Calculating Expected Values From A Frequency Matrix

 A frequency table contains nearly the same information as a transition matrix for a stochastic process and can also be used to estimate the expected value of a succeeding state given an initial state. For the monthly global land anomalies the values ranged between x_i = -2 and x_f = 2 and this interval was subdivided into 20 parts as follows. The center of each sub-interval is given by x_j.

In the following table the entry in a row shows the number of times a value, x', of the row followed the value, x, of its column. You may recognize the similarity of this table to those that Galton used for the inheritance of characteristics of children from those of their parents.

The frequencies can be used to compute the expected value of x' by summing the product of x' and the probability or relative frequency for a column using the formula below.

The differences of the expected values of x' from the initial x values were used to estimate the drift function.