AI-GEOSTATS: Standard deviation, Variance
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 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 -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
On 29/11/02 5:32 AM, "Digby Millikan" <[EMAIL PROTECTED]> 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 There are in fact situations where you might want to use the mean absolute deviation from the mean (madogram) since any squared term would amplify the effect of outliers or other extreme values on the spatial correlation measure. For example, a range might be better defined from an inspection of the madogram instead of the variogram. -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
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.EngGeolite Mining SystemsU4/16 First Ave.,Payneham South SA 5070Australia.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 absolutedeviation from the mean. It rings a bell that intergral calculus hassomethingto do with it e.g. related to formulea for a normal distribution . M.David states the variogram uses the squared term as it makes calculationseasier, as it would being related to statistical parameters such asvariance,covariance similarly, A.Journel informed me, as Donald exaplained Krigingis 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 comprehensiveAssume a population with ANY distribution of elements. Then randomly selecta number of sample elements from the population to characterize theunderlying population. That distribution of sample elements ALWAYS tendstoward a normal [Gaussian] distribution. And the mean and standard deviationof the sample distribution are unbiased representations of the mean andstandard deviation of the underlying population.WDAend //===
Re: AI-GEOSTATS: Standard deviation, Variance
> 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. Things have obviously changed since I was a lad. I was taught that the Central Limit Theorem was a theorem NOT a law. There are distributions which do not conform to this behaviour and (alas for us) the lognormal is one of them. The Central Limit theorem also does not apply to mixed distributions or in cases of non-stationarity. Mind you, neither does geostatistics Isobel Clark http://geoecosse.bizland.com/news.html __ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
Isobel, > NOT a law. There are distributions which do not > conform to this behaviour and (alas for us) the > lognormal is one of them. > Is this the reason for transforming the data (only upto page 14). At the moment I am thinking kriging minimizes the variance of the sampling distribution as I am also reading a book on classical statistics. Is this distribution common in elements other than gold and uranium. > > The Central Limit theorem also does not apply to mixed > distributions or in cases of non-stationarity. Mind > you, neither does geostatistics > John Sturgul was my lecturer in mine evaluation, I think he mentioned your 1979 book in that course, but I did use it as a reference for a project I did on geostatistics. Thanks again, 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 -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
I find this fascinating. Apparently what I said is almost entirely wrong. What I said was 'I was taught that...' I do not recollect Don Myers being in my classrooms as an undergraduate (or during my MSC for that matter). You know, I welcome criticism, especially when I get things wrong. I have a big problem with people who do not actually read what I write but react at some visceral level to what they think I said. Also, I must be really stupid, because the comments given by Don include the statement " If any of the conditions in the theorem are not satisfied then the theorem may not apply. " Which, I am fairly sure, is what I was trying to say. Isobel Clark http://uk.geocities.com/drisobelclark/resume --- "Donald E. Myers" <[EMAIL PROTECTED]> wrote: > Regrettably the following statement by I. Clark is > almost entirely wrong > See below for a correct statement of the CLT, the > problem in part is > simply carelessness in terminology and replacing > correct > statements/formulations by sort of heuristic ones > (which are not correct) > Donald E. Myers > http://www.u.arizona.edu/~donaldm > *** > Isobel Clark wrote: > > >>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. > >> > > > > > *** > > CLT > Let X_1,, X_n be a sequence of independent, > identically distributed > random variables with common mean m and common > standard deviation > sigma. Let Z_n be defined as a normalized sum > > Z_n = [S_n - m]/ (sigma/sqt root of n), > S_n = [Z_1 > +.+ X_n]/n > > S_n is the sample mean > > Let F_n(z) be the cumulative probability > distribution function for Z_n > and let G(z) be the cumulative probability > distribution function for the > standard Normal,. Then F_n(z) --> G(z) as n > increases. > > Note two things about this statement, (1) the > theorem does not say how > "fast" the cdf for Z_n approaches the standard > Normal, (2) the speed of > convergence depends on z. Also the speed of > convergence depends on the > distribution type of the X_i's > > If any of the conditions in the theorem are not > satisfied then the > theorem may not apply. The convergence in this > theorem is what is called > "convergence in distribution", this is one of the > weakest forms of > convergence for a sequence of random variables. > There are theorems that > will give estimates or bounds on the speed of > convergence. There are > also special cases of this theorem that are somewhat > simpler such as the > the Normal approximation to the Binomial > > The simplest proof of the theorem above uses > characteristic functions > (Fourier Transforms of the densities). > __ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
Apologise, The original email, > > >>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. was not written by Isobel, it came from W.D. Allen on sci.stat.math which I posted in the summary of my replies. Thankyou both for your help in this matter, I am currently reading Practical Geostatistics 2000 and have ordered the statistics books as recommended by Donald. 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 -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
Isobel did not write the careless paragraph about the central limit theorem (CLT) Don replied to, as pointed out by Digby. I wish to add something to what Don said about the conditions under which the CLT applies, and that people usually miss in considering the universality of the CLT. See below. >> Let X_1,, X_n be a sequence of independent, >> identically distributed >> random variables with common mean m and common >> standard deviation >> sigma. Let Z_n be defined as a normalized sum >> >> Z_n = [S_n - m]/ (sigma/sqt root of n), >> S_n = [Z_1 >> +.+ X_n]/n >> >> S_n is the sample mean >> >> Let F_n(z) be the cumulative probability >> distribution function for Z_n >> and let G(z) be the cumulative probability >> distribution function for the >> standard Normal,. Then F_n(z) --> G(z) as n >> increases. >> >> Note two things about this statement, (1) the >> theorem does not say how >> "fast" the cdf for Z_n approaches the standard >> Normal, (2) the speed of >> convergence depends on z. Also the speed of >> convergence depends on the >> distribution type of the X_i's Note also the sum operation. The CLT, more precisely called the Additive CLT, applies to sums of pairwise independent random variables as n tends to infinity. But if the operation is multiplication with equal-signed r.v., then convergence in distribution is towards the lognormal, not the normal. It might well be that when considering natural phenomena, multiplicative processes be more or equally common than additive ones, as we oftenly observed skewed continuous data. Rubén http://webmail.udec.cl -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: Standard deviation, Variance
Thanks to Rubén and Digby for pointing out what I had misunderstood about Don Myers' email. It had not occurred to me (duh) that the lines starting '>' would be read as being from me rather than part of a forwarded email. Another score on the dumb side. Apologies for the strong reaction to Don's email if (on this occasion) he was not criticising my contribution. Isobel http://uk.geocities.com/drisobelclark __ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Fwd: Re: AI-GEOSTATS: Standard deviation, Variance
Thank you Isobel --- "Donald E. Myers" <[EMAIL PROTECTED]> wrote: > Date: Fri, 06 Dec 2002 12:05:54 -0700 > From: "Donald E. Myers" <[EMAIL PROTECTED]> > To: Isobel Clark <[EMAIL PROTECTED]> > Subject: Re: AI-GEOSTATS: Standard deviation, > Variance > > I stand corrected on mistakingly attributing the > "description" of the > CLT to you > > Donald Myers > > Isobel Clark wrote: > > >Thanks to Rubén and Digby for pointing out what I > had > >misunderstood about Don Myers' email. > > > >It had not occurred to me (duh) that the lines > >starting '>' would be read as being from me rather > >than part of a forwarded email. > > > >Another score on the dumb side. Apologies for the > >strong reaction to Don's email if (on this > occasion) > >he was not criticising my contribution. > > > >Isobel > > > >http://uk.geocities.com/drisobelclark > > > > > >__ > >Do You Yahoo!? > >Everything you'll ever need on one web page > >from News and Sport to Email and Music Charts > >http://uk.my.yahoo.com > > > >-- > >* To post a message to the list, send it to > [EMAIL PROTECTED] > >* As a general service to the users, please > remember to post a summary of any useful responses > to your questions. > >* To unsubscribe, send an email to > [EMAIL PROTECTED] with no subject and "unsubscribe > ai-geostats" followed by "end" on the next line in > the message body. DO NOT SEND Subscribe/Unsubscribe > requests to the list > >* Support to the list is provided at > http://www.ai-geostats.org > > > > > > __ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
