https://gssc.esa.int/navipedia/index.php?action=history&feed=atom&title=Weighted_Least_Square_Solution_%28WLS%29
Weighted Least Square Solution (WLS) - Revision history
2026-04-28T14:42:19Z
Revision history for this page on the wiki
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https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15130&oldid=prev
Gema.Cueto at 16:38, 3 April 2020
2020-04-03T16:38:47Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:38, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math> \displaystyle \|{\mathbf <ins style="font-weight: bold; text-decoration: none;">{</ins>Y}-{\mathbf <ins style="font-weight: bold; text-decoration: none;">{</ins>\hat{Y<ins style="font-weight: bold; text-decoration: none;">}</ins>}\|^{^{ 2}}_{_{\mathbf <ins style="font-weight: bold; text-decoration: none;">{</ins>W<ins style="font-weight: bold; text-decoration: none;">}}}</ins>}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2}}_{_{\mathbf {W}}}}} =\sum{w_i(y_i-\hat{y}_i)^2}</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15129&oldid=prev
Gema.Cueto at 16:38, 3 April 2020
2020-04-03T16:38:08Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:38, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2}}_{_{\mathbf {W}}}}}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2}}_{_{\mathbf {W}}}}<ins style="font-weight: bold; text-decoration: none;">} =\sum{w_i(y_i-\hat{y}_i)^2</ins>}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15128&oldid=prev
Gema.Cueto at 16:37, 3 April 2020
2020-04-03T16:37:29Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:37, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2}}}}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2<ins style="font-weight: bold; text-decoration: none;">}}_{_{\mathbf {W}</ins>}}}}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15127&oldid=prev
Gema.Cueto at 16:36, 3 April 2020
2020-04-03T16:36:22Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:36, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2}}}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{ 2<ins style="font-weight: bold; text-decoration: none;">}</ins>}}}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15126&oldid=prev
Gema.Cueto at 16:34, 3 April 2020
2020-04-03T16:34:43Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:34, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{2}}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|<ins style="font-weight: bold; text-decoration: none;">^{</ins>^{ 2<ins style="font-weight: bold; text-decoration: none;">}</ins>}}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15125&oldid=prev
Gema.Cueto at 16:33, 3 April 2020
2020-04-03T16:33:42Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:33, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{2}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{2<ins style="font-weight: bold; text-decoration: none;">}</ins>}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15124&oldid=prev
Gema.Cueto at 16:33, 3 April 2020
2020-04-03T16:33:14Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:33, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^2</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^<ins style="font-weight: bold; text-decoration: none;">{</ins>2<ins style="font-weight: bold; text-decoration: none;">}</ins></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15123&oldid=prev
Gema.Cueto at 16:30, 3 April 2020
2020-04-03T16:30:53Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:30, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^<ins style="font-weight: bold; text-decoration: none;">2</ins></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15122&oldid=prev
Gema.Cueto at 16:30, 3 April 2020
2020-04-03T16:30:22Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:30, 3 April 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^<del style="font-weight: bold; text-decoration: none;">{^{2}}}</del></math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
</table>
Gema.Cueto
https://gssc.esa.int/navipedia/index.php?title=Weighted_Least_Square_Solution_(WLS)&diff=15121&oldid=prev
Gema.Cueto at 16:29, 3 April 2020
2020-04-03T16:29:13Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:29, 3 April 2020</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The measurements quality can be incorporated in the fitting criteria by introducing a symmetric, positive-definite weighting matrix <math>{\mathbf W}</math> and re-defining the "best-fit" as the condition <ref group="footnotes"> Notice that, if <math>{\mathbf W}</math> is a diagonal matrix, thence, the norm <math>\displaystyle \|{\mathbf Y}-{\mathbf \hat{Y}\|^{^{ 2}}_{_{\mathbf W}} =\sum{w_i(y_i-\hat{y}_i)^2}</math>, where the terms in the sums are weighted by the diagonal elements <math>w_i</math>. Thence, it follows that the measurements associated to large <math>w_i</math> values will contribute more to the adjustment.</ref>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{2}}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math> \displaystyle \|{\mathbf {Y}-{\mathbf {\hat{Y}}\|^{^{2<ins style="font-weight: bold; text-decoration: none;">}</ins>}}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
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