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- Author or Editor: Richard W. Katz x
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Abstract
A probabilistic model for the sequence of daily amounts of precipitation is proposed. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation. Methods are given for computing the distribution of the maximum amount of daily precipitation and the distribution of the total amount of precipitation. The application of this model is illustrated by an example, using State College, Pennsylvania, precipitation data.
Abstract
A probabilistic model for the sequence of daily amounts of precipitation is proposed. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation. Methods are given for computing the distribution of the maximum amount of daily precipitation and the distribution of the total amount of precipitation. The application of this model is illustrated by an example, using State College, Pennsylvania, precipitation data.
Abstract
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A compound Poisson process is proposed as a stochastic model for the total economic damage associated with hurricanes. This model consists of two components, one governing the occurrence of events and another specifying the damages associated with individual events. In this way, damage totals are represented as a “random sum,” with variations in total damage being decomposed into two sources, one attributable to variations in the frequency of events and another to variations in the damage from individual events. The model is applied to the economic damage, adjusted for societal vulnerability, caused by North Atlantic hurricanes making landfall in the continental United States. The total number of damaging storms per year is fitted reasonably well by a Poisson distribution, and the monetary damage for individual storms is fitted by the lognormal. The fraction of the variation in annual damage totals associated with fluctuations in the number of storms, although smaller than the corresponding fraction for individual storm damage, is nonnegligible. No evidence is present for a trend in the rate parameter of the Poisson process for the occurrence of storms, and only weak evidence for a trend in the mean of the log-transformed damage from individual storms is present. Stronger evidence exists for dependence of these parameters, both occurrence and storm damage, on the state of El Niño.
Abstract
A compound Poisson process is proposed as a stochastic model for the total economic damage associated with hurricanes. This model consists of two components, one governing the occurrence of events and another specifying the damages associated with individual events. In this way, damage totals are represented as a “random sum,” with variations in total damage being decomposed into two sources, one attributable to variations in the frequency of events and another to variations in the damage from individual events. The model is applied to the economic damage, adjusted for societal vulnerability, caused by North Atlantic hurricanes making landfall in the continental United States. The total number of damaging storms per year is fitted reasonably well by a Poisson distribution, and the monetary damage for individual storms is fitted by the lognormal. The fraction of the variation in annual damage totals associated with fluctuations in the number of storms, although smaller than the corresponding fraction for individual storm damage, is nonnegligible. No evidence is present for a trend in the rate parameter of the Poisson process for the occurrence of storms, and only weak evidence for a trend in the mean of the log-transformed damage from individual storms is present. Stronger evidence exists for dependence of these parameters, both occurrence and storm damage, on the state of El Niño.
Abstract
The Richardson model is a popular technique for stochastic simulation of daily weather variables, including precipitation amount, maximum and minimum temperature, and solar radiation. This model is extended to include two additional variables, daily mean wind speed and dewpoint, because these variables (or related quantities such as relative humidity) are required as inputs for certain ecological/vegetation response and agricultural management models. To allow for the positively skewed distribution of wind speed, a power transformation is applied. Solar radiation also is transformed to make the shape of its modeled distribution more realistic. A model identification criterion is used as an aid in determining whether the distributions of these two variables depend on precipitation occurrence. The approach can be viewed as an integration of what is known about the statistical properties of individual weather variables into a single multivariate model.
As an application, this extended model is fitted to weather data in the Pacific Northwest. To aid in understanding how such a stochastic weather generator works, considerable attention is devoted to its statistical properties. In particular, marginal and conditional distributions of wind speed and solar radiation are examined, with the model being capable of representing relationships between variables in which the variance is not constant, as well as certain forms of nonlinearity.
Abstract
The Richardson model is a popular technique for stochastic simulation of daily weather variables, including precipitation amount, maximum and minimum temperature, and solar radiation. This model is extended to include two additional variables, daily mean wind speed and dewpoint, because these variables (or related quantities such as relative humidity) are required as inputs for certain ecological/vegetation response and agricultural management models. To allow for the positively skewed distribution of wind speed, a power transformation is applied. Solar radiation also is transformed to make the shape of its modeled distribution more realistic. A model identification criterion is used as an aid in determining whether the distributions of these two variables depend on precipitation occurrence. The approach can be viewed as an integration of what is known about the statistical properties of individual weather variables into a single multivariate model.
As an application, this extended model is fitted to weather data in the Pacific Northwest. To aid in understanding how such a stochastic weather generator works, considerable attention is devoted to its statistical properties. In particular, marginal and conditional distributions of wind speed and solar radiation are examined, with the model being capable of representing relationships between variables in which the variance is not constant, as well as certain forms of nonlinearity.
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.
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.
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.
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.
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.
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.
Abstract
Most climate impact studies rely on changes in means of meteorological variables, such as temperature, to estimate potential climate impacts, including effects on agricultural production. However, extreme meteorological events, say, a short period of abnormally high temperatures, can have a significant harmful effect on crop growth and final yield. The characteristics of daily temperature time series, specifically mean, variance and autocorrelation, are analyzed to determine possible ranges of probabilities of certain extreme temperature events [e.g., runs of consecutive daily maximum temperatures of at least 95°F (35°C)] with changes in mean temperature of the time series. The extreme temperature events considered are motivated primarily by agricultural concerns, particularly, the effects of high temperatures on corn yields in the U.S. Corn Belt. However, runs of high temperatures can also affect, for example, energy demand or morbidity and mortality of animals and humans.
The relationships between changes in mean temperature and the corresponding changes in the probabilities of these extreme temperature events are quite nonlinear, with relatively small changes in mean temperature sometimes resulting in relatively large changes in event probabilities. In particular, the likelihood of occurrence of a run of five consecutive daily maximum temperatures of at least 95°F under a 3°F (1.7°C) increase in the mean (holding the variance and autocorrelation constant) is about three times greater than that under the current climate at Des Moines, Moreover, by allowing either the variance or the autocorrelation as well as the mean to change, this likelihood of a run event varies over a relatively wide range of values. These changes in the probabilities of extreme events need to be taken into consideration in order to obtain realistic estimates of the impact of climate changes such as increases in mean temperature that may arise from increases in atmospheric carbon dioxide concentration.
Abstract
Most climate impact studies rely on changes in means of meteorological variables, such as temperature, to estimate potential climate impacts, including effects on agricultural production. However, extreme meteorological events, say, a short period of abnormally high temperatures, can have a significant harmful effect on crop growth and final yield. The characteristics of daily temperature time series, specifically mean, variance and autocorrelation, are analyzed to determine possible ranges of probabilities of certain extreme temperature events [e.g., runs of consecutive daily maximum temperatures of at least 95°F (35°C)] with changes in mean temperature of the time series. The extreme temperature events considered are motivated primarily by agricultural concerns, particularly, the effects of high temperatures on corn yields in the U.S. Corn Belt. However, runs of high temperatures can also affect, for example, energy demand or morbidity and mortality of animals and humans.
The relationships between changes in mean temperature and the corresponding changes in the probabilities of these extreme temperature events are quite nonlinear, with relatively small changes in mean temperature sometimes resulting in relatively large changes in event probabilities. In particular, the likelihood of occurrence of a run of five consecutive daily maximum temperatures of at least 95°F under a 3°F (1.7°C) increase in the mean (holding the variance and autocorrelation constant) is about three times greater than that under the current climate at Des Moines, Moreover, by allowing either the variance or the autocorrelation as well as the mean to change, this likelihood of a run event varies over a relatively wide range of values. These changes in the probabilities of extreme events need to be taken into consideration in order to obtain realistic estimates of the impact of climate changes such as increases in mean temperature that may arise from increases in atmospheric carbon dioxide concentration.
