By Petr Mandl
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Extra resources for Analytical treatment of one-dimensional Markov processes
The region sampled, R, is divided by perpendicular bisectors between the N sampling points into polygons or tiles, Vi , i ¼ 1; 2; . . ; N, such that in each polygon all points are nearer to its enclosed sampling point xi than to any other sampling point. e. zÃ ðx0 Þ ¼ zðxi Þ. The weights are & 1 if xi 2 Vi ; ð3:2Þ li ¼ 0 otherwise: The shortcomings of the method are evident; each prediction is based on just one measurement, there is no estimate of the error, and information from neighbouring points is ignored.
Thus, at a point x a property, ZðxÞ, is treated as a random variable with a mean, m, a variance, s 2 , and higher-order moments, and a cumulative distribution function (cdf). It has a full probability distribution, and it is from this that the actual value is drawn. If we know approximately what that distribution might be we can estimate values at unrecorded places from data in the neighbourhood and put errors on our estimates. A Stochastic Approach to Spatial Variation 49 Most environmental variables, such as the soil’s pH and potassium concentration, are continuous.
The curves are of the (lognormal) probability density. smaller; the lower fence is defined analogously. Again, skew is seen to be removed by taking logarithms. 3(a)–(b) shows the cumulative distributions plotted on the probability scale and as normal equivalent deviates, respectively. 3(c)–(d) shows the same graphs for log10 K. These graphs are close to the normal line, and clearly transformation to logarithms yields a near-normal distribution in this instance. , 1982). 2, taking logarithms normalizes the data very effectively.
Analytical treatment of one-dimensional Markov processes by Petr Mandl