Re: AI-GEOSTATS: Standard deviation, Variance

[Digby Millikan]

Hello,  My inital enquiry about why variance is used as a basis for geostatistics appears to be the tip of the iceberg of many reasons and a large amount of complex mathematical theory. I have included a summary of further postings I have received concerning this matter and will later write a short summary of these reasons based on the emails received.   Regards Digby Millikan B.Eng Geolite Mining Systems U4/16 First Ave., Payneham South SA 5070 Australia. Ph: +61 8 84312974 [EMAIL PROTECTED] http://www.users.on.net/digbym //=======================================================  Digby Millikan wrote:   Hello,   I was wondering if someone can tell me about statistical parameters, why standard deviation and variance is used as opposed to mean absolute deviation from the mean. It rings a bell that intergral calculus has something to do with it e.g. related to formulea for a normal distribution .  M.David states the variogram uses the squared term as it makes calculations easier, as it would being related to statistical parameters such as variance, covariance similarly, A.Journel informed me, as Donald exaplained Kriging is Least Squared Error. Thanks in advance, Regards Digby Millikan //====================================================== This question comes up from time to time in statistics and it is likely  that the answer pertains to optimization.  The variance is a second  moment, i.e., it is related to a sum of squares. Problems pertaining to  sums of squares arise in a number of places  (e.g., moment of inertia,  PCA,  energy) but part of the reason for the emphasis on squares as  opposed to absolute values probably has to do with differentiation. The  absolute value function is not differentiable at zero whereas the sum of  squares is differentiable. Moreove when optimizing a sum of squares one  obtains a system of linear equations,  to optimize a function involving  the absolute value does not lead to a nice analytic solution. Note that  Newton used squares in his landmark study on errors.   The absolute value is not exactly a first moment but it certainly is not  a second moment. Consequently if one constructs an objective function  using absolute values as opposed to squares it will behave differently.   The absolute deviation probably more naturally relates to the median  (than to the mean).   In summary I don't think there is an absolute answer to your question  and you may get different answers/explanations from different people but  I think all will include some of the ideas above.   Donald E. Myers //======================================================= Virgil wrote;   Partly because way back in the days when calculators and computers were people, there were nicely developed shortcuts for calculating means and variances which were not available for medians and mean absolute deviations (MADs). Secondly, the theoretical analysis of Gaussian distributions was easier to develop in terms of means and variances than in terms of medians and MADs, and, originally, Gaussian were, by far, the most studied of the continuous distributions in the early days of statistics. Then Gossett developed the Student distributions, again strongly dependent on means and variances. //======================================================= The reason is simple and comprehensive.... Assume a population with ANY distribution of elements. Then randomly select a number of sample elements from the population to characterize the underlying population. That distribution of sample elements ALWAYS tends toward a normal [Gaussian] distribution. And the mean and standard deviation of the sample distribution are unbiased representations of the mean and standard deviation of the underlying population. WDA end //=======================================================