AI-GEOSTATS: Random Functions

[Digby Millikan]

Dear list,   Thanks for the replies about random functions and variables Z(x,w), I thought of a good example for w which may represent grades of a material and Z(x,w) could represent dollar values, if one for instance were modeling multiple grades. Another example may be calorific values of coal from ash and sulphur contents. Of course the conversion to dollar values from grade values is normally performed after the kriging process and not before, however an example of the application of a function to a number of variables, prior to modeling is provided, which may find some applications in modeling of spatial or time series data.   Replies;       Hello Digby,   recall that a random variable (not spatial!) may be defined as a function from an event space "Omega" to the real space "R". "Omega" is a set containing all possible outcomes of your random phenomenon, and the random variable "just" attaches a real number to each possible event. Take a cube (also known as die :-) and paint each side with a diferent colour; throw the die. The results might be for instance: "red", "green", "blue", "yellow", "magenta" and "cyan": this set of outcomes form up "Omega". Now write a number in each side (from 1 to 6, following the same order of colours above). Then you have defined a function ("red"->1, "green"->2,... "cyan"->6) from the colour outcome to the real space: this function is your random variable, which is usually denoted as "X(w)", being "X" a number and "w" a color.   Now, a random function in geostatistics is just a random variable which does not only depend on the event space ("Omega"), but also on the physical space (the volume occupied by the deposit "D"). This is why we may denote it as: Z(x,w), being "Z" your grade (probably), "x" its spatial position and "w" its "randomness". In short, "w" just says that your "Z" is random.   Hope to have hit your question...   :-) Raimon       The second statement is in fact the more precise version of the first, provided the second one is properly interpreted.   First of all, for any particular value of w, Z(x,w) is a particular function of x. The values of w do not need to be of any particular kind -- all that is needed is that they serve to index the functions Z(x,w). The set of all possible values of w can be denoted by W, say (this bit is not explicit in what you say).   So the set of all available functions of x is     {Z(x,w) for w in W}   Next, Z(x,w) will be a random function of x if w has been chosen at random from amongst the values in W. To do this, you need to have a probability distribution defined on W. That is why W is called the "sample space". So a probability distribution P is defined on W.   Thus to "realise" a random function from {Z(x,w) for w in W} when the probability distribution P is defined on W, use P to sample a value at random from W and let this value be w. Then the random function is Z(x,w).   Does this help?   Best wishes, Ted.   -------------------------------------------------------------------- E-Mail: (Ted Harding) <[EMAIL PROTECTED]> Fax-to-email: +44 (0)870 094 0861 Date: 22-Jul-06                                       Time: 14:07:46 ------------------------------ XFMail ------------------------------