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Karl-Ivar Ivarsson
,
Rune Joelsson
,
Erik Liljas
, and
Allan H. Murphy

Abstract

This paper describes new operational and experimental forecasting programs at the Swedish Meteorological and Hydrological Institute (SMHI) designed to provide users with more detailed and more useful weather forecasts. User groups currently served by these programs include construction contractors, farmers, electric power companies, street and highway departments, and ski resorts. The programs represent a major component of a SMHI-wide effort to develop products to meet the needs of the public and private sectors in Sweden for meteorological and hydrological information.

An important feature of these programs is that many of the forecasts are expressed in probabilistic terms, and some results of the probability forecasting components of four programs are presented here. These subjective forecasts specify the likelihood of occurrence of various precipitation, wind speed, temperature, and cloud amount events, and they generally involve relatively short lead times and/or valid periods. The probabilistic forecasts of measurable precipitation are found to be reasonably reliable and definitely skillful. Some forecasts of larger precipitation amounts and the wind speed forecasts for shorter lead times also demonstrate positive skill, and the probabilistic temperature forecasts appear to be quite reliable. On the other hand, most of the experimental and operational probability forecasts reveal some degree of overforecasting, which tends to increase as lead time increases and as the climatological probability of the event decreases. As a result, the wind speed forecasts for longer lead times, some forecasts of precipitation amount, and the cloud amount forecasts exhibit negative skill.

Some factors that may have contributed to the deficiencies in the forecasters' performance are identified. The need to refine various components of the forecasting system is emphasized, and current efforts to implement such refinements at SMHI are outlined.

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Richard W. Katz
,
Allan H. Murphy
, and
Robert L. Winkler

Abstract

The methodology of decision analysis is used to investigate the economic value of frost (i.e., minimum temperature) forecasts to orchardists. First, the fruit-frost situation and previous studies of the value of minimum temperature forecasts in this context are described. Then, after a brief overview of decision analysis, a decision-making model for the fruit-frost problem is presented. The model involves identifying the relevant actions and events (or outcomes), specifying the effect of taking protective action, and describing the relationships among temperature, bud loss, and yield loss. A bivariate normal distribution is used to model the relationship between forecast and observed temperatures, thereby characterizing the quality of different types of information. Since the orchardist wants to minimize expenses (or maximize payoffs) over the entire frost-protection season and since current actions and outcomes at any point in the season are related to both previous and future actions and outcomes, the decision-making problem is inherently dynamic in nature. As a result, a class of dynamic models known as Markov decision processes is considered. A computational technique called dynamic programming is used in conjunction with these models to determine the optimal actions and to estimate the value of meteorological information.

Some results concerning the value of frost forecasts to orchardists in the Yakima Valley of central Washington are presented for the cases of red delicious apples, bartlett pears, and elberta peaches. Estimates of the parameter values in the Markov decision process are obtained from relevant physical and economic data. Twenty years of National Weather Service forecast and observed temperatures for the Yakima key station are used to estimate the quality of different types of information, including perfect forecasts, current forecasts, and climatological information. The orchardist's optimal actions over the frost-protection season and the expected expenses associated with the use of such information are determined using a dynamic programming algorithm. The value of meteorological information is defined as the difference between the expected expense for the information of interest and the expected expense for climatological information. Over the entire frost-protection season, the value estimates (in 1977 dollars) for current forecasts were $808 per acre for red delicious apples, $492 per acre for bartlett pears, and $270 per acre for elberta peaches. These amounts account for 66, 63, and 47%, respectively, of the economic value associated with decisions based on perfect forecasts. Varying the quality of the minimum temperature forecasts reveals that the relationship between the accuracy and value of such forecasts is nonlinear and that improvements in current forecasts would not be as significant in terms of economic value as were comparable improvements in the past.

Several possible extensions of this study of the value of frost forecasts to orchardists are briefly described. Finally, the application of the dynamic model formulated in this paper to other decision-making problems involving the use of meteorological information is mentioned.

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Barbara G. Brown
,
Richard W. Katz
, and
Allan H. Murphy

Abstract

A general approach for modeling wind speed and wind power is described. Because wind power is a function of wind speed, the methodology is based on the development of a model of wind speed. Values of wind power are estimated by applying the appropriate transformations to values of wind speed. The wind speed modeling approach takes into account several basic features of wind speed data, including autocorrelation, non-Gaussian distribution, and diurnal nonstationarity. The positive correlation between consecutive wind speed observations is taken into account by fitting an autoregressive process to wind speed data transformed to make their distribution approximately Gaussian and standardized to remove diurnal nonstationarity.

As an example, the modeling approach is applied to a small set of hourly wind speed data from the Pacific Northwest. Use of the methodology for simulating and forecasting wind speed and wind power is discussed and an illustration of each of these types of applications is presented. To take into account the uncertainty of wind speed and wind power forecasts, techniques are presented for expressing the forecasts either in terms of confidence intervals or in terms of probabilities.

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Barbara G. Brown
,
Richard W. Katz
, and
Allan H. Murphy

Abstract

The use of a concept called a precipitation “event” to obtain information regarding certain statistical properties of precipitation time series at a particular location and for a specific application (e.g., for modeling erosion) is described. Exploratory data analysis is used to examine several characteristics of more than 31 years of primitive precipitation events based on hourly precipitation data at Salem, Oregon. A primitive precipitation event is defined as one or more consecutive hours with at least 0.01 inches (0.25 mm) of precipitation. The characteristics of the events that are considered include the duration, magnitude, average intensity and maximum intensity of the event and the number of hours separating consecutive events.

By means of exploratory analysis of the characteristics of the precipitation events, it is demonstrated that the marginal (i.e., unconditional) distributions of the characteristics are positively skewed. Examination of the conditional distributions of some pairs of characteristics indicates the existence of some relationships among the characteristics. For example, it is found that average intensity and maximum intensity are quite dependent on the event duration. The existence and forms of these relationships indicate that the assumption commonly made in stochastic models of hourly precipitation time series that the intensities (i.e., hourly amounts within an event) are independent and identically distributed must be violated. Again using exploratory data analysis, it is shown that the hourly intensities at Salem are, in fact, stochastically increasing and positively associated within a precipitation event.

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Allan H. Murphy
and
Carl-Axel S. Staël von Holstein

Abstract

Some forecasters apparently subscribe to a model of the subjective probability forecasting process in which their judgments are expressed in terms of “second-order” probabilities. First, we briefly consider the nature of these second-order probabilities and describe the second-order model, and then we demonstrate that strictly proper scoring rules encourage forecasters who subscribe to the second-order model to make their forecasts correspond to their expected judgments.

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Cameron R. Peterson
,
Kurt J. Snapper
, and
Allan H. Murphy

An experiment was conducted in which forecasters expressed temperature forecasts in terms of intervals of variable width and fixed probability. The use of such intervals, called credible intervals, permits forecasters to describe the uncertainty inherent in their temperature forecasts in a meaningful, quantitative way. The results of the experiment indicate that forecasters can use credible intervals to quantify this information, information which may be important to potential users of these forecasts. Several recommendations are made regarding credible interval temperature forecasting on an operational basis.

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Thomas R. Stewart
,
Richard W. Katz
, and
Allan H. Murphy

This paper reports some results of a descriptive study of the value of weather information used by fruit growers in the Yakima Valley of central Washington to decide when to protect their orchards against freezing temperatures. Specifically, the study provides data concerning the decision-making procedures of individual orchardists, the growers' use of weather information including frost (i.e., minimum temperature) forecasts, and the dimensions of the value of such forecasts.

Results from the descriptive study regarding the orchardists' information-processing and decision-making procedures are compared with the procedures included in a previous prescriptive study of the fruit-frost problem in the same geographical area (Katz et al., 1982). The prescriptive study employed a dynamic decision-making model and yielded estimates of the economic value of frost forecasts under the assumption (inter alia) that the orchardists' decisions were based solely on these forecasts. On the other hand, the descriptive study with which the current paper is primarily concerned indicates that growers use temperature and dew point observations available after the frost forecast has been issued, as well as the frost forecasts themselves, to make frost protection decisions. Furthermore, while the results of the descriptive study show that the grower makes a series of decisions to protect or not to protect during the night, the model assumed that an irreversible commitment is made early in the night. The results of an initial effort to modify the original prescriptive model in accordance with the descriptive findings to obtain more realistic estimates of the value of frost forecasts also are reported in this paper.

Some implications of this study for the further development of prescriptive models of the decision-making process in the fruit-frost context and in other weather-information-sensitive contexts are discussed.

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Barbara G. Brown
,
Richard W. Katz
, and
Allan H. Murphy

The so-called fallowing/planting problem is an example of a decision-making situation that is potentially sensitive to meteorological information. In this problem, wheat farmers in the drier, western portions of the northern Great Plains must decide each spring whether to plant a crop or to let their land lie fallow. Information that could be used to make this decision includes the soil moisture at planting time and a forecast of growing-season precipitation. A dynamic decision-making model is employed to investigate the economic value of such forecasts in the fallowing/planting situation.

Current seasonal-precipitation forecasts issued by the National Weather Service are found to have minimal economic value in this decision-making problem. However, relatively modest improvements in the quality of the forecasts would lead to quite large increases in value, and perfect information would possess considerable value. In addition, forecast value is found to be sensitive to changes in crop price and precipitation climatology. In particular, the shape of the curve relating forecast value to forecast quality is quite dependent on the amount of growing-season precipitation.

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Allan H. Murphy
,
Yin-Sheng Chen
, and
Robert T. Clemen

Abstract

In this paper we investigate the interrelationships between objective and subjective temperature forecasts. An information-content approach is adopted within the overall context of a general framework for forecast verification. This approach can be used to address questions such as whether the subjective forecasts contain information regarding the corresponding observed temperatures that is not included in the objective forecasts. Two methods of analysis are employed: 1) ordinary least squares regression analysis and 2) a Bayesian information-content analysis.

Maximum and minimum temperature forecasts formulated operationally for six National Weather Service offices during the period 1980–86 are analyzed. Results produced by the two methods are quite consistent and can be summarized as follows: 1) the subjective forecasts contain information not included in the objective forecasts for all cases (i.e., stratifications) considered and 2) the objective forecasts contain information not included in the subjective forecasts in a substantial majority of these cases. Generally, the incremental information content in the subjective forecasts considerably exceeds the incremental information content in the objective forecasts. The implications of these results for operational short-range temperature forecasting are briefly discussed.

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Yin-Sheng Chen
,
Martin Ehrendorfer
, and
Allan H. Murphy

Abstract

This paper investigates the relationship between the quality and value of forecast in the context of a generalized N-action, N-event model of the cost-loss ratio situation. The forecasts of interest are imperfect categorical forecasts, calibrated according to past performance and represented by multidimensional sets of conditional and predictive probabilities. Forecasts quality is measured by the ranked probability score (RPS), a natural measure of the accuracy of forecasts in the context of this model. The measure of value is the difference between the expected expense associated with climatological information and the expected expense associated with imperfect forecasts. Thus, climatological and perfect information define lower and upper bounds, respectively, on the quality and value of the imperfect forecasts.

Quality-value relationships are explored in the three-action, three-event situation, using brute form and mathematical programming methods. Numerical results are presented for several specific cases. In all cases, the relationships are described by envelopes of values rather than by single-valued functions, indicating that a range of forecast value is generally associated with a given level of forecast quality (and vice versa). The existence of these envelopes reveals two important deficiencies in scalar (i.e., one-dimensional) measures of forecast quality, such as the RPS, when they an used as surrogates for measures of value: 1) these quality measures generally provide only imprecise estimates of forecast value and 2) increases in forecast quality, as reflected by such measures may actually be associated with decreases in forecast value.

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