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Book4_Ch10_Python_Codes/Bk4_Ch10_01.py

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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch10_01.py
+
+import seaborn as sns
+import numpy as np
+import matplotlib.pyplot as plt 
+import pandas as pd  
+from sklearn.datasets import load_iris
+
+# A copy from Seaborn
+iris = load_iris()
+
+X = iris.data
+y = iris.target
+
+feature_names = ['Sepal length, x1','Sepal width, x2',
+                 'Petal length, x3','Petal width, x4']
+
+# Convert X array to dataframe
+X_df = pd.DataFrame(X, columns=feature_names)
+
+#%% Original data, X
+
+X = X_df.to_numpy();
+
+# Gram matrix, G and orthogonal basis V
+
+G = X.T@X
+D, V = np.linalg.eig(G)
+
+#%%
+
+def heatmap(Matrices,Titles,Ranges,Equal_tags):
+    
+    M1 = Matrices[0]
+    M2 = Matrices[1]
+    M3 = Matrices[2]
+    
+    Title_1 = Titles[0]
+    Title_2 = Titles[1]
+    Title_3 = Titles[2]
+    
+    fig, axs = plt.subplots(1, 5, figsize=(12, 3))
+    
+    plt.sca(axs[0])
+    ax = sns.heatmap(M1,cmap='RdYlBu_r',
+                     vmin = Ranges[0][0], 
+                     vmax = Ranges[0][1],
+                     cbar=False,
+                     xticklabels=False,
+                     yticklabels=False)
+    
+    if Equal_tags[0] == True:
+        ax.set_aspect("equal")
+        
+    plt.title(Title_1)
+    
+    plt.sca(axs[1])
+    plt.title('=')
+    plt.axis('off')
+    
+    plt.sca(axs[2])
+    ax = sns.heatmap(V,cmap='RdYlBu_r',
+                     vmin = Ranges[1][0], 
+                     vmax = Ranges[1][1],
+                     cbar=False,
+                     xticklabels=False,
+                     yticklabels=False)
+    if Equal_tags[1] == True:
+        ax.set_aspect("equal")
+    plt.title(Title_2)
+    
+    plt.sca(axs[3])
+    plt.title('@')
+    plt.axis('off')
+    
+    plt.sca(axs[4])
+    ax = sns.heatmap(V.T,cmap='RdYlBu_r',
+                     vmin = Ranges[2][0], 
+                     vmax = Ranges[2][1],
+                     cbar=False,
+                     xticklabels=False,
+                     yticklabels=False)
+    
+    if Equal_tags[2] == True:
+        ax.set_aspect("equal")
+    plt.title(Title_3)
+
+#%%
+
+def plot_four_figs(X,v_j,idx):
+
+    # Fig 1: X@v_j = z_j
+    
+    z_j = X@v_j
+    Titles = ['$X$',
+              '$v_' + str(idx) + '$',
+              '$z_' + str(idx) + '$']
+    
+    Ranges = [[-2,11],
+              [-1,1],
+              [-2,11]]
+    
+    Equal_tags = [False,True,False]
+    heatmap([X,v_j,z_j],Titles,Ranges,Equal_tags)
+
+    # Fig 2: z@v_j.T = X_j
+    X_j = z_j@v_j.T
+    Titles = ['$z_' + str(idx) + '$',
+              '$v_' + str(idx) + '^T$',
+              '$X_' + str(idx) + '$']
+    
+    Ranges = [[-2,11],
+              [-1,1],
+              [-2,11]]
+    
+    Equal_tags = [False,True,False]
+    
+    heatmap([z_j,v_j.T,X_j],Titles,Ranges,Equal_tags)
+
+    # Fig 3: T_j = v_j@v_j.T
+    T_j = v_j@v_j.T
+    
+    Titles = ['$v_' + str(idx) + '$',
+              '$v_' + str(idx) + '^T$',
+              '$T_' + str(idx) + '$']
+    
+    Ranges = [[-1,1],
+              [-1,1],
+              [-1,1]]
+    
+    Equal_tags = [True,True,True]
+    
+    heatmap([v_j,v_j.T,T_j],Titles,Ranges,Equal_tags)
+
+    
+    # Fig 4: X@T_j = X_j
+    
+    T_j = X@T_j
+    
+    Titles = ['$X$',
+              '$T_' + str(idx) + '$',
+              '$X_' + str(idx) + '$']
+    
+    Ranges = [[-2,11],
+              [-1,1],
+              [-2,11]]
+    
+    Equal_tags = [False,True,False]
+    
+    heatmap([X,T_j,X_j],Titles,Ranges,Equal_tags)
+
+
+#%% First basis vector
+
+v1 = V[:, 0].reshape((-1, 1))
+
+plot_four_figs(X,v1,1)
+
+#%% Second basis vector
+
+v2 = V[:, 1].reshape((-1, 1))
+
+plot_four_figs(X,v2)
+
+#%% Third basis vector
+
+v3 = V[:, 2].reshape((-1, 1))
+
+plot_four_figs(X,v3)
+
+#%% Fourth basis vector
+
+v4 = V[:, 3].reshape((-1, 1))
+
+plot_four_figs(X,v4)
+