Detection of Hydrological Changes – Nonparametric Approaches Professor Ke-Sheng Cheng Dept. of Bioenvironmental Systems Engineering National Taiwan University

Parametric vs nonparametric Parametric statistics Body of statistical methods based on an assumed model for the underlying population from which the data was sampled. Inference is about some parameter (mean, variance, correlation) of the population• If model is incorrect the inference may be misleading. Example: Classical t-test assumes normal populations.

Nonparametric statistics Body of statistical methods that relax the assumptions about the underlying population model. Typically statistics based on ranks or simple counts. Can be used for both ordinal and nominal data.

MWP test for change point detection

Let us suppose the Xi are independent Bernoulli random variables.

Assessment on the performance of MWP test In general, the time of change occurrence is detected at a time that occurs later than the real time of change occurrence. [The problem of late detection.]

Mann-Kendall test for trend detection The Mann-Kendall trend test is derived from a rank correlation test for two groups of observations. In the Mann-Kendall trend test, the correlation between the rank order of the observed values and their order in time is considered. The null hypothesis for the Mann-Kendall test is that the data are independent and randomly ordered, i.e. there is no trend or serial correlation structure among the observations.

However, in many real situations the observed data are autocorrelated However, in many real situations the observed data are autocorrelated. The autocorrelation in observed data will result in misinterpretation of trend tests results. Positive serial correlation among the observations would increase the chance of significant answer, even in the absence of a trend.

The original Mann-Kendall trend test

Under the null hypothesis that X and Y are independent and randomly ordered, the statistic S tends to normality for large n, with mean and variance given by:

If the values in Y are replaced with the time order of the time series X, i.e. 1,2,...,n, the test can be used as a trend test. In this case, the statistic S reduces to that given in Eq. (5):

Kendall (1955) gives a proof of the asymptotic normality of the statistic S. The significance of trends is tested by comparing the standardized test statistic Z = with the standard normal variate at the desired significance level.

Autocorrelated data series If X is normally distributed with mean  and variance 2, then (xj – xi) will also be normally distributed with mean zero and variance 22.

It can be shown that for autocorrelated data series var(S) is the same as that of independent series, i.e.,

The assumption of normally distributed X was used to derive the above results, but in fact the test is nonparametric and does not depend on the distribution of X.

The effect of autocorrelation on the Mann-Kendall trend test Fig. 1 shows two time series X and Y each of length n = 100 observations. Visual inspection of the two time series would not indicate a large difference in the apparent trends for the two series. In fact, series X is stationary white noise, while series Y is generated as an AR(1) series with  = 0.4 using series X as the input noise. Both series are stationary without trend.

Autocorrelation between the ranks of the observations

Approximate formula for calculating V(S)