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Book4_Ch14_Python_Codes/Streamlit_Bk4_Ch14_04.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
+###############
+
+import plotly.graph_objects as go
+import streamlit as st
+import numpy as np
+import plotly.express as px
+import pandas as pd
+import sympy
+
+def bmatrix(a):
+    """Returns a LaTeX bmatrix
+
+    :a: numpy array
+    :returns: LaTeX bmatrix as a string
+    """
+    if len(a.shape) > 2:
+        raise ValueError('bmatrix can at most display two dimensions')
+    lines = str(a).replace('[', '').replace(']', '').splitlines()
+    rv = [r'\begin{bmatrix}']
+    rv += ['  ' + ' & '.join(l.split()) + r'\\' for l in lines]
+    rv +=  [r'\end{bmatrix}']
+    return '\n'.join(rv)
+
+with st.sidebar:
+    
+    st.latex(r'''
+             A = \begin{bmatrix}
+    a & b\\
+    b & c
+    \end{bmatrix}''')
+    
+    a = st.slider('a',-2.0, 2.0, step = 0.05, value = 1.0)
+    b = st.slider('b',-2.0, 2.0, step = 0.05, value = 0.0)  
+    c = st.slider('c',-2.0, 2.0, step = 0.05, value = 1.0)
+    
+#%%
+
+theta_array = np.linspace(0, 2*np.pi, 36)
+
+X = np.column_stack((np.cos(theta_array), 
+                     np.sin(theta_array)))
+
+# st.write(X)
+A = np.array([[a, b],
+              [b, c]])
+
+st.latex(r'''z^Tz = 1''')
+st.latex(r'''x = Az''')
+
+st.latex('A =' + bmatrix(A))
+
+X_ = X@A
+
+#define symbolic vars, function
+x1,x2 = sympy.symbols('x1 x2')
+y1,y2 = sympy.symbols('y1 y2')
+x = np.array([[x1,x2]]).T
+y = np.array([[y1,y2]]).T
+
+Q = np.linalg.inv(A@A.T)
+D,V = np.linalg.eig(Q)
+D = np.diag(D)
+
+st.latex(r'Q = \left( AA^T\right)^{-1} = ' + bmatrix(np.round(Q, 3)))
+
+st.latex(r'''Q = V \Lambda V^{T}''')
+st.latex(bmatrix(np.around(Q, decimals=3)) + '=' + 
+         bmatrix(np.around(V, decimals=3)) + '@' + 
+         bmatrix(np.around(D, decimals=3)) + '@' + 
+         bmatrix(np.around(V.T, decimals=3)))
+
+f_x = x.T@np.round(Q, 3)@x
+f_y = y.T@np.round(D, 3)@y
+
+from sympy import *
+st.write('The formula of the ellipse:')
+st.latex(latex(simplify(f_x[0][0])) + ' = 1')
+
+st.write('The formula of the transformed ellipse:')
+st.latex(latex(simplify(f_y[0][0])) + ' = 1')
+
+#%% 
+color_array = np.linspace(0,1,len(X))
+# st.write(color_array)
+X_c = np.column_stack((X_, color_array))
+df = pd.DataFrame(X_c, columns=['x1','x2', 'color'])
+
+#%% Scatter
+
+fig = px.scatter(df, 
+                 x="x1", 
+                 y="x2", 
+                 color='color',
+                 color_continuous_scale=px.colors.sequential.Rainbow)
+
+
+
+fig.update_layout(
+    autosize=False,
+    width=500,
+    height=500)
+
+
+fig.add_hline(y=0, line_color = 'black')
+fig.add_vline(x=0, line_color = 'black')
+fig.update_layout(coloraxis_showscale=False)
+fig.update_xaxes(range=[-3, 3])
+fig.update_yaxes(range=[-3, 3])
+
+st.plotly_chart(fig)
+
+
+