Which predictions should be subject to a random component and which can be held 

 to their mean estimate is a choice that depends on the nonlinearity of the effects of 

 variation in subsequent calculations. If assessing the variability of the outcome 

 per se were one of the objectives of the modeling, then most of the prediction equations 

 would require a random component. To obtain the expected value of the random process, 

 the whole sequence of computations would be repeated with different random errors and 

 the process averaged over the replications. One of the drawbacks to this approach is 

 that the volume of computations is very large if it is applied to the solution of all 

 prediction equations that make up the overall model. Another drawback is that little 

 is known of the serial correlation that ought to characterize the successive values of 

 the random variables. Are large positive deviations from expected diameter increment 

 more likely to be associated with large positive deviations from expected height 

 increment — or with expected changes in crown dimensions? 



The approach used in the present version of this program may be considered in 

 Monte Carlo terms, a "swindle." The purpose is to produce a prognosis that overall is 

 the result of averaging many replications of the random process without actually having 

 to carry out the replications. 



Random Error in Tree Development 



The program assigns all random effects to the distribution of errors of prediction 

 of the logarithm of basal area increment. Basal area increment was selected to carry 

 the stochastic variation because the effects of differing diameter growth rates ramify 

 in highly nonlinear ways through most of the remaining components and variables such as 

 percentile in the basal area distribution, relative stocking, the height increment 

 model, and the crown development model. This distribution is asstmied to be Normal, 

 with a mean of zero. The variance of this Normal Distribution is computed as a 

 weighted average of two estimates; the first such estimate is derived from the 

 regression analysis that developed the prediction function and the second estimate is 

 the standard deviation of the differences between the actually recorded growth (trans- 

 formed to the logarithm of basal area increment) for the sample trees in the population 

 and their corresponding regression estimates. The weights assigned to these two esti- 

 mates are 100 for the prior component of error, and the number of growth-sample trees 

 in the stand for the second component of error (Mehta 1972) . 



The effects of modifying the predicted change in tree d.b.h. by a random variable 

 can then be carried into other predicted changes in tree characteristics by using the 

 change in d.b.h. as an independent variable in each successive model. The random 

 variable associated with each tree record is saved until the following cycle so that 

 the appropriate serial correlation can be preserved in the distribution of the random 

 variables . 



The rsindom component of change in tree d.b.h. is treated in two ways, depending on 

 how many tree-records^ make up the stand being projected. When there are many tree 

 records, the effects of any one random deviation on the growth rate of one tree would 

 be blended with many other trees. Consequently, the stand totals should be quite 

 stable estimates. Accordingly, a random deviate from the specified distribution is 

 added to the logarithm of basal area increment. Because of the logarithmic 

 transformation, the effect on predicted diameter increment is multiplicative. 



IVhen the stand is represented by relatively few sample trees, however, a different 

 strategy is used. In order to increase the number of replications of the random effects, 

 each tree record is augmented by two additional records. These new records duplicate 

 all characteristics of the tree except the predicted change in d.b.h. and the number of 

 trees per acre represented by the source tree record. The trees-per-acre value of the 

 source record is reduced to 60 percent of its current value. The two new records are 

 given 15 and 25 percent of the source value; thus, the three records together still 

 represent the same number of trees per acre. 



12 



