Homogeneity tests for Regional Frequency Analysis.
ADbootstrap.test (x, cod, Nsim=500, index=2)
HW.tests (x, cod, Nsim=500)
DK.test (x, cod)
discordancy (x, cod)
criticalD ()vector representing data from many samples defined with cod
array that defines the data subdivision among sites
number of regions simulated with the bootstrap of the original region
if index=1 samples are divided by their average value;
if index=2 (default) samples are divided by their median value
ADbootstrap.test and DK.test test gives its test statistic and its distribution value \(P\).
If \(P\) is, for example, 0.92, samples shouldn't be considered heterogeneous with significance level minor of 8%.
HW.tests gives the two Hosking and Wallis heterogeneity measures \(H_1\) and \(H_2\); following Hosking and Wallis (1997), the region under analysis can therefore be regarded as ‘acceptably homogeneous’ if \(H_1<1\), ‘possibly heterogeneous’ if \(1 \leq H_1 < 2\), and ‘definitely heterogeneous’ if \(H \geq 2\).
discordancy returns the discordancy measure \(D\) of Hosking and Wallis for all sites.
Hosking and Wallis suggest to consider a site discordant if \(D \geq 3\) if \(N \geq 15\) (where \(N\) is the number of sites considered in the region). For \(N<15\) the critical values of \(D\) can be listed with criticalD.
The Hosking and Wallis heterogeneity measures
The idea underlying Hosking and Wallis (1993) heterogeneity statistics is to measure the sample variability of the L-moment ratios and compare it to the variation that would be expected in a homogeneous region. The latter is estimated through repeated simulations of homogeneous regions with samples drawn from a four parameter kappa distribution (see e.g., Hosking and Wallis, 1997, pp. 202-204). More in detail, the steps are the following: with regards to the \(k\) samples belonging to the region under analysis, find the sample L-moment ratios (see, Hosking and Wallis, 1997) pertaining to the \(i\)-th site: these are the L-coefficient of variation (L-CV), $$t^{(i)}=\frac{\frac{1}{n_i}\sum_{j=1}^{n_i}\left(\frac{2(j-1)}{(n_i-1)}-1\right)Y_{i,j}}{\frac{1}{n_i}\sum_{j=1}^{n_i}Y_{i,j}}$$ the coefficient of L-skewness, $$t_3^{(i)}=\frac{\frac{1}{n_i}\sum_{j=1}^{n_i}\left(\frac{6(j-1)(j-2)}{(n_i-1)(n_i-2)}-\frac{6(j-1)}{(n_i-1)}+1\right)Y_{i,j}}{\frac{1}{n_i}\sum_{j=1}^{n_i}\left(\frac{2(j-1)}{(n_i-1)}-1\right)Y_{i,j}}$$ and the coefficient of L-kurtosis $$t_4^{(i)}=\frac{\frac{1}{n_i}\sum_{j=1}^{n_i}\left(\frac{20(j-1)(j-2)(j-3)}{(n_i-1)(n_i-2)(n_i-3)}-\frac{30(j-1)(j-2)}{(n_i-1)(n_i-2)}+\frac{12(j-1)}{(n_i-1)}-1\right)Y_{i,j}}{\frac{1}{n_i}\sum_{j=1}^{n_i}\left(\frac{2(j-1)}{(n_i-1)}-1\right)Y_{i,j}}$$ Note that the L-moment ratios are not affected by the normalization by the index value, i.e. it is the same to use \(X_{i,j}\) or \(Y_{i,j}\) in Equations.
Define the regional averaged L-CV, L-skewness and L-kurtosis coefficients, $$t^R = \frac{\sum_{i=1}^k n_i t^{(i)}}{ \sum_{i=1}^k n_i}$$ $$t_3^R =\frac{ \sum_{i=1}^k n_i t_3^{(i)}}{ \sum_{i=1}^k n_i}$$ $$t_4^R =\frac{ \sum_{i=1}^k n_i t_4^{(i)}}{\sum_{i=1}^k n_i}$$ and compute the statistic $$V = \left\{ \sum_{i=1}^k n_i (t^{(i)} - t^R )^2 / \sum_{i=1}^k n_i\right\} ^{1/2}$$
Fit the parameters of a four-parameters kappa distribution to the regional averaged L-moment ratios \(t^R\), \(t_3^R\) and \(t_4^R\), and then generate a large number \(N_{sim}\) of realizations of sets of \(k\) samples. The \(i\)-th site sample in each set has a kappa distribution as its parent and record length equal to \(n_i\). For each simulated homogeneous set, calculate the statistic \(V\), obtaining \(N_{sim}\) values. On this vector of \(V\) values determine the mean \(\mu_V\) and standard deviation \(\sigma_V\) that relate to the hypothesis of homogeneity (actually, under the composite hypothesis of homogeneity and kappa parent distribution).
An heterogeneity measure, which is called here \(HW_1\), is finally found as $$\theta_{HW_1} = \frac{V - \mu_V}{\sigma_V}$$ \(\theta_{HW_1}\) can be approximated by a normal distributed with zero mean and unit variance: following Hosking and Wallis (1997), the region under analysis can therefore be regarded as ‘acceptably homogeneous’ if \(\theta_{HW_1}<1\), ‘possibly heterogeneous’ if \(1 \leq \theta_{HW_1} < 2\), and ‘definitely heterogeneous’ if \(\theta_{HW_1}\geq2\). Hosking and Wallis (1997) suggest that these limits should be treated as useful guidelines. Even if the \(\theta_{HW_1}\) statistic is constructed like a significance test, significance levels obtained from such a test would in fact be accurate only under special assumptions: to have independent data both serially and between sites, and the true regional distribution being kappa.
Hosking and Wallis (1993) also give alternative heterogeneity measures (that we call \(HW_2\) and \(HW_3\)), in which \(V\) is replaced by: $$V_2 = \sum_{i=1}^k n_i \left\{ (t^{(i)} - t^R)^2 + (t_3^{(i)} - t_3^R)^2\right\}^{1/2} / \sum_{i=1}^k n_i$$ or $$V_3 = \sum_{i=1}^k n_i \left\{ (t_3^{(i)} - t_3^R)^2 + (t_4^{(i)} - t_4^R)^2\right\}^{1/2} / \sum_{i=1}^k n_i$$ The test statistic in this case becomes $$\theta_{HW_2} = \frac{V_2 - \mu_{V_2}}{\sigma_{V_2}}$$ or $$\theta_{HW_3} = \frac{V_3 - \mu_{V_3}}{\sigma_{V_3}}$$ with similar acceptability limits as the \(HW_1\) statistic. Hosking and Wallis (1997) judge \(\theta_{HW_2}\) and \(\theta_{HW_3}\) to be inferior to \(\theta_{HW_1}\) and say that it rarely yields values larger than 2 even for grossly heterogeneous regions.
The bootstrap Anderson-Darling test
A test that does not make any assumption on the parent distribution is the Anderson-Darling (\(AD\)) rank test (Scholz and Stephens, 1987). The \(AD\) test is the generalization of the classical Anderson-Darling goodness of fit test (e.g., D'Agostino and Stephens, 1986), and it is used to test the hypothesis that \(k\) independent samples belong to the same population without specifying their common distribution function.
The test is based on the comparison between local and regional empirical distribution functions. The empirical distribution function, or sample distribution function, is defined by \(F(x)=\frac{j}{\eta}, x_{(j)}\leq x < x_{(j+1)}\), where \(\eta\) is the size of the sample and \(x_{(j)}\) are the order statistics, i.e. the observations arranged in ascending order. Denote the empirical distribution function of the \(i\)-th sample (local) by \(\hat{F}_i(x)\), and that of the pooled sample of all \(N = n_1 + ... + n_k\) observations (regional) by \(H_N (x)\). The \(k\)-sample Anderson-Darling test statistic is then defined as $$\theta_{AD} = \sum_{i=1}^k n_i \int _{{\rm all}\ x} \frac{[\hat{F}_i (x) - H_N (x) ]^2}{H_N (x) [ 1 - H_N (x) ] } dH_N (x)$$
If the pooled ordered sample is \(Z_1 < ... < Z_N\), the computational formula to evaluate \(\theta_{AD}\) is: $$\theta_{AD} = \frac{1}{N} \sum_{i=1}^k \frac{1}{n_i}\sum_{j=1}^{N-1} \frac{(N M_{ij} - j n_i)^2 }{j (N-j)}$$ where \(M_{ij}\) is the number of observations in the \(i\)-th sample that are not greater than \(Z_j\). The homogeneity test can be carried out by comparing the obtained \(\theta_{AD}\) value to the tabulated percentage points reported by Scholz and Stephens (1987) for different significance levels.
The statistic \(\theta_{AD}\) depends on the sample values only through their ranks. This guarantees that the test statistic remains unchanged when the samples undergo monotonic transformations, an important stability property not possessed by \(HW\) heterogeneity measures. However, problems arise in applying this test in a common index value procedure. In fact, the index value procedure corresponds to dividing each site sample by a different value, thus modifying the ranks in the pooled sample. In particular, this has the effect of making the local empirical distribution functions much more similar to the other, providing an impression of homogeneity even when the samples are highly heterogeneous. The effect is analogous to that encountered when applying goodness-of-fit tests to distributions whose parameters are estimated from the same sample used for the test (e.g., D'Agostino and Stephens, 1986; Laio, 2004). In both cases, the percentage points for the test should be opportunely redetermined. This can be done with a nonparametric bootstrap approach presenting the following steps: build up the pooled sample \(\cal S\) of the observed non-dimensional data. Sample with replacement from \(\cal S\) and generate \(k\) artificial local samples, of size \(n_1, \dots ,n_k\). Divide each sample for its index value, and calculate \(\theta^{(1)}_{AD}\). Repeat the procedure for \(N_{sim}\) times and obtain a sample of \(\theta^{(j)}_{AD}\), \(j=1,\dots , N_{sim}\) values, whose empirical distribution function can be used as an approximation of \(G_{H_0}(\theta_{AD})\), the distribution of \(\theta_{AD}\) under the null hypothesis of homogeneity. The acceptance limits for the test, corresponding to any significance level \(\alpha\), are then easily determined as the quantiles of \(G_{H_0}(\theta_{AD})\) corresponding to a probability \((1-\alpha)\).
We will call the test obtained with the above procedure the bootstrap Anderson-Darling test, hereafter referred to as \(AD\).
Durbin and Knott test
The last considered homogeneity test derives from a goodness-of-fit statistic originally proposed by Durbin and Knott (1971). The test is formulated to measure discrepancies in the dispersion of the samples, without accounting for the possible presence of discrepancies in the mean or skewness of the data. Under this aspect, the test is similar to the \(HW_1\) test, while it is analogous to the \(AD\) test for the fact that it is a rank test. The original goodness-of-fit test is very simple: suppose to have a sample \(X_i\), \(i = 1, ..., n\), with hypothetical distribution \(F(x)\); under the null hypothesis the random variable \(F(X_i)\) has a uniform distribution in the \((0,1)\) interval, and the statistic \(D = \sum_{i=1}^n \cos[2 \pi F(X_i)]\) is approximately normally distributed with mean 0 and variance 1 (Durbin and Knott, 1971). \(D\) serves the purpose of detecting discrepancy in data dispersion: if the variance of \(X_i\) is greater than that of the hypothetical distribution \(F(x)\), \(D\) is significantly greater than 0, while \(D\) is significantly below 0 in the reverse case. Differences between the mean (or the median) of \(X_i\) and \(F(x)\) are instead not detected by \(D\), which guarantees that the normalization by the index value does not affect the test.
The extension to homogeneity testing of the Durbin and Knott (\(DK\)) statistic is straightforward: we substitute the empirical distribution function obtained with the pooled observed data, \(H_N(x)\), for \(F(x)\) in \(D\), obtaining at each site a statistic $$D_i = \sum_{j=1}^{n_i} \cos[2 \pi H_N(X_j)]$$ which is normal under the hypothesis of homogeneity. The statistic \(\theta_{DK} = \sum_{i=1}^k D_i^2\) has then a chi-squared distribution with \(k-1\) degrees of freedom, which allows one to determine the acceptability limits for the test, corresponding to any significance level \(\alpha\).
Comparison among tests
The comparison (Viglione et al, 2007) shows that the Hosking and Wallis heterogeneity measure \(HW_1\) (only based on L-CV) is preferable when skewness is low, while the bootstrap Anderson-Darling test should be used for more skewed regions. As for \(HW_2\), the Hosking and Wallis heterogeneity measure based on L-CV and L-CA, it is shown once more how much it lacks power.
Our suggestion is to guide the choice of the test according to a compromise between power and Type I error of the \(HW_1\) and \(AD\) tests. The L-moment space is divided into two regions: if the \(t_3^R\) coefficient for the region under analysis is lower than 0.23, we propose to use the Hosking and Wallis heterogeneity measure \(HW_1\); if \(t_3^R > 0.23\), the bootstrap Anderson-Darling test is preferable.
traceWminim, roi, KAPPA, HW.original.
# NOT RUN {
data(hydroSIMN)
annualflows
summary(annualflows)
x <- annualflows["dato"][,]
cod <- annualflows["cod"][,]
split(x,cod)
#ADbootstrap.test(x,cod,Nsim=100) # it takes some time
#HW.tests(x,cod) # it takes some time
DK.test(x,cod)
fac <- factor(annualflows["cod"][,],levels=c(34:38))
x2 <- annualflows[!is.na(fac),"dato"]
cod2 <- annualflows[!is.na(fac),"cod"]
ADbootstrap.test(x2,cod2,Nsim=100)
ADbootstrap.test(x2,cod2,index=1,Nsim=200)
HW.tests(x2,cod2,Nsim=100)
DK.test(x2,cod2)
discordancy(x,cod)
criticalD()
# }
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