Non-stationary approach to at-site flood frequency modelling II. Weighted least squares estimation

doi:10.1016/S0022-1694(01)00398-5

Journal of Hydrology

Volume 248, Issues 1–4, 15 July 2001, Pages 143-151

https://doi.org/10.1016/S0022-1694(01)00398-5 Get rights and content

Abstract

This is the second of the three-part paper and generalises the least squares method to the weighted least squares (WLS) method in order to deal with the trend in the first two moments. The generalised method applies when the assumption of constant variance does not hold and the functional form of a trend in the variance is given. In the generalised method, the parameters of trend in the mean and variance are estimated simultaneously. To keep the weights as power functions of variances only, the restrictions on distribution functions are formulated, which, in fact, are not difficult to fulfil in hydrological studies. It is shown that the WLS method coincides with the maximum likelihood method in the case of the normal distribution.

Introduction

For detecting a trend, standard statistical parametric and non-parametric techniques are used, which are based on the theory of hypothesis testing. The advantage of a non-parametric approach is that no assumption is made about the trend model and the functional form of the distribution function of random variables (e.g. Mitosek, 1992). If no tendency is detected with respect to the mean value, one can investigate a change in the variance by Mann's test (Mann, 1945). However, if the mean happens to be nonstationary there is neither a possibility of detecting trend in the variance nor investigating nonstationarity of the two first moments simultaneously. A great disadvantage of the non-parametric tests applied in hydrology is that they require much longer series than do parametric tests. Moreover, hydrologists expect that the information on statistically significantly increasing (or decreasing) tendency in the mean or the variance is supplemented by quantification of recorded trend and its prediction. The probability distribution function with time-variant parameters is the preferable output of trend investigation for further use in hydraulic design and water resources planning.

In the parametric approach, it is the form of a trend in the mean value that should be assumed. A linear or higher degree polynomial model is usually adopted for this purpose. Its coefficients are estimated by the least squares (LS) method and the parametric tests are used to verify the statistical significance of the trend (e.g. Giakoumakis and Baloutsos, 1997). However, most parametric tests require the characteristics under verification to be normally distributed, which rarely holds in hydrology. While estimating a trend in the mean by the LS method, no allowance is usually made for the time-variant variance. In this paper — the second of the three-part paper — a generalisation of the LS method for time-variable variance is presented which does not require as rigorous a distribution assumption as does the ML method. In fact, since water resources systems are supposed to perform satisfactorily during extreme hydrological conditions, such as exceptionally dry or wet years, a time-change of the variance may be even more important than that of the mean. For example, the difference ΔQ (m³/s) in the mean of flood frequency distribution results in the difference ΔQ in peak flow with a return period of 100 years (Q_1%), while the same difference in the standard deviation (SD) makes the difference 2.33ΔQ in Q_1%, if the normal distribution is assumed.

It should be stressed that even if an investigation of trend in only the mean is of concern, heteroscedasticity has to be taken into account. Therefore investigating a long time series of unknown character, one assumes a most general case, i.e. of functionally non-related trends in the first two statistical moments, and then may proceed to simpler cases ending at the stationary case, and finally identifying the model showing the best fit to the time series. The investigation can comprise various functional forms of trend, e.g. linear, parabolic or the periodic functions of time, while keeping in mind the need for parameter parsimony. Statistical significance of detected trends is not tested but the model showing the best fit of all competing models is considered here as the optimal one.

Section snippets

Method of moments as the LS method

The method of moments (MOM) is used to estimate the mean and other moments of time-series when all observations are considered to come from the same distribution, while the LS method is usually applied when this condition no longer holds. Since the subject of the study is the estimation of the time-variant moments, it is worth looking at the MOM from the point of view of the LS method.

Let the problem be to estimate, by the LS method, the moments of the probability distribution of the random

Non-identical parent distributions

So far, we have been considering the problem of estimation when all the observations come from the same underlying distribution. We will now examine the situation when this condition no longer holds. In general, the acceptability of the LS method depends on the properties of the estimators to which it leads. In a situation usually appearing in the investigation of trend in hydrological time series, it has such a property, even in small samples, that it provides unbiased estimators, linear in

Unequal known variances

In order to examine the influence of the time-variant variance on the properties of the estimator of the mean, a simple case will be considered first when elements of the sample differ in the variances which are known. It can be exemplified by the estimation of the mean from a sample with unequal accuracy of data. Let us place the weight γ_t⁽¹⁾ to every term S_t⁽¹⁾ of the sum of squares in Eq. (1) $∑ WS_{t}^{(1)} = ∑ t=1 T γ_{t}^{(1)} (x_{t} − m ̂)^{2} .$ Then, from the LS method we get: $m ̂ = 1 ∑ t=1 T γ_{t}^{(1)} ∑ t=1 T γ_{t}^{(1)} x_{t}$ For unbiased

Unknown variances

To use Eq. (19) (or Eq. (14a)) we need to know σ²(x_t) for t=1,2,…,T. This rarely holds in practice. On the other hand, in order to estimate unknown variances, the functional form of a trend in variance should be assumed. If so and the time-dependent mean $m ̂_{t}$ is also known, then the only problem left will be to estimate the parameters of time dependent variance (Eq. (10a)). To this end, the WLS method shall be applied this time in respect to the second central moment. We have: $∑ WS_{t}^{(2)} = ∑ t=1 T γ_{t}^{(2)} ϵ_{t}$

Derivation of weighting factors

To solve , , the number of unknown variables would be reduced to those describing the time-dependent mean and variance. However, neither the variance of the second central moment nor the fourth central moment of x_t in Eq. (22) is known. In order to express $μ ̂_{4} (x_{t})$ in terms of lower order moments, we have to presuppose the knowledge of the functional form of the distribution(s) generating the time series, which would contain less than four parameters. Similarly, to derive the coefficient of

The normality assumption

The LS method, while conceptually quite distinct from the ML method, coincides with the ML method in the case of normally distributed observations. If the distributions of x_t about their expectations are independently normal with the same variance, as in the trend model A, the LS estimator will be identical with the ML estimator of time-variant mean m_t. It will be shown here that the above identity holds for the WLS estimators.

The log likelihood function in a time series from normal

Concluding remarks

To deal with the trend in the first two moments, the LS method has been generalised to the case where the assumption of constant variance does not hold and functional form of a trend in the variance is given. Its generalisation is the WLS method where the parameters of trend in the mean and variance are estimated simultaneously. Each of the two weighted squares functions, i.e. for time-variant mean and variance, contains both variables and therefore they have to be minimised jointly. The first

Acknowledgements

The work reported in this study was supported in part by the Polish Committee of Scientific Research (KBN) grant No. PO4D 056 17, ‘Revision of applicability of the parametric methods for estimation of statistical characteristics of floods’. This support is gratefully acknowledged.

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Non-stationary approach to at-site flood frequency modelling II. Weighted least squares estimation

Abstract

Introduction

Section snippets

Method of moments as the LS method

Non-identical parent distributions

Unequal known variances

Unknown variances

Derivation of weighting factors

The normality assumption

Concluding remarks

Acknowledgements

A new look at the statistical model identification

IEEE Trans. Automat. Contr.

Probability Theory and Mathematical Statistics

Investigation of trend in hydrological time series of the Evinos river basin

Hydrol. Sci. J.

Statistical methods in Hydrology and Meteorology

The advanced theory of statistics