Difference between revisions of "Sampling And Statistics"

Latest revision as of 00:26, 25 March 2013

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

About sampling forest data

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 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):
$\hat{Y} = A\times\hat{Y}_{ha}$

Adaption to prognosis model

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
y_ij^pred = predicted value (tree volume or tree biomass),

w_ij = 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 average values, such as mean diameter and mean height

Using sample trees for calibration

Estimation of the variance within a stand

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

@@ Line 1: / Line 1: @@
+THIS IS ONLY A STUB... MAY CONTAIN SOME NONSENSE
 ==About sampling forest data==
+Assume we want to estimate some ''population parameters'' for a forest stand (called [[Definition:Treatment unit|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==
-===Basics===
+Assume we have have a systematic sample of ''n'' plots (called [[Definition:Reference unit|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 ''Y<sub>ha</sub>'', which is computed as: <br>
-Assume we have sampled ''n'' [[Definition:Sample plot|plots]] in a stand (called [[Definition:Treatment unit|treatment unit]] in Heureka), and that all trees have been measured in each plot. 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 ''Y<sub>ha</sub>'', which is computed as: <br>
-<math>\hat{Y}_{ha}} = \frac{\sum_{i=1}^n{\hat{y}_i}}{\sum_{i=1}^n{a_i}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1a)
+<math>\hat{Y}_{ha} = \frac{\sum_{i=1}^n{\hat{y}_i}}{\sum_{i=1}^n{a_i}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1a)
 where
@@ Line 12: / Line 14: @@
 We then compute the estimated value of  ''Y'' by simply multiplying with ''A'' (assuming that ''A'' is known):<br>
-<math>\hat{Y}} = A\times\hat{Y}_{ha}}</math>
+<math>\hat{Y} = A\times\hat{Y}_{ha}</math>
 ===Adaption to prognosis model===
-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 (m<sup>2</sup>)/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 equation (1a) above to
+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 (m<sup>2</sup>)/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
-<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>
+<math>{Y}_{ha}^{pred} = \sum_{i=1}^n{w_i\frac{y_{i}^{pred}}{a_i}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1b)
-where
+<br>
-<br><math>\frac{\hat{y}_i}{a_i}</math> = the variable value per area unit, and
+<br>where
-<br><math>p_i=\frac{a_i}{\sum_{i=1}^n{a_i}}</math> = the plot weight
+<br>
+<br><math>{y}_{i,ha}^{pred}=\frac{\hat{y}_i}{a_i}=</math> the predicted variable value per area unit at plot ''i'', and
+<br>
+<br><math>w_i=\frac{a_i}{\sum_{i=1}^n{a_i}}=</math> the plot weight
 ===Estimation of plot values from tree measurements===
-The variable <math>\hat{y}_i</math> is computed from the ''m'' number of trees registered on the plot:
+The (predicted) value of variable <math>y_{i,ha}^{pred}</math> is computed from the ''m'' number of trees registered on the plot:
-<br><math>\hat{y}_i=\sum_{j=1}^mp_{ij}y_{ij}</math>
+<br><math>y_{i,ha}^{pred}=\sum_{j=1}^mw_{ij}y_{ij}^{pred}</math>
+<br>
 <br>where
-<br> ''y<sub>ij</sub>'' = value (tree volume or tree biomass),
+<br> ''y<sub>ij</sub><sup>pred</sup>'' = predicted value (tree volume or tree biomass),
-<br> ''p<sub>ij</sub>'' = 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 m<sup>2</sup> = 0.0314159 ha) then the tree weight is 1/0.0314159 = 31.831)
+<br>
+<br> ''w<sub>ij</sub>'' = 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 m<sup>2</sup> = 0.0314159 ha) then the tree weight is 1/0.0314159 = 31.831)
-===Using sample trees for calibration===
 ==Estimation of average values, such as mean diameter and mean height==
+==Using sample trees for calibration==
 ==Estimation of the variance within a stand==