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

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 \(\alpha = 0.05\).

Function `notrend_test`

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

Consider the following pair of hypotheses

\(H_0\): no trend

\(H_1\): linear trend

that can be tested specifically using t-test.

Assuming the time series may be autocorrelated (which is the usual case with observational data), we apply sieve-bootstrap version of the **t-test**, by adapting the approach of Noguchi, Gel, and Duguay (2011):

```
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
```

The large \(p\)-value correctly indicates that there is not enough evidence to reject the hypothesis of no trend in `X0`

in favor of the alternative hypothesis of a linear trend.

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

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

indicating that the null hypothesis of no trend could be rejected and hypothesis of a linear trend could be accepted for `X1`

and `X2`

. While `X3`

has a trend (based on the way it was simulated and the time series plot above), the alternative hypothesis of a linear trend does not fit in this case, so the test for linear trend (t-test) failed to reject the null hypothesis.

Since a linear trend is also a monotonic trend, we may expect seeing similar results when testing the following pair of hypotheses

\(H_0\): no trend

\(H_1\): monotonic trend

using Mann–Kendall test.

Apply **Mann–Kendall test**, also with the sieve-bootstrap enhancement for potentially autocorrelated data; \(p\)-values are shown below:

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

indicating that the null hypothesis of no trend could be rejected and hypothesis of a monotonic trend could be accepted for `X1`

and `X2`

. For `X0`

and `X3`

, the null hypothesis could not be rejected, because `X0`

does not have a trend, and `X3`

has a trend that does not match the alternative hypothesis.

If the interest is in testing for any, potentially non-monotonic trend, consider testing the following pair of hypotheses

\(H_0\): no trend

\(H_1\): any trend

using local regression-based WAVK test (Wang, Akritas, and Van Keilegom 2008).

Apply **WAVK test**, also with the sieve-bootstrap enhancement for potentially autocorrelated data:

```
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
```

The results indicate that WAVK test was correct in non-rejecting the null hypothesis for `X0`

, and correctly rejected it for the time series with trends `X1`

, `X2`

, and `X3`

.

Lyubchich, Gel, and El-Shaarawi (2013) originally implemented *hybrid* bootstrap to this test statistic, available from the `wavk_test`

function described in the next section.

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
```

Lyubchich, V., Y. R. Gel, and A. El-Shaarawi. 2013. “On Detecting Non-Monotonic Trends in Environmental Time Series: A Fusion of Local Regression and Bootstrap.” *Environmetrics* 24 (4): 209–26. https://doi.org/10.1002/env.2212.

Noguchi, K., Y. R. Gel, and C. R. Duguay. 2011. “Bootstrap-Based Tests for Trends in Hydrological Time Series, with Application to Ice Phenology Data.” *Journal of Hydrology* 410 (3): 150–61. https://doi.org/10.1016/j.jhydrol.2011.09.008.

Wang, L., M. G. Akritas, and I. Van Keilegom. 2008. “An ANOVA-Type Nonparametric Diagnostic Test for Heteroscedastic Regression Models.” *Journal of Nonparametric Statistics* 20 (5): 365–82.