How should trends in hydrological extremes be estimated?

Robin T Clarke

Instituto de Pesquisas Hidraulicas

Universidade Federal do Rio Grande do Sul (UFRGS)

Porto Alegre – RS

Brazil

Keywords: Trend estimation; hydrological extremes; extreme value distributions; Theil-

Sen.

This article has been accepted for publication and undergone full peer review but has not beenthrough the copyediting, typesetting, pagination and proofreading process which may lead todifferences between this version and the Version of Record. Please cite this article as an‘Accepted Article’, doi: 10.1002/wrcr.20485

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Summary.

A comparison of six procedures for estimating the linear trend parameter β in

annual maximum one-day river flows at five sites in southern Brazil showed marked

differences between, on the one hand, estimates obtained by incorporating trend into the

GEV location parameter with all parameters estimated by maximum likelihood (ML); and

on the other hand, estimates found by least squares, trend-removal prior to fitting the

GEV by ML, boot-strap sampling, and Theil-Sen estimation. ML estimates of trend were

considerably smaller than those given by all other procedures. The same was true where

trend had been incorporated into the Gumbel location parameter. Where 95% confidence

intervals were calculated for the “true” trend β by different procedures, some confidence

intervals bracketed zero (indicating that the trend was not “significant” at the 5% level)

but there was no consistency between results from different procedures; Theil-Sen

confidence intervals always bracketed zero, confidence intervals given by de-trending

never did. It is concluded that not only do different estimation procedures give different

measures of trend uncertainty, as reported elsewhere, but the estimated trends themselves

may differ, and the paper suggests an explanation why this may occur. Some

philosophical issues relating to estimation of trend in climatological and hydrological

extremes are discussed, and it is concluded that selection of a method to estimate trend

must depend on context.

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Introduction.

Recent decades have seen a surge in analyses of hydrological and climatological

data to seek evidence of trends brought about by anthropogenic influences. In terms of

trends in river flows, the international literature reports many recent studies of trends in

annual and seasonal flows [e.g., Hannaford and Buys, 2012; Burn et al., 2010; Wilson et

al., 2010; Fiala et al., 2010; Novotny and Stefan, 2007, Hodgkins and Dudley, 2006]

some of which have included analyses of trends in high flows [Marsh and Harvey 2012;

MacDonald, 2012; Petrow and Merz, 2009; Hannaford and Marsh, 2008; Svensson et

al., 2006; Robson et al., 1998]. The present paper addresses some issues relating to the

analysis of trends in high flows, in a region where land-use change from native forest to

arable cropping is likely to be at least as influential on extreme river flows as any existing

or potential climate change over the last eighty years. In particular, the paper is concerned

with the apparently simple issue of how to estimate trends in annual maximum one-day

river flows, although the methods discussed also have relevance to the estimation of

trends in annual maximum rainfalls of any given duration, and trends in annual maxima

of climatological variables.

There is a very extensive literature on the use of Extreme Value (EV) distributions

for describing the variability amongst “block maxima” (such as the series obtained by

abstracting maximum values during periods or blocks – typically years) with the

Generalized Extreme Value (GEV) distribution given by

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G (q) = exp{- [ 1 + ξ ( q – μ ) / σ ] -1/ξ } 1 + ξ ( q – μ ) / σ > 0; σ > 0 (1)

where μ, σ, ξ are parameters of location, scale and shape respectively; q denotes the

annual maximum one-day flow. The methods of Hosking and Wallis [1997] based on

estimation of the GEV parameters (μ, σ, ξ) by L-moments have proved simple to use, and

software is also widely available for fitting GEV distributions by maximum likelihood

(ML). Coles [2001, Chapter 6] has described how, in the presence of trend in the series of

block maxima, the GEV distribution can be adapted to estimate trends in any of the

parameters, typically and most commonly by fitting the modified distribution GEV

(μ (t), σ, ξ) where μ (t) = α + β t ; extensions to GEV (μ (t), σ (t), ξ(t)) are also possible.

In all such approaches, it is assumed that annual maxima are statistically independent.

This paper considers only the simpler model GEV (α + β t , σ, ξ), with particular

emphasis on the estimation of β, and on the uncertainty in this estimate as measured by

its 95% confidence interval. The linear trend parameter β is estimated by ML using the

statistical package GenStat [VSN International, 2012]; other packages (e.g., ismev,

http://www.ral.ucar.edu/~ericg/softextreme.php, Heffernan and Stephenson, 2013, and

related packages evd, evdbayes, lmom, POT) allow more general GEV models to be fitted

with link functions relating parameters to predictors [Coles, 2001, Section 6.1] , which

may include other predictors as well as time. Also, since the GEV distribution reduces to

the widely-used Gumbel distribution when the GEV shape parameter ξ is zero, the paper

includes this too, using a form in which the Gumbel parameter μ is μ (t) = α + β t.

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A GEV distribution with time-variant location parameter is not the only way to

estimate trend parameters when annual maxima are statistically independent. The many

alternatives include bootstrap estimation [e.g., Venables and Ripley, 1999], Theil-Sen

distribution-free estimation [e.g., Hannaford and Buys, 2012], two methods that are used

in this paper. Ordinary least squares also provides a valid estimate of the trend parameter

β, but the fact that extreme flows are commonly heteroscedastic, with larger extremes

having greater variances, means that confidence limits for β calculated on the

assumption that the residual variance σe2 is constant will no longer be valid. Another

estimate of β explored in this paper is given by using least squares to estimate the trend

coefficient β, then calculating the residuals {ε t} to which a GEV (or Gumbel) is fitted,

the location parameter then being no longer time-dependent. An advantage of such a

procedure is that ML estimation requires a search over the three-dimensional (μ, σ, ξ)

space instead of the four-dimensional (μ, σ, ξ, β): a considerable advantage if many

sequences are to be analysed for trend Other methods of trend estimation (M-estimators;

least median of squares (LMS); least trimmed squares (LTS); S-estimation; MM-

estimation) might have been included, but most have drawbacks [Venables and Ripley,

1999], and in any case, are not widely used by hydrologists and climatologists. Nor does

the paper include the estimation of trends in “peaks-over-threshold” (PoT) models, which

can also be modelled, in very general forms, by the ismev software referred to above.

Thus the purpose of the paper is to compare estimates ^

β of the trend coefficient

β obtained from a number of possible estimation procedures, and to compare their

uncertainties as measured by these estimates’ approximate confidence intervals. The

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following sections describe the data used, the analytical procedures by which they were

analysed, and the results. A discussion follows, and conclusions are stated.

Data.

The primary data source was measurements of daily mean flow over the period

1940 to 2010 at five gauging stations in the southernmost Brazilian state of Rio Grande

do Sul, a region where clearance of native forest and land-use change to agriculture has

been extensive over the period of record. For one site, 14 de Julho, the record was even

longer: 1931 to 2010, but with the two years 2004 and 2005 missing. The five sites were

used because they are the longest in the region, and because they are almost complete.

Table 1 give details of location, upstream drainage area, and mean annual rainfall for the

five sites. From the records of daily mean flow, the maximum daily flow in each year of

record was abstracted. Figure 1 shows a plot of the five sequences of annual maximum

flows for the period 1940-2010, standardized by subtracting each sequence mean and

dividing by each sequence’s standard deviation, so that the vertical axis is dimensionless.

Visual inspection suggests a slight positive trend in annual maxima over the period at

some sites, as well as substantial cross-correlation between sites. When each record is

divided into four roughly equal periods, a Bartlett test shows evidence of variance

heterogeneity at four of the five sites (χ2 values 9.32, 8.85, 8.08, 13.32, 6.60; P-values

0.025, 0.031, 0.044, 0.004, 0.086, all with three degrees of freedom); but where variances

differ significantly, they do not show an increasing trend with time. Variance

heterogeneity was also found where data were divided into four groups by magnitude, as

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well as by time. Modelling the variance of annual maxima, and of cross-correlation, are

not discussed further in this paper, which focuses on the estimation of linear trend in the

measured variable.

Analytical method.

The analysis is to compare estimates of trend, and the uncertainties in those estimates (as

measured by 95% confidence intervals), when different analytical procedures were used

for trend estimation. Because the flow sequences at the five gauge sites are annual

maxima, it is appropriate to assume that serial correlation is absent, and this was

quantified by calculating the correlogram and a portmanteau test for each sequence [Box

et al., 2009]. For the site 14 de Julho, Figure 2 gives a Q-Q plot comparing sequence

quantiles with quantiles of a standard GEV distribution. Figures for the other four sites

are similar, and are not shown. In all cases, quantiles of the flow sequence lay

comfortably within the 95% confidence interval for the Q-Q plot, suggesting that a GEV

distribution, possibly with super-imposed linear trend, should be an adequate description

of the statistical characteristics of annual maximum flows. ML procedures [Coles, 2001,

Chapter 6] were used to estimate the in-built linear trend of the form μ (t) = α+ β t (Coles

also used an exponential trend in the dispersion parameter σ, but this was not explored in

this paper). Τhus the following procedure was used to calculate an estimate ^

β of a linear

trend parameter β, at each of the five sites in Table 1:

Method 1: G (q) = exp{- [ 1 + ξ ( q – α − β t ) / σ ] -1/ξ (2)

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with the four parameters α, β, σ and ξ estimated by ML. Standard errors for estimates of

the parameters were calculated from the information matrix 12 ]/log[ −∂∂−∂ jiL θθ evaluated

at the maximum of the log-likelihood function log L (where ji θθ , are the four parameters

taken in pairs) and were used to calculate large-sample confidence intervals for the trend

β. The mean of the distribution in (2) is

α + β t + σ [Γ (1 + ξ ) – 1 ] / ξ (3)

which provides a linear trend with trend coefficient β. It is shown below that, at each of

the five sites, the shape parameter ξ was not large relative to its standard error, suggesting

that the simpler Gumbel distribution, with linear trend incorporated, might also be an

acceptable model for the purposes of this study. Therefore, the GEV reduces to:-

Method 2: G (q) = exp{- exp [- ( q – α − β t ) / σ ] } (4)

which was also used to estimate the trend parameter β , and the standard error of^

β , from

which approximate 95% confidence intervals were calculated . These estimates are also

shown in Table 2.

As a third method, a linear trend was estimated at each site, using the same expression as

that used in ordinary linear regression, with

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^

β = = =

−−N

i

N

iiii ttttq

1 1

2__

)(/)( (5)

where ti is the i-th year and N is the number of years in the sequence (i =1…N). For any

linear trend model such that q t = α + β t + (possibly, a function of model parameters

which does not involve the data q t) + ε t, the expression in (5) is an unbiased estimator of

β, whatever the distribution of the residuals ε t , whether or not residuals are uncorrelated,

and whether or not the residuals ε t have homoscedastic variance. After estimating the

trend β using (5), the sequence was “de-trended” and the GEV distribution given by (1)

was fitted by ML to give estimates of the remaining three parameters α, σ, ξ. The

variance of a random variable Q with a GEV distribution is (σ / ξ)2 (g 2 - g1 2) where gk =

Γ (1 + k ξ), k= 1,2, so that if the N years are consecutive, the standard error of the trend

estimate ^

β given by (5) is the square root of

var[^

β ] = 12 (σ / ξ)2 (g 2 - g1 2) / [N (N 2 – 1 )] (6)

As mentioned, an advantage of this procedure is that, if the trend can be estimated

efficiently by least squares, estimation of the GEV parameters then requires a search over

the three-dimensional space of α, σ, ξ instead of the four dimensional space for α, σ, ξ ,

β.

Therefore the following method was also used:-

Method 3: Trend estimated by ^

β = 12 =

N

i 1

qi { i - (N+1) / 2 } /[N (N 2 – 1 )]

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with standard error given by the square root of (6), and with σ, ξ estimated from the de-

trended data sequence.

The variance of a random variable Q with a Gumbel distribution is σ2 π2 / 6, so

that a fourth estimator is given by

Method 4: Trend estimated by ^

β = 12 =

N

i 1

qi { i - (N+1) / 2 } /[N (N 2 – 1 )]

with standard error the square root of var[^

β ] = 2 σ 2 π2 / [N (N 2 – 1 )].

In both Methods 3 and 4, the de-trended sequences no longer consist of

statistically-independent values, because the estimated residuals are linear functions of

the original data sequence { q t }, t = 1…N. It can be shown [e.g., Johnson and Wichern ,

2007] that the de-trended residuals, denoted by r*, are given by r* = (I- H) q where H is

the ‘hat’ matrix H = X (XTX)-1XT, I is the unit matrix, and X is the N×2 matrix with 1’s in

the first column and the year numbers in the second. Thus the covariance between the r-

th and s-th residuals, obtained after de-trending when data are from a GEV distribution, is

[ ≠ srii ,,

h r i h s i – (2 – h r r – h s s) h r s]. (g 2 - g1 2) (σ / ξ)2 (7)

In the case of the Gumbel distribution, the expression in square brackets is multiplied by

σ2 π2 / 6. This covariance structure differs from the covariance structure of (for example)

a lag-one auto-regression, so the type of portmanteau test used to test for serial

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correlation between values in the original data sequence is no longer appropriate. Thus,

whilst the estimate of trend given by (5) is valid, the presence of correlation between

residuals after de-trending means that the usual log-likelihood function is incorrect. This

may affect estimates of the parameters μ, σ, ξ of the GEV distribution, and hence the

standard error of the trend estimate given by (5).

Methods 1 to 4 above are all based on probability distributions for the sequences

of annual maximum flows. To compare their performance with methods which do not use

such assumptions, two distribution-free procedures were used: namely, a bootstrap

estimate of trend, and the Theil-Sen estimate. Thus:

Method 5: For a data-set of length N years, N pairs of values (t i , q i) were drawn with

replacement, where t i is the year number and q i (i = 1…N) is the annual maximum for

that year. The slope (given by (5) ) and intercept of the sample were calculated, and the

procedure was repeated 600 times; all simulations and boot-strap calculations used 600

repetitions, since this number was recommended by Wilcox [2012] for calculating

confidence intervals for the Theil-Sen trend estimate described below. Quantiles

corresponding to probabilities 0.025 and 0.975 then gave a 95% confidence interval for

the trend β. This procedure is equivalent to a randomly-weighted regression [Venables

and Ripley, 1999]; as an alternative, they suggest (Section 6.6) model-based re-sampling

in which the residuals about the fitted regression are re-sampled. However the calculated

residuals r* do not then have the correct variance, or even the same variance as each

other. Corrections are possible [Venables and Ripley, 1999] but model-based re-sampling

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was not pursued further in this paper. Methods 3, 4 and 5 all estimate trend by the

expression in (5).

Method 6 estimated trend by means of the Theil-Sen distribution-free procedure.

Trends were calculated as follows: (i) at each site (except for 14 de Julho), the set of

M=70×71/2=2485 differences (qk - qi)/(k - i) for i < k. The median of these differences

gave the Theil-Sen estimate of trend β; (ii) 600 samples of size N annual maxima were

drawn, with replacement, from the record of length N years, and the Theil-Sen estimate of

trend was calculated from each of the 600 samples. Quantiles corresponding to

probabilities 0.025 and 0.975 then gave a 95% confidence interval for the trend β . For

the site 14 de Julho, where two years were missing, the longest period of unbroken record

(1931-2003) was used, giving M = 2701. This procedure differs from that given by

Wilcox [2011] who re-sampled the M differences with replacement. It also differs from

the variant of the Theil-Sen estimator given by Siegel [1982] who determined, for each

sample point (ti, qi), the median mi of the slopes (qj − qi)/(tj − ti) of lines through that

point, and then calculated the overall estimator as the median of these medians.

Results.

Table 2(a) shows estimates of the linear trend parameter estimated by these six

methods. The table shows that differences between estimates of trend obtained by

Method 1 (fitting a GEV by ML, with trend incorporated in the location parameter) and

Method 3 (de-trending by least squares, before fitting a GEV distribution to the de-

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trended series) are large. Trend estimates from Method 3 are in all cases much larger than

the ML estimates, and in one case (14 de Julho) are several times larger. The same is true

for trend estimates obtained by Method 2 (Gumbel, with trend incorporated in the

location parameter) and Method 4 (Gumbel fitted after de-trending). Where the location

parameters of GEV and Gumbel distributions have trend incorporated, Gumbel estimates

of the trend coefficient β are always larger than those from fitting the GEV distribution,

despite the fact that the shape parameter ξ is not large, relative to its standard error, at any

of the five sites, as shown at the bottom of Table 2(a). For Site 14 de Julho, Figure 3

compares the trend estimated by fitting a GEV with trend parameter incorporated

(Method 1), with the least-squares estimate of trend (Methods 3, 4 and 5); visually, the

least-squares fit looks the better fit. Both Theil-Sen estimates of trend β, and the least-

squares estimate of β, are very considerably greater than estimates of β obtained from

both GEV and Gumbel distributions, with or without de-trending. As reported [e.g., Frei,

2011], the Theil-Sen estimates of β are smaller than the ^

β found by least squares.

Table 2(b) shows 95% upper and lower confidence limits, and the width of the

95% confidence intervals, for linear trends estimated by the six methods at the five sites.

For fits by ML, it is assumed that large-sample properties hold so that the approximate

95% confidence intervals are given by ± 2×SE, where the standard error SE is found as

described above. Where the confidence interval brackets zero, the estimated trends are

consistent with a null hypothesis of zero trend; where zero is not bracketed, the null

hypothesis of zero trend would be rejected. In terms of statistical significance, therefore,

results given by the five methods are very different. For the site 14 de Julho, the

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confidence interval from Method 1 (GEV with incorporated trend) brackets zero, and so

does the confidence interval for Method 2 (Gumbel with incorporated trend), Method 5

(bootstrapped linear regression) and Method 6 (Theil-Sen), whilst the ‘detrended’

methods (Methods 3 and 4) both indicated that the null hypothesis of zero trend should be

rejected. At all five sites, Theil-Sen confidence intervals bracketed zero, showing that

Method 6 always gave estimates of trend consistent with the zero-trend hypothesis.

Confidence-interval widths often differed substantially between methods at any given

site, but no consistent pattern emerges; the widths of confidence intervals where data are

de-trended are usually greater than where trend is built into a distribution’s location

parameter, and the differences are sometimes large.

By comparing the upper and lower confidence limits with the least-squares

estimates of trend shown in Table 2(a), it can also be seen that, at three of the five sites,

the least-squares estimate of trend lay outside the confidence interval for the ML

estimate; at two of the five sites, the Theil-Sen estimate lay outside the confidence

interval for the ML estimate. Interpretation of this result requires caution, however,

because both the least-squares and Theil-Sen estimates are also subject to uncertainty,

and because records at the five sites are to some extent cross-correlated.

Table 2(c) shows how estimates of the location and scale parameters, α (= μ) and

σ, differ when they are estimated from GEV distributions with the trend parameter

incorporated, and from GEV distributions after linear trend has been removed (i.e.,

Methods 1 and 3). Estimates of the GEV distribution’s α (i.e., the location parameter of

the de-trended residuals) are consistently greater when data are de-trended, but the

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differences are not large; the same is true of estimates of σ, the scale parameter, but here

the differences between Methods 1 and 3 are substantially greater. For the Gumbel

distribution, differences between estimates of α with and without trend removal (Methods

2 and 4) are very small; estimates of the scale parameter are greater where data are de-

trended, but the increases are smaller than those for the GEV distribution. De-trending

has a marked effect on estimates of the shape parameter ξ, shown in Table 2(a); these,

and their standard errors, were substantially reduced where the data were de-trended

before fitting a GEV.

The main point emerging from Tables 2(a), (b) and (c) is that when a trend

coefficient is incorporated into the location parameter (whether GEV or Gumbel) of

annual maximum one-day flows, the estimates of trend were markedly different from any

estimates of trend obtained by least squares, or by Theil-Sen, or even fitting by eye (see

Fig. 3). To explore this in greater detail, samples of size 70 were drawn from five GEV

distributions with parameter values equal to those given in Table 2 (in the case of the

gauge 14 de Julho, for example, the GEV parameters were α = 2522, σ = 1063, ξ =

0.1251, β = 5.626). Six hundred samples were drawn in each case. For each sample, the

trend β was estimated (i) by ML (Method 1), (ii) by ordinary least squares (giving the

estimated trend for Methods 3, 4 and 5), and (iii) by the Theil-Sen estimator (Method 6).

Table 3 gives statistics derived from the 600 generated samples. Comparisons by t-test

between the means of the ^

β ’s over all 600 simulations and the “true” trends shown at

the top of Table 3, show that none of the three methods shows evidence of major bias,

and indeed theory shows that the ML estimates ^

β are unbiased when the sample of years

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is large. Provided that the trend is linear, least-squares estimates ^

β are unbiased whatever

the length of record. Theory also shows that, given certain regularity conditions [Kendall

et al., 1983] no other estimation procedure will give estimates^

β with smaller variance

than ML estimates when the length of record is sufficiently long (when ML estimates are

“asymptotically efficient”). Table 3 also shows standard deviations (SDs) of the 600

samples at the five sites. The SD for ML estimates ^

β is substantially smaller than the

SDs of either least squares or Theil-Sen estimates, and the SDs of the Theil-Sen estimates

are substantially smaller than the SDs given by least squares. Figure 4 shows a scatter-

diagram for the Site 14 de Julho; the greater variability in the least-squares estimate

“pulls” the line towards the horizontal, giving a slope (0.393) considerably less than 45o.

Correlations between the estimates ^

β obtained from the 600 samples by ML,

least squares and Theil-Sen are shown at the bottom of Table 3; the correlation between

ML and least-squares estimates of trend is lowest at all sites, ranging from about 0.5 to

0.75. Correlation between ML and Theil-Sen estimates is considerably greater, whilst

correlations between least-squares and Theil-Sen estimates are in the range 0.82 to 0.92.

These correlations, together with the asymptotic efficiency of ML estimates

mentioned above, suggest an explanation for the differences between ML and other trend

estimates shown in Table 2(a). The argument is as follows. Consider the probability

distributions of the ML and least-squares estimates of trend (given, in the case of Figure

4, by the projections of points on to the vertical and horizontal axes respectively). The

dispersion of the former (ML) distribution will be less than that of the latter because of

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ML asymptotic efficiency; both distributions will be centred about the true trend value,

however, because both ML and least-squares estimators are unbiased (the former in large

samples, the latter in samples of any size, when the trend is truly linear). Hence, when the

sample of years in a record of annual maxima gives a trend that is large and positive

when estimated by least squares, the ML estimate of trend will be smaller (closer to the

central value common to both distributions) but still positive (because ML and least-

squares estimates are positively correlated). The converse must also be true: when the

sample of years in the record of annual maxima shows a trend that is small or even

negative when estimated by least squares (i.e., the least squares estimate lies in the left-

hand tail of its probability distribution), the ML estimate will be greater (again, closer to

the central value common to both distributions). A similar argument is developed by

substituting ‘Theil-Sen estimator’ for ‘least-squares estimator’.

Discussion.

The trends in the estimation methods described above are linear trends, and it

could be argued that changes may be episodic rather than linear. The many methods for

modeling episodic changes over time include piece-wise or “broken stick” regression,

with or without continuity at the break-points, the number of which may be known or

unknown [Khodadadi and Asgharian, 2008; Toms and Lesperance, 2003]. Erdman and

Emerson [2007] following Barry and Hartigan [1993] used Bayesian methods to estimate

posterior distributions of parameters θ i {i=1…b} where a data sequence {Xi}, i=1…n is

modeled as b “blocks” with the i-th block specified by θ i which may be (for example) a

distribution mean or a regression coefficient. In the physical context of hydrological

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extremes, however, results from any analysis of episodic trends will be more acceptable

where changes are known to have occurred at specific times, and from specific causes,

rather than found by using a computer algorithm to determine the number of change

points and the change magnitudes. As far as is known, data used to illustrate methods

described above were not subject to any such specific changes; in the absence of such

knowledge, a linear trend gives a simple description of the way that annual maxima

changed over the period of record.

In an important paper on the calculation of trends and their uncertainties, Cohn

and Lins [2005] explored aspects of the statistical significance of trends in

hydroclimatological time series, concluding that “while trend magnitude can be

determined with little ambiguity, the corresponding statistical significance … is less

certain because significance depends critically on the null hypothesis, which in turn

reflects subjective notions about what one expects to see”. They also concluded that “it

may be preferable to acknowledge that the concept of statistical significance is

meaningless when discussing poorly-understood systems.” The results presented above

suggest that uncertainty is a characteristic not only of the statistical significance of a

trend, as Cohn and Lins state, but also of its magnitude. Two sets of procedures for

estimating trend in annual maximum one-day flows, both arguably valid, have been

shown to give very different results in some specific cases: one set, based on the almost

universal assumption in frequency analyses that annual maxima are statistically

independent with some kind of extreme-value distribution (GEV or Gumbel), and another

set which assumed statistical independence but no distributional form (bootstrap, Theil-

Sen). Table 2(a) shows that estimates of trend given by the two sets of procedures can be

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markedly different. Furthermore, as confirmation of the statement from Cohn and Lins

[2005] quoted above, the two sets of procedures lead (in the case of the five records here

considered) to different conclusions about the statistical significance of trends, shown by

Table 2(b).

It can be argued [Koutsoyiannis, 2006] that analysing time series of hydrological

or climatic data for trends is in any case illogical, and that “a stochastic approach

hypothesizing stationarity and simultaneously admitting a scaling behaviour reproduces

climatic trends (considering them as large-scale fluctuations) in a manner that is logically

consistent.” Such an approach, based on the well-known Hurst coefficient, “does not

require the separation of the time series into two or more components, so it does not

attempt to de-trend the original series. It admits that the existence of trends is the normal

behaviour of real world time series” [Koutsoyiannis, 2006]. However a careful reading of

his text suggests that Koutsoyiannis’ criticism is levelled principally at the fallacy of

concluding that such “trends” are deterministic. But the term “deterministic” must imply

that present behaviour determines a system’s future behaviour; in the present context

where annual extreme one-day flows are analysed over a period of 70 to 80 years,

changes – where they exist – are almost certainly the consequence of land-use change

from native forest to arable cropping, with no “deterministic” interpretation about future

behaviour possible. It can therefore be argued that in such circumstances, a linear trend

coefficient is a useful summary of past behaviour (although it can never be more than

that; certainly no extrapolation to future years is possible). In a sense, a trend coefficient

b and H coefficient are complementary entities: the former is a statistical summary of

time-series behaviour over a specified period of the past, and the latter is a statistical

20

summary of future long-term behaviour over an indeterminate (but long) period; neither

conveys information about deterministic influences that caused, or will cause, variability

observed in the time-series.

It has been mentioned that the Methods 3 and 4, which de-trended the sequence of

annual maxima before fitting GEV or Gumbel distributions to the de-trended residuals,

introduces a correlation (although not a serial correlation) amongst them. The issue of

introduced correlation appears to be a much wider issue than in the present paper. Where,

for example, future “scenarios” of annual climatic extremes are produced by Global

Climate Models (GCMs), the annual (or seasonal) extremes in the scenario will all be

functions of GCM parameters and their initial values, and will remain so however many

“members” are calculated in the scenario. A similar argument would appear to hold

where coupled climate-hydrological models are used to produce future scenarios of

extreme flows. Further research is required to explore whether such introduced

correlations are important or whether they can be safely ignored.

In conclusion, we return to the title of this paper: how should trends in hydrologic

extremes be estimated? The answer is that the method must depend on the context, and

no single method can be recommended for all circumstances. We illustrate with three

cases: (i) where the context requires exploration of the causes of trend (i.e., determination

of whether certain predictors explain the trend fully, partially, or not at all) then a

parametric approach is required, with assumptions – to be verified subsequently – about

underlying probability distributions that allow the use of likelihood theory for hypothesis

21

testing (or, in a Bayesian context, for calculating the posterior probability distribution of a

trend). This would rule out the use of distribution-free methods such as bootstrapping or

Theil-Sen estimation. If, however, (ii) the context requires the exploration of a regional

average trend of, say, annual maximum n-day rainfalls measured at P sites within the

region, distribution-free methods could be used; the trend could be estimated by either

bootstrapping or Theil-Sen at each of the P sites, with standard geostatistical methods

used to calculate the regional trend and its uncertainty. Again (iii) if it were required to

estimate the difference between trends at two or more sites, so that estimating differences

between trends is more important than obtaining good estimates of the trend at each site

separately, then any of the methods compared above could be used (although if the

uncertainty in the differences were also to be estimated, this would complicate the use of

Theil-Sen and bootstrap methods). Whichever methods are used, the estimates of trend

may differ from method to method; the uncertainty as measured by confidence intervals

may differ; and (with due reminder of the warning of Cohn and Lins [2005] referred to

above) if significance-testing is required, conclusions about significance may vary from

one method to another.

Conclusion.

A comparison of six procedures for estimating trend β in annual maximum one-

day river flows at five sites showed marked differences between, on the one hand,

estimates obtained by incorporating trend into the GEV location parameters; and on the

other hand, estimates found by (i) trend removal by least squares prior to fitting the GEV,

22

(ii) boot-strap sampling, and (iii) Theil-Sen estimation. The same was true where trend

had been incorporated into the Gumbel location parameter. Comparison of approximate

95% confidence intervals for β given by the different methods, showed that 95%

confidence limits given by the different methods bracketed zero for some of the six

methods, whilst for others, they did not, so that different conclusions about statistical

significance would be drawn according to which method was used to estimate the trend.

Acknowledgement.

The author is grateful to Professor Juan Martin Bravo for making daily flow records

available, and to anonymous reviewers for constructive comments.

23

References.

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Table 1(a): Descriptions of drainage basins, Rio Grande do Sul, Southern Brazil

Gauge site: Code River: Latitude: Longitude: Area(km2): Annual P

14 de Julho 284 R. Antas -29.0648 -51.6749 12835 1613 1

Castro Alves 98 R. Antas -29.0056 -51.3844 7784 1617 2

Chapeco 94 R. Uruguai -27.1416 -53.0449 52949 1650 3

Ernestina 110 R. Uruguai -28.5556 -52.5456 1047 1646 4

Ita 92 R. Uruguai -27.2767 -52.3822 43954 1623 5

Periods: 1944-2010 1; 1944-2010 2; 1957-2010 3; 1930-1985 4; 1958-2008 5

Table 1(b): Summary statistics of annual maximum one-day flows. Q1 and Q3 are the

first and third quantiles:

Min: Q1: Median: Q3: Max:

14 de Julho 1170 2045 2938 3910 6912

Castro Alves 559 1097 1499 2391 4507

Chapeco 1434 5946 8583 12731 25466

Ernestina 54 154 221 326 729

Ita 1446 5067 7821 11332 25544

27

Table 2(a): Trends (m3s-1yr-1)obtained when trend is incorporated in location parameter

of GEV and Gumbel distributions: i.e., GEV(μ + β t, σ, ξ) and Gumbel (μ + β t, σ )

(Methods 1 and 2),; be removing trend and fitting GEV(μ , σ, ξ) and Gumbel (μ , σ ) to

the residuals (Methods 3 and 4); by bootstrapped Linear Regression (Method 5) with 600

boot-strapped samples, and by using the Theil-Sen estimate (Method 6) with 600 boot-

strapped samples.

Site:

14 de Julho Castro Alves Chapeco Ernestina Ita

Method:

1 5.63 9.80 58.00 1.94 44.61

2 8.47 12.68 62.47 2.11 47.45

3 ]

4 ] 20.88 18.99 86.50 3.85 60.00

5 ]

6 15.63 17.55 79.50 3.67 59.42

Estimates of shape parameter ξ in GEV(μ + β t, σ, ξ), and of ξ in GEV(μ , σ, ξ) after de-

trending:

Method 1 0.1251 0.2249 0.0894 0.0372 0.1426

SE(ξ ) ± 0.1345 ± 0.1306 ± 0.1171 ± 0.1220 ± 0.1311

Method 3: -0.0725 0.0597 0.0340 -0.1133 0.1202

SE(ξ ) ± 0.1083 ± 0.1029 ± 0.0996 ± 0.0716 ± 0.1262

28

Table 2(b): Approximate 95% lower and upper confidence limits (denoted by L, U), and

width of confidence intervals. Units: (m3s-1yr-1).

Site:

14 de Julho Castro Alves Chapeco Ernestina Ita

Method 1:

L -5.64 3.32 17.76 0.27 11.47

U 16.89 16.26 98.24 3.60 77.75

Width: 22.53 12.94 80.48 3.33 66.28

Method 2:

L -3.20 4.92 19.37 0.78 8.89

U 20.15 20.43 105.57 3.44 86.01

Width: 23.35 15.51 86.20 2.66 77.12

Method 3:

L 9.48 11.93 38.08 2.36 19.22

U 32.34 25.41 135.72 5.36 98.78

Width: 22.86 13.48 97.64 3.10 79.56

Method 4:

L 7.79 10.31 32.78 2.29 12.64

U 34.03 27.03 141.02 5.43 105.36

Width: 26.03 16.72 108.24 3.14 92.72

Method 5:

29

L -5.53 7.80 36.80 2.03 17.20

U 37.24 31.07 133.90 5.62 104.60

Width: 42.37 23.27 97.10 3.59 87.40

Method 6:

L -15.54 -7.82 -51.74 -1.43 -47.45

U 17.02 8.66 50.19 1.44 49.64

Width: 32.56 16.48 101.93 2.87 97.09

30

Table 2(c): Estimates of location parameter α (= μ) and scale parameter σ, from methods

with and without de-trending by least squares. Values of ξ, where relevant, are shown at

the bottom of Table 2(a).

Method 1 (GEV, trend parameter incorporated in location parameter):

α: 2522 1364 7514 195.9 6392

σ: 1063 566.9 3670 106.4 3143

Method 2 (Gumbel, trend parameter incorporated in location parameter):

α: 2598 1439 7699 198.3 6647

σ: 1129 636.5 3818 108.2 3364

Method 3 (GEV, data de-trended):

α: 2639 1418 7622 205.0 6429

σ: 1221 644.8 3838 120.1 3200

Method 4 (Gumbel, data de-trended):

α: 2593 1438 7694 197.8 6646

σ: 1151 660.2 3889 118.1 3383

31

Table 3: Statistics derived from 600 simulated samples, at each of the 5 sites, drawn from

GEV(α + β t, σ, ξ), with parameters α , σ, ξ, β given in Table 2. Means, standard

deviations and standard errors of means, are given for 600 maximum-likelihood estimates ^

β ML, 600 least-squares estimates ^

β LS , and 600 Theil-Sen estimates ^

β TS. Correlations

between maximum-likelihood and least-squares estimates are denoted by r ML,LS .

Similarly for r ML,TS and r TS,LS.

14 de Julho Castro Alves Chapeco Ernestina Ita

“True” value: 5.626 9.795 58.00 1.938 44.610

Mean ^

β ML 5.742 9.727 58.757 1.931 44.122

SE(Mean) ± 0.248 ±0.125 ±0.883 ± 0.027 ± 0.698

SD(^

β ML ) ±6.083 ±3.059 ±21.612 ±0.659 ±17.083

Mean ^

β LS 6.094 9.608 56.901 1.929 44.437

SE(Mean) ± 0.388 ±0.265 ±1.291 ± 0.035 ± 1.258

SD(^

β LS ) ±9.496 ±6.489 ±31.588 ±0.853 ±30.797

Mean^

β TS 5.879 9.704 57.546 1.931 44.619

SE(Mean) ± 0.310 ± 0.169 ± 1.041 ± 0.027 ± 0.698

SD(^

β TS ) ±7.589 ±4.138 ±25.469 ±0.753 ±22.367

32

r ML,LS 0.613 0.472 0.686 0.751 0.557

r ML,TS 0.791 0.706 0.827 0.858 0.757

r TS,LS 0.877 0.820 0.832 0.917 0.863

33

Figure legends.

Figure 1: Annual maximum one-day flows at five sites, standardized by subtracting the

mean of the sequence and dividing by its standard error.

Figure 2: Site 14 de Julho: quantile-quantile plot when GEV distribution is fitted.

Figure 3. Site 14 de Julho: trends estimated from GEV distribution with trend parameter

incorporated (Method 1: broken black line), and by least-squares fit (Methods 3, 4 and 5:

broken red line). Symbols: YML: trend by maximum likelihood; YR: year; YLS: least

squares; Q: annual maximum discharge.

Figure 4 : Estimate b of trend coefficient β in 600 samples of size 70 “years” drawn

from a GEV distribution with parameters equal to those estimated for the site 14 de Julho.

Red line shows the 45o line. Ordinates along vertical axis are the estimates of β when

trend was incorporated into the GEV distribution; abscissa along the horizontal axis are

estimates of β found be least squares (as for boot-strapped estimate, and de-trended

values subsequently fitted to GEV(α, σ, ξ) ).

34

Figure 1: Annual maximum one-day flows at five sites, standardized by subtracting the

mean of the sequence and dividing by its standard error.

35

Figure 2: Site 14 de Julho: quantile-quantile plot when GEV distribution is fitted.

36

Figure 3. Site 14 de Julho: trends estimated from GEV distribution with trend parameter

incorporated (Method 1: broken black line), and by least-squares fit (Methods 3, 4 and 5:

broken red line). Symbols: YML: trend by maximum likelihood; YR: year; YLS: least

squares; Q: annual maximum discharge.

37

Figure 4 : Estimate b of trend coefficient β in 600 samples of size 70 “years” drawn

from a GEV distribution with parameters equal to those estimated for the site 14 de Julho.

Red line shows the 45o line. Ordinates along vertical axis are the estimates of β when

trend was incorporated into the GEV distribution; abscissa along the horizontal axis are

estimates of β found be least squares (as for boot-strapped estimate, and de-trended

values subsequently fitted to GEV(α, σ, ξ) ).