Growth Stratification Applied to Prognosis of Diametric Structure in Araucaria Forest ()
vironment [16] [17] , plus the characteristics related to different stages of forest succession in a phytogeographical unit.
Taking as basis for stratifying the average growth rate 
and the standard deviation between the species (SMPI = 0.071 cm×year−1), the limits of the strata were defined as: E1 for the values of
; E2 for values of
; E3 for the values of 
and E4 for the values of
, as shown in Table2
The stratum of greater representativeness was the E2, corresponding to 44% of the species evaluated, followed by E3 (27% of species), E4 (17% of species) and E1 (12% of species). The growth mean values of the respective strata were:
;
;
;
.
The variance for the population without stratification resulted in
, a value nine times higher than the stratified variance
. The strata 1 to 4 presented variances of:
;
; 
and 
. Consequently, groups of greater growth homogeneity were obtained with the application of stratification inside the total population.
Each stratified data set followed the normal distribution, evaluated by the Shapiro-Wilk test (W) at 95% probability. The effective number of degrees of freedom calculated by Satterthwaite’s method was 41 (ne = 40.4). The variances were not homogeneous (heteroscedasticity), with rejection of the null hypothesis H0 by Hartley’s test (Fcalc = 5.49 > Ftab;0.01 = 2.97). The data set was then subjected to Box-Cox’s transformation, with a value of 
equal to −3.5, which led to the lowest value for Fcalc, ensured the homogeneity of variances (Fcalc = 2.58 <
Table 2. Stratification of diameter growth by species in Araucaria Forest (E).
Ftab;0.01 = 2.97) and remained normally distributed. Consequently, detected heteroscedasticity of the variances within strata in which the total population of 
was allocated, the application of Box and Cox transformation, using the prospective simulation proposed by Draper and Smith to identify the most appropriate root to be used in the process, resulted in total efficiency to get homoscedasticity of variances (Figure 1).
By means of analysis of variance, it can be observed that there are statistical differences between the strata (Table 3). Therefore, it is inferred that the auto ecological characteristics of each species that composes the plant community favor decisively the formation of groups with different diameter growth. Consequently, the stratification of diameter growth reduced the variance nine times, enabling the formation of groups of greater homogeneity and statistically different between themselves.
The Tukey’s test indicated statistical differences between means of the strata (Table 4). Assuming that the differences between strata represent species with different growth potential, the choice of species belonging to a particular stratum may favor the accomplishment of the objectives of forest management, where growth has an important meaning. Consequently, the allocation of species in strata favors the decisions in forestry projects, in which the diameter increment is one of the relevant variables for the establishment of activities and delineation of targets.
Whereas the models of production meets a set of criteria that assist in decision making for the forest planning [18] , it was evaluated the prognosis of the diametric structure from the stratified data and without stratification, by means of the methodology of ratio of change, which is based on diameter’s growth and migration of them between diameter classes, unlike the Markov chain, which considers only the enumeration of individuals that move between classes. The predicted values in each stratum and total, which represents the sum of stratified estimates, are presented in Figure 2.
Meyer et al. in their studies that support the sustainable management, about the diameter distribution in hete-
Figure 1. Behavior of the value Fmax as a function of the values of λ1 tested.
Table 3. Analysis of variance for stratification in Araucaria Forest.
**Significant at 1% level of probability.
Table 4. Tukey’s test at 95% probability between the strata means.
*Means followed by same letter do not differ from each other by Tukey’s test at 95% probability.
Figure 2. Projected frequency distribution per stratum and for the population.
rogeneous all age forests, refer to a pattern of decreasing exponential distribution to the extent to which diameter classes advance [19] . However, such a pattern was only found in the sum of stratified subpopulations, which generated the projected distribution. Each stratum presented differentiated settings in their distributions, justified by the auto ecological and synecological characteristics of the species of each stratum.
The K-S test (Dcalc = 0.00949 < Dtab;0.95 = 0.01855) attested goodness of fit of projected values from stratified data, however the projected values from no stratified data did not present goodness of fit to the observed distribution (Dcalc = 0.09826 > Dtab;0.95 = 0.01855). Therefore, results obtained with greater homogeneity between diameter growth values within strata tend to provide better estimates because the variations of growth between species are minimized with the group formation, allowing a refinement of estimates when using the results of the stratified parameters. This confirms our formulated hypotheses. The distributions of projected and observed frequencies are presented in Table5
De Liocourt considers that in a balanced forest, the diametric distribution in successive classes (k) is derived from a constant geometric series called “Quotient of De Liocourt” (q), essential parameter for the forest management proposal [20] , in addition to structural evaluations of plant community. In perfect balanced forests q assumes a constant value, indicating a pattern of decreasing exponential distribution in the forest [21] .
The prognosis of the diametric structure, in these circumstances, in addition to estimate with accuracy the forest density, must maintain values equally accurate for obtaining the quotient q, indicating an approximate results for quotients of distribution of frequencies for the forest structure, which can be observed in Table6
In both projected distributions and in the observed one, the q quotients, minimum and maximum, remained in the same classes, k4/k5 and k7/k8 respectively. The lowest accuracy between the projected and observed ratios was found in the estimation without stratification, for the lower and upper diameter classes, reaffirming what was observed in the distribution of frequencies. Generally speaking, disregarding the lower and the upper diameter classes without stratification, there is similarity between the estimated distributions and the observed one, essentially when performed with stratification. Consequently, the prognosis of diametric distribution for the year 2008 resulted more accurate when calculated from stratified data, evaluated by the goodness of fit attested at 95% probability by the application of the Kolmogorov-Smirnoff test in relation to data observed in 2008; the same condition occurred in the evaluation of the Quotient of Liocourt (Figure 3).
Other methodologies can be developed to fit the models of prognosis, providing greater accuracy in their results. In heterogeneous forests, for example, it will be appropriate to separate groups formed by species or botanical families. The application of probability density functions can also improve the prognosis, favoring a distribution with greater similarity between estimated and observed values of the forest.
4. Conclusions
The auto ecological characteristics of each species that composes the plant community favor decisively the formation of groups with different diameter growth and the stratification of diameter growth reduced the variance nine times, enabling the formation of groups statistically different of greater homogeneity.
The stratification of diameter growth approached the predicted results to those observed in the sample population, allowing the forest manager to work with the best estimators in preparation of management proposals.
The methodology of ratio of change, which is based on diameter’s growth and migration of them between
Figure 3. Distribution of De Liocourt quotients for observed and predicted values.
Table 5. Prognosis of the number of individuals per diameter class generated from the values with and without stratification, and observed values of the diametric distribution of Araucaria Forest, for the year 2008.
Table 6. Values for the quotient of De Liocourt from the prognosis of diametric distribution and observed values of Araucaria Forest, for the year 2008.
*Values obtained with all population data.
diameter classes showed to be very efficient to make prognosis in native forests.
The prognosis of diametric distribution for the year 2008 resulted more accurately when calculated from stratified data, evaluated by the goodness of fit attested at 95% probability by the application of the Kolmogorov-Smirnoff test in relation to data observed in 2008.
The evaluation of the coefficient of De Liocourt behaved better in stratification using diameter growth by classes when compared to the results obtained without stratification.
NOTES
*Corresponding author.