Rainfall probability charts have been used to quantify the effect of the Southern Oscillation Index (SOI) on rainfall for many years. To better understand the effect of the SOI phases, we discuss forming confidence intervals on the predicted rainfall quantiles using percentile bootstrap methods

Conover

DiCiccio

Efron

Hall

Hammer

Johns

Künsch

Lange

McCown

Meinke

Quinn

Singh

Stone

Troup

Wilks

Willcocks

Zwiers

English

Dunn, Peter K.

University of Southern Queensland ePrints

Bootstrap Confidence Intervals for PredictedRainfall QuantilesPeter K. DunnDepartment of Mathematics and ComputingUniversity of Southern Queensland, Toowoombaemail: dunn@usq.edu.auTelephone: +61 7 4631 2772Fax: +61 7 4631 1775February 25, 2003AbstractRainfall probability charts have been used to quantify the effect ofthe Southern Oscillation Index on rainfall for many years. To betterunderstand the effect of the SOI phases, we discuss forming confidence1intervals on the predicted rainfall quantiles using percentile bootstrapmethods.Keywords: bootstrap; southern oscillation; confidence intervals; rain-fall.1 IntroductionThe relationship between the Southern Oscillation Index (SOI) and rainfallhas been well known for many years (for example, Troup (1965), Quinn andBurt (1972) and others). Using the SOI, Stone and Auliciems (1992) con-structed five phases of the SOI which can then be used to study the effect ofthe SOI. The use of phases has proved to be far more constructive than usingjust the raw SOI itself. The phases have since been used to study the effectson cropping (for example, Meinke et al., 1996), frost (for example, Willcocksand Stone, 2000 and Stone et al., 1996a), assessing planting opportunities(Stone and McKeon, 1993), wheat crop management (Hammer et al. 1996),the effects of sorghum ergot on sorghum production (Meinke and Ryley 1997)and other phenomena. In addition, the SOI phases have been used in theAPSIM computer simulation software (McCown et al. (1995)) to model the2growing of crops.Despite the successful use of the method, one shortcoming, however, isthat no confidence bands have been placed on the quantiles used in suchapplications. Indeed, the determination of confidence limits for quantileson rainfall is a difficult, if not impossible, problem analytically. Confidencebands would also serve to indicate the accuracy of the results reported inthe rainfall probability charts. Using the bootstrap method, however, suchconfidence intervals can be generated using computer simulation.In this paper, we add bootstrap confidence intervals to the rainfall prob-ability charts (introduced by Stone and Auliciems (1992)) commonly usedto show the effects of the SOI phases on rainfall quantiles. In the past,Kruskall-Wallis or Kolmogorov-Smirnoff non-parametric tests (see, for ex-ample, Conover (1999)) on medians have been used to quantify the effectsof the SOI (for example, Stone and Auliciems 1992). Confidence intervalsprovide more information, and enable the results to potentially be carriedinto calculation based on the SOI phases.In the next section, we briefly discuss the SOI phases. We then dis-cuss the bootstrap method in Section 3, and different bootstrap methods forconfidence intervals in Section 4. In Section 5, we examine rainfall proba-3bility charts produced using the bootstrap confidence intervals. Finally, inSection 6, we conclude and make further comments.2 The SOI PhasesMany attempts have been made to quantify the relationship between theSOI and rainfall. The novel approach taken by Stone and Auliciems (1992)is based on a principal components analysis and cluster analysis of the SOI.This results in the monthly SOI and the changes in the monthly SOI beingused to classify the SOI trend into five different “phases”. Broadly speaking,the five phases correspond to the SOI falling rapidly, staying consistently neg-ative, staying consistently near zero, staying consistently positive, and risingrapidly. Based on historical records, the rainfall (or any other phenomenon)at any location with sufficient records can be gathered for each of the fivedifferent phases, and the distributions of rainfall for these phases can be dis-played using boxplots or rainfall probability charts. These charts usuallyplot the proportion of years for which the rainfall exceeds given levels. For amore complete discussion of the SOI phases, see Stone and Auliciems (1992).Stone et al. (1996b) recently applied the phases to global rainfall prediction,4and to the effect of the SOI on rainfall in the whole of Australia (Stone etal. (1999).In seasonal rainfall prediction, the SOI phases are usually used to predictthe total rainfall in the three months following the month in which the averagemonthly SOI is calculated. We then define the pth rainfall quantile, rp,as that value of rainfall below which a proportion p of years have receivedgreater rainfall. That is, the pth rainfall quantile of rainfall R is defined asPr(R > rp) = p. Note that this is different to the usual statistical definitionof quantiles, where we would have Pr(R < rp) = p.3 The Bootstrap MethodThe bootstrap method is a technique used for determining, among otherthings, the accuracy of statistics. It relies heavily on computer simulations.Traditionally, standard errors have been calculated using well known formu-lae often based on assumptions that are not satisfied or only approximatelysatisfied, or perhaps worse, where it is not known if the assumptions holdor not. The use of bootstrap methods overcomes these problems. For anexcellent, easy to understand introduction to the subject, the reader is re-5ferred to Efron and Tibshirani (1993) or DiCiccio and Efron (1996). Moremathematical details can be found, for example, in Hall (1992). In essence,the bootstrap method relies on resampling with replacement from the givensample and calculating the required statistic from these repeated samples.The values of the statistic from the repeated sampling can then be used togenerate standard errors and confidence intervals for the statistic. It maybe clear to some readers that the bootstrap method shares come commonelements with other computational methods, such as the jackknife and cross-validation. An illustrative, easy to follow example can be found in Efron andTibshirani (1993), Chapter 1.One of the assumptions made, even in the bootstrap, is that the data areindependent. See Zwiers (1990), Remark 2.2 of Singh (1981), for example;Wilks (1997), Ku¨nsch (1989) and Liu and Singh (1992) discuss blockingbootstrap as a solution. The SOI series and the rainfall series themselvesnaturally exhibit serial correlation. However, the current application doesnot use the straight SOI or rainfall series, but selections from these thatsatisfy certain criteria of SOI phase and month. For example, if we areinterested in the rainfall distribution for the three months starting in Julybased on a June with an SOI Phase 5, our entire data set would comprise6the three month rainfall totals from the 28 years 1890, 1891, 1894, 1895,1898, 1899, 1907, 1908, 1913 and so on, and serial correlation is not an issue.Indeed, the autocorrelation function of the rainfall for the above years showsno significant autocorrelations.4 Applying the Bootstrap MethodMore formally, we suppose a random sample of size n, x = (x1, x2, . . . , xn)from an unknown population distribution F . We let the parameter of interestbe θ = t(F ), estimated from the sample x. We wish to find an estimate of θ(say θˆ) using the information in the sample x; we let this statistic of interest,then, be θˆ = s(x). We then define a bootstrap sample as x∗ as a randomsample of size n drawn with replacement from the given sample x. Fromthis bootstrap sample, the statistic of interest can again be determined; thisbootstrap replication of θˆ determined from x∗ is then denoted by θˆ∗. Thebootstrap estimate of the standard error of θ is then the standard error of θˆfor the datasets of size n randomly drawn with replacement from Fˆ .In the context of confidence intervals, there are many bootstrap meth-ods available. The percentile-t method uses the sample itself to generate7the equivalent of t-scores in the usual normal theory confidence intervals.These intervals are transformation respecting (Efron and Tibshirani (1993),page 175). If an estimate is transformation respecting, the confidence in-terval of the transformation of the parameter of interest, m = T (θ), is just[mlow, mupp] = [T (θˆlow), T (θˆupp)]. In practice, we can see how the confidenceintervals of rainfall quantiles would be used in further dependent studies (incropping simulations for example), and the transformation respecting prop-erty may be of some importance.The bias-corrected and accelerated version, usually abbreviated to BCa,allows for bias correction, and also introduces a skewness adjustment. Amajor shortcoming of the BCa method is that of undercoverage with smallsamples and large values of α (Hall, 1992, §3.10.5). It is quite possible thatsmall samples will arise in this application (for example, when a particularSOI phase has not historically occurred very often in a particular month).The BCa method is also transformation respecting and second order accurate(see Efron and Tibshirani, 1993, §14.3 for further details). The percentilemethod is only first order accurate.A further method is the approximate bootstrap confidence intervals, thatattempt to reduce the computation involved. This method, however, uses a8Taylor series expansion and hence requires that θˆ = s(x) be smooth, whichis not the case for quantiles.There are numerous methods for confidence intervals; we have just pre-viewed some of those. For the present application, we must be aware thequantiles are not smooth, and in particular, finding confidence for quantilesclose to 0 and 100 must be treated carefully. For example, Johns (1988) usesthe concept of importance resampling to reduce the number of bootstrapsamples that need to be taken to produce desirable properties.We decide that the percentile-t method is the best compromise to use inthe current application.The idea of percentile intervals is reasonably simple. Suppose we generatenB bootstrap samples from our data, and for each sample we determine thestatistic of interest, θˆ∗ = s(x∗). (The statistics of interest in our applicationare the rainfall quantiles.) If we let the distribution function of θˆ∗ be Fˆ (θˆ∗),we can hence determine the 1− 2α confidence interval as[Fˆ−1(α), Fˆ−1(1− α)].9We can then define Fˆ−1(α) = θˆ∗(α), and write the confidence interval as[θˆ∗(α), θˆ∗(1−α)].Essentially, we determine the statistic for each bootstrap replication, and the100(1−α) confidence interval is the αth and (1−α)th empirical percentiles.5 Application of the Bootstrap to RainfallQuantilesThe percentile bootstrap method discussed in the previous section can nowbe applied to the forming of rainfall quantiles based on phases of the SOI,and displayed using rainfall probability charts. The calculations themselvesare relatively straight forward. The calculations were done in matlab (TheMathWorks Inc (1997)) but could also have be done in C++, FORTRAN,S-Plus (MathSoft (1997)) or something else. matlab was chosen since thecode is more transparent.For the purpose of an example, we use the rainfall records at Charlevilleto demonstrate the technique. In the Figures, we plot the rainfall probability10Figure 1: The 95% Confidence Intervals on the Rainfall Percentiles for Pre-dicted Rainfall from August to October Based on a Phase 2 SOI in thePrevious July.Figure 2: The 95% Confidence Intervals on the Rainfall Percentiles for Pre-dicted Rainfall from September to November Based on a Phase 2 SOI in thePrevious June.charts with the 95% confidence intervals supplied in dotted lines. We selecta few cases to show the types of plots that result. In practice, we found thatat least 1000 bootstrap samples were needed for reasonably stable bootstrapconfidence intervals, in line with the discussion in Efron and Tibshirani (1993)§19.3.It is clear from the figures that there is some measure of confidence thatwe can place on the predicted rainfall quantiles. For example, using Figure 1,the probability of obtaining at least 100mm of rain from August to Octoberbased on a Phase 2 SOI in July is about 45%. We can also add some measureof confidence to the prediction, stating that the 95% confidence interval isbetween approximately 30% and 60%. These confidence intervals could po-tentially be useful in modelling situations and computer simulations based onFigure 3: The 95% Confidence Intervals on the Rainfall Percentiles for Pre-dicted Rainfall from March to April Based on a Phase 3 SOI in the PreviousFebruary.11Figure 4: The 95% Confidence Intervals on the Rainfall Percentiles for Pre-dicted Rainfall from November to January Based on a Phase 2 SOI in thePrevious October.Figure 5: The 95% Confidence Intervals on the Rainfall Percentiles for Pre-dicted Rainfall from January to March Based on a Phase 4 SOI in the Pre-vious December.the SOI, as discussed earlier, to introduce a confidence interval into forecasts.6 Conclusion and DiscussionIn this paper, we have discussed using SOI phases to predict rainfall patterns.We have introduced the notion of adding confidence limits on predicted rain-fall quantiles using the percentile form of the bootstrap. The results provideuseful bounds on the predicted rainfall quantiles based on SOI phases, whichserve to indicate the accuracy of the rainfall probability charts used by Stoneand Auliciems (1992). The methods could also be used to provide boundson other quantities that are based on SOI phases.The concept of importance resampling (see Lange (1999), Johns (1988)or Efron and Tibshirani (1993)) could also be used to increase the efficiencyof the procedure when finding quantiles with small or large α.127 AcknowledgementsThe author would like to thank the two reviewers, whose comments and rec-ommendations substantially improved this paper. We also thank the Queens-land Department of Primary Industries for the data.References1. Conover WJ. 1999. Practical Nonparametric Statistics, 3rd edition.John Wiley and Sons: New York.2. DiCiccio TJ, Efron B. 1996. Bootstrap confidence intervals. StatisticalScience. 11(3): 189–228.3. Efron B, Tibshirani RJ. 1993. An Introduction to the Bootstrap. Chap-man and Hall: New York.4. Hall P. 1992. The Bootstrap and Edgeworth Expansion. Springer-Verlag: New York.5. Hammer GL, Holzworth DP, Stone RC. 1996. The value of skill inseasonal climate forecasting to wheat crop management in a region with13high climatic variability. Australian Journal of Agricultural Research.47: 717–737.6. Johns MV. 1988. Importance sampling for bootstrap confidence inter-vals. Journal of the American Statistical Association. 83(403): 709–714.7. Ku¨nsch HR. 1989. The jackknife and the bootstrap for general station-ary observations. The Annals of Statistics. 17: 1217–1241.8. Lange K. 1999. Numerical Analysis for Statisticians. Springer-Verlag:New York.9. Liu RY, Singh, K. 1992. Moving blocks bootstrap captures weak de-pendence, in Lepage R, Billard L (eds) Exploring the Limits of theBootstrap. John Wiley: New York, 225–248.10. MathSoft (pub). 1997. S-Plus 4 Guide to Statistics. Data AnalysisProducts Division, MathSoft: Seattle.11. The MathWorks Inc (pub). 1997. The matlab Reference Guide ver-sion 5, MA:Author.12. McCown RL, Hammer GL, Hargreaves JNG, Holzworth D, Huth NI.141995. APSIM: an agricultural production system simulation model foroperational research. Mathematics and Computers in Simulation. 39:225–231.13. Meinke H, Stone RC, Hammer GL. 1996. SOI phases and climatic riskto peanut production: A case study for northern Australia. Interna-tional Journal of Climatology. 16: 783–789.14. Meinke H, Ryley M. 1997. Effect of sorghum ergot on grain sorghumproduction: a preliminary analysis. Australian Journal of AgriculturalResearch. 48:1241–1247.15. Quinn WH, Burt WV. 1972. Use of southern oscillation in weatherprediction. Journal of Applied Meteorology. 11: 616–628.16. Stone RC, Auliciems A. 1992. SOI phase relationships with rainfall ineastern Australia. International Journal of Climatology. 12: 625–636.17. Stone RC, McKeon GM. 1993. Prospects for using weather predictionto reduce pasture establishment risk. Tropical Grasslands. 27: 406–413.18. Stone RC, Nicholls N, Hammer G. 1996a. Frost in northeast Australia:15Trends and influences of phases of the southern oscillation. Journal ofClimate. 9(8): 1896–1909.19. Stone RC, Hammer GL, Marcussen T. 1996b. Prediction of global rain-fall probabilities using phases of the southern oscillation index. Nature.384: 252–255.20. Stone, R, Marcussen, T, Partridge, I. 1999. Seasonal Rainfall in Aus-tralia based on Phases of the SOI. DPI Information Series CON00997,Queensland Department of Primary Industries: Toowoomba21. Troup AJ. 1965. The “Southern Oscillation”. Quarterly Journal of theRoyal Meteorological Society. 91: 490–506.22. Wilks, DS. 1997. Resampling Hypothesis Tests for AutocorrelatedFields. Journal of Climate. 10: 65–82.23. Willcocks, JR, Stone, RC. 2000. Frost Risk in Eastern Australia andthe Influence of the Southern Oscillation. DPI Information Series QI00001,Queensland Department of Primary Industries: Toowoomba.24. Zwiers FW. 1990. The Effect of Serial Correlation on Statistical In-ferences Made with Resampling Procedures. Journal of Climate. 3:160 100 200 300 400 500 600 70000.10.20.30.40.50.60.70.80.91July Phase 2 Predicting Rainfall forThree Months Starting AugustRainfall (in mm)Probabilitiesn=26; bstrap samples: 1000; α= 0.051452–1461.170 50 100 150 200 250 30000.10.20.30.40.50.60.70.80.91June Phase 2 Predicting Rainfall forThree Months Starting JulyRainfall (in mm)Probabilitiesn=24; bstrap samples: 1000; α= 0.050 100 200 300 400 500 600 700 800 90000.10.20.30.40.50.60.70.80.91February Phase 3 Predicting Rainfall forThree Months Starting MarchRainfall (in mm)Probabilitiesn=20; bstrap samples: 1000; α= 0.05180 50 100 150 200 250 30000.10.20.30.40.50.60.70.80.91October Phase 2 Predicting Rainfall forThree Months Starting NovemberRainfall (in mm)Probabilitiesn=33; bstrap samples: 1000; α= 0.050 100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.91December Phase 4 Predicting Rainfall forThree Months Starting JanuaryRainfall (in mm)Probabilitiesn=22; bstrap samples: 1000; α= 0.0519

An Introduction to the Bootstrap. Chapman and Hall:

Bootstrap con intervals.

Eect of sorghum ergot on grain sorghum production: a preliminary analysis.

Frost in northeast Australia: 15Trends and in uences of phases of the southern oscillation.

Frost Risk

Importance sampling for bootstrap con intervals.

Moving blocks bootstrap captures weak dependence,

Numerical Analysis for Statisticians.

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Prediction of global rainfall probabilities using phases of the southern oscillation index.

Prospects for using weather prediction to reduce pasture establishment risk.

Resampling Hypothesis Tests for Autocorrelated Fields.

S-Plus 4 Guide to Statistics. Data Analysis Products Division,

SOI phase relationships with rainfall in eastern Australia.

SOI phases and climatic risk to peanut production: A case study for northern Australia.

The \Southern Oscillation&quot;.

The Bootstrap and Edgeworth Expansion.

The Eect of Serial Correlation on Statistical Inferences Made with Resampling Procedures.

The jackknife and the bootstrap for general stationary observations. The Annals of Statistics.

The MathWorks Inc (pub).

The value of skill in seasonal climate forecasting to wheat crop management in a region with 13high climatic variability.

Bootstrap confidence intervals for predicted rainfall quantiles

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