SOCR EduMaterials ModelerActivities MixtureModel 1 - Revision history

IvoDinov: /* Model Fitting */

2009-12-19T22:10:10Z

‎Model Fitting

IvoDinov: /* Model Fitting */

2009-12-19T22:06:52Z

‎Model Fitting

IvoDinov: /* Model Fitting */

2009-12-19T22:06:10Z

‎Model Fitting

IvoDinov: /* Model Fitting */

2009-12-14T02:45:42Z

‎Model Fitting

IvoDinov: /* Model Fitting */

2009-12-14T02:44:30Z

‎Model Fitting

IvoDinov: /* Model Fitting */

2009-12-14T02:41:38Z

‎Model Fitting

IvoDinov: /* Model Fitting */ formula typo

2009-12-14T02:40:52Z

‎Model Fitting: formula typo

IvoDinov: /* Model Fitting */ added the Z test fragment at the end of the sub-section

2009-12-14T02:38:01Z

‎Model Fitting: added the Z test fragment at the end of the sub-section

IvoDinov at 06:26, 20 June 2009

2009-06-20T06:26:00Z

IvoDinov: /* See also */

2009-01-22T17:35:08Z

‎See also

@@ Line 18: / Line 18: @@
 ===Model Fitting===
-Now go back to the SOCR [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html Modeler] browser (where you did the data sampling). Choose Mixed-Model-Fit from the drop-down list in the left panel. <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig4.jpg|400px]]</center>
+Now go back to the SOCR [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html Modeler] browser (where you did the data sampling). Choose [http://www.socr.ucla.edu/htmls/mod/MixFit_Modeler.html Mixed-Model-Fit] from the drop-down list in the left panel. <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig4.jpg|400px]]</center>
 * We will now try to fit a 2-component mixture of Gaussian (Normal) distributions to this Bimodal Laplace distribution (of the generated sample). You may need to click the '''Re-Initialize''' button a few times. The [http://repositories.cdlib.org/socr/EM_MM Expectation-Maximization algorithm] used to estimate the mixture distribution parameters is unstable and will produce somewhat different results for different initial conditions. Hence, you may need to re-initialize the algorithm a few times until a visually satisfactory result is obtained. <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig5.jpg|400px]]</center>
 * Notice the quantitative results of this mixture model fitting protocol (in the '''Results''' panel). Recall that we sampled 100 observations from Laplace distribution with mean of zero (not Normal Gaussian, which we could also have done and the fit would have been much better, of course) and then another 100 observations from Laplace distribution with mean = 20.0. In this case, the reported estimates of the means of the two Gaussian mixtures are 0 and 22 (pretty close to the original/theoretical means). We could have also fit in a mixture of 3 (or more) Gaussian mixture components, if we had a reason to believe that the data distribution is tri- (or higher-)modal, and therefore, requires a multi-modal mixture fit.

@@ Line 24: / Line 24: @@
 * There are statistical tests added to assess:
 ** how statistically significant are the mean values of any pair of Gaussian models <math>\left \{N(\mu_1,\sigma_1^2), N(\mu_2,\sigma_2^2)\right\}</math> in the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
-** And how goot the model is as a whole, using the [[SOCR_EduMaterials_AnalysisActivities_KolmogorovSmirnoff | Kolmogorov-Smirnoff Test]].
+** And how good the overall mixture model fir tot he data is, using the [[SOCR_EduMaterials_AnalysisActivities_KolmogorovSmirnoff | Kolmogorov-Smirnoff Test]].
 ===Caution===

@@ Line 22: / Line 22: @@
 * Notice the quantitative results of this mixture model fitting protocol (in the '''Results''' panel). Recall that we sampled 100 observations from Laplace distribution with mean of zero (not Normal Gaussian, which we could also have done and the fit would have been much better, of course) and then another 100 observations from Laplace distribution with mean = 20.0. In this case, the reported estimates of the means of the two Gaussian mixtures are 0 and 22 (pretty close to the original/theoretical means). We could have also fit in a mixture of 3 (or more) Gaussian mixture components, if we had a reason to believe that the data distribution is tri- (or higher-)modal, and therefore, requires a multi-modal mixture fit.
 <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig6.jpg|400px]]</center>
-* There are statistical tests added to assess how statistically significant are the mean values of any pair of Gaussian models (<math>\left \{N(\mu_1,\sigma_1^2), N(\mu_2,\sigma_2^2)\right\}</math> in the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
+* There are statistical tests added to assess:
 ===Caution===

@@ Line 22: / Line 22: @@
 * Notice the quantitative results of this mixture model fitting protocol (in the '''Results''' panel). Recall that we sampled 100 observations from Laplace distribution with mean of zero (not Normal Gaussian, which we could also have done and the fit would have been much better, of course) and then another 100 observations from Laplace distribution with mean = 20.0. In this case, the reported estimates of the means of the two Gaussian mixtures are 0 and 22 (pretty close to the original/theoretical means). We could have also fit in a mixture of 3 (or more) Gaussian mixture components, if we had a reason to believe that the data distribution is tri- (or higher-)modal, and therefore, requires a multi-modal mixture fit.
 <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig6.jpg|400px]]</center>
-* There are statistical tests added to assess how statistically significant are the mean values of any pair of Gaussian models (<math>N(\mu_1,\sigma_1^2)</math> and <math>N(\mu_2,\sigma_2^2)</math>) in the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
+* There are statistical tests added to assess how statistically significant are the mean values of any pair of Gaussian models (<math>\left \{N(\mu_1,\sigma_1^2), N(\mu_2,\sigma_2^2)\right\}</math> in the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
 ===Caution===

@@ Line 22: / Line 22: @@
 * Notice the quantitative results of this mixture model fitting protocol (in the '''Results''' panel). Recall that we sampled 100 observations from Laplace distribution with mean of zero (not Normal Gaussian, which we could also have done and the fit would have been much better, of course) and then another 100 observations from Laplace distribution with mean = 20.0. In this case, the reported estimates of the means of the two Gaussian mixtures are 0 and 22 (pretty close to the original/theoretical means). We could have also fit in a mixture of 3 (or more) Gaussian mixture components, if we had a reason to believe that the data distribution is tri- (or higher-)modal, and therefore, requires a multi-modal mixture fit.
 <center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig6.jpg|400px]]</center>
-* There are statistical tests added to assess how statistically significant are the mean values of any pair of Gaussian models part of the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
+* There are statistical tests added to assess how statistically significant are the mean values of any pair of Gaussian models (<math>N(\mu_1,\sigma_1^2)</math> and <math>N(\mu_2,\sigma_2^2)</math>) in the Mixture Distributions. Normal Z tests are use for this assessment <math>\left \{ Z_o=\frac{\mu_1-\mu_2}{\sqrt{\frac{\sigma_1^2}{N_1}+ \frac{\sigma_2^2}{N_2}}} \sim N(0,1^2)\right \}</math>.
 ===Caution===

@@ Line 29: / Line 29: @@
 * [http://socr.ucla.edu/htmls/dist/Mixture_Distribution.html SOCR Mixture-Distribution applet]
 * [[SOCR_EduMaterials_Activities_2D_PointSegmentation_EM_Mixture | SOCR 2D Mixture Modeling Activity]]
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