This chapter examines potential agronomic and economic impacts of several climate change and adaptation scenarios at farm-level. The analysis was based on a protocol that links climatic, agronomic, and economic models to form an integrated model. The protocol was applied to two representative farms of important grain regions of Midwest: Southwestern Minnesota and Southeastern Nebraska. The farm-level analysis will consist of examining the potential agronomic and economic impacts of several climate change and adaptation scenarios. The integrated model, which was developed by Kaiser, consists of a stochastic weather generator, a dynamic crop yield simulation model, and a farm-level linear programming model. One advantage of using a stochastic weather generator is that changes in variability of temperature and precipitation can also be imposed. Agricultural adaptation to a changing climate will occur in several forms, including technical innovations, changes in agricultural land areas, and changes in use of irrigation. Three types of farm-level adaptation to climate change are considered in this analysis.

The Dawid–Sebastiani score (DSS) is a theoretically attractive tool for evaluating the accuracy of multivariate ensemble forecasts, wherein each ensemble member produces a vector of forecasts. It has been little used to date mostly because its computation requires calculation and inversion of the ensemble covariance matrix for each forecast occasion. These matrices are not invertible unless the ensemble size is larger than the dimension of the forecast vectors. Moreover, these matrices are poorly estimated unless the ensemble size is quite large relative to the dimension of the vectors. This article describes application to the inverse correlation matrix in the DSS of a relatively recently developed regularization procedure called the graphical lasso (“glasso”), which not only suppresses the sampling variability of the score for relatively small ensemble sizes but also allows its computation even when the ensemble size is smaller than the dimension of the forecast and observation vectors. Use of glasso regularization with the DSS, and its performance in comparison to the Energy Score and the Variogram Score, are illustrated using a novel statistical model for multivariate ensemble forecasts which allows separate manipulation of bias, univariate calibration, multivariate calibration, and multivariate correlation characteristics.

I think the paper still has serious problems, and is not acceptable in current form.Most problematic is the selective presentation of results, for (one assumes) only seasons that show best results.No attempt has been made to account for multiple testing in evaluation of the "significant" results among many evaluations, even for the presented results.This is so even though the Wilks (2016) False Discovery Rate paper is cited in the introduction on page 3.This concern was also voiced by Reviewer 2 of the original submission.Consequently, meaningful forecast skill has been demonstrated.

Forecast quality is evaluated through examination and manipulation of the joint frequency distribution of forecasts and their corresponding observations. Full examination of this distribution allows diagnosis of various strengths and weaknesses in a set of forecasts. The various available scalar summary measures are convenient simplifications but inevitably discard some information.

Statistical forecasts of atmospheric behavior are often made using both linear and nonlinear regression methods. Regression methods are also used to post-process dynamical forecasts, in order to correct systematic biases. Ensemble forecasting is a Monte-Carlo approach to use of deterministic dynamical models, in which effects of chaos may be accounted for.

Principal component analysis is a linear data transformation defined in terms of the eigenvectors of their covariance matrix, which provides maximal data compression. Often useful insights into multivariate data arise from interpretation of these transformations. In meteorology and climatology this method is most often applied to time series of spatial fields, although it is also applicable in other settings.

A review of the portions of probability useful for understanding experimental design and analysis. The material in this section is intended as a review of the topic of probability as covered in the prerequisite course (36-201 at CMU). The material in gray boxes is beyond what you may have previously learned, but may help the more mathematically minded reader to get a deeper understanding of the topic. You need not memorize any formulas or even have a firm understanding of this material at the start of the class. But I do recommend that you at least skim through the material early in the semester. Later, you can use this chapter to review concepts that arise as the class progresses. For the earliest course material, you should have a basic idea of what a random variable and a probability distribution are, and how a probability distribution defines event probabilities. You also need to have an understanding of the concepts 3.1 Definition(s) of probability We could choose one of several technical definitions for probability, but for our purposes it refers to an assessment of the likelihood of the various possible outcomes in an experiment or some other situation with a " random " outcome. Note that in probability theory the term " outcome " is used in a more general 19

Quantitative evaluation of the flatness of the verification rank histogram can be approached through formal hypothesis testing. Traditionally, the familiar chi(2) test has been used for this purpose. Recently, two alternatives-the reliability index (RI) and an entropy statistic (omega)-have been suggested in the literature. This paper presents approximations to the sampling distributions of these latter two rank histogram flatness metrics, and compares the statistical power of tests based on the three statistics, in a controlled setting. The chi(2) test is generally most powerful (i.e., most sensitive to violations of the null hypothesis of rank uniformity), although for overdispersed ensembles and small sample sizes, the test based on the entropy statistic omega is more powerful. The RI-based test is preferred only for unbiased forecasts with small ensembles and very small sample sizes.

Discrimination is the process of finding rules to best sort groups of data vectors of known group membership based on their joint characteristics, which rules can be applied to classify future data vectors of unknown group membership. When the data vector is observed before the group membership will be known, the method can be used for forecasting. Probabilistic classifications can be computed by applying Bayesian ideas.

The MVN generalizes the familiar normal (“bell-curve”) distribution to vector data. It is mathematically tractable and possesses several other very useful characteristics. Good alternative multivariate parametric distributions are few, but often the MVN can be used with non-normal data for both inference and simulation after transformations. The multivariate generalization of the familiar t-test provides a framework for statistical inferences about vector means.

Parametric distributions are theoretical mathematical forms, which are often useful for compactly representing variations and uncertainty in data. Different classes of these distributions are appropriate for discrete and continuous data. These characterizations can provide the mathematical structure for inferences about the data-generating processes. Generation of random numbers from these distributions provides the basis for statistical simulation.

Multivariate statistics (pertaining to the joint behavior of multiple variables) requires the use of linear (“matrix”) algebra as a notational and computational tool. This Chapter contains a brief overview of the elements of linear algebra needed for the remainder of Part III of this book, and their use in representing random vectors and matrices.

The relative-frequency view of probability leads to statistical inferences using hypothesis tests and confidence intervals. Parametric tests target inference on distribution parameters, whereas nonparametric tests may relate to any sample statistic of interest. Special problems arise for correlated data, and for multiple simultaneous inferences (“field significance”).