Detectability of Trends in Long Time Series of River Flow Data - A Run Up Effect

8/12/2019 Detectability of Trends in Long Time Series of River Flow Data - A Run Up Effect

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Detectability of trends in long time series of river flow data a run-up effect

Maciej Radziejewski1, Zbigniew W. Kundzewicz2, Dariusz Graczyk3

One of important practical problems in analysis of long time series of hydrological

records is to evaluate the detectability of trends in observed data, especially wherethe change is not strong, while the natural variability is considerable (e. g., in caseof river flow records). Study of a run-up effect was carried out, whose results allowan expert to assess how strong a change (gradual trend or abrupt jump) must be andhow long it must take (run-up duration) before it can be detected by alternative

tests available in the Hydrospect software package. A set of generated, trend-freedata mimicking the river flow records, has been contaminated by controlled

addition of a change / trend. Analysis of change detectability for differentconditions of natural variability of the process and of the contaminating component(its amplitude, intensity, and run-up time) has been made and the performance ofdifferent tests has been compared. The results are of broad applicability in thesearch for a climate change signature in hydrological data.

Introduction

The importance of change detection has never been as widely recognized as in the recent

years. The great question of climate change involves searching for traces of changes in

various natural processes measured numerically. Such changes might be very subtle, easy to

overlook in their beginning, but likely become larger and larger in the long run. Even a

relatively strong, abrupt change may be difficult to distinguish from the process natural

variability if it has occurred only recently. On the other hand, one may be able to detect a

small change if it affects a longer period of time in a systematic way. Simple rules like theabove, when put in a quantitative form, may serve as guidelines as to what kind of changes

one may hope to detect in a time series of observations affected (in part) by a change in the

underlying process.

Statement of the problem

In this study a simplified view of a process with changes is taken, namely as a composition of

a random process with independent values (base process) and a deterministic signal (change).

The base process adheres to the normal distribution and changes are introduced by adding the

deterministic signal to the base process. Although the results are of pure mathematical nature,

they were obtained with time series of river flows in mind, with year as the time unit of the

process. The estimates of detectability of changes of given type, duration and intensity (inrelation to the process standard deviation) may serve as rules of thumb for processes with

distribution reasonably close to normal.

In reality, changes of various aspects of the process distribution are possible and the base

process distribution may be different. However, it is not possible to examine every

1Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna, Poland, and Research

Centre for Agricultural and Forest Environment, Polish Academy of Sciences, Pozna, Poland,

[email protected] Centre for Agricultural and Forest Environment, Polish Academy of Sciences, Pozna, Poland, ph. (+

48 61) 8 475 601, fax (+ 48 61) 8 473 668, [email protected] Centre for Agricultural and Forest Environment, Polish Academy of Sciences, Pozna, Poland, ph. (+

48 61) 8 475 601, fax (+ 48 61) 8 473 668, [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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conceivable distribution and there is no unique best candidate. Moreover, it is not the shape

of the base process distribution that determines changes detectability, but the way in which

changes are related to this distribution. For example, it may be shown, that the results

analyzed here transform directly to the case of a process with a log-normal distribution,

subject to multiplicative changes. The log-normal case may fit better to the analysis of river

flows, where negative values are impossible and the zero value does not occur in the case ofperennial large rivers, such as Warta. It must be noted, that the assumption of independence

may be invalid in the study of observed annual mean flows, although it is often taken for

granted. This caveat does not affect the validity of presented results, only their applicability to

the case of river flows.

Random time series of different lengths were generated and contaminated with changes of

controlled shape, intensity and time-span. The mean and the standard deviation of the base

process have no influence on the detectability of changes as long as the intensity of changes is

understood as an appropriate multiple of the base process standard deviation. Therefore, the

random (base) series values were drawn independently from the normal distribution, with

mean and standard deviation equal to zero and one, respectively. Changes of two shapes,namely gradual (linear) trend and an abrupt, step change were added to these series in a

controlled way, so that the series before the change point is unaffected and linear component

of assumed gradient, or jump component of assumed (constant) magnitude is added from the

change point onwards. The period unaffected by changes in the series will be called the

reference period and the remaining period the change period. The length of the change

period will also be called run-up time in the case of gradual changes.

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Fig. 1 Time series with a linear trend starting in its midpoint.

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Fig. 2 Time series with a step change in its midpoint.


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An example of a generated series with a linear trend (base period of 30 years, change period

of 30 years, intensity of 0.05 standard deviation per year) is shown in Fig. 1, while an

example of a generated series with jump-shaped changes (base period 30 years, change period

30 years, magnitude 1 standard deviation) is shown in Fig. 2.

Methods of change detectionA large number of tests can be used for change detection in long time series of hydrological

records (cf. Kundzewicz, Robson, 2000, Radziejewski, Kundzewicz, 2000). In the present

study, five tests have been used, which do not assume any specific marginal distribution, such

as: 1. Mann-Kendall's test (e.g. Press, 1992), 2. Spearmans rank correlation (ibid.), 3. Normal

scores linear regression (ibid.), 4. Distribution-free CUSUM (Chiew, McMahon, 1993), and

5. Cumulative deviations (Buishand, 1982).

Since only the classic case of time series with independent values is analyzed, one may

make use of formulae for the significance level available for these tests. The notion of

significance level allows one to put all the results on the common scale between 0% and

100%. The convention used in this study means that significance close to 100% denotesstrong changes (i.e. one refers to detection of changes on the 95% significance level, not on

the 5% level, as in the other convention). Additionally, for tests Nos. 1-3, the direction of

changes is detected and a signed value of significance (between 100% and 100%) is used

accordingly.

For the example time series presented in Fig. 1 the tests Nos. 1-3 detect changes at the 99%

significance level. The test No. 5 achieves a result at 97.5% significance while the use of the

test No. 4 results in only 80% significance. Similarly in case of the time series presented in

Fig. 2 the first three tests result in significance levels at least 98.9%, which goes down to

about 98.4% for the test No. 5 and 65.8% for the test No. 4.

Analyses of detectability

Significance at the level of 99% means that the probability of obtaining a test result like this

or greater (in absolute value) in case when there are no changes (i.e., just due to a coincidental

arrangement of base process values) is exactly 1%. While one would definitely view 99% as a

fairly high significance level, it is important to remember that, on average, 1 in 100 random

time series may exhibit changes at this level by pure chance. This phenomenon is

particularly visible in detectability studies, where changes are controlled. If no changes have

been introduced to the base process, one may get, for a single realization, any value of

significance between 0% and 100% (or between 100% and 100%) with equal probability.

When a strong increasing trend is introduced, the significance level is usually quite high, butin theory any random arrangement of the process values (even one resulting in significance

99.9%) is possible, it may only be very improbable.

Thus one has to speak about the distributionof significance levels under given conditions and

compute detectability for a large number (10 000) of realizations of the base process to obtain

information about the distribution for each set of parameters. One can look at the mean, the

median, and the 25% as well as 75% quartiles of such distribution as functions of the type of

changes, reference period length, base period length, changes intensity and test number. It can

be demonstrated that the mean significance level is a fairly good measure of detectability.

Obviously, an important task to undertake is to reduce the number of variables.


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One may ask a question, how long a linear trend of intensity of 0.05 of the standard deviation

per year has to last before its presence in a time series can be detected. A complete answer can

be seen in a graph like Fig. 3(a), which presents, in the box-plot format, the three quartiles

(25%, median, 75%), and the mean of the significance distribution, as functions of the run-up

time. The reference period was 30 years. The quick shift from low to very high detectability

with the increase of the run-up time is not surprising, when one considers that not only changeduration but also its overall magnitude becomes greater.

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change period length [years]

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Mann-Kendall's test, median

Mann-Kendall's test, mean

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Fig. 3 The distribution of significance levels obtained with the Mann-Kendalls test for time series with (a) a

linear trend of 0.05 standard deviation, and (b) a step change by 1 standard deviation. 30-years reference period

and varied after-change period duration. The boxes span from the lower to the upper quartile of the significance

level distribution.


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A similar function for jump-shaped changes, with a magnitude of one standard deviation (Fig.

3(b)), is much gentler. For example, in case of 20-years change duration, the lower quartile is

96% and the median is 99.2%. This means that changes will usually be detected at least at the

96% significance level, but in the majority of cases at the 99% level. The mean significance is

below those values, at 95.1%, and goes up only to about 98.5%, while the lower quartile may

reach 99.3% when the after-change time is about 55 years. What is more surprising, andspecific for jump-shaped changes, is that detectability would actually decrease if the after-

change period is prolonged even further, because then the reference period would not be

sufficiently long.

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linear trend of 0.05 standard deviation, and (b) a step change by 1 standard deviation. 60 years total series length,

varied reference period and after-change period duration. The boxes span from the lower to the upper quartile of

the significance level distribution.


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If the total series length is fixed to, say, 60 years, one addresses another problem: Given a

time series of 60 years of data, what are the chances of detecting changes depending on the

time when the changes started? In case of fairly strong changes (0.05 standard deviation per

year and a jump by 1 standard deviation), the answer is given in Figs. 4(a) and 4(b). A weaker

linear trend, of 0.02 standard deviation per year (Fig. 5(a)), could only be detected in such a

time series if it spanned over most of the examined 60-year window. A jump by 0.5 standarddeviation will only be detected occasionally (Fig. 5(b)), if it occurs near to the middle of the

series. One might be disappointed by this last result, but it must be noted that 60 values

constitute still a relatively short time series.

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linear trend of 0.02 standard deviation, and (b) a step change by 0.5 standard deviation. 60 years total series

length, varied reference period and after-change period duration. The boxes span from the lower to the upper

quartile of the significance level distribution.


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Figure 6 offers an example of a 60 year-series with such a jump. The inhomogeneity of the

series is visible to the human eye, but one could hardly assert it with certainty. The tests show

changes correspondingly, only at a statistically insignificant level of 70-80%.

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Fig. 6 Time series with a step change by 0.5 standard deviation in the midpoint.

Jump detectability might have been slightly better with a test designed specifically for jump-

shaped changes. However, in practice, one seldom knows the type of changes expected.

One can see from these plots that the significance level distribution tends to be skewed

towards 100% for upwards-directed changes. In fact, the mean significance level is often

below the lower quartile, notably more so in case of better-detectable changes. Hence it tends

to be a more rigorous measure of detectability than the quartiles. On the other hand it is well-

correlated with the quartiles and has a natural interpretation in terms of unconditional

probability: it is precisely the probability that a random series, not contaminated by changes,

would yield a smaller test result than a random contaminated series with these parameters.

Hence, from now on, the study will be restricted to the mean significance level under given

conditions.

Consolidated results

The study of the dependence of the mean significance level on intensity of change reveals thatfor tests Nos. 1-3, under various combinations of other parameters, the relationship is

described remarkably well by the appropriately scaled logistic function (e. g. Fig.

Intensity2Mean_Lin - Intensity2Mean_Jump) described by the formula

)exp()exp(

)exp()(

axax

axxf

-+

= ,

where ais a horizontal stretch parameter. Apart from the general asymptotic characteristics

this function has an adequate behavior for small values and the correct thickness of tails.

This means that one looses virtually no information if one evaluates just one parameter of this

function: the intensity threshold, at which the mean significance crosses 95%. For tests Nos.


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4-5 the function in question is of a different character but still the intensity threshold defined

as above is a meaningful and useful measure of changes which one can hope to detect.

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Spearmans rank

correlation, meansignificance

Logistic function

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regression, mean

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Logistic function

Fig. 7 Mean significance as a function of the intensity of changes. (a) Linear trend detectability by Spearman's

rank correlation. 30 years reference, 20 years run-up time. (b) Jump-shaped changes detectability by normalscores linear regression. 30 years reference, 40 years after the change.

Table 1 shows the approximate intensity thresholds for linear changes, for all the tests and

different period lengths, with the reference period of length 10, 30 and 50, respectively. The

results were computed independently several times and the differences were usually below a

1% relative error. Analogous results for jump-shaped changes are presented in Table 2.


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Table 1 Intensity of a linear trend necessary to achieve a 95% mean significance level. Intensity is given as a

multiple of the base process standard deviation per year.

Reference

period

Run-up

time

Test No. 1 Test No. 2 Test No. 3 Test No. 4 Test No. 5

10 0.23 0.23 0.22 0.30 0.26

20 0.09 0.08 0.08 0.11 0.1030 0.05 0.05 0.05 0.06 0.05

40 0.03 0.03 0.03 0.04 0.04

50 0.02 0.02 0.02 0.03 0.03

60 0.02 0.02 0.02 0.02 0.02

10

70 0.01 0.01 0.01 0.02 0.02

10 0.28 0.28 0.26 0.40 0.26

20 0.08 0.08 0.08 0.10 0.08

30 0.05 0.05 0.04 0.05 0.05

40 0.03 0.03 0.03 0.04 0.03

50 0.02 0.02 0.02 0.03 0.02

60 0.02 0.02 0.02 0.02 0.02

30

70 0.01 0.01 0.01 0.02 0.01

10 0.38 0.37 0.31 0.81 0.28

20 0.09 0.09 0.09 0.11 0.09

30 0.05 0.05 0.05 0.06 0.05

40 0.03 0.03 0.03 0.04 0.03

50 0.02 0.02 0.02 0.03 0.02

60 0.02 0.02 0.02 0.02 0.02

50

70 0.01 0.01 0.01 0.02 0.01

Table 2 Magnitude of a jump necessary to achieve a 95% mean significance level. Magnitude is given as amultiple of the base process standard deviation.

Reference

period

Run-up

time

Test No. 1 Test No. 2 Test No. 3 Test No. 4 Test No. 5

10 1.65 1.57 1.57 1.80 1.68

20 1.49 1.44 1.42 1.64 1.45

30 1.56 1.50 1.47 1.69 1.42

40 1.67 1.60 1.55 1.83 1.45

50 1.84 1.75 1.66 2.06 1.49

60 2.02 1.91 1.77 2.48 1.55

10

70 2.25 2.09 1.88 1.60

10 1.55 1.49 1.46 1.74 1.42

20 1.00 0.99 0.97 1.16 0.98

30 0.86 0.86 0.84 1.00 0.86

40 0.82 0.81 0.79 0.95 0.80

50 0.80 0.79 0.78 0.93 0.7760 0.79 0.79 0.77 0.91 0.76

30

70 0.80 0.80 0.78 0.91 0.75

10 1.82 1.73 1.64 2.15 1.49

20 1.01 0.99 0.97 1.14 0.94

30 0.80 0.79 0.78 0.93 0.77

40 0.71 0.70 0.69 0.84 0.69

50 0.66 0.66 0.65 0.78 0.65

60 0.64 0.64 0.62 0.75 0.62

50

70 0.63 0.63 0.61 0.73 0.60

The infinity sign in Table 2 means that in an 80-year-long time series a jump which occurred

10 years after the start of the series will not be detected as change by test No. 4 at the 95%level, regardless of the jumps magnitude, because the reference period is too short.


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A linear trend apparently needs to be quite strong (0.1 of the base process standard deviation)

to be detectable within 20 years after its initiation. The length of the reference period seems to

have no major influence on the trends detectability. All the tests gave similar results for

linear trends, except for the case of a long (50 years) reference period and short change

period, when the test No. 4 needed distinctly higher thresholds. It does not have to be a

disadvantage of a test. Indeed, in this case a large part of the time series in question ishomogeneous.

In case of jump-shaped changes, the case of a short reference period and a long after-change

period is equivalent to the opposite situation, of a jump occurring close to the end of the

series. In order to detect a jump of moderate magnitude (one standard deviation or less) one

needs at least a total of 50 years of data and the jump must be close to the midpoint of the

series. The test No. 4 is, again, much less sensitive to changes spanning a very small part of

the series.

Concluding remarks

As results of change detection in hydrological records are, contrary to signature in suchmeteorological data as temperature, quite ambiguous, it is important to conduct

methodological studies, so that the role of the run-up, and the time of occurrence of the

change in a time series is adequately assessed. This study corroborates the corollary of

Kundzewicz, Robson (2002) that indeed a reliable detection of change is only possible when a

long time series of records (with a sufficiently long time after the onset of change) is

available. The time of necessary run-up depends on the natural variability of the process and

the amplitude of the change, yet sterile numerical experiment is needed to achieve meaningful

quantification. It is believed that the results of the present study shed some light on this

problem of considerable practical importance in hydrology.

Acknowledgements

The research reported in the present study has been conducted within the grant No.

6P04E05520 of the Scientific Research Committee, Republic of Poland.

References

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Hydrol. 58: 1127.

Chiew F. H. S. and McMahon T. A. 1993. Detection of trend or change in annual flow of

Australian rivers.Int. J. of Climatology13: 643653.

Kundzewicz Z. W. and Robson A. (eds). 2000. Detecting Trend and Other Changes in

Hydrological Data. World Climate Programme Water, World Climate Programme Data

and Monitoring, WCDMP-45, WMO/TD No. 1013, Geneva.

Kundzewicz Z. W. and Robson A. 2002. Change detection in river flow series a guided

tour. In: Bonnel, M., Sampurno Bruijnzeel & Kirby, C. (eds)Forests-Water-People in the

Humid Tropics. Cambridge University Press (in press).

Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. 1992.Numerical Recipes in C:

The Art of Scientific Computing, 2nded. Cambridge University Press.

Radziejewski M. and Kundzewicz Z. W. 2000. Hydrospect software for detecting changes

in hydrological data. Appendix 2 in: Kundzewicz, Z. W., Robson, A. (eds)Detecting Trend

and Other Changes in Hydrological Data; World Climate Programme Water, World

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Detectability of Trends in Long Time Series of River Flow Data - A Run Up Effect

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