Update README.md

Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>

scikit-learn/pca/README.md

@@ -11,6 +11,8 @@
 
 Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
 
+## Test data
+
 ```python
 import numpy as np
 import matplotlib.pyplot as plt
@@ -36,7 +38,105 @@ data_r = data @ roation_matrix
 
 plt.plot(data[:, 0], data[:, 1], "b.")
 plt.plot(data_r[:, 0], data_r[:, 1], "r.")
+plt.show()
 ```
 
 ![image0](image0.png)
 
+## Train and use PCA​ [sklearn.decomposition.PCA](https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA)
+
+```python
+class sklearn.decomposition.PCA(n_components=None, *, copy=True, whiten=False, svd_solver='auto', tol=0.0, iterated_power='auto', n_oversamples=10, power_iteration_normalizer='auto', random_state=None)
+```
+
+> Principal component analysis (PCA).
+> 
+> Linear dimensionality reduction using Singular Value Decomposition of the data to project it to a lower dimensional space. The input data is centered but not scaled for each feature before applying the SVD.
+> 
+> It uses the LAPACK implementation of the full SVD or a randomized truncated SVD by the method of Halko et al. 2009, depending on the shape of the input data and the number of components to extract.
+> 
+> It can also use the scipy.sparse.linalg ARPACK implementation of the truncated SVD.
+> 
+> Notice that this class does not support sparse input. See TruncatedSVD for an alternative with sparse data.
+
+Attributes
+
+> **explained_variance_** : ndarray of shape (n_components,)
+> The amount of variance explained by each of the selected components. The variance estimation uses n_samples - 1 degrees of freedom.
+> 
+> Equal to n_components largest eigenvalues of the covariance matrix of X.
+
+> **singular_values_** : ndarray of shape (n_components,)
+> The singular values corresponding to each of the selected components. The singular values are equal to the 2-norms of the n_components variables in the lower-dimensional space.
+
+Methods:
+
+```python
+fit(X, y=None)
+```
+
+> **X** : array-like of shape (n_samples, n_features)
+> 
+> Training data, where n_samples is the number of samples and n_features is the number of features.
+
+```python
+transform(X)
+```
+
+> Apply dimensionality reduction to X.
+> 
+> X is projected on the first principal components previously extracted from a training set.
+
+> **X** : array-like of shape (n_samples, n_features)
+> 
+> New data, where n_samples is the number of samples and n_features is the number of features.
+
+
+We rotate the red cloud back. This creates the black cloud. ​This fits nicely with the original data (blue cloud).
+
+**Be aware that the sign of an ​individual axis can switch!!!​**
+
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.decomposition import PCA
+
+rng = np.random.default_rng(1)
+
+a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
+a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
+data_a = np.concatenate((a_x, a_y), axis=1)
+
+b_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
+b_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
+data_b = np.concatenate((b_x, b_y), axis=1)
+
+data = np.concatenate((data_a, data_b), axis=0)
+
+angle = -0.3
+
+roation_matrix = np.array(
+    [[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]]
+)
+data_r = data @ roation_matrix
+
+
+pca = PCA(n_components=2)
+
+# Train
+pca.fit(data_r)
+
+print(pca.explained_variance_ratio_)  # -> [0.52996112 0.47003888]
+print(pca.singular_values_)  # -> [287.55360494 270.80938189]
+
+# Use
+transformed_data = pca.transform(data_r)
+
+
+plt.plot(data[:, 0], data[:, 1], "b.")
+plt.plot(data_r[:, 0], data_r[:, 1], "r.")
+plt.plot(transformed_data[:, 0], transformed_data[:, 1], "k.")
+plt.show()
+```
+
+![image1](image1.png)