To examine the performance of the raw dependence estimator
and the regional min-stable model using real annual maxima rainfall problem to more manageable proportions we examined various small sub-regions of rain gauges. Firstly, we looked at sub-regions created by distributed within the sub-region. If the sub-region contains separate groups of gauges then the fit of the model can be noticeably degraded.
From the previous section we know that the raw dependence estimator can be rather inaccurate and so we can only examine the modelled dependence results in a very general way. One interesting result that requires comment is the relative lack of raw dependence estimates greater than 2 even for sub-regions containing very widely dispersed sites that
results for one particular sub-region. This sub-region was defined by that gauge 7, the central gauge, has the worst fit of all the gauges. We find that the mean squared difference betwen the raw and modelled model that relates inter-site dependence to the distance between the
gauges. This is a reasonable model to use to study the dependence in rainfall regions but it is probably inappropriate to use such a model to analyse flood flow data. However, the use of the raw dependence estimator and some suitable min-stable process w ould be a perfectly feasible way of studying inter-site dependence in the flood flow data.
C O N C L U S IO N S
We conclude this work with a brief summary of what has been achieved and suggest some ways in which these results may be used and perhaps improved. A new method for studying dependence structures in hydrological regions has been developed using the properties of min- stable processes. The uses of these min-stable processes have been examined and we have drawn some conclusions about their merits and limitations.
We consider first a limitation of min-stable processes as a hydrological model. In this work all we have attempted is to model the dependence structures in hydrological regions. Is it not proposed that we should use min-stable processes as a full regional estimation technique.
This is in part due to the fact that despite knowing what form a min- stable survivor function must have - see equation 4.9 - it has not yet proved possible to generate min-stable distributions that have continuous density functions for more than two variables. Indeed the number of bi-variate distributions found is not great. However, we may attempt to circumvent this problem by assuming that some min-stable process governs the behaviour of the regions data and then studying the inter-site dependence between all pairs of sites. Using the inter-site dependence in this way we need a method for fitting a min-stable process.
A fitting method has been developed and its properties assessed using
simulation studies. The raw estimator on which the estimation technique was based was found to be rather inaccurate - both variance and bias tended to be large. Hence in practice if we wish to make any dependence structures. These dependence structures also have the advantage that they arise from the extreme value nature of the data that the sites are non-identically distributed. Hence a min-stable process is an extremely useful tool if we wish to generate data from a non-
models in measuring the level of inter-site dependence in real data. The best way of improving the accuracy of the modelled dependence would be to find a better raw dependence estimator. Hence it may be profitable to search for some new dependence estimator. The fact that we have only managed to produce continuous min-stable distributions in two dimensions is a serious weakness and the search for distributions in higher dimensions should continue. If this search is successful it may at som e stage prove possible to develop a full min-stable regional estimation technique.
At present very little is known about the effects of inter-site dependence on the regional estimation techniques now in use. The use of min-stable processes provides a simple and elegant method for assessing these effects by simulation. These simulation studies could use the very wide of dependence structures that min-sable processes can model to assess the effects of inter-site dependence over a range of dependence structures that may occur in the real data. This work could make a major contribution to the confidence we have in the regional estimates and may be regarded as work of high priority.
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f \ m . M i . 0 Z)
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Comparing the raw dependence estimator and the modelled dependence for a real rainfall sub-region.
Gauge N u m b e r T A B L E 1
1 2 3 4 5 6 7 8 9
1 R a w 1
M o d 1
2 R a w 1.50 1 M o d 1.47 1
3 R a w 1.47 1.01 1 M o d 1.48 1.00 1
4 R a w 1.49 1.25 1.14 1 M o d 1.62 1.06 1.04 1
5 R a w 1.45 1.21 1.19 1.21 1 M o d 1.56 1.06 1.04 1.00 1
6 R a w 1.46 1.33 1.34 1.40 1.32 1 M o d 1.42 1.08 1.06 1.04 1.02 1
7 R a w 1.60 1.47 1.44 1.50 1.44 1.45 1 M o d 1.31 1.13 1.11 1.10 1.07 1.01 1
8 R a w 1.63 1.55 1.58 1.39 1.37 1.20 1.60 1 M o d 1.73 1.63 1.58 1.42 1.39 1.30 1.27 1
9 R a w 1.29 1.52 1.48 1.41 1.46 1.40 1.58 1.38 1 M o d 1.33 1.60 1.57 1.53 1.47 1.31 1.21 1.18 1
10 R a w 1.22 1.46 1.44 1.38 1.43 1.46 1.50 1.42 1.17 M o d 1.19 1.51 1.48 1.49 1.43 1.27 1.16 1.27 1.02
For the locations of the gauges see Figure 11.
M e a n s u m of squared differences = 0.0042.
Comparing the r a w dependence estimator and the modelled
For the locations of the gauges see Figure 11.
M e a n s u m of squared differences 0.0017.
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