Weighted least squares method

The principle of the WLS method is based on the minimisation of sums of weighted squared deviations of observed and estimated moments, where the weights are reciprocals of their expected values (Strupczewski & Kaczmarek, 1998). The WLS, being conceptually quite distinct from the ML-method, coincides with the ML method in the case of normally distributed data. In this case a simple presentation of the WLS as a problem of the ML- estimation is possible.

The log-likelihood function in a time series subject to normal distribution with time- variable parameters has a form:

log

∑ ∑

The conditions of maximum log L are:

( )

Note that both sets of equations contain both time-dependent mean (mt) and variance (σt), i.e. they need to be solved jointly unless the standard deviation can be assumed to be constant. The WLS method covers four classes of trend, i.e.:

A - trend in the mean only, which is the common least squares problem of trend estimation B - trend in the standard deviation only;

C - trend both in the mean and standard deviation being functionally related, e.g. by a constant value of the variation coefficient (cv);

D - non-related trend in the mean and standard deviation.

The most basic case is that of time-invariable moments, is called the stationary option (S). In this case, equations (13.2)-(l3.3) reduce to the MOM equations.

The test of goodness-of-fit based on Akaike Information Criterion serves to identify an optimum model in a class of competing models of trend:

AIC = -21nML + 2k (13.4) where ML denotes the maximum likelihood for the model and k is the number of fitted

parameters. The best approximating model is the one, which achieves the minimum AIC value in the class of competing models.

Equations (13 .2)-( 13.3) derived here for the ML-estiniation of time-variable parameters of normally distributed variables are equivalent to the WLS equations and as such they remain valid for other distribution functions providing that certain conditions are fulfilled. Table 13.1 shows applicability of equations (13.2)-(13.3) for various two- and three-parameter distribution functions and various classes of trend. A constant value of the coefficient of variation is accepted for the class C. Six pdfs were examined, namely: Normal (N), two- parameter Lognormal (LN2), three-parameter Lognormal (LN3), Gamma (P2), three- parameter Pearson (P3) and GEV of the first type, i.e. Fisher-Tippett of the first type (FT).

Shading indicates applicability of the WLS method for the case, while dark colour denotes the cases of equal weights, i.e. where both the LS method or MOM are applicable. An absence of shading indicates cases where equation (13.3) does not hold. The constraints of the WLS method do not restrict, in any significant way, its climatological or hydrological application. This is because the lack of a prior information on the functional form of the probability distribution gives a certain freedom of a distribution choice. For example, instead of the two- parameter Lognormal (LN2) or Gamma distribution (P2), three-parameter distributions with time invariant coefficient of asymmetry can be assumed (e.g. LN3 or P3).

With this approach, the WLS method is effectively applicable in all the above cases.

Table 13.1. Applicability of the Weighted Least Squared Estimation (WLS) of trend for various types of probability distribution and various classes of trend. c is the coefficient of assymetry.

A rough assessment of the type of probability distribution can be made using observed time series. This should be followed by a more detailed analysis performed on the series reduced to stationary conditions, e.g. assuming a normal distribution for the series (Y1,Y2,…Yt,..), where Yt = (Xt — mt)/σt is the standardised variable.

For example, consider the case of a class D with trend of linear form, i.e. mt=a+bt and σt=c+dt

Then equations (13.2) and (13.3) take the following form:

∑ [

∑ ∑

while the AIC formula reads

AIC = - 1n ML + 8 (13.4)

To solve equations (13.2a-b) and (13.3a-b), a gradient method was applied with moment estimates of the stationary case as the starting vector.

Strupczewski & Mitosek (1998) demonstrate the importance of relaxing the assumption of homoscedasticity in an investigation of linear trend in annual peak flow series of Polish rivers. Out of 39 series covering the common period of 192 1-1990, the stationary model (S) was preferred in 14 cases, while the A, B, C and D models were found to be the best in 1, 6, 17 and 1 cases, respectively. The predominant identification of the class C model (with constant coefficient of variation) gives evidence of a strong positive correlation of trends in the mean and in the standard deviation. According to the AIC, allowing for time-variable variance may significantly improve the fit of a model. In fact, it also affects the estimated trend in the mean. The average difference in the gradient of trend in the mean between class A (time-constant variance) and classes C and D is 26% for the 18 series in the C and D classes above. Analysing equation (13.4), one can see that its first term accounts for the criterion of a good statistical fit and its value is growing with the series length. The second term represents the doctrine of parameter parsimony in AIC. Therefore, the longer the time series the higher a chance of selection of multiparameter forms of trend and the class D instead of C.

13.4 Conclusions

The presented method enables estimation of the trend in the two first moments of time series and identification of optimum trend model from a set of competing models. For selection of alternative forms of trends in both moments both visual properties of a time series and its length should be taken into account. The results of the case study corroborate the need to account for time-dependent variance when investigating trend in time series of hydroclimatological data.

Acknowledgement 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.

References

Strupczewski, W.G. & Feluch,, W., 1998. Investigation of trend in annual peak flow series. I.

Maximum likelihood estimation. Proceedings of the Second International Conf on Climate and Water, Espoo, 241-250.

Strupczewski, W.G. & Kaczmarek, Z., 1998. Investigation of trend in annual peak flow series. II. Weighted Least Squares estimation. Proceedings of the Second International Conf on Climate and Water, Espoo, 25 1-263.

Strupczewski, W.G. & Mitosek, H.T., 1998. Investigation of trend in annual peak flow series.

ifi. Flood analysis of Polish rivers. Proceedings of the Second International Conf on Climate and Water, Espoo, 264-272.

Appendix 1

GUIDELINES: SOFTWARE