Merge pull request #6 from Machine-Learning-Foundations/dev_kons

math fixes and note about pegel axis

Author: kgasenzer

README.md

@@ -34,13 +34,13 @@ With n=2,
 
 $$\mathbf{A}_2^{\dagger}\mathbf{b} = \mathbf{x}$$
 
-will produce the coefficients for a straight line. Evaluate your first-degree polynomial via ax+b.
+will produce the coefficients for a straight line. Evaluate your first-degree polynomial via $ax+b$.
 Plot the result using `matplotlib.pyplot`'s `plot` function.
 
 
 #### Fitting a Polynomial to a function:
 The straight line above is insufficient to model the data. Using your 
-implementation of `set_up_point_matrix`, set n=300 (to set up a square matrix) and fit the polynomial
+implementation of `set_up_point_matrix` set $m=300$ (to set up a square matrix) and fit the polynomial
 by computing
 
 $$\mathbf{A}^{\dagger}\mathbf{b} = \mathbf{x}_{\text{fit}}.$$
@@ -66,30 +66,35 @@ into the form:
 $$ \mathbf{A} = \mathbf{U} \Sigma \mathbf{V}^T 
 $$
 
-In the SVD-Form computing, the inverse is simple. Swap U and V  and replace every of the m singular values with it's inverse
+In the SVD-Form computing, the inverse is simple. Swap $U$ and $V$  and replace every of the m singular values with it's inverse
 
 $$1/\sigma_i .$$
 
+This results in the matrix 
+```math
+\Sigma^\dagger = \begin{pmatrix}
+      \sigma_1^{-1} & & & \\\\
+      &  \ddots & \\\\
+      &  & \sigma_m^{-1} \\\\ \hline
+      & 0 &
+\end{pmatrix}
+```
+
 A solution to the overfitting problem is to filter the singular values.
 Compute a diagonal for a filter matrix by evaluating:
 
 $$f_i = \sigma_i^2 / (\sigma_i^2 + \epsilon)$$
 
-The idea is to compute a loop over i for all of the m singular values.
-Roughly speaking, multiplication by f underscore i will filter a singular value when
+The idea is to compute a loop over $i$ for all of the m singular values.
+Roughly speaking multiplication by $f_i$ will filter a singular value when
 
 $$\sigma_i \lt \epsilon .$$
 
 Apply the regularization by computing:
 
 
 $$
-    \mathbf{x}_r= \mathbf{V} \mathbf{F} \begin{pmatrix}
-      \sigma_1^{-1} & & & \\\\
-      &  \ddots & \\\\
-      &  & \sigma_n^{-1} \\\\ \hline
-      & 0 &
-    \end{pmatrix}
+    \mathbf{x}_r= \mathbf{V} \mathbf{F} \mathbf{\Sigma}^\dagger
     \mathbf{U}^T \mathbf{b}
 $$
 
@@ -98,14 +103,14 @@ with
 
 $$\mathbf{V} \in \mathbb{R}^{n,n}, \mathbf{F} \in \mathbb{R}^{n,n}, \Sigma^{\dagger} \in \mathbb{R}^{n,m}, \mathbf{U} \in \mathbb{R}^{m,m} \text{ and } \mathbf{b} \in \mathbb{R}^{m,1}.$$
 
-Setting n=300 turns A into a square matrix. In this case, the zero block in the sigma-matrix disappears.
+Setting $m=300$ turns $A$ into a square matrix. In this case, the zero block in the sigma-matrix disappears.
 Plot the result for epsilon equal to 0.1, 1e-6, and 1e-12.
 
 #### Model Complexity (Optional):
 Another solution to the overfitting problem is reducing the complexity of the model.
 To assess the quality of polynomial fit to the data, compute and plot the Mean Squared Error (Mean Squared Error (MSE) measure how close the regression line is to data points) for every degree of polynomial up to 20.
 
-MSE can be calculated using the following equation, where N is the number of samples, $y_i$ is the original point and $\hat{y_i}$ is the predictied output.
+MSE can be calculated using the following equation, where $N$ is the number of samples, $y_i$ is the original point and $\hat{y_i}$ is the predictied output.
 $$MSE=\frac{1}{N} \sum_{i=1}^{N} (y_i-\hat{y_i})^2$$
 
 Are the degree of the polynomial and the MSE linked?
@@ -127,8 +132,7 @@ Plot the result. Compute the zero. When do the regression line and the x-axis in
 
 #### Fitting a higher order Polynomial:
 
-Re-using the code you wrote for the proof of concept task, fit a polynomial of degree 20 to the data.
-Study the scaling of the timestamp values. Is the range suitable for stable numerical computations?
+Re-using the code you wrote for the proof of concept task, fit a polynomial of degree 20 to the data. Before plotting have a closer look at `datetime_stamps` and its values and scale the axis appropriately.
 Plot the result.
 
 #### Regularization: