Difference between revisions of "Sampling And Statistics"

Revision as of 13:52, 30 August 2008

About sampling forest data

Estimations of total values from plot sample data

Assume we have sampled n plots in a treatment unit (stand), and that all trees have been measured in each plot. The stand has area A and we want to estimate some stand variable Y. We are also interested in Y per area unit, denoted as Y_ha, which is computed as:

$\hat{Y}_{ha}} = \frac{\sum_{i=1}^n{\hat{y}_i}}{\sum_{i=1}^n{a_i}}$ (1a)

where
$\hat{y}_i=$ estimated value in plot i, and
$a_i=$ inventoried area of plot i, in our case given in ha units

We then compute the estimated value of Y by simply multiplying with A (assuming that A is known):
$\hat{Y}} = A\times\hat{Y}_{ha}}$

In a prognosis, growth is computed for each plot separately, one period at a time (although treatments may be coordinated between plots). A prognosis model often include some variables that describe the density of the tree cover. For example, total basal area and stem density are common variables in growth functions, expressed as basal area per ha (m²)/ha) and number of trees per ha (trees/ha). Therefore, it is practical to keep all density variables at the plot level to a per ha-value. We can rewrite the equation (1a) above to

$\hat{Y}_{ha}} = \sum_{i=1}^n{p_i\frac{\hat{y}_i}{a_i}}$ (1b)
where
$\frac{\hat{y}_i}{a_i}$ = the variable value per area unit, and
$p_i=\frac{a_i}{\sum_{i=1}^n{a_i}}$ = the plot weight (similar to inclusion probability)

Estimation of plot values from tree measurements

The variable $\hat{y}_i$ is computed from the m number of trees registered on the plot:
$\hat{y}_i=\sum_{j=1}^mp_{ij}y_{ij}$
where
y_ij = value (tree volume or tree biomass),
p_ij = tree weight, or expansion factor (cf. inclusion probability), for the number of trees the tre record represent a a per ha-basis. For example, if the plot area is 0.0314159 ha (given a plot radius of 10 m and hence a plotarea of 314.159 m² = 0.0314159 ha) then the tree weight is 1/0.0314159 = 31.831)

Using sample trees for calibration

Estimation of mean diameter and mean height

Estimation of the variance within a stand

The following formula is valid also when sample plots have unequal size.

@@ Line 18: / Line 18: @@
 <math>\hat{Y}_{ha}} = \sum_{i=1}^n{p_i\frac{\hat{y}_i}{a_i}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1b) <br>
 where
-<br><math>\frac{y_i}{a_i}</math> = the variable value per area unit, and
+<br><math>\frac{\hat{y}_i}{a_i}</math> = the variable value per area unit, and
 <br><math>p_i=\frac{a_i}{\sum_{i=1}^n{a_i}}</math> = the plot weight (similar to inclusion probability)