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Forecast–Decision Theory

1.1 Decision Problem

1.1.1 Decision

Nothing is more difficult, and therefore more valuable and admirable, than to be able to decide.

(Napoléon Bonaparte)

What makes deciding difficult? Whereas a precise answer depends on the problem at hand, major sources of the decisional difficulty, and the accompanying decisional stress, may be grouped under four headings. (i) Complexity of the situation in the context of which one must decide. (ii) Multiplicity of objectives one wants to achieve. (iii) Multiplicity of alternative courses of action one can pursue. (iv) Uncertainty about the outcome, when it depends not only upon one’s decision, but also upon inputs beyond one’s control.

Since its origin in the eighteenth century, decision theory has developed a coherent logical-mathematical framework and effective analytical tools for dealing with all major sources of the decisional difficulty. This book focuses on mathematical models for solving basic decision problems in which uncertainty constitutes the major difficulty.

1.1.2 Uncertainty

Uncertainty is ubiquitous. Its existence is increasingly recognized. And the advantages of taking it into account in decision making, rather than ignoring it and relying on deterministic models (mental or mathematical), are progressively winning the argument among professionals of many disciplines. Here is a handful of examples.

Example 1.1 (Accreditation Board for Engineering and Technology (ABET))

In the 1990s, the ABET, which every 6 years reviews and accredits each undergraduate engineering degree program in the USA, began to require at least one probability and statistics course in each engineering curriculum, regardless of the discipline (e.g., computer, electrical, civil, mechanical). The premise was that every contemporary engineer must acquire at least a rudimentary appreciation of uncertainty and its quantification.

Example 1.2 (American Meteorological Society (AMS))

In 2002, the AMS issued a statement that endorsed making and disseminating probabilistic forecasts of weather elements (e.g., precipitation amount, temperature), in lieu of, or in addition to, traditional deterministic forecasts, which give only a single number (a so-called best estimate). The statement argued that forecasts in probabilistic terms “would allow the user to make decisions based on quantified uncertainty with resulting economic and social benefits”.

Example 1.3 (Secretary of the US Department of the Treasury)

In his 1999 commencement address at the University of Pennsylvania, Robert E. Rubin, former Secretary of the Treasury, recalled from his early career on Wall Street an incident in which he lost a lot of money on a stock. But another security trader, who had believed with “absolute certainty” that particular events would occur and had purchased a large volume of the same stock, “lost an amount way beyond reason — and his job”, when his belief turned out to be wrong. Rubin’s advice to the young graduates: “Reject absolute answers and recognize uncertainty … then all decisions become matters of judging the probability of different outcomes, and the costs and benefits of each.”

1.2 Forecast–Decision System

1.2.1 Structure

To conquer the difficulty of making a rational decision under uncertainty, decision theory offers a way of structuring the problem as follows. The two major activities, quantifying uncertainty and making decisions, can be conceptualized as being performed by a forecast–decision system (F–D system) — a cascade coupling of two components (Figure 1.1): the forecast system (in short, the forecaster), and the decision system (in short, the decider). The coupling can be analyzed in each phase of the system’s life: the design, the operation, the evaluation.

1.2.2 Design: Requirements

The design phase involves (i) specification of requirements for each system component, and (ii) formulation of models and procedures for the forecaster and the decider.

Requirements for decision system

The design begins with the decision system, which should meet the requirements of a client who wants to make rational decisions. To identify the requirements, four basic questions should be asked: (i) What is the purpose of the system? (ii) What is the decision to be made? (iii) What is the outcome of concern to the decider? (iv) What is the future input which is beyond the control of the decider and which, together with the decision, determines the outcome? When this input is uncertain at the decision time, it constitutes a random variable (in short, a variate), and it must be forecasted.

Requirements for forecast system

The design of the forecast system should meet the requirements of the decision system (Figure 1.1) with respect to (i) the predictand — the variate to be forecasted; (ii) the lead time of the forecast — the time that elapses from the moment the forecast is made to the moment the input occurs or can be observed; (iii) the forecast frequency — when the decision is to be made repeatedly (e.g., every day, week, month). Additional requirements may be specified, for instance, regarding information which should be used to prepare the forecast.

Inasmuch as many books are devoted to the general problem of system design, this book does not address any further the first phase of the design process. Instead, system requirements are either specified, or implied through assumptions or problem descriptions, and the text concentrates on the second phase, which is modeling.

1.2.3 Design: Models

Given system requirements, models and procedures must be developed. This is accomplished in two steps. (i) A verbal description of the forecast problem or the decision problem is transformed into a mathematical formulation, which consists of symbols defining variables, sets, and functions (basically the notation which is given meaning in the context of the problem at hand). A well-defined mathematical formulation has two advantages. First, it provides a structure which bestows the clarity and precision on the modeler’s thought process. Second, it prescribes a decomposition which facilitates the next step. (ii) Detailed modeling of functions, estimation of parameters, and writing of procedures are undertaken, separately for each system component.

Figure 1.1 Forecast–decision system — a cascade coupling of the forecast system with the decision system in three phases of the system’s life: (i) design (coupling through system requirements), (ii) operation (coupling through input–output), and (iii) evaluation (coupling through forecast verification). The topics covered in the book are: judgmental forecasting (Chapters 4, 9), statistical forecasting (Chapters 5, 10), verification of forecasts (Chapters 6, 11), decision making (Chapters 7, 8, 12, 13, 14, 15).

Forecaster

The major task is to model (to quantify) uncertainty about the predictand in terms of a distribution function. This probabilistic forecaster may be (i) a human expert preparing forecasts judgmentally based on the available information (quantitative and qualitative), or (ii) a statistical model calculating a forecast based on the realization of a quantitative predictor (a variate which is stochastically dependent on the predictand).

Decider

There are three major tasks. (i) To model (to assess) the preferences of a rational decider over possible outcomes in terms of a criterion function (a utility function, a profit function, an opportunity loss function). (ii) To model the outcome function that maps the decision and the input into an outcome. (iii) To formulate a decision procedure that integrates the probabilistic forecast, the criterion function, and the output function; and to find an optimal decision — one that optimizes (maximizes or minimizes) the expected value of the criterion function.

1.2.4 Operation

The forecaster prepares a probabilistic forecast of the predictand. Next, the decider makes the optimal decision under uncertainty as quantified by the forecast (Figure 1.1). The forecast and the realization of the predictand (once it becomes known) should be archived for the purpose of verification.

1.2.5 Evaluation

Periodically, a set of forecasts should be verified against actual realizations of the predictand. The purpose is to evaluate, and to track over time, the performance of the forecaster with respect to the needs of the decider (Figure 1.1). The Bayesian verification measures quantify two attributes of probabilistic forecasts (calibration and informativeness) and provide (i) feedback to the forecaster, and (ii) statistics to the decider regarding the calibration, the informativeness, and the economic value of forecasts.

1.2.6 Coupling

The above overview of the design, operation, and evaluation phases of the F–D system reveals the intrinsic nature of the...