Merge branch 'main' into dev_kons

Author: kgasenzer

README.md

@@ -12,27 +12,27 @@ Feel free to read more about nox at https://nox.thea.codes/en/stable/ .
 
 ### Part 1: Proof of concept
 Use `b = pandas.read_csv('./data/noisy_signal.tab')` to load a noisy signal.
-The first part will be concerned with modeling this signal using polynomials.
+The first part will be concerned with modelling this signal using polynomials.
 
 #### Regression:
 Linear regression is usually a good first step. Start by implementing the function
 `set_up_point_matrix` from the `src/regularization.py` module. 
-The function should produce polynomial-coordinate matrices $\mathbf{A}_m$ of the form:
+The function should produce polynomial-coordinate matrices $\mathbf{A}_n$ of the form:
 
 $$
-\mathbf{A}_m = 
+\mathbf{A}_n = 
 \begin{pmatrix}
-          1       & a_1^1    & a_1^2  & \dots & a_1^{m-1}  \\\\ 
-          1       & a_2^1    & a_2^2  & \dots & a_2^{m-1}  \\\\
-          1       & a_3^1    & a_3^2  & \dots & a_3^{m-1}  \\\\
+          1       & a_1^1    & a_1^2  & \dots & a_1^{n-1}  \\\\ 
+          1       & a_2^1    & a_2^2  & \dots & a_2^{n-1}  \\\\
+          1       & a_3^1    & a_3^2  & \dots & a_3^{n-1}  \\\\
           \vdots  & \vdots   & \vdots  & \ddots & \vdots \\\\ 
-          1       & a_n^1    & a_n^2  & \dots & a_n^{m-1}  \\\\
+          1       & a_m^1    & a_m^2  & \dots & a_m^{n-1}  \\\\
    \end{pmatrix}
 $$
 
-With m=2,
+With n=2,
 
-$$\mathbf{A}_m^{\dagger}\mathbf{b} = \mathbf{x}$$
+$$\mathbf{A}_2^{\dagger}\mathbf{b} = \mathbf{x}$$
 
 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.
@@ -101,19 +101,19 @@ $$
 
 with
 
-$$\mathbf{A} \in \mathbb{R}^{m,n}, \mathbf{U} \in \mathbb{R}^{m,m}, \mathbf{V} \in \mathbb{R}^{n,n}, \mathbf{F} \in \mathbb{R}^{m,m}, \Sigma^{\dagger} \in \mathbb{R}^{n,m} \text{ and } \mathbf{b} \in \mathbb{R}^{n,1}.$$
+$$\mathbf{V} \in \mathbb{R}^{n,n}, \mathbf{F} \in \mathbb{R}^{n,n}, \Sigma^{\dagger} \in \mathbb{R}^{n,m}, \mathbf{U} \in \mathbb{R}^{m,m} \text{ and } \mathbf{b} \in \mathbb{R}^{m,1}.$$
 
 Setting $m=300$ 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 upto 20.
+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.
 $$MSE=\frac{1}{N} \sum_{i=1}^{N} (y_i-\hat{y_i})^2$$
 
-From the plot, estimate the optimal degree of polynomial and fit the polynomial with this new degree and compare the regression.
+Are the degree of the polynomial and the MSE linked?
 
 ### Part 2: Real data analysis
 Now we are ready to deal with real data! Feel free to use your favorite time series data or work with the Rhine level data we provide.
@@ -123,7 +123,8 @@ Data source: https://pegel.bonn.de.
 #### Regression:
 The `src/pegel_bonn.py` file already contains code to pre-load the data for you.
 Make the Rhine level measurements your new vector $\mathbf{b}$.
-Generate a matrix A with m=2 using the timestamps for the data set and compute 
+
+Generate a matrix A with n=2 using the timestamps for the data set and compute 
 
 $$\mathbf{A}^{\dagger}\mathbf{b}.$$
 
@@ -134,9 +135,7 @@ Plot the result. Compute the zero. When do the regression line and the x-axis in
 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:
-Something happened around the year 2000. To investigate further, focus on the data from 2000 onward and
-filter the singular values.
+Focus on the data from the year 2000 onward and filter the singular values.
 Matrix A is not square in this case. Consequently, a zero block must appear in your singular value matrix. 
 Plot filtered eigen-polynomials using epsilon equal to 0.1, 1e-3, 1e-9.