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- Author or Editor: Irving I. Gringorten x

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## Abstract

Previous models for estimating the conditional probability of an event have used, as the condition, an initial categorized event such as *no rain* or *overcast* at time zero. But initial conditions frequently are observed and known in greater detail, and these observed values can replace the categories in determining conditional probabilities. A model that has as its underlying assumption the “Ornstein-Uhlenbeck” process is applicable to this problem. It uses the antecedent quantitatively without loss of information and with surprising simplicity.

## Abstract

Previous models for estimating the conditional probability of an event have used, as the condition, an initial categorized event such as *no rain* or *overcast* at time zero. But initial conditions frequently are observed and known in greater detail, and these observed values can replace the categories in determining conditional probabilities. A model that has as its underlying assumption the “Ornstein-Uhlenbeck” process is applicable to this problem. It uses the antecedent quantitatively without loss of information and with surprising simplicity.

## Abstract

Single-station climatic probability of a meteorological condition can be estimated from a set of observations taken at the station. The probability, however, of the same condition prevailing along a line of sight or line of travel or in an area or fraction thereof is not so readily obtained. The aproach to this problem has been to model the spatial variability and to determine the values of model parameters. In the absence of an analytical solution a simulation technique involving random number generation was used to obtain answers that are presented in graphical form. Two models have been developed, one effective with upper air temperature fields. The other, and more interesting model, is effective with mesoscale phenomena (such as rainfall) in areas ranging from a few hundred square kilometers up to 100 000 km^{2}.

## Abstract

Single-station climatic probability of a meteorological condition can be estimated from a set of observations taken at the station. The probability, however, of the same condition prevailing along a line of sight or line of travel or in an area or fraction thereof is not so readily obtained. The aproach to this problem has been to model the spatial variability and to determine the values of model parameters. In the absence of an analytical solution a simulation technique involving random number generation was used to obtain answers that are presented in graphical form. Two models have been developed, one effective with upper air temperature fields. The other, and more interesting model, is effective with mesoscale phenomena (such as rainfall) in areas ranging from a few hundred square kilometers up to 100 000 km^{2}.

## Abstract

After a review of the methods, advanced most recently, of estimating extreme values from data samples it is concluded that the straightforward method of moments offers the advantages of stability and simplicity without bias. The sample mean and standard deviation are computed, but the observations need not be ordered and the equation of estimate need not differ with sample size. A few charts are readily constructed, or used out of this text, to yield answers to the operational risks involved in a given period of time.

## Abstract

After a review of the methods, advanced most recently, of estimating extreme values from data samples it is concluded that the straightforward method of moments offers the advantages of stability and simplicity without bias. The sample mean and standard deviation are computed, but the observations need not be ordered and the equation of estimate need not differ with sample size. A few charts are readily constructed, or used out of this text, to yield answers to the operational risks involved in a given period of time.

## Abstract

The map is proposed for worldwide climatological statistics, to depict accurately the area covered by some specified meteorological condition or element. Since it is square, a grid overlay divides the map into small squares, each covering exactly the same amount of global area. The map is centered on the north pole where it is conformal. The parallels of latitude in each of the four quadrants of the square are represented by elliptical arcs that change from circular shape at either pole to a straight line at the equator.

Except for Antarctica no continent is split or divided in this projection. The Northern Hemisphere is presented without interruptions or discontinuities of direction. The map's four quadrants can be resembled to place the south pole and the whole of Antarctica at the center of the representation, as an interim step in the drawing of isopleths in the Southern Hemisphere.

## Abstract

The map is proposed for worldwide climatological statistics, to depict accurately the area covered by some specified meteorological condition or element. Since it is square, a grid overlay divides the map into small squares, each covering exactly the same amount of global area. The map is centered on the north pole where it is conformal. The parallels of latitude in each of the four quadrants of the square are represented by elliptical arcs that change from circular shape at either pole to a straight line at the equator.

Except for Antarctica no continent is split or divided in this projection. The Northern Hemisphere is presented without interruptions or discontinuities of direction. The map's four quadrants can be resembled to place the south pole and the whole of Antarctica at the center of the representation, as an interim step in the drawing of isopleths in the Southern Hemisphere.

## Abstract

Many studies of the joint frequency of the initial and final conditions of weather elements such as cloud cover, visibility, rainfall or temperature attest to the importance of the initial event as a predictor of the later event. Most efforts have involved the actual collection of the data in contingency tables but there is a strong need for an analytical tool to estimate the conditional probabilities from more readily available climatic frequencies.

By assuming the Markov process, and with the help of published tables detailing the bivariate normal distribution, a succinct two-parameter model, using the climatic frequencies in a single equation, has been developed to estimate conditional probabilities, of both frequent and rare events, within a few percentage points. The two parameters have been charted as direct functions of the probability of the initial event and the temporal persistence of the element.

## Abstract

Many studies of the joint frequency of the initial and final conditions of weather elements such as cloud cover, visibility, rainfall or temperature attest to the importance of the initial event as a predictor of the later event. Most efforts have involved the actual collection of the data in contingency tables but there is a strong need for an analytical tool to estimate the conditional probabilities from more readily available climatic frequencies.

By assuming the Markov process, and with the help of published tables detailing the bivariate normal distribution, a succinct two-parameter model, using the climatic frequencies in a single equation, has been developed to estimate conditional probabilities, of both frequent and rare events, within a few percentage points. The two parameters have been charted as direct functions of the probability of the initial event and the temporal persistence of the element.

## Abstract

To test the skill of a forecaster the rule for the score *S*, for the quantitative forecast of Temperature or a similar variable, becomes *S* = −ln(1 − *P*
_{1}
*P*
_{2} − 1 where *P*
_{1} is the cumulative climatic frequency of the forecast value *T*
_{
F
}, or the cumulative climatic frequency of the subsequently verified value *T*
_{
v
}, whichever is smaller. The value *P*
_{2} is the greater of these two frequencies. Such frequencies must be made conditional to the initial state of the weather in order to properly reward forecasters for recognizing future changes in the weather. For the quantitative forecast of precipitation, or similar variables, there are several alternate formulas for skill scores, each formula depending upon whether or not any precipitation is forecast or observed, or both forecast and observed.

This system of scoring assures that unskilled strategies, such as the forecasting of the mart frequent values, or persistence forecasting, will net the forecaster an expected average of zero. For individual accurate forecasts the rewards are greatest but still depend on the frequency or infrequency of the verified events. For inaccurate forecasts the rewards can be positive or negative, depending upon the sign and amount of change that is predicted, as well as the subsequent verification.

## Abstract

To test the skill of a forecaster the rule for the score *S*, for the quantitative forecast of Temperature or a similar variable, becomes *S* = −ln(1 − *P*
_{1}
*P*
_{2} − 1 where *P*
_{1} is the cumulative climatic frequency of the forecast value *T*
_{
F
}, or the cumulative climatic frequency of the subsequently verified value *T*
_{
v
}, whichever is smaller. The value *P*
_{2} is the greater of these two frequencies. Such frequencies must be made conditional to the initial state of the weather in order to properly reward forecasters for recognizing future changes in the weather. For the quantitative forecast of precipitation, or similar variables, there are several alternate formulas for skill scores, each formula depending upon whether or not any precipitation is forecast or observed, or both forecast and observed.

This system of scoring assures that unskilled strategies, such as the forecasting of the mart frequent values, or persistence forecasting, will net the forecaster an expected average of zero. For individual accurate forecasts the rewards are greatest but still depend on the frequency or infrequency of the verified events. For inaccurate forecasts the rewards can be positive or negative, depending upon the sign and amount of change that is predicted, as well as the subsequent verification.

## Abstract

If an operation is influenced by the weather, each course of action should result in a profit, cost, or loss depending upon the subsequent development or state of the weather. To delve into the problem of relating the weather forecast to its operational usefulness, this paper defines an *income matrix*, which is essentially a table of numerical values of the utility of each course of action followed by each state of the weather. The probabilities of the several states of the weather, arranged in a single-columned matrix, are multiplied with the figures in the income matrix to give the expected gain or loss from each course of action. A decision, then, is generally the choice of that course of action whose expected mean operational value is a maximum. If one or more operations are not easily analyzed, the suggestion is advanced that the decision-making process is influenced by the increase of the probability of an event above its mean or climatic frequency.

Probability figures, however, are only estimates. It is desirable, therefore, to devise a forecasting scheme so as to minimize the effects of errors in these estimates. A least-squares method is applied to yield optimum probability estimates which are shown to be affected by the operation.

## Abstract

If an operation is influenced by the weather, each course of action should result in a profit, cost, or loss depending upon the subsequent development or state of the weather. To delve into the problem of relating the weather forecast to its operational usefulness, this paper defines an *income matrix*, which is essentially a table of numerical values of the utility of each course of action followed by each state of the weather. The probabilities of the several states of the weather, arranged in a single-columned matrix, are multiplied with the figures in the income matrix to give the expected gain or loss from each course of action. A decision, then, is generally the choice of that course of action whose expected mean operational value is a maximum. If one or more operations are not easily analyzed, the suggestion is advanced that the decision-making process is influenced by the increase of the probability of an event above its mean or climatic frequency.

Probability figures, however, are only estimates. It is desirable, therefore, to devise a forecasting scheme so as to minimize the effects of errors in these estimates. A least-squares method is applied to yield optimum probability estimates which are shown to be affected by the operation.

## Abstract

General expressions for the expected score for accuracy, score for skill and for operational value of the forecasts are developed and discussed. The expressions are then applied to the special case of two-class predictors and predictand, and an example is given to illustate how one set of probability forecasts can meet the operational requirements better than another set, even though both sets of forecasts are equally accurate and equally skillful.

## Abstract

General expressions for the expected score for accuracy, score for skill and for operational value of the forecasts are developed and discussed. The expressions are then applied to the special case of two-class predictors and predictand, and an example is given to illustate how one set of probability forecasts can meet the operational requirements better than another set, even though both sets of forecasts are equally accurate and equally skillful.

## Abstract

The article is limited to consideration of a forecasting program in which a forecaster chooses as his forecast one event out of a finite group of well-defined, mutually exclusive events, for example, “rain” or “no rain.” If a table of scores is prepared, from which a forecaster's score can be obtained for each of his forecasts, the sum total of a set of scores will become normally distributed as the number of forecasts is made large. It is possible to set confidence limits on the score total or on the difference of two score totals. But it is necessary to assume that the forecasts by which a forecaster's performance is judged are independent of each other and that the days constitute a fair sample of the climatology of his station.

## Abstract

The article is limited to consideration of a forecasting program in which a forecaster chooses as his forecast one event out of a finite group of well-defined, mutually exclusive events, for example, “rain” or “no rain.” If a table of scores is prepared, from which a forecaster's score can be obtained for each of his forecasts, the sum total of a set of scores will become normally distributed as the number of forecasts is made large. It is possible to set confidence limits on the score total or on the difference of two score totals. But it is necessary to assume that the forecasts by which a forecaster's performance is judged are independent of each other and that the days constitute a fair sample of the climatology of his station.