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Daniel S. Wilks

Abstract

Probabilistic quantitative precipitation forecasts (PQPFS) for discrete amount classes can be formulated as the product of precipitation probabilities (as issued in PoP forecasts) and the climatological probabilities of precipitation in specified categories conditional on the occurrence of measurable precipitation. Such forecasts, derived from historical subjective PoP forecasts, are investigated here for the 12–24 hour projection using two types of conditional climatologies. The first involves simply the conditional probabilities of precipitation in selected amount classes given that at least 0.254 mm (0.01″) occurred. The second conditional climatology involves three distributions for the same events, conditional both on the occurrence of measurable precipitation and on the magnitude of the PoP forecast. In this later case, separate conditional distributions are tabulated for occasions when “low” (0%–20%), “moderate” (30%–50%), and “high” (60%–100%) subjective PoP forecast were issued.

Comparison of the conditional climatological distributions shows clearly that forecast periods for which “high” PoPs were issued are characterized by distributions having higher probabilities of larger amounts and lower probabilities of smaller amounts, as compared to both distributions conditioned on other PoP categories and distributions not conditioned on the PoP forecasts. The reverse is true of distributions conditioned on “low” PoP values. These are seen to be general properties of PoP forecasts issued for the conterminous United states.

Forecast results are presented for Brownsville, Houston, and San Antonio, Texas, and for the period February 1981 to June 1982. This choice allows comparison of the present results with those of a subjective PQPF experiment conducted at these stations during this period. Surprisingly, the present forecasts show skill comparable to or somewhat better than that exhibited by the subjective PQPFs in the Texas experiment and the corresponding model output statistics (MOS) PQPFS. Use of climatological precipitation amount distributions conditioned on the magnitude of the subjective PoP forecasts improves overall skill worn only modestly, but (importantly) in a way that allows more frequent use of higher probabilities for the larger precipitation amount categories.

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Daniel S. Wilks

Abstract

A simple approach to long-range forecasting of monthly or seasonal quantities is as the average of observations over some number of the most recent years. Finding this “optimal climate normal” (OCN) involves examining the relationships between the observed variable and averages of its values over the previous one to 30 years and selecting the averaging period yielding the best results. This procedure involves a multiplicity of comparisons, which will lead to misleadingly positive results for developments data. The statistical significance of these OCNs are assessed here using a resampling procedure, in which time series of U.S. Climate Division data are repeatedly shuffled to produce statistical distributions of forecast performance measures, under the null hypothesis that the OCNs exhibit no predictive skill. Substantial areas in the United States are found for which forecast performance appears to be significantly better than would occur by chance.

Another complication in the assessment of the statistical significance of the OCNs derives from the spatial correlation exhibited by the data. Because of this correlation, instances of Type I errors (false rejections of local null hypotheses) will tend to occur with spatial coherency and accordingly have the potential to be confused with regions for which there may be real predictability. The “field significance” of the collections of local tests is also assessed here by simultaneously and coherently shuffling the time series for the Climate Divisions. Areas exhibiting significant local tests are large enough to conclude that seasonal OCN temperature forecasts exhibit significant skill over parts of the United States for all seasons except SON, OND, and NDJ, and that seasonal OCN precipitation forecasts are significantly skillful only in the fall. Statistical significance is weaker for monthly than for seasonal OCN temperature forecasts, and the monthly OCN precipitation forecasts do not exhibit significant predictive skill.

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Daniel S. Wilks

Abstract

A recursive solution for optimal sequences of decisions given uncertainty in future weather events, and forecasts of those events, is presented. The formulation incorporates a representation of the autocorrelation that is typically exhibited. The general finite-horizon dynamic decision–analytic framework is employed, with the weather forecast for the previous decision period included as a state variable. Serial correlation is represented through conditional probability distributions of the forecast for the current decision period, given the forecast for the previous period. Autocorrelation of the events is represented by proxy through the autocorrelation of the forecasts. The formulation is practical to implement operationally, and efficient in the sense that the weather component can be represented through a single state variable.

A compact representation of the required conditional distributions, based on an autoregressive model for forecast autocorrelation, is presented for the em of 24-h probability of precipitation forecasts. Parameters describing operationally available precipitation forecasts are given. The overall procedure is illustrated for the case of these forecasts in the context of the generalized cost/loss ratio problem.

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Daniel S. Wilks

Abstract

The three multivariate statistical methods of canonical correlation analysis, maximum covariance analysis, and redundancy analysis are compared with respect to their probabilistic accuracy for seasonal forecasts of gridded North American temperatures. Derivation of forecast error covariance matrices for the methods allows a probabilistic formulation for the forecasts, assuming Gaussian predictive distributions. The three methods perform similarly with respect to probabilistic forecast accuracy as reflected by the ranked probability score, although maximum covariance analysis may be preferred because of its slightly better forecast skill and calibration. In each case the forecast accuracy for North American seasonal temperatures compares favorably to results from previously published studies. In addition, two alternative approaches are compared for alleviating the cold biases in the forecasts that derive from ongoing climate warming. Adding lagging 15-yr means to forecast temperature anomalies improved forecast accuracy and reduced the cold bias in the forecasts, relative to using the more conventional lagging 30-yr mean.

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Daniel S. Wilks

Abstract

Principal component analysis (PCA), also known as empirical orthogonal function (EOF) analysis, is widely used for compression of high-dimensional datasets in such applications as climate diagnostics and seasonal forecasting. A critical question when using this method is the number of modes, representing meaningful signal, to retain. The resampling-based “Rule N” method attempts to address the question of PCA truncation in a statistically principled manner. However, it is only valid for the leading (largest) eigenvalue, because it fails to condition the hypothesis tests for subsequent (smaller) eigenvalues on the results of previous tests. This paper draws on several relatively recent statistical results to construct a hypothesis-test-based truncation rule that accounts at each stage for the magnitudes of the larger eigenvalues. The performance of the method is demonstrated in an artificial data setting and illustrated with a real-data example.

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Daniel S. Wilks

Abstract

A new method for estimating absorption and runoff at a point on the basis of total daily precipitation and the absorption capacity of the soil is proposed. The method is based on a statistical characterization of the variation of precipitation rates over the course of a day. Short-duration precipitation rates are modeled as Weibull random variates. Distribution parameters are estimated using maximum likehood methods by regarding precipitation from “dry” intervals as censored data.

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Daniel S. Wilks

Abstract

A method for fitting parameters of the gamma distribution to data containing some zero values using maximum likelihood methods is presented. The procedure is based on a conceptual model of the data having resulted from a censoring process so that the number, but not the numerical values of observations failing below a detection limit are known. For the case of precipitation data, this detection limit is related to the threshold value for reporting occurrence or nonoccurrence. The procedure is shown to provide parameter estimates that are more efficient (i.e., precise) than those obtained using the method of moments.

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Daniel S. Wilks

Abstract

Full exposition of the performance of a set of forecasts requires examination of the joint frequency distribution of those forecasts and their corresponding observations. In settings involving probability forecasts, this joint distribution has a high dimensionality, and communication of its information content is often best achieved graphically. This paper describes an extension of the well-known reliability diagram, which displays the joint distribution for probability forecasts of dichotomous events, to the case of probability forecasts for three disjoint events, such as “below,” “near,” and “above normal.” The resulting diagram, called the calibration simplex, involves a discretization of the 2-simplex, which is an equilateral triangle. Characteristics and interpretation of the calibration simplex are illustrated using both idealized verification datasets, and the 6–10- and 8–14-day temperature and precipitation forecasts produced by the U.S. Climate Prediction Center.

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Daniel S. Wilks

Abstract

The performance of the method of Hughes and Sangster (1979) for combining precipitation probabilities pertaining to standard 12-h forecast periods is examined for 100 stations in the conterminous United States. Precipitation probabilities for both 24- and 36-h periods are investigated. Although originally derived for a small number of stations by using data from a limited time period, the original formulation is found to be remarkably robust. There is a tendency for overforecasting in the lower half of the probability range, which is most pronounced for the 36-h forecasts. A modification of the original procedure is suggested which largely corrects this problem.

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Daniel S. Wilks and Megan McKay

Abstract

A procedure is developed to estimate extreme-value statistics for snowpack water equivalent (SWE) using historical snow depth measurements at cooperative observer stations in the northeastern United States. The method specifies “pseudodensities” that allow transformation of the statistical distribution of the deepest snow-packs to the distribution of extreme SWE values at a location. These pseudodensities vary according to characteristics of the local snow climatology, the geographic location, and the values of the snow depth data themselves. The performance of a suite of theoretical probability distributions for representing the resulting distributions of extreme SWE is also investigated, and it is concluded that five-parameter Wakeby distributions provide the best representations for the region overall. The results suggest that previous estimates for extreme SWE values may underestimate the wettest snowpacks at northern and/or higher-elevation locations, but may yield overestimates in warmer portions of the northeastern United States.

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