Variation analysis of precipitation during past 286 years in

Variation analysis of precipitation during past 286 years in
Hydrol. Process. 27, 2934–2943 (2013)
Published online 22 June 2012 in Wiley Online Library
( DOI: 10.1002/hyp.9388
Variation analysis of precipitation during past 286 years in
Beijing area, China, using non-parametric test and wavelet
Miao Li,1,2 Jun Xia,1*,† Zhi Chen,3 Dejuan Meng1,2 and Chongyu Xu4
Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences & Natural Resources Research, Chinese Academy
of Sciences, Beijing 100101, China
Graduate University of Chinese Academy of Sciences, Beijing 100039, China
Department of Building, Civil & Environmental Engineering, Concordia University, Montreal, Canada
Department of Geosciences, University of Oslo, P.O. Box 1047 Blindern, N-0316 Oslo, Norway
Non-parametric methods including Mann–Kendall (M–K) test, continuous wavelet transform (CWT) and discrete wavelet transform
analysis are applied in this paper to detect the trend and periodic trait of precipitation data series in Beijing area where the data set spans
nearly 300 years from 1724 to 2009. First, the trend of precipitation variables is elaborated by the M–K test (Sequential M–K test). The
results show that there is an increasing trend (the value of this trend is 1.98) at the 5%-significance level and there are not turning points in
the whole data series. Then, CWT and wavelet variance are used to check for significant periodic characteristics of data series. In the plots
of wavelet transform coefficients and figure of wavelet variance, some periodic events affect the trend of the annual total precipitation
series in Beijing area. 85-year, 35-year and 21-year periodic events are found to be the main periodic series of long-term precipitation data,
and they are all statistically significant. Moreover, the results of non-parametric M–K test are exhibited on seven different combinations of
discrete wavelet components. D5 (32-year periodicity) periodic component is the effective and significant component on data. It is
coincident with the result (35-year periodic event as one part of main periodicity) by using CWT analysis. Moreover, approximation mode
shows potential trend of the whole data set because it is the residuals as all periodicities are removed from data series. Thus, the mode
A + D5 is responsible for producing a real basic structure of the trend founded on the data. Copyright © 2012 John Wiley & Sons, Ltd.
long-term data series; Mann–Kendall test; wavelet analysis; precipitation; Beijing area
Received 19 August 2011; Accepted 27 April 2012
The need for better information about the variability exhibited
by climatic and hydrological process has increased as a result
of the changing climate (Kalra and Ahmad, 2011). The
complexity of climatic and hydrological phenomena with nonstationary characteristics leading to diverse periodic events
occurred in different time periods has been paid close attention
by meteorologists and hydrologists (Buishand et al., 1988;
Andreo et al., 2006; Partal and Kahya, 2006; Jiang et al., 2007;
Kang and Lin, 2007; Huang et al., 2008; Kumar et al., 2009;
Dibike et al., In press; Zhang et al., 2011). Detecting the period
of observed data is used to be as a meaningful method to study
climatic and hydrological changes. Both time series of adequate
length and mathematical tools are necessary to optimize the
results. Thus, the long-term time series are important because
they permit to study climatic changes using the real data or to
reconstruct longer series using different mathematical tools.
In all factors of climatic and hydrological domain, the
alteration of precipitation with increasing and decreasing trend
*Correspondence to: Jun Xia, Key Laboratory of Water Cycle and Related
Land Surface Processes, Institute of Geographic Sciences and Natural Resources
Research, Chinese Academy of Sciences, Beijing 100101, China.
E-mail: [email protected]
Present address: State Key Laboratory of Water Resources and
Hydropower Engineering, Wuhan University, 430072, China
Copyright © 2012 John Wiley & Sons, Ltd.
has attracted more attention as the changing climate has caused
several catastrophic flood and drought events which already
caused much disaster around the world to both nature and
human beings (Karl and Knight, 1997; Zhang et al., 2009; Shen
et al., 2011). Thus, it is indispensable to study the hydrologic
regime especially the alternation of precipitation for a long-term
data series. It would be remiss not to mention some notable
works that have emphasized the changes in climatic and
hydrological process especially precipitation (Lettenmaier
et al., 1994; Hennessy et al., 1999; Luis et al., 2000).
Lettenmaier et al. (1994) examined spatial patterns of mean
streamflow and precipitation in the continental United States
and found a significant upward trend in both during
September–December. Hennessy et al. (1999) used the Kendall
tau test to detect trends in seasonal and total rainfall for 379
stations spread over the whole of Australia from 1910 to 1995
and compared the results using linear regression tests. They
concluded that changes are significant for total rainfall but show
insignificant changes during different seasons. Luis et al.
(2000) used non-parametric Mann–Kendall (M–K) and
Spearman’s rho tests to evaluate the trends in rainfall for 97
rain gauge stations in the region of Valencia (eastern Spain).
They observed a decrease in annual rainfall and showed a
significant increase in the interannual variability. Moreover,
discrete wavelet transform (DWT) analysis recently was also
used to detect the period of hydro-meteorological factors. Partal
and Küçük (2006) studied the precipitation of 64 years in
Marmara region, Turkey through using DWT analysis. The
data set was decomposed into series of approximation and five
details following Mallat’s algorithm, and they found 16 years
periodic component (DW4) was the effective component and
was responsible for producing a real trend founded on the data.
Xing et al. (2010) found that it is a fast and effective way to
detect and compare the similarity of different hydrological
series by using the DWT analysis.
There are lots of studies to focus upon the time series trends
analysis of precipitation (Widmann and Schar, 1997; Partal and
Kahya, 2006; Mekis and Vincent, 2011) Widmann and Schar
(1997) detected 113 series of the long-term record during
1901–1990 spread all over Switzerland. They found a
wintertime increase in precipitation by up to 30 percent per
100 years in the western and northern parts of Switzerland, and
this trend is statistically significant at the 90 percent level. In
most parts of southeastern Switzerland, winter precipitation
increased as well but by a smaller rate and at a slightly lower
statistical significance level. Partal and Küçük (2006) determined trends in the long-term annual mean and monthly total
precipitation series (1929–1993) with 96 stations across Turkey
using non-parametric methods. They found that a noticeable
decrease in the annual mean precipitation was observed mostly
in western and southern Turkey.
Beijing is the political, economic and cultural centre of
China, with a history of over 2500 years and the residents of
more than 20 million. Several researches have investigated the
trend of precipitation in Beijing area. Li et al. (2010) studied the
precipitation variation in Beijing area, and they considered that
the trend of precipitation in Beijing area is linearly declined in
the latest 51 years, and some other studies also came to the same
conclusions (Xie et al., 2000; Kuang et al., 2000). For studying
the periodic fluctuation of precipitation in Beijing area, Li et al.
(2010) found three periods of 7-year, 14-year and 26-year as
main periods of 51-year data series. Xie et al. (2000) studied
wavelet characteristics of climate change in Beijing area since
the 19th century, and they concluded that in summer, 35-year
period of precipitation was found to be a main period in around
150 years data series, and in spring and autumn, the periodicity
of around 20 years controlled the whole series. Kuang et al.
(2000) determined three dominating periods of precipitation,
6–7 years, 21–22 years and 35–36 years, using data of 117
years in North China Plain. However, all these researches
mentioned above did not consider the precipitation variations of
much longer time series. It still remains unknown about the
alternation of increasing and decreasing of precipitation of
nearly 300 years in Beijing area. Therefore, it will have positive
and far-reaching significance to take a research on trend and
fluctuation of precipitation in large time scales of 300 years for
well-known Beijing city with such a long history.
This study aimed to detect the possible trends of annual
precipitation data series by using the non-parametric methods
such as the M–K test and the wavelet analysis. First, the M–K
test was applied to detect the trend and turning points of longterm precipitation data series in Beijing area, and then the
continuous Morlet wavelet analysis is applied to find the
periodicity oscillation strength of series and the methods of
wavelet variance and wavelet power spectrum are used to
Copyright © 2012 John Wiley & Sons, Ltd.
Figure 1. The annual total precipitation plot of the Beijing meteorological
station in China
determine the main periods and its statistical significance.
Finally, we made a discrete wavelet analysis based on the scales
2m (m = 1, . . . , 7) to find which periodicities are mainly
responsible for the trend of measurement series.
Annual total precipitation data which cover a time series
of 286 years (from 1724 to 2009) in Beijing, China
were examined for trend and wavelet detection (Figure 1).
The Beijing meteorological station is located at 39º 480
latitude and 116º 280 longitude which mostly controlled by
semiarid climate.
Data consistency check. The total precipitation data for
180 years (1724–1904) in Beijing area has been compiled by
Chinese meteorologists1 (1975) according to the ‘Clear and
Rain Records’ in the Forbidden City in the mid-1970s, and it
extended sequence of precipitation in Beijing for 117 years.
Linking up all the data with observed modern instrumental
precipitation data (1841–2009) acquired from the Beijing
Meteorological Station, China Meteorological Administration
made precipitation data series in Beijing reaching up to 286
In the precipitation data series for 286 years, the compiling
method of the former 117-year precipitation data is as follows:
First, both series of number of hours for precipitation and
quantity of precipitation of the overlapped sections are found
according to the two precipitation series between the number of
hours for precipitation in the ‘Clear and Rain Records’ and the
quantity of precipitation from instrumental observation in
Beijing station. Then, a relationship between them is found;
Second, according to the established relationship, the hours for
precipitation of the former 117 years based on ‘Clear and Rain
Records’ are calculated into precipitation data. Hence, it can be
considered that two data series of precipitation are consistent.
Institute of China Academy of Meteorology. 250 years Precipitation in
Beijing (1724–1973). 1975.
Hydrol. Process. 27, 2934–2943 (2013)
The details of specific conversion method for reconstructing annual precipitation and the relevant statistical
parameters are presented in Appendix.
Data standardized. Before the DWT analysis, a process
of data standardized should be carried out by the
following equation.
x’t ¼
ðxt xÞ
where x’t and xt are the standardized precipitation data and
are the
the observed data for t years, respectively; x and s
mean value and the mean square deviation of precipitation
data series, respectively.
Mann–Kendall test
The rank-based non-parametric M–K statistical test (Mann,
1945; Kendall, 1975) has been commonly used for trend
detection (Yue and Wang, 2002; Zheng et al., 2007) because
of its robustness for non-normally distributed and censored
data, which are frequently encountered in hydroclimatic time
series. This method defines the test statistic Z as
> VarðSÞ
0 if S ¼ 0
: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i f S < 0 >
n1 X
sgn xj xi and
Wavelet analysis
Wavelet analysis is recently paid more attention as a useful
method and is applied in wide ranges of fields in science and
engineering. Compared to the Fourier transform, the wavelet
transform analysis is considered to be more effective in
studying non-stationary time series. Moreover, the wavelet
transformation variance which expresses the energy of
fluctuations with distribution of time scales could distinguish
the main time scales of time series.
Continuous wavelet analysis. Assuming a continuous
time series x(t), t 2 (1, 1), a wavelet function c() that
depends on a non-dimensional time parameter can be
written as
t t
cðÞ ¼ cðt; sÞ ¼ s1=2 c
where t is time, tis the time step in which the window
function is iterated and s 2 (0, 1) is the wavelet scale.
c() must have zero mean and be localized in both time
and Fourier space (Meyer, 1993). The continuous wavelet
transform (CWT) is given by the convolution of x(t) with
a scaled and translated c(),
nðn 1Þð2n þ 5Þ Var ðSÞ ¼
i¼1 jiþ1
where n is the data record length and xi and xj are the
sequential data values. The function sgn(x) is defined as
if x > 0
sgnðxÞ ¼ 0
if x ¼ 0
1 i f x < 0
Equation (3) gives the standard deviation of S with
correction for ties in the data, with ei denoting the number
of ties of extent i. The upward or downward trend in the
data is statistically significant if |Z| > u1 a/2, where u1 a/2
is the (1 a/2) quantile of the standard normal distribution
(Kendall, 1975) and when a = 0.05 and u1 a/2 = 1.96. A
positive Z indicates an increasing trend in the time series,
and a negative Z indicates a decreasing trend.
Sequential Mann–Kendall test
Also, the sequential values u(t) from the progressive
analysis of the M–K test (Sneyers, 1990) were determined
in order to see the change of trend with time. u(t) is similar
Copyright © 2012 John Wiley & Sons, Ltd.
to the Z value. Herein, u(t) is a standardized variable that
has zero mean and unit standard deviation. Therefore, its
sequential behavior fluctuates around the zero level. u(t) is
the same as the Z values that are found from the first to the
last data point (Partal and Kahya, 2006).
iði 1Þð2i þ 5Þ
W ðt; sÞ ¼ s1=2
xðt Þc
t t
where * indicates the conjugate complex function. By
smoothly varying both s and t, one can construct a twodimensional picture of wavelet power,|W(t, s)|2, showing
the frequency (or scale) of peaks in the spectrum of x(t), and
how these peaks change with time (Drago and Boxall,
2002). The lower scales refer to a compressed wavelet, and
the abrupt changes or high frequency component of a
signal can be identified by the lower scales. On the
contrary, slowly progressing occurrences or low-frequency
component of the signal should be presented by the
corresponding coefficients of the higher scales composed
by the stretched version of a wavelet.
Wavelet variance. Taking an integral to square of wavelet
coefficients in time domain could get the wavelet variance.
VarðsÞ ¼
Wf ðs; tÞ2 dt
Hydrol. Process. 27, 2934–2943 (2013)
Process of wavelet variance with scaletcan get a map of
wavelet variance. It reflects the distribution of fluctuation
energy with time scales, thus it could determine the main
time scale in a time series which is called the main period
(Heng et al., 2002; Lu et al., 2010).
Discrete wavelet transform. It takes amount of work
when calculating the wavelet coefficients at every possible
scale, and it also generates lots of data. Obviously, it is
more efficient and will not lose accuracy when using
powers of two (dyadic scales and positions) to replace
calculating every scale of wavelet coefficients which is
called DWT that has the form of
t t
t nt0 sm
¼ s0 c
Where m and n are integers that control, respectively, the
wavelet dilation (scale) and translation (time), s0 is a
specified fixed dilation step greater than 1 and t0 is the
location parameter and must be greater than zero. From this
equation, it can be seen that the translation step nt0 sm
depends on the dilation sm
0 . The most common (and
simplest) choices for the parameters s0 and t0 are 2 and 1
(time steps), respectively. This power of two logarithmic
scaling of the translations and dilations is known as dyadic
grid arrangement and is the simplest and most efficient case
for practical purpose (Mallat, 1989). For a discrete time
series xi, where xi occurs at discrete time i (i.e. here integer
time steps are used), the DWT becomes
Wm;n ¼ 2m=2
N 1
xi cð2m i nÞ
where Wm, n is wavelet coefficient for the discrete wavelet
of scale s = 2m and location t = 2mn. Therefore, the DWT
informs variation in a time series at different scales and
location as defined above.
Mann–Kendall results
The M–K test is applied to detect the trend and turning
points of precipitation data series during past 286 years in
Beijing area. The values calculated by M–K test algorithm
are shown by the respective solid line, dotted line in
Figure 2, and the horizontal dashed lines correspond to the
confidence limits at the 5%-significance level (1.96). If
the solid line passes over the dashed line, it means there is a
statistically significant trend of increasing or decreasing.
The value of M–K test of the whole data series is 1.98,
which presents a significant increasing trend for long-term
precipitation data series in Beijing. The solid line
calculated by the sequential M–K test in Figure 2
demonstrates temporal changes of the annual precipitation
in Beijing area. It is clearly seen that the long-term data
series is divided into two parts on 1777. Before this year, a
decreasing trend of precipitation happened in Beijing area.
Copyright © 2012 John Wiley & Sons, Ltd.
Figure 2. Statistics values using Mann–Kendall test for Beijing precipitation data series
It is beyond the threshold of significant level U0.05 = 1.96
over the period of 1745–1767, and even beyond the
significant level U0.001 = 2.56 from 1747 to 1762. Therefore, it shows the significant trend of precipitation in
Beijing area. However, meaningful long-term trends are
identified in increasing trend from in end of 1760s to 2009,
and the trend always keeps increasing. For a very long time
period, the trend of precipitation exhibited significance
because the values of M–K test are more than 1.96 and even
more than 2.56. From the first decade of the 21st century, it
became insignificant (the values of M–K test are less than
1.96), but it is still an increasing trend according to the
M–K values of precipitation in Beijing area.
There are two points of intersection in 1767 and 1999 in
Figure 2 which are zoomed in and shown in Figure 3 for a
clear seeing. Because two intersect points are not at the
thresholds of significant level (U0.05 = 1.96) which
represents as dashed lines in Figure 2 and Figure 3,
therefore, both two intersect points are not turning points.
Continuous-time Morlet wavelet analysis
Before the wavelet transform, the preprocessing of data
should be carried out by anomaly for filtering one-year
natural cycle of precipitation series.
Multiple time scale analysis of precipitation. The
continuous Morlet wavelet analysis is applied, which
allows the completion of time-scale representation of
localized and transient phenomena occurring at different
time scales. The time-scale discrimination is achieved in a
more satisfactory way than time-frequency decompositions
such as the windowed Fourier method. In Figure 4, the real
part time-frequency distribution of Morlet wavelet transform coefficients of the whole precipitation data series in
Beijing area is shown. It is a time-scale plot of the signal
where the x-axis represents position along the signal (time),
the y-axis represents a periodicity scale and the solid
contour represents the positive phase which means the
coefficients of real part is greater than or equal to 0.0; and
Hydrol. Process. 27, 2934–2943 (2013)
Figure 3. Mann–Kendall values of intersect points zoomed in of Figure 2
Figure 4. Real part time-frequency distribution from Morlet wavelet
transform coefficients
Figure 5. Modulus square time-frequency distribution from Morlet
wavelet transform coefficients
the dotted contour indicates a negative phase which means
coefficients of real part is less than 0.0. Figure 4 clearly
shows the fluctuation characteristics of the real part of the
wavelet transform coefficients, which reflects alternative
variation of precipitation (more than normal or less than
normal) in Beijing area. The fluctuations of time scale for
21-year, 35-year and 85-year are distinctly shown in
Figure 4 in which positive and negative phase happened
with an alternation pattern.
The modulus square time-frequency distribution through
the Morlet wavelet transform is shown in Figure 5, and it
represents the signal strength of characteristic time scales.
There are three primary time scales of fluctuation variation
giving impact on precipitation in the whole time domain.
The time scale of 85–95 years exhibits the strongest signal
whose oscillation centre is around 1790, and it happens
throughout the entire time domain. It is also the most
prevailing time scale affecting the future precipitation in
Beijing area. It has a stronger signal in time scale of 30–35
years whose oscillation centres are located at 1870 and
1960, respectively, and the time scale of 30–35 years
happened primarily after 1830. The time scale of 20–25
years which happened mainly after 1850 is relatively weak,
and the oscillation centres are in 1885 and 1980.
In order to make a further illustration of the fluctuation
characteristics of precipitation in Beijing area, we made three
cutting lines parallel to x-axis on the values mentioned above
(21 years, 35 years and 85 years) respectively in Figure 4; and
then the intersection points on each cutting line with Figure 4
formed a time series of real parts in Morlet wavelet
coefficients in the period of 1724–2009; and at last, we
added the three time series in the annual precipitation
anomaly series, as shown in Figure 6. It is found that, for a
specific year, different time scales were studied, different
conclusions whether the precipitation is over or short would
be got. That is because in different time scales, the
corresponding phase of the same year is diverse. In essence,
small-scale alternating variations are nested into large-scale
ones with macro-structure. It is also proved the necessity to
study the dynamic trend of precipitation by combining with
the time scales.
As shown in Figure 6, the dynamic variations and
mutation points of precipitation can be distinguished
through the wavelet coefficients line of real part of all the
Copyright © 2012 John Wiley & Sons, Ltd.
Hydrol. Process. 27, 2934–2943 (2013)
Figure 6. The annual precipitation anomaly in Beijing area and real part
variability of wavelet transformation
main periods. The amplitude of 21-year time scale becomes
gradually larger in the period of 1724–1860 and stabilizes
in the period of 1860–2009. For 35-year time scale, the
amplitude in the period of 1724–1820 is not strong and
takes a significant increase after 1820. In addition, the
fluctuation of 85-year time scale is stable in the entire
calculation domain. However, it shows a slow trend after
1970. For the alternation of increasing and decreasing
precipitation, taking 85-year time scale as an example,
there exists a positive real part wavelet phase at periods of
1773–1816, 1861–1902 and 1944–1984 with sufficient
precipitation, while negative phase happened in the periods
of 1729–1772, 1817–1860, 1903–1943 and 1985–2009
with scarce rainfall. Thus, it shows alternative characteristics with decreasing and increasing in the whole time
series of precipitation in Beijing area. According to the
periodical features, it can be concluded that the period of
2009–2030 is at a less precipitation one.
The primary cycle analysis of precipitation time series.
The wavelet variance is applied to confirm the main periods
of precipitation data series. The wavelet coefficients
calculated are substituted into Equation (6) to calculate
the wavelet variance of annual precipitation anomaly
series, and the plot (Figure 7) of wavelet transformation
Figure 7. Wavelet transformation variance
Copyright © 2012 John Wiley & Sons, Ltd.
variance would be drawn with the wavelet transformation
variance Var as the vertical axis and the time scales a as the
horizontal axis. The wavelet transformation variance
reflects the energy of fluctuations with distribution of time
scales, those with stronger energy (peaks in Figure 7) are
called the main periods. Three peaks of the wavelet
variance are marked in Figure 7 whose values are located at
the scale of 21 years, 35 years and 85 years. The apex value
of wavelet variance of scale is 85 years, and it means the
strongest oscillation appears in an 85 year period, and it is
the uppermost period of precipitation series throughout 286
years. In addition, the second and third periods are 35 years
and 21 years, respectively.
The wavelet power spectrum of precipitation data series
of 286 years in Beijing area were shown in Figure 8 to
detect the statistically significance of main periodicities of
85 years, 35 years and 21 years. In Figure 8, the dashed line
represents the threshold values of 95% confidence level
and obviously the spectral density of 33–200 years (values
of frequency are from 0.005 to 0.03) surpass the dashed line
(95% confidence level), which proves that the periodicities
of 85 years and 35 years are statistically significant.
Meanwhile, the spectral density of 20 years (value of
frequency is from 0.05) surpasses the dashed line, and the
main period of 21 years is also statistically significant.
In previous studies shown in Table I, 6–7 years, for the
short series around 50 years, 7 years, 14 years and 26
years were found to control the whole series and for
longer precipitation data series (117 years and 153 years),
21–22 years and 35–36 years were found to be the main
periods in Beijing area. In fact, the results for longer time
series above took a consistency with the part of results in
this study (35 years and 21 years as the main period
events). However, because of the different length of data
series acquired for this study and others, a slightly
different but significant conclusion could be reached.
From a view of longer data series of nearly 300 years,
85-year periodic event of precipitation is identified as the
strongest main period controlling the whole data series in
Beijing area. This has never been detected in previous
Figure 8. Wavelet power spectrum of 286 years precipitation data series in
Beijing area
Hydrol. Process. 27, 2934–2943 (2013)
Table I. Documented results of wavelet transform analysis in Beijing (BJ) and North China Plain (NCP)
Methods for WT
Main periods
Mexican hat
Mexican hat
Mexican hat
6–7 years, 21–22 years,35–36 years
35 years
15 years
7 years, 14 years, 26 years
21 years, 35 years, 85 years
Kuang et al. (2000)
Xie et al. (2000)
Niu et al. (2004)
Li et al. (2010)
this study
work on rainfall data series. Moreover, in the CWT analysis, a
time scale of 284 years showed with an alternating positive
and negative phase in Figure 4. Although it is not clear
because of its non-closed contour of phase, and much longer
data series is needed to make a validation, it is certain that
85-year period is the strongest one as the peak value in the
figure of wavelet variance (Figure 7).
Trend analysis results of wavelet components
Decomposition of precipitation data. The Mallat’s
algorithm is used to decompose the standardized data
according to Equation (9). The whole process includes
some successive filtering steps. First, the standardized
signal is decomposed into an approximation and accompanying detail, and then the iterating algorithm mentioned
above with successive approximation is decomposed in
turn, so that the standardized signal is divided into many
lower resolution components (Mallat, 1989).
Figure 9 presents the time series of seven components,
which include 2-year mode (D1), 4-year mode (D2), 8-year
mode (D3), 16-year mode (D4), 32-year mode (D5),
64-year mode (D6), 128-year mode (D7) and the
approximation acquired from Equation (8) for standardized
precipitation data series. The D transform coefficients
elaborate precipitation variations and interannual scale at
different periods of the whole long-term data set. Table II is
the results of M–K test of different D transform and total
mean square errors (MSEs). The third column of Table II
represents the D combination values of M–K test which
surpass the significant level a = 0.05(1.96) from the whole
128 models with different D components. The M–K values
of A + D5 model is 5.51 which indicates the increasing
trend of precipitation data series is most significant and also
the A + D2 + D3 + D5 + D7 model is the nearest D
components compared to the trend of standardized series.
As found by continuous Morlet wavelet transform and
wavelet variation, 35-year period is the one of main periods
Figure 9. Time series of (D1) 2 year mode, (D2) 4 year mode, (D3) 8 year mode, (D4) 16 year mode, (D5) 32 year mode, (D6) 64 year mode, (D7) 128
year mode and approximation mode in precipitation series
Copyright © 2012 John Wiley & Sons, Ltd.
Hydrol. Process. 27, 2934–2943 (2013)
Table II. Mann–Kendall (M–K) test significant results of different
D transform and total mean square errors between data and
different combinations of different subseries of data
Model combinations
M–K test
Original time series
A + D3
A + D4
A + D5
A + D7
A + D2 + D5
A + D3 + D4
A + D3 + D5
A + D3 + D7
A + D4 + D5
A + D4 + D7
A + D5 + D7
A + D2 + D3 + D5
A + D2 + D5 + D6
A + D2 + D5 + D7
A + D3 + D4 + D5
A + D3 + D4 + D7
A + D3 + D5 + D7
A + D4 + D5 + D7
A + D2 + D3 + D5 + D7
A + D3 + D4 + D5 + D7
MSE, total mean square error; R correlation between data and D models.
for the whole long-term precipitation data set. Thus, we
find that the study of series by DWT reveals which
component is responsible for the whole precipitation data
set. In all the 128 D combinations, total 20 D combinations
are beyond the significance positive level, and 13
combinations include D5 periodic component. Thus, we
consider that D5 (32-year periodicity) periodic component
is the effective one on the data and is responsible for
producing a real trend found in the data.
Comparison of trends
In this part, we discuss the relation between the results of
sequential M–K test from different DWT coefficient of D
combinations and standardized precipitation time series
calculated from observed data set in Beijing area. The
fourth and fifth columns of Table II show total MSE and
correlation coefficients R for these correlations. It is found
that MSE value of A + D2 + D3 + D5 + D7 model is 0.12
which means this model is the nearest to the M–K trend
values of standardized data and it is parallel to M–K test
statistics. Moreover, models of A + D3 + D5, A + D2 +
D3 + D5, A + D2 + D5 + D7 which all include D5 component show the best results (less MSE values and higher
R values). When every periodic component is separately
focused, the value of R between D5 and standardized data
reaches 0.88. All these analysis above reveal that D5 is the
dominant periodic component of precipitation data series
in Beijing area. Moreover, it is expected from Figure 9 that
the approximation mode which is the residuals as all
periodicities are removed from data series shows potential
trend of the whole data set. Thus, we may conclude that
A + D5 mode is responsible for the real whole precipitation
data series (periodicity and trend)
As shown in Figure 10, there are also good match to the
values of sequential M–K test on models of A + D3 + D5,
Figure 10. Sequential values of the Mann–Kendall test on original values and D transform values of Beijing precipitation series. Solid lines represent the
original series and the dotted lines mean D transform series
Copyright © 2012 John Wiley & Sons, Ltd.
Hydrol. Process. 27, 2934–2943 (2013)
A + D2 + D5 + D7 and A + D2 + D3 + D5 + D7 and
standardized data. It exhibits a new opinion with values
on the wavelet coefficient computed by D transform. In this
figure, the sequential values of M–K test, both observed
data and D components, are illustrated by the respective
solid and dotted lines. Only are best match periodic
components (least MSE and highest R) shown in Figure 10
for precipitation series in Beijing meteorological station.
The sequential M–K test values of models A + D3 + D5,
A + D2 + D5 + D7 and A + D2 + D3 + D5 + D7 are rather
similar to that of observed data. The sequential M–K values
of A + D5 are harmonious with one of the data. These
figures together with Table II present that decadal periodic
changes (30–35 years) are responsible for forming
fluctuating trend at Beijing. Table II and Figure 10 could
clearly indicate that the A + D5 component dominated the
whole long-term precipitation data series from the view of
periodicity and trend, and the DWT analysis well explains
the basic structure of the trend.
Non-parametric methods including the M–K test, the CWT
and the DWT analysis are applied in this paper to detect the
trend and periodic traits of precipitation data series of
nearly 300 years in Beijing area.
First, the trend of precipitation is elaborated by the M–K
test (Sequential M–K test). Results show that there is an
increasing trend (the value of this trend is 1.98) at the
5%-significance level. Considering the long-term data
series of about nearly 300 years, an increasing trend is
found from 1786, and it is clearly significant (beyond the
5% significance level) from 1786 to 2003.
Then, CWT and wavelet variance are used to check for
significant periodic characteristics of data series. The
plots of wavelet transform coefficients and figure of
wavelet variance show that some periodic events affect
the trend of the annual total precipitation series in Beijing
area. 85-year, 35-year and 21-year periodic events are
found to be the main periodic series of long-term
precipitation data.
Also, the results of non-parametric M–K test on different
combinations of discrete wavelet components shows that
D5 (32-year periodicity) is the most effective periodic
component on the data and A + D5 mode is responsible for
producing a real basic structure of the trend founded on the
data. It is coincident with the result (35-year periodic event
as one part of main periodicity) by using CWT analysis.
Therefore, it is useful to explain the trend structure of the
precipitation data series based on D components from
DWT method.
In this study, we proved that it is of importance using a
longer enough data series when studying non-stationary
problems in hydro-meteorological domain or climate
changes. In fact, we still could not clarify if the 284-year
periodic event is as one main period of the whole
precipitation data series in Beijing area. Future works are
Copyright © 2012 John Wiley & Sons, Ltd.
needed first to acquire further longer enough data series to
clarify the 284-year periodic event and second to focus on
the extreme drought or flooded problems.
This study was supported by the External Cooperation
program of the Chinese Academy of Sciences, Grant No.
GJHZ1016 & GJHZ06, and the National Basic Research
Program of China (2010CB428406).
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The regression equations and correlation coefficients of
each month are constructed and calculated by using the
overlapped records of 40 years during 1841–1903 from
precipitation hours of ‘Clear and Rain Records’ and
precipitation of observed station respectively, which
Copyright © 2012 John Wiley & Sons, Ltd.
covers period 1841–1843, 1845–1855, 1860–1861,
1869–1884, 1889–1890, 1892–1897 and 1899.
Regression equation goes as follow,
P ¼ a0 þ a1 t1 þ a2 t2
a0 represents constant values; a1 and a2 represent
regression coefficients; t1 means the precipitation hours
of slight rainfall (slight snow); t2 means the precipitation
hours of rainfall (snow).
Regression coefficients, correlation coefficients and
F-test values of reconstruction of precipitation data series for
each month are shown in Table A1. The values of
correlation coefficients of each month are more than 0.68
except in February which value is 0.41, and F-test values are
much higher than statistical significance of F0.05 and F0.01
whose values are 3.25 and 5.24, respectively. Therefore, the
restruction values of precipitation data series are reliable
from the point of view of mathematical statistics.
Table A1. Regression coefficients, correlation coefficients
and F-test values of multiple regression equation for
reconstruction of precipitation
F-test values
Hydrol. Process. 27, 2934–2943 (2013)
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