Statistics for Spatial Data.

A diagram has been devised that can provide a concise statistical summary of how well patterns match each other in terms of their correlation, their root-mean-square difference, and the ratio of their variances. Although the form of this diagram is general, it is especially useful in evaluating complex models, such as those used to study geophysical phenomena. Examples are given showing that the diagram can be used to summarize the relative merits of a collection of different models or to track changes in performance of a model as it is modified. Methods are suggested for indicating on these diagrams the statistical significance of apparent differences and the degree to which observational uncertainty and unforced internal variability limit the expected agreement between model-simulated and observed behaviors. The geometric relationship between the statistics plotted on the diagram also provides some guidance for devising skill scores that appropriately weight among the various measures of pattern correspondence.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JD900719

Summarizing multiple aspects of model performance in a single diagram

Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionF if he or she issues the probabilistic forecast F, rather than G ≠ F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the ...

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Strictly Proper Scoring Rules, Prediction, and Estimation

All currently available climate models predict a near-surface warming trend under the influence of rising levels of greenhouse gases in the atmosphere. In addition to the direct effects on climate--for example, on the frequency of heatwaves--this increase in surface temperatures has important consequences for the hydrological cycle, particularly in regions where water supply is currently dominated by melting snow or ice. In a warmer world, less winter precipitation falls as snow and the melting of winter snow occurs earlier in spring. Even without any changes in precipitation intensity, both of these effects lead to a shift in peak river runoff to winter and early spring, away from summer and autumn when demand is highest. Where storage capacities are not sufficient, much of the winter runoff will immediately be lost to the oceans. With more than one-sixth of the Earth's population relying on glaciers and seasonal snow packs for their water supply, the consequences of these hydrological changes for future water availability--predicted with high confidence and already diagnosed in some regions--are likely to be severe.

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Potential impacts of a warming climate on water availability in snow-dominated regions

A suite of climate change indices derived from daily temperature and precipitation data, with a primary focus on extreme events, were computed and analyzed. By setting an exact formula for each index and using specially designed software, analyses done in different countries have been combined seamlessly. This has enabled the presentation of the most up-to-date and comprehensive global picture of trends in extreme temperature and precipitation indices using results from a number of workshops held in data-sparse regions and high-quality station data supplied by numerous scientists world wide. Seasonal and annual indices for the period 1951-2003 were gridded. Trends in the gridded fields were computed and tested for statistical significance. Results showed widespread significant changes in temperature extremes associated with warming, especially for those indices derived from daily minimum temperature. Over 70% of the global land area sampled showed a significant decrease in the annual occurrence of cold nights and a significant increase in the annual occurrence of warm nights. Some regions experienced a more than doubling of these indices. This implies a positive shift in the distribution of daily minimum temperature throughout the globe. Daily maximum temperature indices showed similar changes but with smaller magnitudes. Precipitation changes showed a widespread and significant increase, but the changes are much less spatially coherent compared with temperature change. Probability distributions of indices derived from approximately 200 temperature and 600 precipitation stations, with near-complete data for 1901-2003 and covering a very large region of the Northern Hemisphere midlatitudes (and parts of Australia for precipitation) were analyzed for the periods 1901-1950, 1951-1978 and 1979-2003. Results indicate a significant warming throughout the 20th century. Differences in temperature indices distributions are particularly pronounced between the most recent two periods and for those indices related to minimum temperature. An analysis of those indices for which seasonal time series are available shows that these changes occur for all seasons although they are generally least pronounced for September to November. Precipitation indices show a tendency toward wetter conditions throughout the 20th century.

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Global observed changes in daily climate extremes of temperature and precipitation

Praise for the First Edition: 'I recommend this book, without hesitation, as either a reference or course text...Wilks' excellent book provides a thorough base in applied statistical methods for atmospheric sciences' - "BAMS" ("Bulletin of the American Meteorological Society"). Fundamentally, statistics is concerned with managing data and making inferences and forecasts in the face of uncertainty. It should not be surprising, therefore, that statistical methods have a key role to play in the atmospheric sciences. It is the uncertainty in atmospheric behavior that continues to move research forward and drive innovations in atmospheric modeling and prediction. This revised and expanded text explains the latest statistical methods that are being used to describe, analyze, test and forecast atmospheric data. It features numerous worked examples, illustrations, equations, and exercises with separate solutions. "Statistical Methods in the Atmospheric Sciences, Second Edition" will help advanced students and professionals understand and communicate what their data sets have to say, and make sense of the scientific literature in meteorology, climatology, and related disciplines. This book presents and explains techniques used in atmospheric data summarization, analysis, testing, and forecasting. Chapters feature numerous worked examples and exercises. Model Output Statistic (MOS) includes an introduction to the Kalman filter, an approach that tolerates frequent model changes. It includes a detailed section on forecast verification, including statistical inference, diagrams, and other methods. It provides an expanded treatment of resampling tests within nonparametric tests. It offers an updated treatment of ensemble forecasting. It provides expanded coverage of key analysis techniques, such as principle component analysis, canonical correlation analysis, discriminant analysis, and cluster analysis. It includes careful updates and edits throughout, based on users' feedback.

Statistical Methods in the Atmospheric Sciences

This article reviews the historical development of statistical weather models, from simple analyses of runs of consecutive rainy and dry days at single sites, through to multisite models of daily precipitation. Weather generators have been used extensively in water engineering design and in agricultural, ecosystem and hydrological impact studies as a means of in-filling missing data or for producing indefinitely long synthetic weather series from finite station records. We begin by describing the statistical properties of the rainfall occurrence and amount processes which are necessary precursors to the simulation of other (dependent) meteorological variables. The relationship between these daily weather models and lower-frequency variations in climate statistics is considered next, noting that conventional weather generator techniques often fail to capture wholly interannual variability. Possible solutions to this deficiency - such as the use of mixtures of slowly and rapidly varying conditioning variabl...

The weather generation game: a review of stochastic weather models

A range of different statistical downscaling models was calibrated using both observed and general circulation model (GCM) generated daily precipitation time series and intercompared. The GCM used was the U.K. Meteorological Office, Hadley Centre's coupled ocean/atmosphere model (HadCM2) forced by combined CO2 and sulfate aerosol changes. Climate model results for 1980–1999 (present) and 2080–2099 (future) were used, for six regions across the United States. The downscaling methods compared were different weather generator techniques (the standard “WGEN” method, and a method based on spell-length durations), two different methods using grid point vorticity data as an atmospheric predictor variable (B-Circ and C-Circ), and two variations of an artificial neural network (ANN) transfer function technique using circulation data and circulation plus temperature data as predictor variables. Comparisons of results were facilitated by using standard sets of observed and GCM-derived predictor variables and by using a standard suite of diagnostic statistics. Significant differences in the level of skill were found among the downscaling methods. The weather generation techniques, which are able to fit a number of daily precipitation statistics exactly, yielded the smallest differences between observed and simulated daily precipitation. The ANN methods performed poorly because of a failure to simulate wet-day occurrence statistics adequately. Changes in precipitation between the present and future scenarios produced by the statistical downscaling methods were generally smaller than those produced directly by the GCM. Changes in daily precipitation produced by the GCM between 1980–1999 and 2080–2099 were therefore judged not to be due primarily to changes in atmospheric circulation. In the light of these results and detailed model comparisons, suggestions for future research and model refinements are presented.

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Statistical downscaling of general circulation model output: A comparison of methods

Multisite generalization of a daily stochastic precipitation generation model

Special care must be exercised in the interpretation of multiple statistical hypothesis tests—for example, when each of many tests corresponds to a different location. Correctly interpreting results of multiple simultaneous tests requires a higher standard of evidence than is the case when evaluating results of a single test, and this has been known in the atmospheric sciences literature for more than a century. Even so, the issue continues to be widely ignored, leading routinely to overstatement and overinterpretation of scientific results, to the detriment of the discipline. This paper reviews the history of the multiple-testing issue within the atmospheric sciences literature and illustrates a statistically principled and computationally easy approach to dealing with it—namely, control of the false discovery rate.

“The Stippling Shows Statistically Significant Grid Points”: How Research Results are Routinely Overstated and Overinterpreted, and What to Do about It

Daniel S. Wilks

Papers

Statistical Methods in the Atmospheric Sciences

The weather generation game: a review of stochastic weather models

Statistical downscaling of general circulation model output: A comparison of methods

Multisite generalization of a daily stochastic precipitation generation model

“The Stippling Shows Statistically Significant Grid Points”: How Research Results are Routinely Overstated and Overinterpreted, and What to Do about It