Wikipedia:Reference desk/Archives/Mathematics/2011 April 3

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April 3

Probability model for rain

How would you go about modelling the probability that if it's raining in suburb A, then it's also raining in suburb B? I imagine we'd need to take into account the displacement of B from A, the area of suburb B, the average size of a raincloud, the probability that any given portion of the cloud is producing rain, and so on. This isn't the science desk, so I'm not particularly interested in fine detailing (i.e. seasonal variations, the time of day); just a rough outline of the mathematics. Thanks. —Anonymous Dissident^Talk 03:02, 3 April 2011 (UTC)

From a purely mathematical standpoint, you probably won't get any closer than postulating that at any given time, there's a certain probability distribution among "no rain", "rain at A", "rain at B", and "rain both places". Instead of four separate probabilities (with the constrain that they must sum to 1), you could also express the parameters as "chance of rain at A" (whether or not it rains at B), "change of rain at B" (whether or not it rains at A), and their covariance.

If you want to know how to find the values of these parameters, pure mathematics cannot help you; for that you'll have to ask meteorologists instead. Unless your city sits on a flat featureless plain, it is likely that the raw distance between A and B is not of much use compared to the detailed topography in their vicinity. –Henning Makholm (talk) 05:57, 3 April 2011 (UTC)

Numerical weather prediction computer models use complex algorithms to predict meteorological conditions based largely on extrapolations of current conditions at another upstream location. However, that's different from modeling the probability of current conditions being a certain way in city A based on those at B. For this, there's probably no more accurate model than a progressively decreasing probability of similar precipitation conditions as distance increases (or the one mentioned by Hening Makholm). Obviously, while the atmosphere is a fluid, there are many more variables involved in the mechanics of surface weather than merely fluid dynamics. That's why even the most advanced and complex supercomputers are unable to predict the state of the fluid that is the atmosphere with much certainty past a relatively immediate timeframe. Juliancolton (talk) 21:21, 8 April 2011 (UTC)

Hessian determinant for 3 or higher-variate functions

Hi:

The entry Second partial derivative test only uses a function of 2 variables to illustrate how Hessian determinant can be used to determine maximums and minimums. I am wondering, for functions of 3 or more variables, what is the proper way of using Hessian determinants to determine the maximums and minimums?

Thanks,

70.31.156.62 (talk) 15:40, 3 April 2011 (UTC)

At a critical point, if the eigenvalues of the Hessian matrix are all negative, then it is a local max. If they are all positive, it's a local min. If they are of mixed (nonzero) sign, then it's a saddle point. If some of the eigenvalues are zero, the test is inconclusive. There is a test for each of these cases that involves computing the leading principal minors for the matrix (see positive-definite matrix). If the determinant of the Hessian is zero, the second derivative test is inconclusive. If the leading minors do not alternate in sign, then the critical point is a saddle point. Otherwise, the critical point is a local maximum or minimum according as the first entry of the matrix is negative or positive, respectively. Sławomir Biały (talk) 18:10, 3 April 2011 (UTC)