TIME SERIES FORECASTING AND ADAPTIVE FILTERS
2.4 Application of recursive filters to the forecasting of streamftow and spatial rainfall data
2.4.2 Case study - Adaptive time series forecasting of image scale statistics
Fig. 2.9: A time series plot of WAR and SMF. The data has been sampled at 5 minute intervals over the duration of a rainfall event lasting 42 hours during the 24th and 25th of February 1996. The observations of the event were made with the SAWS MRL5 radar in Bethlehem,South Africa.
2.4.2 Case study - Adaptive time series forecasting of image
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team in Bethlehem,resultedin controlled releasesfrom the re ervoirsuch that the peakoutflow wasattenuated to 2300 m3. Rainfall acc umulations from the MRL-5 weather radar near Bethlehem,South Africa were produced at hourly intervalsfor selectedsub-catchments of the Vaal and relayed to the DWAF Hydrology group by telephone.The rainfall valueswere then manuallyinput into therainfall runoff model running in the DWAF offices and the resulting flow forecasts used to time releases from the reservoir. This practical solution to a large flood event (with prior warning due to the size of the catchment) relied on the skilled judgement of knowledgeable people. In the case of flash flood forecas ting it is unlikely that theseresourcescan be mobilized or that a largedamexists toallow forsome atten- uation oftheflood.Techniquesthereforerequire more automation and forecasting becomescritical.
Adaptive time ser ies modelfitting
Oneof the advantag esof adapti ve modelsis that the values of the parameters are estimated from recent ob ervations in a recursive manner and the model is thus fitted"on the fly". However, it is necessary to select the number of parameters defining the model in an objective way (for autoregressive models, the model length fixes thenumber of parameters).
A generally accepted measure (Haykin, 200 I),when using autoregressivemod- els, isthe Akaike information criterion ,AIC (Akaike, 1973). TheAIC balancesthe dimension ality mofthe modelrelativeto the samplesizetiby penalizing the sum ofsquares fitasmbecomeslarger. Increasing mincrea e the degree sof freedom in themodelwhich will resultin(spuriously) good fits tothedata asm approaches
ti, However, such close fitswill notnecessarily contain much structural informa- tionand the AIC(a common formulation isgiven below)addressesthis through a penaltyterm thatenforces parsimony in the selected model.
AIC
=
n (log0-2+
1)+
2(m+
1)Here 0-2 isthesum of quared error between the model foreca stsand the data,m
in model selection for non-station ary time series.
1
+
!!!AICc = n (loga2)
+
ti1_ ~n
The choice of the decay factor A, which controls the "memory" of the algo- rithm will alsohaveaneffec ton themodelfit and mustthereforebetaken into ac- count. Therefore,the approach adopted herewasto minimizethe sum ofsquared errors with respect to Afor a numberof different model lengths. The best model length wasthen selected on the basisof the AICccriterion.
As an exampl eof a suitable methodologyfor the applicationof adaptive mod- els to time serie of the kind we wish to forecast,the following is offered . The decay factors Aof two univariate models were selected by minimizing the AICc with respect to Aand the model length for the first200 data points of the WAR and SMFsequences shown in figure 2.9 (total sequence len gth 512 data points).
In both cases it turned out that the optimum model length of each process fitted independently was 1. Figure s2.10 and 2.11 show the variation of the AICc and optimum A'swith increasing model length m.
Figures2.12and 2.14show comparisonsbetween the observedWAR and SMF and the correspondingforecastsout to one hour ahead. The left handpanels show the forecastsproducedusingthe adapti vetime series modeland it isevidentfor the longer leadtimesthat the forecasts can occa ionallybe very unlikely.A pragmatic approach to detectin g and removingtheseoccasion al "outliers" hasbeen adopted.
Ratiosofsuccess ivevaluesof bothWARand SMFwereexamined forthefirst 200 datapoints andathreshold ratio elected that is 2 - 3time ashigh asthehighest ratioobservedduringthisperiod. This ratio isthen applied toscreen forecast val- ues which are very different from the most recent observed value. Any forecas t which doesn 't pass the screening is simply replaced with the most recently ob- erved value (i.e. a persistence forecast). The plots in the right hand panels show the forecasts produced (with the same adaptive model) using this "constrained"
35
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O. : 0.;
~
0.6 0.•; 0.-1
0.:1 10
odcl length
Fig. 2.10: Plots of the AICc and corres ponding optimum Avalue for a variety of modellengths.Theseresult s arefor theWAR time eriesshown in figure 2.9.
version of the forecas ting algorithm.
Scatterplots of theobserved and forecast valuesareshown in figures 2. 13and 2.15. Again the forecastsare madeasfar asan hour ahead and thestandard appli- cation of the adaptive algorithm is shown in the left panel while the constrained version is shown in therighthand panel. In each plot thelinearregressionR2value is alsoshownand thisindicates aclearimprovement for theconstrainedalgorithm at the longer lead times. At shorter lead times the results of both approaches are virtually identic al.
The improvement in the sum of squared errors (achieved by using the con- strained algorithm) is indicated in figure 2. 16 where the y-axis has been plot- ted with a logarithmic scale. Note that WAR is a ratio and thereforethe sum of squarederrors isa dimensionl essquantity.
r.ro
i:.!0 Q iOO Q
"'f. (i1!0
(i(i0 (i·\0
1.00 O.!)"
o.no
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O.'5O. 0 n.i;'
2 :l .\ s (i i 8 9 10 11 \2 \:1 1·\ I;'
odcl Iength
Fig. 2.11: Plots of the AICc and corre ponding optimum>. value for a variety of model lengths. These results are for the SMF time series shown in figure 2.9.
37
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