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Morten Hjorth-Jensen
2021-09-06 22:05:17 +02:00
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@@ -2990,9 +2990,7 @@ eigenvalues ordered in a descending way, that is \( \sigma_i \geq
\sigma_{i+1} \).
<p>
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
</section>
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@@ -2944,9 +2944,7 @@ eigenvalues ordered in a descending way, that is \( \sigma_i \geq
\sigma_{i+1} \).
<p>
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
<p>
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@@ -2949,9 +2949,7 @@ eigenvalues ordered in a descending way, that is \( \sigma_i \geq
\sigma_{i+1} \).
<p>
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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@@ -3928,9 +3928,7 @@
"eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n",
"\\sigma_{i+1}$.\n",
"\n",
"For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.\n",
"Actually, calculating the variance of $\\boldsymbol{X}\\boldsymbol{v}_j$ shows that this quantity is equal to $\\sigma_j^2/n$.\n",
"With a parameter $\\lambda$ we can thus shrink the role of specific parameters. \n",
"For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.\n",
"\n",
"\n",
"## More interpretations\n",
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@@ -2436,9 +2436,7 @@ $\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has
eigenvalues ordered in a descending way, that is $\sigma_i \geq
\sigma_{i+1}$.
For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
Actually, calculating the variance of $\bm{X}\bm{v}_j$ shows that this quantity is equal to $\sigma_j^2/n$.
With a parameter $\lambda$ we can thus shrink the role of specific parameters.
For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
!split
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