data("marginalTrend")

# Perform the mdt test:
mdt <- MDT_test(MarginalTrend, width)
print(mdt)


# CASE 3: No trend
# Sample data:
NoTrend <- generate_data("noTrend")

# Perform the mdt test:
mdt <- MDT_test(NoTrend, width)
print(mdt)

Multivariate Overall Trend Test

Description

Computes the test statistic for the multivariate overall trend (MOT) test and the p-value.

Usage

MOT_test(data, covar = NULL, width, alpha = 0.05, Nbs = 1000)

Arguments

Details

The multivariate overall trend (MOT) test allows to check for trend in the marginals and dependence structure of a multivariate distribution. The moving window technique has been employed in order to take into account the dependence evolution, represented by Kendall's \tau, according to time.

The test evaluates a null hypothesis of no trend in the data series, against an alternative hypothesis of a monotonic trend. Goutali and Chebana (2024) propose the following test statistic:

Let X = (x_1, x_2, ..., x_n) and Y = (y_1, y_2, ..., y_n) be variables in a data series, and T the time order, the test statistic is given by:

T_{MOT}=\frac{1}{3}\left(\tau_n(X,T)^2 + \tau_n(Y,T)^2 + \tau_n(\tau_{nw}(X,Y),T')^2 \right)

where \tau_n is the empirical version of bivariate Kendall's \tau. The series of the Kendall's \tau obtained through moving window is \tau_{nw}(X, Y), and T' is the time order of the moving window series.

The choice of width is a trade-off. A small w increases the number of rolling windows for reliable analysis, while a large w is necessary to have sufficient values to identify the dependence structure. The p-value is computed using a bootstrap procedure.

Value

A named list:

• statistic, computed MOT test statistic.

• p.value, computed p-value computed using bootstrap.

• result, the result of the test. If TRUE the trend is considered significant.

References

Goutali, D., and Chebana, F. (2024). Multivariate overall and dependence trend tests, applied to hydrology, Environmental Modelling & Software, 179, doi:10.1016/j.envsoft.2024.106090

See Also

• kendall.tau: Function from the package VGAM used for computing the bivariate Kendall's \tau.

• rollapply: Function from the package zoo used to apply the moving window technique.

• samp.bootstrap: Function from the resample package, used to generate the samples necessary to perform bootstrapping.

Examples

# CASE 1: Only trend in the dependence structure
# Sample data:
DependenceStructure <- generate_data("dependenceStructure", n = 50)

width <- 10

# Perform the mot test:
mot <- MOT_test(DependenceStructure, covar = NULL, width, alpha = 0.05, Nbs = 1000)
print(mot)


# CASE 2: Only trend in the marginal distributions
# Sample data:
MarginalTrend <- generate_data("marginalTrend", n = 50)

# Perform the mot test:
mot <- MOT_test(MarginalTrend, width = width)
print(mot)


# CASE 3: No trend
# Sample data:
NoTrend <- generate_data("noTrend", n = 50)

# Perform the mot test:
mot <- MOT_test(NoTrend, width = width)
print(mot)

Generate Synthetic data

Description

Synthetic data generated using copulas and marginal distributions, with the purpose of exampling the functions of the package. Three options are given: "noTrend", "marginalTrend", and "dependenceStructure".

The generated "noTrend" data follows a Clayton copula with fixed Kendall Tau (\tau=0.2). "marginalTrend", follows the same copula, however the variables follow a Generalized Extreme Value distribution with fixed scale and shape parameters (\sigma=1, \xi=-0.1), the location is linearly non-stationary with \mu_X = 0.05\cdot t and \mu_Y = 0.07\cdot t. Finally "dependenceStructure" presents trend in the dependence structure, the data was generated from a Clayton copula with a linear non-stationary \tau parameter. For more information we refer the reader to the source material by Goutali and Chebana (2024).

Usage

generate_data(
  trend = c("noTrend", "marginalTrend", "dependenceStructure"),
  n = 100
)

Arguments

Value

A dataset of dimensions n \times 2 with the generated data.

References

Goutali, D., and Chebana, F. (2024). Multivariate overall and dependence trend tests, applied to hydrology, Environmental Modelling & Software, 179, doi:10.1016/j.envsoft.2024.106090

Examples

# NO TREND
generate_data("noTrend", n = 50)

# TREND IN BOTH MARGINALS
generate_data("marginalTrend", n = 50)

# TREND IN DEPENDENCE STRUCTURE
generate_data("dependenceStructure", n = 50)

Component Wise Mann-Kendall Test Statistic

Description

The functions performs the univariate Mann-Kendall test statistic to each variable of a data series.

Usage

mkComponent(data)

Arguments

Details

Let M be a dataset of m components and n observations. The Mann-Kendall's (MK) test statistic for a variable of the dataset M^{(u)} is given by:

M^{(u)} = \sum_{1 \leq i \leq j \leq n} sgn(x_j^{(u)} - x_i^{(u)})

where sgn(\cdot) is the sign function:

sgn(x)=\begin{cases} -1 \quad \text{if } x<0, \\ 0 \quad \text{if } x=0, \\ +1 \quad \text{if } x>0 \end{cases}

This test statistic is normal distributed, with mean and variance:

E(M^{(u)}) = 0

,

\text{var}(M^{(u)}) = \frac{n(n-1)(2n+5)}{18}

Value

A numeric vector with the univariate MK test statistic for each component of the data series.

References

• Hamed, K.H., Rao, A.R., 1998. A modified Mann-Kendall trend test for autocorrelated data. J. Hydrol. 204 (1–4), 182–196.

• Kendall, M., (1975). Rank Correlation Methods; Griffin: London.

Examples

# Sample data (Both marginal distributions have trend):
dataMarginalTrend <- generate_data("marginalTrend", n = 50)

# Perform multivariate MK test on sample data:
mkComponent(dataMarginalTrend)

Univariate Mann-Kendall Test

Description

The functions performs the univariate Mann-Kendall test.

Usage

mkUnivariate(x)

Arguments

Details

The univariate Mann-Kendall (MK) test is used to detect monotonic trends in a univariate data series. It tests the null hypothesis (h_0) of no trend, against an alternative.

Let (x_1, x_2, ..., x_n) be a data series of length n, the MK test statistic is given by:

M = \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} sgn(x_j-x_i)

where sgn(\cdot) is the sign function:

sgn(x)=\begin{cases} -1 \quad \text{if } x<0, \\ 0 \quad \text{if } x=0, \\ +1 \quad \text{if } x>0 \end{cases}

Under H_0 the test statistic is asymptotically normally distributed with mean and variance:

E(M) = 0

\text{Var}(M)=\frac{n(n-1)(n+5)}{18}

Value

A named list

• statistic, the estimated Mann-Kendall test statistics.

• p.value, the estimated p-value for the test.

References

• Hamed, K.H., Rao, A.R., 1998. A modified Mann-Kendall trend test for autocorrelated data. J. Hydrol. 204 (1–4), 182–196.

• Kendall, M., (1975). Rank Correlation Methods; Griffin: London.

See Also

mkComponent: The multivariate extension of this test.

Examples

# Sample data (Both marginal distributions have trend):
dataMarginalTrend <- generate_data("marginalTrend", n = 50)

# Perform two tailed MK test on sample data:
mkUnivariate(dataMarginalTrend[, 1])
mkUnivariate(dataMarginalTrend[, 2])

Trend Tests Plots

Description

Informative plots on the given data series in regards to univariate and multivariate trend testing.

Usage

plotTrend(
  data,
  covar = NULL,
  width,
  graph = c("summary", "variable1", "variable2", "window"),
  color = c("blue", "dark", "green", "warm")
)

Arguments

Details

The function is able to do three plots: "window" returns the scatter plot of Kendall \tau with the moving window technique, "variable1" and "variable2" return the line plots for each variable against the covariate or "Time", finally "summary" returns the previous plot combined in the same display. Four color profiles are available .

Value

The specified plot.

References

Goutali, D., and Chebana, F. (2024). Multivariate overall and dependence trend tests, applied to hydrology, Environmental Modelling & Software, 179, doi:10.1016/j.envsoft.2024.106090

See Also

• kendall.tau : Function from the package VGAM used for computing the bivariate Kendall's \tau.

• rollapply : Function from the package zoo used to apply the moving window technique.

Examples

# Sample data:
dataDependenceStructure <- generate_data("dependenceStructure", 50)
dataMarginalTrend <- generate_data("marginalTrend", 50)
dataNoTrend <- generate_data("noTrend", 50)

# Plot Trend summary:
plotTrend(dataDependenceStructure, covar = NULL, width = 10, graph = "summary", color = "blue")

plotTrend(dataMarginalTrend, covar = NULL, width = 10, graph = "summary", color = "green")

plotTrend(dataNoTrend, covar = NULL, width = 10, graph = "summary", color = "warm")

# Plot a variable
plotTrend(dataMarginalTrend, width = 10, graph = "variable1", color = "green")

# Plot the evolution of Kendall tau
plotTrend(dataDependenceStructure, width = 10, graph = "window", color = "warm")

Univariate Spearman's Rho Test

Description

The functions performs the univariate Spearman's rho test.

Usage

srUnivariate(x)

Arguments

Details

The Spearman's Rank test is a non-parametric trend test based on rank-order, It tests a null hypothesis of no trend against an alternative. Given a data series X = (x_1, x_2, ..., x_n) of length n, the test statistic is given by

D = 1 - \frac{6 \sum_{i=1}^n [R(x_i) - i]^2}{n(n^2 - 1)}

where R(x_i) is the rank of the i-th observation in the data series.

Under the null hypothesis D has asymptotically normal distribution, with E(D)=0, and variance

\text{Var}(D) = \frac{1}{n-1}

Value

A named list

• statistic, the estimated Spearman's rho test statistics

• p.value, the estimated p-value for the test.

References

• Sneyers, R., 1990. On the Statistical Analysis of Series of Observations. World Meteorol. Organ.

Examples

# Sample data (Both marginal distributions have trend):
dataMarginalTrend <- generate_data("marginalTrend", n = 50)

# Perform SR test on sample data:
srUnivariate(dataMarginalTrend[, 1])

srUnivariate(dataMarginalTrend[, 2])

Trend Tests Summary

Description

Performs multivariate and univariate trend tests on the given data series and returns a data frame with the results.

Usage

summaryTrend(data, width, covar = NULL)

Arguments

Details

The function performs the Multivariate Dependence Trend and Multivariate Overall Trend tests as described by Goutali and Chebana (2024), as well as the univariate Mann-Kendall (MK) test to each variable and returns a data frame with the results.

This functions performs the test with the default values for alpha = 0.05 and Nbs = 1000, for a more precise testing you can use the functions described in see also.

Value

The results dataframe, a column for the respective test statistic, a column for the p-value, and four rows each for a test MDT, MOT, and MK for each variable.

References

Goutali, D., and Chebana, F. (2024). Multivariate overall and dependence trend tests, applied to hydrology, Environmental Modelling & Software, 179, doi:10.1016/j.envsoft.2024.106090

See Also

• MDT_test: Function used to perform the MDT test.

• MOT_test: Function used to compute the MOT test.

• mkUnivariate: Function used to perform the univariate MK test.

Examples

# Summary for data with trend in the Dependence Structure:
DependenceStructure <- generate_data("dependenceStructure", 50)
summaryTrend(DependenceStructure, covar = NULL, width = 10)

# Summary for data with trend in the Marginals:
MarginalTrend <- generate_data("marginalTrend", 50)
summaryTrend(MarginalTrend, covar = NULL, width = 10)

# Summary for data without trend:
NoTrend <- generate_data("noTrend", 50)
summaryTrend(NoTrend, covar = NULL, width = 10)