7.3. Statistical analysis

The first question I address is do the y_i's agree with property (2) of the Wiener process. In other words, do the y_i's have the distribution N(rT_i, ²T_i). To test this, I use Liliefor's test for normality (Conover, 1980). This is a Kolmogorov-Smirnoff type of test specific to a population of normal variables with unknown mean and variance. The first step is to estimate the parameters r and , which are then used to determine mean and variance. I use maximum likelihood to estimate the parameters. The likelihood function is

. (7.18)

The maximum likelihood estimator (mle) for r is:

, (7.19)

which is just the average downstream velocity of the individual. To estimate , I plug into the likelihood function and maximize log(L) with respect to numerically, using a downhill simplex method (Press, et al. 1988).

The statistic of Liliefor's test measures the deviation of the observations from a cumulative normal distribution. This statistic is compared to a lookup table to determine the approximate probability. Normality is rejected for small p-values.

The third property of the Wiener process is independent increments. If normality is rejected, then the increments can be transformed to standard normal variables as follows:

. (7.20)

The property of independence of successive increments can be tested for by determining whether the Z_i's are uncorrelated. (In general, showing that two random variables are uncorrelated does not demonstrate independence; in the case of normal random variable, however, it does). The correlation between successive movements can be determined by computing the correlation coefficient

, (7.21)

where n is the number of movement increments observed. To test the null hypothesis of no (or negative) correlation among successive Z_i's versus the alternative hypothesis of positive correlation, the test statistic

(7.22)

is compared to a t distribution with n-2 degrees of freedom (Sokal and Rohlf, 1981). The null hypothesis is rejected for small p-values.

To estimate the parameters for the O-U based model, I follow a similar procedure. I first subtract off the average (time scaled) displacement from the observations. This is identical to the first step above and can be expressed as

. (7.23)

The transformed variable Y' has mean displacement of 0. The likelihood function is

, (7.24)

where f_OU is a bivariate normal distribution with parameters defined in equations (7.14), (7.15), and (7.16). Again, the parameters and are estimated by maximizing the log of (7.24) numerically with respect to the parameters.

The importance of the parameter , which determines correlation in the O-U based model, can be assessed by computing the log likelihood ratios and BIC values. I report the difference between the BIC values for the O-U based model and Wiener drift model. The null model (Wiener drift model) is rejected for positive BIC values.