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Bivariate normal distributions

Available with Geostatistical Analyst license.

Disjunctive kriging requires that the data comes from a bivariate normal distribution. Also, to develop probability and quantile maps, it's assumed that the data comes from a full multivariate normal distribution. To check for a univariate normal distribution, you can use the histogram chart (this does not guarantee that the data comes from a full multivariate normal distribution, but it is often reasonable to assume so if univariate normal distributions are detected using this chart).

Consider the following probability statement:

f(p,h) = Prob[Z(s) ≤ zp, Z(s + h) ≤ zp],

where zp is the standard normal quantile for some probability p.

For example, a familiar standard normal quantile occurs when p = 0.975, then zp = 1.96, and when p = 0.5, then zp = 0, and when p = 0.025, then zp = -1.96. The probability statement above takes a variable Z at location s and another variable Z at some other location s + h and gives the probability that they are both less than zp. This probability statement is a function f(p,h) depending on p (and consequently zp) and h. The function will also depend on the amount of autocorrelation between Z(s) and Z(s + h).

Assume that Z(s) and Z(s + h) have a bivariate normal distribution. If the autocorrelation is known, there are formulas for f(p,h). Suppose h is constant and only p changes. You would expect the function to look like this:

Bivariate distribution for probability
Bivariate distribution for quantile

The second figure looks like a cumulative probability distribution. Now, suppose that p is fixed, and f(p,h) changes with h.

First, suppose that h is very small. In that case, Prob[Z(s) ≤ zp, Z(s + h) ≤ zp] is very nearly the same as Prob[Z(s) ≤ zp] = p. Next, suppose that h is very large. In that case, Prob[Z(s) ≤ zp, Z(s + h) ≤ zp] is very nearly the same as Prob[Z(s) ≤ zp] Prob[Z(s + h) ≤ zp] = p2 (because Z(s) and Z(s + h) are very nearly independent). Thus, for fixed p, you expect f(p,h) to vary between p and p2. Now, considering f(p,h) as a function of both p and the length of h, you might observe something similar to the following figure:

Bivariate distribution for probability and distance

This function can be converted to semivariograms and covariance functions for indicators. If you note that Prob[Z(s) ≤ zp, Z(s + h) ≤ zp] = E[I(Z(s) ≤ zp)xI(Z(s + h) ≤ zp)], where I(statement) is the indicator function—is 1 if statement is true, otherwise it is 0—the covariance function for the indicators for fixed p is

CI(h;p) = f(p,h) –p2,

and the semivariogram for indicators for fixed p is

γI(h;p) = p - f(p,h).

Therefore, you can estimate the semivariogram and covariance function on the indicators of the original data and use these to obtain the expected semivariograms and covariance functions of indicators for various values of p.

Learn more about semivariograms and covariance functions