ZsoltTest - Versionsgeschichte

SAFFER Zsolt am 4. März 2024 um 20:07 Uhr

2024-03-04T20:07:38Z

← Nächstältere Version		Version vom 4. März 2024, 20:07 Uhr
Zeile 246:		Zeile 246:
	Applying the principle of MLE to a conditional likelihood function <math display="inline">p(y\|{\bf x})</math>, the conditional likelihood function from a known distribution family with unknown parameter vector <math display="inline">{\boldsymbol{\theta}}</math> becomes <math display="inline">p(y\|{\bf x}, {\boldsymbol{\theta}})</math>. Thus the ML estimate of the parameter vector <math display="inline">{\bf \theta}</math> in such case is given by		Applying the principle of MLE to a conditional likelihood function <math display="inline">p(y\|{\bf x})</math>, the conditional likelihood function from a known distribution family with unknown parameter vector <math display="inline">{\boldsymbol{\theta}}</math> becomes <math display="inline">p(y\|{\bf x}, {\boldsymbol{\theta}})</math>. Thus the ML estimate of the parameter vector <math display="inline">{\bf \theta}</math> in such case is given by

	<math display="block">{\boldsymbol{\theta}}_{ML} = \arg \max_{{\boldsymbol{\theta}}}\sum_{k=1}^{K} log~ p(y_k\|{\bf x}_k, {\boldsymbol{\theta}}).</math>		<math display="block">{\boldsymbol{\theta}}_{ML} = \arg \max_{{\boldsymbol{\theta}}}\sum_{k=1}^{K} log p(y_k\|{\bf x}_k, {\boldsymbol{\theta}}).</math>

	Now we select the normal distribution as the known distribution family and set the mean of the distribution to be the predicted output value, <math display="inline">\hat{y}</math> and the variance of the distribution to a fixed value <math display="inline">\sigma^2</math>. Then the conditional likelihood function, which in this case a conditional pdf, becomes <math display="inline">p(y\|{\bf x},{\boldsymbol{\theta}}) = N(y\| \hat{y}({\bf x},{\bf w}),\sigma^2)</math>, where <math display="inline">N(y\|\mu,\sigma^2)</math> stands for the pdf of the normal distribution with mean <math display="inline">\mu</math> and variance <math display="inline">\sigma^2</math>. Applying the expression of the pdf of the normal distribution <math display="inline">N(y\|\mu,\sigma^2)= \frac{1}{\sqrt(2 \pi \sigma^2)}exp\left(-\frac{1}{2} \frac{(y-\mu)^2}{\sigma^2} \right)</math>, we get for the ML estimate of the predicted output values, <math display="inline">{\bf \hat{y}}</math>		Now we select the normal distribution as the known distribution family and set the mean of the distribution to be the predicted output value, <math display="inline">\hat{y}</math> and the variance of the distribution to a fixed value <math display="inline">\sigma^2</math>. Then the conditional likelihood function, which in this case a conditional pdf, becomes <math display="inline">p(y\|{\bf x},{\boldsymbol{\theta}}) = N(y\| \hat{y}({\bf x},{\bf w}),\sigma^2)</math>, where <math display="inline">N(y\|\mu,\sigma^2)</math> stands for the pdf of the normal distribution with mean <math display="inline">\mu</math> and variance <math display="inline">\sigma^2</math>. Applying the expression of the pdf of the normal distribution <math display="inline">N(y\|\mu,\sigma^2)= \frac{1}{\sqrt(2 \pi \sigma^2)}exp\left(-\frac{1}{2} \frac{(y-\mu)^2}{\sigma^2} \right)</math>, we get for the ML estimate of the predicted output values, <math display="inline">{\bf \hat{y}}</math>

SAFFER Zsolt am 4. März 2024 um 19:56 Uhr

2024-03-04T19:56:12Z

Änderungen zeigen

SAFFER Zsolt am 4. März 2024 um 15:55 Uhr

2024-03-04T15:55:15Z

← Nächstältere Version		Version vom 4. März 2024, 15:55 Uhr
Zeile 1:		Zeile 1:

			<math display="block">\begin{aligned}
			\nabla_{{\bf w}} \left( \lVert {\bf X}{\bf w} - {\bf y} \rVert_2^2 \right) &&\hspace{-5mm}= \nabla_{{\bf w}} \left({\bf X}{\bf w} - {\bf y} \right)^T \left({\bf X}{\bf w} - {\bf y} \right) = \nabla_{{\bf w}} \left( {\bf w}^T {\bf X}^T{\bf X}{\bf w} - 2 {\bf w}^T {\bf X}^T {\bf y} + {\bf y}^T {\bf y}\right) \nonumber \\
			&&\hspace{-5mm}= 2 {\bf X}^T{\bf X}{\bf w} - 2 {\bf X}^T {\bf y} = 0, \nonumber
			\end{aligned}</math>

			<math display="block">{\mbox{\boldmath$\theta$}}_{ML} = \arg \max_{{\mbox{\boldmath$\theta$}}} p({\bf z}\|{\mbox{\boldmath$\theta$}}) = \arg \max_{{\mbox{\boldmath$\theta$}}} p(z_1,\ldots,z_M\|{\mbox{\boldmath$\theta$}}) = \arg \max_{{\mbox{\boldmath$\theta$}}} \prod_{m=1}^{M} p(z_m\|{\mbox{\boldmath$\theta$}}),</math>

			<math display="block">\begin{aligned}
			{\bf \hat{y}} &&\hspace{-5mm}= \arg \max_{\bf \hat{y}}\sum_{k=1}^{K} log N(y_k\| \hat{y_k}({\bf x_k},{\bf w}),\sigma^2)\nonumber \\
			&&\hspace{-5mm} = \arg \max_{\bf \hat{y}}\sum_{k=1}^{K} log \left(\frac{1}{\sqrt(2 \pi \sigma^2)}exp\left(-\frac{1}{2} \frac{(y_k-\hat{y_k})^2}{\sigma^2} \right)\right) \nonumber \\
			&&\hspace{-5mm}= \arg \max_{\bf \hat{y}} \left(-K log \sigma -\frac{K}{2} log(2 \pi) - \sum_{k=1}^{K} \frac{(y_k-\hat{y_k})^2}{2\sigma^2} \right) = \arg \max_{\bf \hat{y}} \left(-\sum_{k=1}^{K} ((y_k-\hat{y_k})^2\right) \nonumber \\
			&&\hspace{-5mm}= \arg \min_{\bf \hat{y}} \sum_{k=1}^{K} ((y_k-\hat{y_k})^2 = \arg \min_{\bf \hat{y}} \lVert {\bf \hat{y}} - {\bf y} \rVert_2^2 \nonumber.
			\end{aligned}</math>

			<math display="block">\begin{aligned}
			\lambda_{i,j} = \left\{
			\begin{array}{l}
			0, \mbox{~if~} i=j \\
			1, \mbox{~if~} i \neq j
			\end{array}
			\right\}, ~i,j = 1, \ldots, C. \nonumber
			\end{aligned}</math>


			‎<comments /> blablabla



	<span id="einführung-in-computational-intelligence-und-ai"></span>		<span id="einführung-in-computational-intelligence-und-ai"></span>
	= Einführung in Computational Intelligence und AI =		= Einführung in Computational Intelligence und AI =

SAFFER Zsolt am 1. Februar 2024 um 17:16 Uhr

2024-02-01T17:16:11Z

← Nächstältere Version		Version vom 1. Februar 2024, 17:16 Uhr
Zeile 212:		Zeile 212:

	<math display="block">\begin{aligned}		<math display="block">\begin{aligned}
	\nabla_{{\bf w}} \left( \lVert {\bf X}{\bf w} - {\bf y} \rVert_2^2 \right) &&\hspace{-~~5mm~~}= \nabla_{{\bf w}} \left({\bf X}{\bf w} - {\bf y} \right)^T \left({\bf X}{\bf w} - {\bf y} \right) = \nabla_{{\bf w}} \left( {\bf w}^T {\bf X}^T{\bf X}{\bf w} - 2 {\bf w}^T {\bf X}^T {\bf y} + {\bf y}^T {\bf y}\right) ~~\nonumber~~ \\		\nabla_{{\bf w}} \left( \lVert {\bf X}{\bf w} - {\bf y} \rVert_2^2 \right) &&\hspace{-5 mm}= \nabla_{{\bf w}} \left({\bf X}{\bf w} - {\bf y} \right)^T \left({\bf X}{\bf w} - {\bf y} \right) = \nabla_{{\bf w}} \left( {\bf w}^T {\bf X}^T{\bf X}{\bf w} - 2 {\bf w}^T {\bf X}^T {\bf y} + {\bf y}^T {\bf y}\right) \\
	&&\hspace{-~~5mm~~}= 2 {\bf X}^T{\bf X}{\bf w} - 2 {\bf X}^T {\bf y} = 0, ~~\nonumber~~		&&\hspace{-5 mm}= 2 {\bf X}^T{\bf X}{\bf w} - 2 {\bf X}^T {\bf y} = 0,
	\end{aligned}</math>		\end{aligned}</math>

SAFFER Zsolt am 1. Februar 2024 um 17:04 Uhr

2024-02-01T17:04:24Z

Änderungen zeigen

SAFFER Zsolt: Der Seiteninhalt wurde durch einen anderen Text ersetzt: „ffgg“

2024-02-01T16:57:07Z

Der Seiteninhalt wurde durch einen anderen Text ersetzt: „ffgg“

Änderungen zeigen

SAFFER Zsolt am 1. Februar 2024 um 16:51 Uhr

2024-02-01T16:51:07Z

Änderungen zeigen

JUNGBAUER Christoph: Die Seite wurde neu angelegt: „Testseite“

2024-02-01T06:25:16Z

Die Seite wurde neu angelegt: „Testseite“

Neue Seite

Testseite

← Nächstältere Version		Version vom 4. März 2024, 20:07 Uhr
Zeile 246:		Zeile 246:
	Applying the principle of MLE to a conditional likelihood function <math display="inline">p(y\|{\bf x})</math>, the conditional likelihood function from a known distribution family with unknown parameter vector <math display="inline">{\boldsymbol{\theta}}</math> becomes <math display="inline">p(y\|{\bf x}, {\boldsymbol{\theta}})</math>. Thus the ML estimate of the parameter vector <math display="inline">{\bf \theta}</math> in such case is given by		Applying the principle of MLE to a conditional likelihood function <math display="inline">p(y\|{\bf x})</math>, the conditional likelihood function from a known distribution family with unknown parameter vector <math display="inline">{\boldsymbol{\theta}}</math> becomes <math display="inline">p(y\|{\bf x}, {\boldsymbol{\theta}})</math>. Thus the ML estimate of the parameter vector <math display="inline">{\bf \theta}</math> in such case is given by

	<math display="block">{\boldsymbol{\theta}}_{ML} = \arg \max_{{\boldsymbol{\theta}}}\sum_{k=1}^{K} log~ p(y_k\|{\bf x}_k, {\boldsymbol{\theta}}).</math>		<math display="block">{\boldsymbol{\theta}}_{ML} = \arg \max_{{\boldsymbol{\theta}}}\sum_{k=1}^{K} log p(y_k\|{\bf x}_k, {\boldsymbol{\theta}}).</math>

	Now we select the normal distribution as the known distribution family and set the mean of the distribution to be the predicted output value, <math display="inline">\hat{y}</math> and the variance of the distribution to a fixed value <math display="inline">\sigma^2</math>. Then the conditional likelihood function, which in this case a conditional pdf, becomes <math display="inline">p(y\|{\bf x},{\boldsymbol{\theta}}) = N(y\| \hat{y}({\bf x},{\bf w}),\sigma^2)</math>, where <math display="inline">N(y\|\mu,\sigma^2)</math> stands for the pdf of the normal distribution with mean <math display="inline">\mu</math> and variance <math display="inline">\sigma^2</math>. Applying the expression of the pdf of the normal distribution <math display="inline">N(y\|\mu,\sigma^2)= \frac{1}{\sqrt(2 \pi \sigma^2)}exp\left(-\frac{1}{2} \frac{(y-\mu)^2}{\sigma^2} \right)</math>, we get for the ML estimate of the predicted output values, <math display="inline">{\bf \hat{y}}</math>		Now we select the normal distribution as the known distribution family and set the mean of the distribution to be the predicted output value, <math display="inline">\hat{y}</math> and the variance of the distribution to a fixed value <math display="inline">\sigma^2</math>. Then the conditional likelihood function, which in this case a conditional pdf, becomes <math display="inline">p(y\|{\bf x},{\boldsymbol{\theta}}) = N(y\| \hat{y}({\bf x},{\bf w}),\sigma^2)</math>, where <math display="inline">N(y\|\mu,\sigma^2)</math> stands for the pdf of the normal distribution with mean <math display="inline">\mu</math> and variance <math display="inline">\sigma^2</math>. Applying the expression of the pdf of the normal distribution <math display="inline">N(y\|\mu,\sigma^2)= \frac{1}{\sqrt(2 \pi \sigma^2)}exp\left(-\frac{1}{2} \frac{(y-\mu)^2}{\sigma^2} \right)</math>, we get for the ML estimate of the predicted output values, <math display="inline">{\bf \hat{y}}</math>