In gbonte/gbcode: Code from the handbook "Statistical foundations of machine learning"

Shiny dashboard "Statistical foundations of machine learning"

Classification and assessment

Common left panel:

• Number of samples: number of samples of observed dataset

Univariate mixture

Goal: visualize the relation between posterior probability and class conditional density Univariate continuous input ${\bf x}$ and binary class taking two possible values: red and green. Both classes have a gaussian inverse conditional distribution $p({\bf x}=x| y)$

$p(x|{\bf y}=\text{red}) \sim {\mathcal N} (\mu_1, \sigma_1^2)$

$p(x|{\bf y}=\text{green}) \sim {\mathcal N} (\mu_2, \sigma_2^2)$

Top left sliders:

• $\mu_1$: mean of red class conditional density
• $\sigma^2_1$: variance of red class conditional density
• $\mu_2$: mean of green conditional density
• $\sigma^2_2$: variance of green class conditional density
• $P_1$: prior probability $P({\bf y}=\text{red})$

Top right: visualization of red and green class conditional densities together with the posterior probability function $P({\bf y}=\text{red}| x)$.

Suggested manipulation:

• change the class conditional density parameters and see the impact on samples and posterior probability
• change the prior red probability and see the impact on samples and posterior probability

Linear discriminant

Goal: visualize the relation between bivariate class conditional densities and linear discriminant. Bivariate continuous input ${\bf x}$ and binary class taking two possible values: red and green. Both classes have a bivariate gaussian class conditional density $p({\bf x}=x| y)$

$p(x|{\bf y}=\text{red}) \sim {\mathcal N} ([\mu_{1x},\mu_{1y}]^T, \sigma_1^2 I_2)$

$p(x|{\bf y}=\text{green}) \sim {\mathcal N} ([\mu_{2x},\mu_{2y}]^T, \sigma_2^2 I_2)$ where $I_2$ is the diagonal [2,2] matrix.

Top left sliders:

• $\mu_{1x}$: abscissa of mean of red class conditional density
• $\mu_{1y}$: ordinate of mean of red class conditional density
• $\sigma^2_1$: variance diagonal term of the red class conditional covariance
• $\mu_{2x}$: abscissa of mean of green class conditional density
• $\mu_{2y}$: ordinate of mean of green class conditional density
• $\sigma^2_2$: variance diagonal green of the red class conditional covariance
• $P_1$: prior probability $P({\bf y}=\text{red})$

Suggested manipulation:

• change the class conditional density parameters and see the impact on discriminant position
• change the priod red probability and see the impact on discriminant position

Perceptron

Goal: visualize the iteration of the gradient based minimization of the hyperplane misclassification

Bivariate continuous input ${\bf x}$ and binary class taking two possible values: red and green. Both classes have a bivariate gaussian class conditional density $p({\bf x}=x| y)$

$p(x|{\bf y}=\text{red}) \sim {\mathcal N} ([\mu_{1x},\mu_{1y}]^T, \sigma_1^2 I_2)$

$p(x|{\bf y}=\text{green}) \sim {\mathcal N} ([\mu_{2x},\mu_{2y}]^T, \sigma_2^2 I_2)$ where $I_2$ is the diagonal [2,2] matrix.

Top left sliders:

• $\mu_{1x}$: abscissa of mean of red class conditional density
• $\mu_{1y}$: ordinate of mean of red class conditional density
• $\sigma^2_1$: variance diagonal term of the red class conditional covariance
• $\mu_{2x}$: abscissa of mean of green class conditional density
• $\mu_{2y}$: ordinate of mean of green class conditional density
• $\sigma^2_2$: variance diagonal green of the red class conditional covariance
• $P_1$: prior probability $P({\bf y}=\text{red})$
• # steps: number of gradient iteration steps carried out at each click of "Gradient step Nnet" button
• $\eta$: learning rate

Suggested manipulation:

• change the learning rate and see the impact on the evolution of the perceptron hyperplane
• click on the SVM button and see the corresponding separating hyperplane

Assessment

Goal: visualize the relation between ROC curve, PR curve, confusion matrix and classifier threshold. Univariate continuous input ${\bf x}$ and binary class taking two possible values: red (-) and green (+). Both classes have a gaussian inverse conditional distribution $p({\bf x}=x| y)$

$p(x|{\bf y}=\text{red})= p(x|{\bf y}=\text{(-)}) \sim {\mathcal N} (\mu_{(-)}, \sigma_{(-)}^2)$

$p(x|{\bf y}=\text{green})= p(x|{\bf y}=\text{(+)}) \sim {\mathcal N} (\mu_{(+)}, \sigma_{(+)}^2)$

Top left sliders:

• $\mu_{(-)}$: mean of negative class conditional density
• $\sigma^2_{(-)}$: variance of negative class conditional density
• $\mu_{(+)}$: mean of positive conditional density
• $\sigma^2_{(+)}$: variance of positive class conditional density
• $P_{(-)}$: prior probability of negative $P({\bf y}=\text{(-)})$
• thr: threshold of confusion matrix

center left: visualization of threshold and data distribution. The color of the dashed areas identify the classes returned by the classifier

center right: visualization of ROC curve: red dot is the ROC point associated to the threshold. Title contains the TPR, FPR associated to the threshold and the Area under the ROC curve.

bottom left: confusion matrix associated to the threshold together with assessment statistics (TPR, TNT, and so on)

bottom right: visualization of PR curve: red dot is the PR point associated to the threshold

Suggested manipulation:

• change the threshold and see the impact on confusion matrix, assessment statistics, position in the ROC curve, position in the PR curve

• change the class conditional densities parameters (or the prior probability) and see the impact on confusion matrix, assessment statistics, ROC curve and PR curve. for instance what if the two class conditional distribution become closer?

gbonte/gbcode documentation built on Feb. 27, 2024, 7:38 a.m.