Predicting Variability in a Lake Ontario Phosphorus Model

The prediction of a model always has a degree of uncertainty. Because the level of uncertainty is inversely related to the value of information contained in the prediction, there is a need to quantify the uncertainty. One approach to estimate prediction uncertainty is first-order error analysis. In this method, the error in a characteristic (variable or parameter) is defined by its first nonzero moment (the variance). Errors are propagated through the model using first-order terms in the Taylor series, and the variances are then combined to yield the total prediction uncertainty. An alternative approach to model prediction error analysis is Monte Carlo simulation. In this technique, probability density functions are assigned to each characteristic (variable or parameter), reflecting the uncertainty in that characteristic. Then, values are randomly selected from the distribution for each term and inserted into the model, to calculate a prediction. Repeating this process a number of times produces a distribution of predicted values, which reflects the combined uncertainties. These two approaches (first-order error analysis and Monte Carlo simulation) are applied to Lake Ontario data using a steady state mass balance phosphorus model. Comparisons are made which suggest guidelines for the use of each.     

Redundancy,values and public policy

Although the Industrial Relations Act marks the most significant break with the traditional “abstentionist” role of the law in British labour relations, it is true to say that during the last ten years the development of public policy has entailed a growing involvement of lawyers and the law in the world of work. As Professor Kahn-Freund has written:     

Zur Definition und Schätzung von Mittelwerten für zufällige Variable auf der Sphäre

Zusammenfassung Diese Arbeit ist die gekürzte Fassung einer Dissertation, die vom Autor 1968 unter gleichem Titel in Würzburg ver?ffentlicht wurde. Es ist nicht sinnovll, Mittelwerte für zuf?llige Variable auf der Sph?re in der üblichen Weise, wie z. B. für zuf?llige Variable auf der reellen Achse, zu definieren. W. Uhlmann [1964] bediente sich entscheidungstheoretischer Begriffe, um für zirkul?re zuf?llige Variable mittlere Winkel zu definieren, die ihre Richtung unabh?ngig von der Wahl der Null-Richtung beibehalten. Die analoge Invarianzeigenschaft wird für alle hier definierten Mittelwertbegriffe (mittlere Richtung, mittlerer Gro?kreis, Mittelachse, Mittelkreis) gesichert, indem einfache Forderungen an die zu verwendenden Verlustfunktionen gestellt werden. Da als Sch?tzungen für diese Mittelwerte stets die entsprechenden Mittelwerte der empirischen Verteilung auftreten, haben diese auch die gleiche Invarianzeigenschaft. Um die Diskrepanz zwischen einer solchen Sch?tzung und dem zu sch?tzenden Mittelwert zu messen, werden neue Verlustfunktionen eingeführt. Es wird gezeigt, da? alle eingeführten Sch?tzungen bezüglich mindestens einer Verlustfunktion unverf?lscht sind, d. h. der erwartete Verlust wird minimal, wenn wir aus allen in Frage kommenden Objekten gerade den betreffenden Mittelwert gesch?tzt werden lassen. Dieser minimale Verlust wird die Dispersion der Sch?tzung bezüglich dieser Verlustfunktion genannt. Es wird bewiesen, da? alle ermittelten Dispersionen mindestens wien ^−1/2 gegen Null gehen, wenn n gegen Unendlich strebt.

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Summary This paper is shortened from an equally entitled dissertation which has been published by the author in 1968 at Würzburg. For random variables on the sphere it would make no sense to define means in the usual way as it is done e. g. for random variables on the real line. Introducing concepts of decision theory,W. Uhlmann [1964] defined mean angles for circular random variables the direction of which does not depend on the choice of the zero direction. Setting up simple conditions for the loss functions to be used, we ensure that all the means defined in the paper (mean directions, mean great circles, mean axes, mean circles) have the analogous invariance property. The estimators of these means are always the corresponding means of the empirical distribution, defined with respect to the same loss function and therefore they have the invariance property too. To measure the discrepance between an estimator and the estimated mean, new loss functions are introduced. It is shown that all the established estimators are unbiased with respect to at least one loss function, i. e. the expected loss is a minimum, if we take just the mean from all the things in question to be estimated by the regarded estimator. This minimum loss is called the dispersion of the estimator with respect to this loss function. It is proved, that all the calculated dispersions go to zero at least asn ^−1/2, ifn tends to infinity.

     

Methods of measuring the distribution of socio-economic conditions

A metropolitan community was faced with the problem of locating areas of the community in terms of more or less need for leisure-time services. The authors developed a set of methods and techniques to solve this problem. It was noted that these methods differed in some ways from prior efforts of this sort. The advantages of the methods were that they could use readily available data, did not require the use of a computer, and presented a relative level of need that was more than a simple ranking of planning areas. Further, the methods could be used to locate any number of conditions in which one might be interested. In addition to spelling out the details of the original effort, an attempt is made to show how factor analysis could be used as an alternate approach to the first effort. The pros and cons of factor analysis as a substitute method are discussed. The major disadvantage would be if the particular community did not have a large computer available. This is outweighed by the higher levels of accuracy, predictability and comparability provided by factor analysis.