1 Introduction

The majority of studies focus on detection of linear or monotonic trends, using

• classical t-test (for linear trends) or
• rank-based Mann–Kendall test (for monotonic trends)

typically under the assumption of uncorrelated data.

There exist two main problems:

• dependence effect, that is, the issue of inflating significance due to dependent observations when the test is developed for independent data (always check assumptions of the testing method!), and
• change points or regime shifts that affect the linear or monotonic trend hypothesis. For example, when testing the null hypothesis (\(H_0\)) of no trend against the alternative hypothesis (\(H_1\)) of linear trend, using t-test, it is easier to reject \(H_0\) and accept \(H_1\) in case A below, than in case B, and especially hard to reject \(H_0\) in case C. Case C reminds us that a test with proper alternative hypothesis should be chosen, and that non-rejection of \(H_0\) does not mean it is true.

These problems can be addressed by using tests for non-monotonic trends assuming that observations can be autocorrelated.

set.seed(777)
n <- 100
Time <- c(1:n)
X0 <- arima.sim(list(order = c(1, 0, 0), ar = 0.5), n = n, n.start = 100, sd = 0.5)
X1 <- 2*Time/n + X0
X2 <- 2*(Time/n)^0.5 + X0
X3 <- 0.5*(Time - n/2)/n - 6*((Time - n/2)/n)^2 + X0
X <- as.data.frame(cbind(X0, X1, X2, X3))

# Warning: package 'ggplot2' was built under R version 4.2.3

The time series above were simulated:
A) X1 with linear trend,
B) X2 with square root – nonlinear monotonic – trend, and
C) X3 with quadratic – nonlinear non-monotonic – trend,
with stationary autocorrelated innovations X0: \(X0_t = 0.5X0_{t-1} + e_t\), where \(e_t \sim N(0, 0.5^2)\).

Let’s test these time series using the functions from package funtimes, using significance level \(\alpha = 0.05\).

To install and load the package, run

install.packages("funtimes")
library(funtimes)

2 Testing for presence of a trend

Function notrend_test tests the null hypothesis of no trend against different alternatives defined by the corresponding tests.

notrend_test(X0)
# 
#   Sieve-bootstrap Student's t-test for a linear trend
# 
# data:  X0
# Student's t value = -2.6429, p-value = 0.098
# alternative hypothesis: linear trend.
# sample estimates:
# $AR_order
# [1] 1
# 
# $AR_coefficients
#     phi_1 
# 0.4212756

apply(X[,-1], 2, function(x) notrend_test(x)$p.value)
#    X1    X2    X3 
# 0.000 0.002 0.858

apply(X, 2, function(x) notrend_test(x, test = "MK")$p.value)
#    X0    X1    X2    X3 
# 0.057 0.000 0.000 0.929

apply(X, 2, function(x) notrend_test(x, test = "WAVK", 
                                     factor.length = "adaptive.selection")$p.value)
#    X0    X1    X2    X3 
# 0.337 0.000 0.026 0.004

3 Testing a specific parametric form of trend

Function wavk_test is developed for the following goodness-of-fit question (Lyubchich, Gel, and El-Shaarawi 2013):
\(H_0\): trend is of form \(f(\theta,t)\)
\(H_1\): trend is not of form \(f(\theta,t)\)
where \(f\) belongs to a known family of smooth parametric functions, and \(\theta\) are its parameters.

Note Considering \(f(\theta,t)\) being some polynomial function, non-rejection of the null hypothesis means that function \(f(\theta,t)\) or its simpler form (lower-order polynomial) is sufficient for describing the trend in the tested time series.

Note The case of \(f(\theta,t) \equiv 0\) corresponds to testing for no trend (in other words, for a constant trend, same as in the previous section), and the following code differs only in the type of bootstrap used,

• sieve bootstrap in notrend_test (WAVK statistic is calculated on original time series and simulated autoregressive series) and
• hybrid bootstrap in wavk_test (WAVK statistic is calculated on time series after the trend \(f(\theta,t)\) and autoregressive dependence are removed, and on simulated independent normal series)

notrend_test(X0, test = "WAVK", factor.length = "adaptive.selection") # WAVK with sieve bootstrap
# 
#   Sieve-bootstrap WAVK trend test
# 
# data:  X0
# WAVK test statistic = 8.7024, moving window = 4, p-value = 0.37
# alternative hypothesis: (non-)monotonic trend.
# sample estimates:
# $AR_order
# [1] 1
# 
# $AR_coefficients
#     phi_1 
# 0.4212756
wavk_test(X0 ~ 0, factor.length = "adaptive.selection") # WAVK with hybrid bootstrap
# 
#   Trend test by Wang, Akritas, and Van Keilegom (bootstrap p-values)
# 
# data:  X0 
# WAVK test statistic = 0.30965, adaptively selected window = 4, p-value
# = 0.632
# alternative hypothesis: trend is not of the form X0 ~ 0.

To test a linear trend \(f(\theta,t) = \theta_0 + \theta_1 t\), use

wavk_test(X0 ~ t, factor.length = "adaptive.selection")
# 
#   Trend test by Wang, Akritas, and Van Keilegom (bootstrap p-values)
# 
# data:  X0 
# WAVK test statistic = -0.085378, adaptively selected window = 4,
# p-value = 0.98
# alternative hypothesis: trend is not of the form X0 ~ t.

Note that the time sequence t is specified automatically within the function.

For the other time series, \(p\)-values are shown below:

apply(X[,-1], 2, function(x) wavk_test(x ~ t, factor.length = "adaptive.selection")$p.value)
#    X1    X2    X3 
# 0.954 0.786 0.020

The function poly could also be used, for example, test quadratic trend \(f(\theta,t) = \theta_0 + \theta_1 t + \theta_2 t^2\) and show the trend estimates using the argument out = TRUE:

wavk_test(X3 ~ poly(t, 2), factor.length = "adaptive.selection", out = TRUE)
# 
#   Trend test by Wang, Akritas, and Van Keilegom (bootstrap p-values)
# 
# data:  X3 
# WAVK test statistic = -0.097613, adaptively selected window = 4,
# p-value = 0.896
# alternative hypothesis: trend is not of the form X3 ~ poly(t, 2).
# sample estimates:
# $trend_coefficients
# (Intercept) poly(t, 2)1 poly(t, 2)2 
#  -0.4860421  -0.2358495  -4.7102192 
# 
# $AR_order
# [1] 1
# 
# $AR_coefficients
#     phi_1 
# 0.4193298 
# 
# $all_considered_windows
#  Window WAVK-statistic p-value
#       4    -0.09761277   0.896
#       5    -0.47737630   0.816
#       7    -0.47880434   0.860
#      10    -0.12694875   0.780