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))

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 0.05.

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.

For example, the case of \(f(\theta,t) \equiv 0\) corresponds to testing for no trend (in other words, for a constant trend), and the following code differs only in the type of bootstrap used, sieve bootstrap in notrend_test (the statistic is calculated on original time series and simulated autoregressive series) and hybrid bootstrap in wavk_test (the 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")
# 
#   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")
# 
#   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:

sapply(names(X[,-1]), function(x) wavk_test(eval(parse(text = x)) ~ t)$p.value)
#    X1    X2    X3 
# 0.810 0.628 0.008

The function poly could also be used, for example, test quadratic trend \(f(\theta,t) = \theta_0 + \theta_1 t + \theta_2 t^2\)

wavk_test(X3 ~ poly(t, 2), factor.length = "adaptive.selection")
# 
#   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).