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Delete Book4_Ch20_Python_Codes directory
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Iris Series: Visualize Math -- From Arithmetic Basics to Machine Learning
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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# Bk4_Ch20_01.py
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import numpy as np
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import matplotlib.pyplot as plt
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alphas = np.linspace(0, 2*np.pi, 100)
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# unit circle
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r = np.sqrt(1.0)
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z1 = r*np.cos(alphas)
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z2 = r*np.sin(alphas)
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Z = np.array([z1, z2]).T # data of unit circle
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# scale
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S = np.array([[2, 0],
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[0, 0.5]])
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thetas = np.array([0, 30, 45, 60, 90, 120])
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for theta in thetas:
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# rotate
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print('==== Rotate ====')
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print(theta)
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theta = theta/180*np.pi
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R = np.array([[np.cos(theta), -np.sin(theta)],
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[np.sin(theta), np.cos(theta)]])
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# translate
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c = np.array([2, 1])
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X = Z@S@R.T + c;
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Q = R@np.linalg.inv(S)@np.linalg.inv(S)@R.T
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print('==== Q ====')
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print(Q)
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LAMBDA, V = np.linalg.eig(Q)
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print('==== LAMBDA ====')
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print(LAMBDA)
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print('==== V ====')
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print(V)
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x1 = X[:,0]
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x2 = X[:,1]
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fig, ax = plt.subplots(1)
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ax.plot(z1, z2, 'b') # plot the unit circle
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ax.plot(x1, x2, 'r') # plot the transformed shape
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ax.plot(c[0],c[1],'xk') # plot the center
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ax.quiver(0,0,1,0,color = 'b',angles='xy', scale_units='xy',scale=1)
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ax.quiver(0,0,0,1,color = 'b',angles='xy', scale_units='xy',scale=1)
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ax.quiver(0,0,-1,0,color = 'b',angles='xy', scale_units='xy',scale=1)
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ax.quiver(0,0,0,-1,color = 'b',angles='xy', scale_units='xy',scale=1)
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ax.quiver(0,0,c[0],c[1],color = 'k',angles='xy', scale_units='xy',scale=1)
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ax.quiver(c[0],c[1],
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V[0,0]/np.sqrt(LAMBDA[0]),
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V[1,0]/np.sqrt(LAMBDA[0]),color = 'r',
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angles='xy', scale_units='xy',scale=1)
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ax.quiver(c[0],c[1],
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V[0,1]/np.sqrt(LAMBDA[1]),
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V[1,1]/np.sqrt(LAMBDA[1]),color = 'r',
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angles='xy', scale_units='xy',scale=1)
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ax.quiver(c[0],c[1],
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-V[0,0]/np.sqrt(LAMBDA[0]),
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-V[1,0]/np.sqrt(LAMBDA[0]),color = 'r',
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angles='xy', scale_units='xy',scale=1)
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ax.quiver(c[0],c[1],
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-V[0,1]/np.sqrt(LAMBDA[1]),
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-V[1,1]/np.sqrt(LAMBDA[1]),color = 'r',
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angles='xy', scale_units='xy',scale=1)
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plt.axvline(x=0, color= 'k', zorder=0)
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plt.axhline(y=0, color= 'k', zorder=0)
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ax.set_aspect(1)
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plt.xlim(-2,4)
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plt.ylim(-2,4)
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plt.grid(linestyle='--')
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plt.show()
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plt.xlabel('$x_1$')
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plt.ylabel('$x_2$')
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@@ -1,56 +0,0 @@
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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# Bk4_Ch20_02.py
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import numpy as np
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import matplotlib.pyplot as plt
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alphas = np.linspace(0, 2*np.pi, 100)
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# unit circle
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r = np.sqrt(1.0)
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z1 = r*1/np.cos(alphas)
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z2 = r*np.tan(alphas)
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Z = np.array([z1, z2]).T # data of unit circle
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# scale
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S = np.array([[1, 0],
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[0, 1]])
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thetas = np.array([0, 30, 45, 60, 90, 120])
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for theta in thetas:
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# rotate
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print('==== Rotate ====')
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print(theta)
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theta = theta/180*np.pi
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R = np.array([[np.cos(theta), -np.sin(theta)],
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[np.sin(theta), np.cos(theta)]])
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X = Z@S@R.T;
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x1 = X[:,0]
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x2 = X[:,1]
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fig, ax = plt.subplots(1)
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ax.plot(z1, z2, 'b') # plot the unit circle
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ax.plot(x1, x2, 'r') # plot the transformed shape
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plt.axvline(x=0, color= 'k', zorder=0)
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plt.axhline(y=0, color= 'k', zorder=0)
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ax.set_aspect(1)
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plt.xlim(-3,3)
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plt.ylim(-3,3)
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plt.grid(linestyle='--')
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plt.show()
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plt.xlabel('$x_1$')
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plt.ylabel('$x_2$')
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@@ -1,50 +0,0 @@
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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# Bk4_Ch20_03.py
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import numpy as np
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import matplotlib.pyplot as plt
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a = 1.5
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b = 1
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x1 = np.linspace(-3,3,200)
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x2 = np.linspace(-3,3,200)
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xx1,xx2 = np.meshgrid(x1,x2)
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fig, ax = plt.subplots()
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theta_array = np.linspace(0,2*np.pi,100)
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plt.plot(a*np.cos(b*np.sin(theta)),b*np.sin(b*np.sin(theta)),color = 'k')
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colors = plt.cm.RdYlBu(np.linspace(0,1,len(theta_array)))
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for i in range(len(theta_array)):
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theta = theta_array[i]
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p1 = a*np.cos(theta)
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p2 = b*np.sin(theta)
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tangent = p1*xx1/a**2 + p2*xx2/b**2 - p1**2/a**2 - p2**2/b**2
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colors_i = colors[int(i),:]
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ax.contour(xx1,xx2,tangent, levels = [0], colors = [colors_i])
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plt.axis('scaled')
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ax.set_xlim(-3,3)
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ax.set_ylim(-3,3)
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ax.spines['top'].set_visible(False)
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ax.spines['right'].set_visible(False)
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ax.spines['bottom'].set_visible(False)
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ax.spines['left'].set_visible(False)
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ax.axvline(x=0,color = 'k')
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ax.axhline(y=0,color = 'k')
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@@ -1,132 +0,0 @@
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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import streamlit as st
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import plotly.graph_objects as go
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import sympy
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import numpy as np
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from scipy.stats import multivariate_normal
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def bmatrix(a):
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"""Returns a LaTeX bmatrix
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:a: numpy array
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:returns: LaTeX bmatrix as a string
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"""
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if len(a.shape) > 2:
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raise ValueError('bmatrix can at most display two dimensions')
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lines = str(a).replace('[', '').replace(']', '').splitlines()
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rv = [r'\begin{bmatrix}']
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rv += [' ' + ' & '.join(l.split()) + r'\\' for l in lines]
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rv += [r'\end{bmatrix}']
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return '\n'.join(rv)
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with st.sidebar:
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st.latex(r'''
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\Sigma = \begin{bmatrix}
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\sigma_1^2 &
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\rho \sigma_1 \sigma_2 \\
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\rho \sigma_1 \sigma_2 &
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\sigma_2^2
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\end{bmatrix}''')
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st.write('$\sigma_1$')
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sigma_1 = st.slider('sigma_1',1.0, 2.0, step = 0.1)
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st.write('$\sigma_2$')
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sigma_2 = st.slider('sigma_2',1.0, 2.0, step = 0.1)
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st.write('$\u03C1$')
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rho_12 = st.slider('rho',-0.9, 0.9, step = 0.1)
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#%%
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st.latex(r'''
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f(x) = \frac{1}{\sqrt{2\pi} \sigma}
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\exp\left( -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{\!2}\,\right)
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''')
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st.latex(r'''
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f(x) = \frac{1}{\left( 2 \pi \right)^{\frac{D}{2}}
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\begin{vmatrix}
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\Sigma
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\end{vmatrix}^{\frac{1}{2}}}
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\exp\left(
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-\frac{1}{2}
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\left( x - \mu \right)^{T} \Sigma^{-1} \left( x - \mu \right)
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\right)
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''')
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#%%
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x1 = np.linspace(-3,3,101)
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x2 = np.linspace(-3,3,101)
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xx1, xx2 = np.meshgrid(x1,x2)
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pos = np.dstack((xx1, xx2))
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Sigma = [[sigma_1**2, rho_12*sigma_1*sigma_2],
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[rho_12*sigma_1*sigma_2, sigma_2**2]]
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rv = multivariate_normal([0, 0],
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Sigma)
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PDF_zz = rv.pdf(pos)
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#%%
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Sigma = np.array(Sigma)
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D,V = np.linalg.eig(Sigma)
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D = np.diag(D)
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st.latex(r'''\Sigma = \begin{bmatrix}%s & %s\\%s & %s\end{bmatrix}'''
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%(sigma_1**2,
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rho_12*sigma_1*sigma_2,
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rho_12*sigma_1*sigma_2,
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sigma_2**2))
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st.latex(r'''\Sigma = V \Lambda V^{T}''')
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st.latex(bmatrix(Sigma) + '=' +
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bmatrix(np.around(V, decimals=3)) + '@' +
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bmatrix(np.around(D, decimals=3)) + '@' +
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bmatrix(np.around(V.T, decimals=3)))
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#%% Plot 3D surface
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fig_surface = go.Figure(go.Surface(
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x = x1,
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y = x2,
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z = PDF_zz,
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colorscale= 'RdYlBu_r'))
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fig_surface.update_layout(
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autosize=False,
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width=500,
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height=500)
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st.plotly_chart(fig_surface)
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#%% Plot 2D contour
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fig_contour = go.Figure(
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go.Contour(
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z=PDF_zz,
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x=x1,
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y=x2,
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colorscale= 'RdYlBu_r'
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))
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fig_contour.update_layout(
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autosize=False,
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width=500,
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height=500)
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st.plotly_chart(fig_contour)
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