A Non-Symmetric Logit Model and Grouped Predictand Category Development

Logistic regression is an alternative to REEP (Regression Estimation of Event Probabilities) and other techniques for estimating weather event probabilities based on NWP output or other predictors. Logistic regression has the advantage over REEP in that the probability estimates are constrained between the values zero and unity, whereas REEP can “overshoot” these values. While several predictors can be used in the logit solution, many examples in the literature have dealt with only one or two predictors. It may be a detriment in some applications that the curves developed, one for each of possibly several predictand categories (events), are symmetric about the predicted 50% value. This paper will show how the logit curve can easily be made non-symmetric as a function of a predictor, and thereby possibly allow a better fit to the data.

As with REEP, the probabilities estimated by logistic regression for each of several categories of a variable may not be consistent. For instance, the probability of snow > 2 inches may exceed the probability of snow > 1 inch. This can be avoided, as proposed by Wilks (2009), by developing a single equation involving all predictand categories and including another predictor which is a function of the predictand. This predictor imposes a metric which is essentially a “separation constant” for each category. After development, the separation constants can be rolled into the overall equation constant to produce an equation for each category. This effectively, for a single predictor, produces parallel curves separated along the predictor axis, but imposes restrictions on the equations and probabilities produced from them. If a suitable functional form of the separation constants can be found, then this procedure has the advantage of being able to produce from the single equation a probability for any of the essentially infinite number of predictand events that can be defined from a continuous predictand, such as precipitation amount (Wilks 2009). This is a desirable feature. However, the relationship between the predictor(s) and the predictand must be considered in determining the separation constants or functional form. This paper will discuss the implications of this metric, and offer an alternative.

Reference:

Wilks, D. S., 2009: Extending logistic regression to provide full-probability-distribution MOS forecasts. Meteorological Applications, 16, 361-368.