On Detecting Non-Monotonic Trends in Environmental
Time Series: a Fusion of Local Regression and Bootstrap
Vyacheslav Lyubchicha , Yulia R. Gela , Abdel El-Shaarawib,c
a
b
University of Waterloo, Canada; The American University in Cairo, Egypt;
c
National Water Research Institute at the Canada Centre for Inland Waters, Canada
Abstract
In this paper, we propose a new testing procedure for detecting smooth (non)monotonic
trends embedded into a linear noise that possibly does not degenerate to a finite-dimensional
representation, or into a conditionally heteroscedastic (ARCH/GARCH) noise. The proposed
non-parametric trend test is local regression-based, and we develop a flexible and computationally efficient hybrid bootstrap procedure to approximate its finite sample behavior. Since the
proposed trend test does not assume prior knowledge on the dependence structure and probability distribution of the observed process, the new testing procedure is fully data-driven and
robust to misspecification of dependence structure and distributional assumptions, which is of
particular importance for noisy environmental measurements. Moreover, since the proposed
methodology allows to test for monotonic vs. non-monotonic trends and, hence, to assess existence of extremums in the hypothesized trend function, the developed approach may be also
employed for preliminary detection of regime shifts and change points in the observed environmental data series. Our simulation studies indicate competitive performance of the proposed
nonparametric procedure for detection of (non)monotonic trends against conventional trend
tests.
Key Words: trend detection, non-monotonic trend, change points and regime shifts, bootstrap,
climate studies
Introduction
One of the key issues in environmental data analysis, especially climate studies, are identification of a possible change and its attribution to a potential cause. The conventional approach for
assessment of possible change is to employ Student’s t-, Spearman’s ρ, Mann-Kendall’s τ or Sen’s
slope test. However, these classical methods for trend detection are typically associated with two
main problems: the dependence effect and the impact of change points and regime shifts. Despite it
was shown, that bootstrap procedures can extend these methods to be distribution-free and robust
to any order of linear dependence in the observed process (Noguchi et al., 2011; McKitrick and
Vogelsang, 2011), these tests are still limited to monotonic trends.
The impact of change points and regime shifts on the drawn conclusions about trend existence
is not yet well investigated and has received its attention only in the last few years. For instance, as
discussed by Vogelsang and Franses (2005), inclusion of a single change point due to the Pacific Climate Shift of 1977 into the study of global temperatures in the tropical lower- and mid-troposphere
delivers substantially different conclusions than the previously drawn ones. In particular, trends
that have been declared earlier as significant are now found out to be insignificant, and the impact extends far beyond the analysis of temperature patterns and includes an overall troposphere
dynamics, fisheries catch records and other biological indicators (Powell and Xu, 2011). As an
alternative, we can drop the assumption that the potential trend is of linear or even monotonic
form and, instead, test whether the potential trend belongs to some known parametric family of
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smooth non-monotonic functions. Recently, there has been an increasing interest in adapting various nonparametric procedures for non-monotonic trend detection to weakly stationary and weakly
dependent time series. We extend the flexible local regression method of Wang and Van Keilegom
(2007) and Wang et al. (2008) to a more general case and improve finite-sample performance of the
test by employing data-driven and robust bootstrap approach and m-out-of-n selection algorithm
of Bickel et al. (1997). Our simulation studies indicate competitive performance of the proposed
nonparametric procedure for detection of (non)monotonic trends against conventional trend tests.
Testing for Parametric Trends
Let us observe a time series process Yi = µ(i/n) + i , i ∈ [1, n], where µ(t), t ∈ [0, 1], is
an unknown trend function, and i is a weakly stationary time series. We are interested in the
following goodness-of-fit question for a parametric form of trend:
H0 : µ(t) = f (θ, t) vs. H1 : µ(t) 6= f (θ, t).
(1)
Here f (·, t) : R → R belongs to a known family of smooth parametric functions S = f (θ, ·), θ ∈
Θ , and Θ is a set of possible parameter values and a subset of a Euclidean space. In order to
test the trend hypotheses (1) under a more general structure, we extend the local factor method of
Wang and Van Keilegom (2007) and Wang et al. (2008) to infinite-dimensional linear time series
and conditionally heteroscedastic processes, while Wang and Van Keilegom (2007) and Wang et al.
(2008) consider the cases for noise t following either a finite autoregressive model with a known
order p or being unconditionally heteroscedastic. The key idea of the proposed local regression
procedure for trend detection consists of the following three steps:
1. Suppose that there exists no prior information on the noise i . Pre-filter the observed time
series Yt with a linear filter whose order p = p(n) → ∞:
!
!
p
p
X
X
b ti+p−j ) ,
φbj,n f (θ,
φ̂j,n Yi+p−j − f (θ̂, ti+p ) −
Zi = Yi+p −
j=1
j=1
√
where θb is a n-consistent estimator of θ under H0 , and φ̂p = (φ̂1,n , . . . , φ̂p,n )0 satisfies the
empirical Yule-Walker system of equations.
2. Now, since the proposed test statistic is based on developing an artificial balanced one-way
ANOVA with n categories, we treat each distinct time point i as a category.
3. Construct a local window Wi around each i such that Wi contains kn nearest values, where
kn → ∞ at an appropriate rate as n → ∞.
Hence, the trend test statistic (named WAVK after the first letters of authors’ surnames who first
proposed the procedure, i.e., Wang, Akritas and Van Keilegom, 2008) takes the form
2
2 .
n X
kn n X
MST
kn X
1
Vi,j − V̄i. ,
WAVK = Fn =
=
V̄i. − V̄..
MSE
n − 1 i=1
n(kn − 1) i=1 j=1
where {Vi1 , . . . , Vikn } = {Zj : j ∈ Wi } denote the kn pre-filtered observations in the i-th group, V̄i.
and V̄.. are the mean of the i-th group and grand mean, respectively.
Our hybrid resampling procedure is implemented as follows:
1. Under H0 of (1), estimate the hypothetical trend parameters θ̂ of f (θ̂, t) and parameters of an
approximating AR(p(n)) filter whose order p = p(n) is selected, for example, by the Bayesian
Information Criterion (BIC), and let Zi be the estimated residuals.
2. Calculate the observed trend test statistic WAVK0 based on Zi .
3. Construct s2 , the sample scale estimator of Zi .
4. Simulate B samples of normally distributed n × 1-vectors Zi∗ from MVN 0, s2 I , where I is a
n × n-identity matrix, and construct B bootstrapped test statistics WAVK∗1 , . . . , WAVK∗B .
2
5. The bootstrap p-value is the proportion of |WAVK∗1 |, . . . , |WAVK∗B | which exceed |WAVK0 |.
Simulation Study
To justify our approach, we run a number of simulation studies on size and power of the proposed
trend testing procedure using various synthetic data. In particular, we use four time series models
(AR(2), ARMA(1,1), ARCH(1) and GARCH(1,1)) with three different distributions of innovations,
i.e., Normal, Student’s t5 and Tukey Contaminated Normal distribution.
In order to select an optimal window size kn , we propose to employ the m-out-of-n subsampling
algorithm (Bickel et al., 1997) outlined as follows:
1. Let kn (j) = [q j n], j ∈ Z+ and 0 < q < 1, where [x] is the greatest
integer not exceeding x.
∗
2. For each kn (j), find a bootstrap
distribution
L
WAVK
.
kn (j) 3. Let k̂n (j) = arg minkn (j) ρ L∗ WAVKkn (j) , L∗ WAVKkn (j+1) , where ρ is a some metric consistent with convergence in law.
4. Employ k̂n (j) for trend detection.
In our simulations studies, n = 100, 200, 300, 500 and we set q = 3/4 to run the kn (j) = [q j n]
selection procedure over j = 8, . . . , 11. We find that the choice of q has no strong impact on the
results in our study, however, for a fixed q, the range for j depends on the sample size.
As the simulations results indicate (not reported here), the estimated Type I error of the WAVK
test is well maintained, i.e., close to the nominal level α = 0.05, across all dependence structures
that fall within the considered framework of linear and (G)ARCH processes respectively, and the
performance improves with increase of the sample size. Remarkably, test has high power for detecting smooth linear and non-monotonic trends, particularly polynomial and sinusoidal. Hence, it
can be recommended for analysis of a broad range of noisy environmental data.
Conclusion
The new robust and data-driven diagnostic procedure for identifying trends of possibly nonmonotonic form allows to simultaneously account for the effect of possible regime shifts and change
points in the observed environmental time series, which is of particular importance in view of the
raising interest on the impact of regime shifts on the conclusion of presence of climate change. The
proposed testing procedure is robust to misspecification of the second order dependence structure
in the observed data as well as across a wide range of probability distributions, even for small
and moderate samples of observations. The developed method can be also employed for initial
assessment of presence and number of possible change points. Our simulation studies on size and
power of the test indicate that the new diagnostic procedure is competitive in respect to conventional
trend tests and, hence, can be recommended for analysis of a broad range of noisy environmental
data.
References
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