http://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&feed=atom&action=historyApproaches to populations of sample plots - Revision history2024-03-29T12:39:59ZRevision history for this page on the wikiMediaWiki 1.19.3http://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=16600&oldid=prevPmagdon at 11:26, 14 June 20232023-06-14T11:26:06Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:4.8-fig69.png|right|thumb|300px|'''Figure  2''' An identical “forest area” subdivided in two different ways in  square sample plots of the same basic size. Right: plot fragments occur  along the border  line. The total of “number of stems” is obviously  identical in both cases; but this is not the case for the per-plot  parametric mean and variance (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:4.8-fig69.png|right|thumb|300px|'''Figure  2''' An identical “forest area” subdivided in two different ways in  square sample plots of the same basic size. Right: plot fragments occur  along the border  line. The total of “number of stems” is obviously  identical in both cases; but this is not the case for the per-plot  parametric mean and variance (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:4.8-fig70.png|right|thumb|300px|'''Figure  3''' Illustration of approaches for plot populations for the same  example population. Left: discrete population of square sample plots  with defined positions. Right: each point in the area is a sampling  element the value of which is determined by the surrounding trees  (sample plot). Each point has a value which is here indicated through  the cloud of points; in addition a trend surface is given (Kleinn and  Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F.  Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die  Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ    176(4):68-74.</ref>).]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:4.8-fig70.png|right|thumb|300px|'''Figure  3''' Illustration of approaches for plot populations for the same  example population. Left: discrete population of square sample plots  with defined positions. Right: each point in the area is a sampling  element the value of which is determined by the surrounding trees  (sample plot). Each point has a value which is here indicated through  the cloud of points; in addition a trend surface is given (Kleinn and  Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F.  Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die  Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ    176(4):68-74.</ref>).]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed <del class="diffchange diffchange-inline">are </del>plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed <ins class="diffchange diffchange-inline">area </ins>plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[population|sampling frame]], in the other case, there are various [[Fixed area plots at the stand boundary|border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[population|sampling frame]], in the other case, there are various [[Fixed area plots at the stand boundary|border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td></tr>
</table>Pmagdonhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=9939&oldid=prevWikiSysop: /* Approach 2 */2013-10-28T10:42:52Z<p><span dir="auto"><span class="autocomment">Approach 2</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If the tree ''k'' has an attribute value <math>y_k</math> (for example basal area in <math>m^2</math>; or simply number of trees which is <math>y_k=1</math> for each tree, obviously) we imagine this value distributed evenly over the inclusion zone. Geometrically, that is a disk with the inclusion zone as base area <math>a_k</math> and a height which is defined by <math>y_k</math> as</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If the tree ''k'' has an attribute value <math>y_k</math> (for example basal area in <math>m^2</math>; or simply number of trees which is <math>y_k=1</math> for each tree, obviously) we imagine this value distributed evenly over the inclusion zone. Geometrically, that is a disk with the inclusion zone as base area <math>a_k</math> and a height which is defined by <math>y_k</math> as</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>d_k=\frac{y_k}{a_k}<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>d_k=\frac{y_k}{a_k}\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of <math>d_k</math> is constant over the entire inclusion zone and may be interpreted as density value. For the variable number of stems,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of <math>d_k</math> is constant over the entire inclusion zone and may be interpreted as density value. For the variable number of stems,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>d_k=\frac{1}{a_k}<del class="diffchange diffchange-inline">,</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>d_k=\frac{1}{a_k}\,</math></div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>because <math>y_k=1</math><del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>because <math>y_k=1</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of one sample element in the infinite population of sample points results from the sum of all such density values ''d'' which are present at the particular position <math>x1_i</math>, <math>x2_i</math>:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of one sample element in the infinite population of sample points results from the sum of all such density values ''d'' which are present at the particular position <math>x1_i</math>, <math>x2_i</math>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>d(x1_i;x2_i)=\sum_{(x1_i,x2_i){\cap}E_k}d_k=\sum_{(x1_i;x2_i){\cap}E_k}\frac{y_k}{a_k}<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>d(x1_i;x2_i)=\sum_{(x1_i,x2_i){\cap}E_k}d_k=\sum_{(x1_i;x2_i){\cap}E_k}\frac{y_k}{a_k}\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>That defines eventually the infinite population of points which is indicated in Figure 3, right, as a cloud of points. The population total <math>\tau</math> is then the integral over the entire area of all inclusion zones</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>That defines eventually the infinite population of points which is indicated in Figure 3, right, as a cloud of points. The population total <math>\tau</math> is then the integral over the entire area of all inclusion zones</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\tau=\iint_{x_2,x_1}\,d(x1,x2)\,dx1,dx2<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\tau=\iint_{x_2,x_1}\,d(x1,x2)\,dx1,dx2\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Observe that this implies to integrate also outside the areal sampling frame of sample points where inclusion zones of border trees are outside the defined inventory area. This leads to the border correction issue which is dealt with in [[Fixed area plots at the stand boundary]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Observe that this implies to integrate also outside the areal sampling frame of sample points where inclusion zones of border trees are outside the defined inventory area. This leads to the border correction issue which is dealt with in [[Fixed area plots at the stand boundary]].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From that infinite population, a sample of size ''n'' is selected. From a sample point ''i'' with the grid coordinates <math>x1_i</math>, <math>x2_i</math> the estimated total <math>\hat\tau_i</math> of forest area ''A'' derives from the [[Sampling with unequal selection probabilities#The Hansen-Hurwitz-estimator|Hansen-Hurwitz-estimator]] from</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From that infinite population, a sample of size ''n'' is selected. From a sample point ''i'' with the grid coordinates <math>x1_i</math>, <math>x2_i</math> the estimated total <math>\hat\tau_i</math> of forest area ''A'' derives from the [[Sampling with unequal selection probabilities#The Hansen-Hurwitz-estimator|Hansen-Hurwitz-estimator]] from</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\hat\tau_i=\frac{d(x1_ix2_i)}{f(x1_ix2_i)}<del class="diffchange diffchange-inline">,</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\hat\tau_i=\frac{d(x1_ix2_i)}{f(x1_ix2_i)}\,</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>f()</math> is the selection probability. Assuming independent random sampling,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>f()</math> is the selection probability. Assuming independent random sampling,</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>for all sample points. It follows</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>for all sample points. It follows</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\bar\tau_i=A\sum_{x1_i,x2_i){\cap}E_k}\frac{y_k}{a_k}<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\bar\tau_i=A\sum_{x1_i,x2_i){\cap}E_k}\frac{y_k}{a_k}\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>One may interpret this estimator also such that each observation <math>y_k</math> is expanded with the plot-specific expansion factor</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>One may interpret this estimator also such that each observation <math>y_k</math> is expanded with the plot-specific expansion factor</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>EF_k=\frac{A}{a_k}<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>EF_k=\frac{A}{a_k}\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From a sample of ''n'' randomly selected sample points, the total is eventually estimated as</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From a sample of ''n'' randomly selected sample points, the total is eventually estimated as</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\bar\tau=\frac{1}{n}\sum_{i}^n\bar\tau_i<del class="diffchange diffchange-inline">.</del>\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\bar\tau=\frac{1}{n}\sum_{i}^n\bar\tau_i\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the [[Fixed area plots at the stand boundary#The mirage technique|mirage technique]] for border plot correction) than with the discrete plot population approach.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the [[Fixed area plots at the stand boundary#The mirage technique|mirage technique]] for border plot correction) than with the discrete plot population approach.</div></td></tr>
</table>WikiSysophttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=9462&oldid=prevFehrmann at 12:13, 26 October 20132013-10-26T12:13:28Z<p></p>
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</table>Fehrmannhttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=8697&oldid=prevPbecksc: /* Approach 2 */2013-04-04T14:33:48Z<p><span dir="auto"><span class="autocomment">Approach 2</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The [[Inclusion probability|inclusion zone concept]] <del class="diffchange diffchange-inline">shows which </del>bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The [[Inclusion probability|inclusion zone concept]] bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The inclusion zone concept and the infinite population approach belong together. Each tree ''k'' has its inclusion zone <math>E_k</math>. While the size of this inclusion zone can be taken as a measure for the selection probability of that particular tree ''k'', we may also follow a straightforward geometric interpretation that helps understanding the estimation, which is, in fact, based on unequal probability sampling (see also [[sampling with unequal selection probabilities]]).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The inclusion zone concept and the infinite population approach belong together. Each tree ''k'' has its inclusion zone <math>E_k</math>. While the size of this inclusion zone can be taken as a measure for the selection probability of that particular tree ''k'', we may also follow a straightforward geometric interpretation that helps understanding the estimation, which is, in fact, based on unequal probability sampling (see also [[sampling with unequal selection probabilities]]).</div></td></tr>
</table>Pbeckschttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=8664&oldid=prevWikiSysop: /* Approach 2 */2013-03-28T07:57:19Z<p><span dir="auto"><span class="autocomment">Approach 2</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[<del class="diffchange diffchange-inline">The </del>inclusion zone concept]] shows which bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The </ins>[[<ins class="diffchange diffchange-inline">Inclusion probability|</ins>inclusion zone concept]] shows which bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The inclusion zone concept and the infinite population approach belong together. Each tree ''k'' has its inclusion zone <math>E_k</math>. While the size of this inclusion zone can be taken as a measure for the selection probability of that particular tree ''k'', we may also follow a straightforward geometric interpretation that helps understanding the estimation, which is, in fact, based on unequal probability sampling (see also [[sampling with unequal selection probabilities]]).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The inclusion zone concept and the infinite population approach belong together. Each tree ''k'' has its inclusion zone <math>E_k</math>. While the size of this inclusion zone can be taken as a measure for the selection probability of that particular tree ''k'', we may also follow a straightforward geometric interpretation that helps understanding the estimation, which is, in fact, based on unequal probability sampling (see also [[sampling with unequal selection probabilities]]).</div></td></tr>
</table>WikiSysophttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=8663&oldid=prevWikiSysop: /* Approach 1 */2013-03-28T07:55:54Z<p><span dir="auto"><span class="autocomment">Approach 1</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[population|sampling frame]], in the other case, there are various [[border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[population|sampling frame]], in the other case, there are various [[<ins class="diffchange diffchange-inline">Fixed area plots at the stand boundary|</ins>border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td></tr>
</table>WikiSysophttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=8662&oldid=prevWikiSysop: /* Approach 1 */2013-03-28T07:55:02Z<p><span dir="auto"><span class="autocomment">Approach 1</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[sampling frame]], in the other case, there are various [[border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[<ins class="diffchange diffchange-inline">population|</ins>sampling frame]], in the other case, there are various [[border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Approach 2==</div></td></tr>
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</table>WikiSysophttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=5608&oldid=prevAspange: /* Approach 2 */2011-03-09T21:07:52Z<p><span dir="auto"><span class="autocomment">Approach 2</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\bar\tau=\frac{1}{n}\sum_{i}^n\bar\tau_i.\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\bar\tau=\frac{1}{n}\sum_{i}^n\bar\tau_i.\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the mirage technique for border plot correction) than with the discrete plot population approach.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the <ins class="diffchange diffchange-inline">[[Fixed area plots at the stand boundary#The </ins>mirage technique<ins class="diffchange diffchange-inline">|mirage technique]] </ins>for border plot correction) than with the discrete plot population approach.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td></tr>
</table>Aspangehttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=5607&oldid=prevAspange: /* Approach 2 */2011-03-09T21:06:21Z<p><span dir="auto"><span class="autocomment">Approach 2</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From that infinite population, a sample of size ''n'' is selected. From a sample point ''i'' with the grid coordinates <math>x1_i</math>, <math>x2_i</math> the estimated total <math>\hat\tau_i</math> of forest area ''A'' derives from the [[Sampling with unequal selection probabilities#The Hansen-Hurwitz-estimator|Hansen-Hurwitz-estimator]] from</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>From that infinite population, a sample of size ''n'' is selected. From a sample point ''i'' with the grid coordinates <math>x1_i</math>, <math>x2_i</math> the estimated total <math>\hat\tau_i</math> of forest area ''A'' derives from the [[Sampling with unequal selection probabilities#The Hansen-Hurwitz-estimator|Hansen-Hurwitz-estimator]] from</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\hat\tau_i=\frac{d(x1_ix2_i)}{<del class="diffchange diffchange-inline">f8x1_ix2_i</del>)},\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\hat\tau_i=\frac{d(x1_ix2_i)}{<ins class="diffchange diffchange-inline">f(x1_ix2_i</ins>)},\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>f()</math> is the selection probability. Assuming independent random sampling,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>f()</math> is the selection probability. Assuming independent random sampling,</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>for all sample points. It follows</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>for all sample points. It follows</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:\bar\tau_i=A\sum_{x1_i,x2_i){\cap}E_k}\frac{y_k}{a_k}.\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<ins class="diffchange diffchange-inline"><math></ins>\bar\tau_i=A\sum_{x1_i,x2_i){\cap}E_k}\frac{y_k}{a_k}.\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>One may interpret this estimator also such that each observation <math>y_k</math> is expanded with the plot-specific expansion factor</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>One may interpret this estimator also such that each observation <math>y_k</math> is expanded with the plot-specific expansion factor</div></td></tr>
</table>Aspangehttp://wiki.awf.forst.uni-goettingen.de/wiki/index.php?title=Approaches_to_populations_of_sample_plots&diff=5606&oldid=prevAspange: /* General observations */2011-03-09T21:04:58Z<p><span dir="auto"><span class="autocomment">General observations</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>__TOC__</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>__TOC__</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">==General observations==</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig68.png|right|thumb|300px|'''Figure 1''' Illustration of  approach 1 for plot populations: subdivision of the forest area into  sample plots of identical shape and size, here: square and hexagonal  sample plots. Such a subdivision is also possible for rectangles and  some types of triangles (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig69.png|right|thumb|300px|'''Figure  2''' An identical “forest area” subdivided in two different ways in  square sample plots of the same basic size. Right: plot fragments occur  along the border  line. The total of “number of stems” is obviously  identical in both cases; but this is not the case for the per-plot  parametric mean and variance (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig70.png|right|thumb|300px|'''Figure  3''' Illustration of approaches for plot populations for the same  example population. Left: discrete population of square sample plots  with defined positions. Right: each point in the area is a sampling  element the value of which is determined by the surrounding trees  (sample plot). Each point has a value which is here indicated through  the cloud of points; in addition a trend surface is given (Kleinn and  Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F.  Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die  Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In [[forest inventory]] field [[:Category:Sampling design|sampling]], the sampling elements that we select and observe are [[Sample plot|sample plots]]. Consequently, the [[population]] from which we sample is a population of sample plots.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In [[forest inventory]] field [[:Category:Sampling design|sampling]], the sampling elements that we select and observe are [[Sample plot|sample plots]]. Consequently, the [[population]] from which we sample is a population of sample plots.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">=</del>==Approach 1<del class="diffchange diffchange-inline">=</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==Approach 1==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig68.png|right|thumb|300px|'''Figure 1''' Illustration of    approach 1 for plot populations: subdivision of the forest area into    sample plots of identical shape and size, here: square and hexagonal    sample plots. Such a subdivision is also possible for rectangles and    some types of triangles (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig69.png|right|thumb|300px|'''Figure  2''' An identical “forest area” subdivided in two different ways in  square sample plots of the same basic size. Right: plot fragments occur  along the border  line. The total of “number of stems” is obviously  identical in both cases; but this is not the case for the per-plot  parametric mean and variance (Kleinn and Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F. Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ  176(4):68-74.</ref>).]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[File:4.8-fig70.png|right|thumb|300px|'''Figure  3''' Illustration of approaches for plot populations for the same  example population. Left: discrete population of square sample plots  with defined positions. Right: each point in the area is a sampling  element the value of which is determined by the surrounding trees  (sample plot). Each point has a value which is here indicated through  the cloud of points; in addition a trend surface is given (Kleinn and  Vilcko 2005<ref  name="kleinn_vilcko2005">Kleinn  C. und F.  Vilčko. 2005. Ein  Vergleich von zwei methodischen Konzepten  für die  Grundgesamtheit von  Probeflächen bei Waldinventuren. AFJZ    176(4):68-74.</ref>).]]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>An intuitive approach is to imagine this population as the set of sample plots that covers the area of interest completely. This is depicted in Figure 1 for the two examples of square and hexagonal fixed area sample plots. The population consists of a discrete number of plots (see Figure 3, left). The sampling process is to select some of these plots. However, this approach does pose a series of interpretation and analysis problems; it works only for [[fixed area plots]] and within the class of fixed are plots only for some plot shapes. For the most frequently used circular plot, for example, this concept can not be applied.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[sampling frame]], in the other case, there are various [[border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Also, once the [[Forest Definition|forest]] area of interest is defined (the area sampling frame), it is not clearly defined as well how the plots come to lie. Figure 2 shows one and the same population over which two different grids of plots are overlaid. In one case the square plots fit perfectly into the square area [[sampling frame]], in the other case, there are various [[border plots]] with smaller size. The per-plot mean and variance will be different between the two sub-divisions; this is certainly an undesirable property of a population concept that the population parameters are not clearly defined. We conclude that this simple concept of populations of sample plots is not well suited as a model for forest inventory sampling.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>   </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>   </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">=</del>==Approach 2<del class="diffchange diffchange-inline">=</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==Approach 2==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[The inclusion zone concept]] shows which bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[The inclusion zone concept]] shows which bases on the idea to select points from the area sampling frame around which the sample [[Tree Definition|trees]] are being selected along a defined plot design. This is a completely different view at the population of sample plots. It is some times called “the [[infinite population approach]]” because the population consists of the infinite number of points within the areal sampling frame. Each point possesses a characteristic which is being observed. However, that value is not being observed at the point itself, but it derives from the sample trees which are tallied around that sample point according to the rules that are defined by the plot design.</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 33:</td>
<td colspan="2" class="diff-lineno">Line 31:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of one sample element in the infinite population of sample points results from the sum of all such density values ''d'' which are present at the particular position <math>x1_i</math>, <math>x2_i</math>:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The value of one sample element in the infinite population of sample points results from the sum of all such density values ''d'' which are present at the particular position <math>x1_i</math>, <math>x2_i</math>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>d(x1_i;x2_i)=\sum_{(x1_i<del class="diffchange diffchange-inline">;</del>x2_i){\cap}E_k}d_k=\sum_{(x1_i;x2_i){\cap}E_k}\frac{y_k}{a_k}.\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>d(x1_i;x2_i)=\sum_{(x1_i<ins class="diffchange diffchange-inline">,</ins>x2_i){\cap}E_k}d_k=\sum_{(x1_i;x2_i){\cap}E_k}\frac{y_k}{a_k}.\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>That defines eventually the infinite population of points which is indicated in Figure 3, right, as a cloud of points. The population total <math>\tau</math> is then the integral over the entire area of all inclusion zones</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>That defines eventually the infinite population of points which is indicated in Figure 3, right, as a cloud of points. The population total <math>\tau</math> is then the integral over the entire area of all inclusion zones</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 41:</td>
<td colspan="2" class="diff-lineno">Line 39:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Observe that this implies to integrate also outside the areal sampling frame of sample points where inclusion zones of border trees are outside the defined inventory area. This leads to the border correction issue which is dealt with in [[Fixed area plots at the stand boundary]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Observe that this implies to integrate also outside the areal sampling frame of sample points where inclusion zones of border trees are outside the defined inventory area. This leads to the border correction issue which is dealt with in [[Fixed area plots at the stand boundary]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>From that infinite population, a sample of size n is selected. From a sample point i with the grid coordinates <del class="diffchange diffchange-inline">x1i</del>, <del class="diffchange diffchange-inline">x2i </del>the estimated total <del class="diffchange diffchange-inline"> </del>of forest area A derives from the Hansen-Hurwitz-estimator <del class="diffchange diffchange-inline">(this </del>estimator <del class="diffchange diffchange-inline">is dealt with in chapter 5.9) </del>from , where f() is the selection probability. Assuming independent random sampling, <del class="diffchange diffchange-inline"> </del>for all sample points. It follows . One may interpret this estimator also such that each observation <del class="diffchange diffchange-inline">yk </del>is expanded with the plot-specific expansion factor <del class="diffchange diffchange-inline">EFk</del>=A<del class="diffchange diffchange-inline">/ak</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>From that infinite population, a sample of size <ins class="diffchange diffchange-inline">''</ins>n<ins class="diffchange diffchange-inline">'' </ins>is selected. From a sample point <ins class="diffchange diffchange-inline">''</ins>i<ins class="diffchange diffchange-inline">'' </ins>with the grid coordinates <ins class="diffchange diffchange-inline"><math>x1_i</math></ins>, <ins class="diffchange diffchange-inline"><math>x2_i</math> </ins>the estimated total <ins class="diffchange diffchange-inline"><math>\hat\tau_i</math> </ins>of forest area <ins class="diffchange diffchange-inline">''</ins>A<ins class="diffchange diffchange-inline">'' </ins>derives from the <ins class="diffchange diffchange-inline">[[Sampling with unequal selection probabilities#The </ins>Hansen-Hurwitz-estimator<ins class="diffchange diffchange-inline">|Hansen-Hurwitz-</ins>estimator<ins class="diffchange diffchange-inline">]] </ins>from</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\hat\tau_i=\frac{d(x1_ix2_i)}{f8x1_ix2_i)}</ins>,<ins class="diffchange diffchange-inline">\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>where <ins class="diffchange diffchange-inline"><math></ins>f()<ins class="diffchange diffchange-inline"></math> </ins>is the selection probability. Assuming independent random sampling,</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>f(x1,x2)=\frac{1}{A}\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>for all sample points. It follows</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:\bar\tau_i=A\sum_{x1_i,x2_i){\cap}E_k}\frac{y_k}{a_k}</ins>.<ins class="diffchange diffchange-inline">\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>One may interpret this estimator also such that each observation <ins class="diffchange diffchange-inline"><math>y_k</math> </ins>is expanded with the plot-specific expansion factor</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>EF_k</ins>=<ins class="diffchange diffchange-inline">\frac{</ins>A<ins class="diffchange diffchange-inline">}{a_k}</ins>.<ins class="diffchange diffchange-inline">\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">From a sample of ''n'' randomly selected sample points, the total is eventually estimated as</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">From a sample of </del>n <del class="diffchange diffchange-inline">randomly selected sample points, the total is eventually estimated as </del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\bar\tau=\frac{1}{</ins>n<ins class="diffchange diffchange-inline">}\sum_{i}^n\bar\tau_i</ins>.<ins class="diffchange diffchange-inline">\,</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the mirage technique for border plot correction) than with the discrete plot population approach.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With this infinite population approach, most properties of plots and other issues in forest inventory sampling can be much better described (including the mirage technique for border plot correction) than with the discrete plot population approach.</div></td></tr>
</table>Aspange