# Sampling And Statistics

THIS IS ONLY A STUB... MAY CONTAIN SOME NONSENSE

Assume we want to estimate some population parameters for a forest stand (called treatment unit in Heureka). The population consists of all trees. We want to estimate, by plot sampling, some population totals (volume, number of trees), and some population means (mean diameter, mean age, mean height). In forestry sampling, it is common to use systematic sampling as sampling design.

## Estimations of total values from plot sample data

Assume we have have a systematic sample of n plots (called reference units in Heureka) in a stand, and that all trees have been measured in each plot. Plots that intersect a stand border are either reflected or split. The stand has area A and we want to estimate some variable Y, such as total volume. We are also interested in Y per area unit, denoted as Yha, 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 includes 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 replace estimatation notations to notations for predicted values, and rewrite equation (1a) to

${Y}_{ha}^{pred} = \sum_{i=1}^n{w_i\frac{y_{i}^{pred}}{a_i}}$          (1b)

where

${y}_{i,ha}^{pred}=\frac{\hat{y}_i}{a_i}=$ the predicted variable value per area unit at plot i, and

$w_i=\frac{a_i}{\sum_{i=1}^n{a_i}}=$ the plot weight

### Estimation of plot values from tree measurements

The (predicted) value of variable $y_{i,ha}^{pred}$ is computed from the m number of trees registered on the plot:
$y_{i,ha}^{pred}=\sum_{j=1}^mw_{ij}y_{ij}^{pred}$

where
yijpred = predicted value (tree volume or tree biomass),

wij = tree weight, or expansion factor, for the number of trees the tree record represents per area unit. 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)

## Estimation of the variance within a stand

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