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Delete Book4_Ch13_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_Ch13_01.py
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import numpy as np
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import matplotlib.pyplot as plt
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A = np.array([[1.25, -0.75],
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[-0.75, 1.25]])
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xx1, xx2 = np.meshgrid(np.linspace(-8, 8, 9), np.linspace(-8, 8, 9))
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num_vecs = np.prod(xx1.shape);
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thetas = np.linspace(0, 2*np.pi, num_vecs)
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thetas = np.reshape(thetas, (-1, 9))
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thetas = np.flipud(thetas);
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uu = np.cos(thetas);
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vv = np.sin(thetas);
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fig, ax = plt.subplots()
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ax.quiver(xx1,xx2,uu,vv,
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angles='xy', scale_units='xy',scale=1,
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edgecolor='none', facecolor= 'b')
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plt.ylabel('$x_2$')
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plt.xlabel('$x_1$')
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plt.axis('scaled')
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ax.set_xlim([-10, 10])
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ax.set_ylim([-10, 10])
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ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
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ax.set_xticks(np.linspace(-10,10,11));
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ax.set_yticks(np.linspace(-10,10,11));
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plt.show()
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# Matrix multiplication
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V = np.array([uu.flatten(),vv.flatten()]).T;
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W = V@A;
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uu_new = np.reshape(W[:,0],(-1, 9));
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vv_new = np.reshape(W[:,1],(-1, 9));
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fig, ax = plt.subplots()
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ax.quiver(xx1,xx2,uu,vv,
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angles='xy', scale_units='xy',scale=1,
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edgecolor='none', facecolor= 'b')
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ax.quiver(xx1,xx2,uu_new,vv_new,
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angles='xy', scale_units='xy',scale=1,
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edgecolor='none', facecolor= 'r')
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plt.ylabel('$x_2$')
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plt.xlabel('$x_1$')
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plt.axis('scaled')
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ax.set_xlim([-10, 10])
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ax.set_ylim([-10, 10])
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ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
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ax.set_xticks(np.linspace(-10,10,11));
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ax.set_yticks(np.linspace(-10,10,11));
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plt.show()
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fig, ax = plt.subplots()
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ax.quiver(xx1*0,xx2*0,uu,vv,
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angles='xy', scale_units='xy',scale=1,
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edgecolor='none', facecolor= 'b')
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ax.quiver(xx1*0,xx2*0,uu_new,vv_new,
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angles='xy', scale_units='xy',scale=1,
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edgecolor='none', facecolor= 'r')
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plt.ylabel('$x_2$')
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plt.xlabel('$x_1$')
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plt.axis('scaled')
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ax.set_xlim([-2, 2])
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ax.set_ylim([-2, 2])
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ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
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ax.set_xticks(np.linspace(-2,2,5));
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ax.set_yticks(np.linspace(-2,2,5));
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plt.show()
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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_Ch13_02.py
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import numpy as np
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import matplotlib.pyplot as plt
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def visualize(X_circle,X_vec,title_txt):
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fig, ax = plt.subplots()
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plt.plot(X_circle[0,:], X_circle[1,:],'k',
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linestyle = '--',
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linewidth = 0.5)
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plt.quiver(0,0,X_vec[0,0],X_vec[1,0],
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angles='xy', scale_units='xy',scale=1,
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color = [0, 0.4392, 0.7529])
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plt.quiver(0,0,X_vec[0,1],X_vec[1,1],
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angles='xy', scale_units='xy',scale=1,
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color = [1,0,0])
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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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plt.ylabel('$x_2$')
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plt.xlabel('$x_1$')
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ax.set_aspect(1)
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ax.set_xlim([-2.5, 2.5])
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ax.set_ylim([-2.5, 2.5])
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ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
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ax.set_xticks(np.linspace(-2,2,5));
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ax.set_yticks(np.linspace(-2,2,5));
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plt.title(title_txt)
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plt.show()
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theta = np.linspace(0, 2*np.pi, 100)
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circle_x1 = np.cos(theta)
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circle_x2 = np.sin(theta)
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V_vec = np.array([[np.sqrt(2)/2, -np.sqrt(2)/2],
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[np.sqrt(2)/2, np.sqrt(2)/2]])
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X_circle = np.array([circle_x1, circle_x2])
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# plot original circle and two vectors
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visualize(X_circle,V_vec,'Original')
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A = np.array([[1.25, -0.75],
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[-0.75, 1.25]])
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# plot the transformation of A
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visualize(A@X_circle, A@V_vec,'$A$')
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#%% Eigen deomposition
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# A = V @ D @ V.T
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lambdas, V = np.linalg.eig(A)
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D = np.diag(np.flip(lambdas))
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V = V.T # reverse the order
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print('=== LAMBDA ===')
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print(D)
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print('=== V ===')
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print(V)
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# plot the transformation of V.T
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visualize(V.T@X_circle, V.T@V_vec,'$V^T$')
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# plot the transformation of D @ V.T
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visualize(D@V.T@X_circle, D@V.T@V_vec,'$\u039BV^T$')
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# plot the transformation of V @ D @ V.T
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visualize(V@D@V.T@X_circle, V@D@V.T@V_vec,'$V\u039BV^T$')
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# plot the transformation of A
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visualize(A@X_circle, A@V_vec,'$A$')
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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_Ch13_03.py
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import numpy as np
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import matplotlib.pyplot as plt
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theta = np.deg2rad(30)
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r = 0.8 # 1.2, scaling factor
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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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S = np.array([[r, 0],
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[0, r]])
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A = R@S
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# A = np.array([[1, -1],
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# [1, 1]])
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Lamb, V = np.linalg.eig(A)
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theta_array = np.arange(0,np.pi*2,np.pi*2/18)
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colors = plt.cm.rainbow(np.linspace(0,1,len(theta_array)))
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fig, ax = plt.subplots()
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for j, theat_i in enumerate(theta_array):
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# initial point
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x = np.array([[5*np.cos(theat_i)],
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[5*np.sin(theat_i)]])
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plt.plot(x[0],x[1],
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marker = 'x',color = colors_j,
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markersize = 15)
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# plot the initial point
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x_array = x
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for i in np.arange(20):
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x = A@x
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x_array = np.column_stack((x_array,x))
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colors_j = colors[j,:]
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plt.plot(x_array[0,:],x_array[1,:],
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marker = '.',color = colors_j)
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plt.axis('scaled')
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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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###############
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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 numpy as np
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import plotly.express as px
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import pandas as pd
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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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A = \begin{bmatrix}
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a & b\\
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c & d
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\end{bmatrix}''')
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a = st.slider('a',-2.0, 2.0, step = 0.1, value = 1.0)
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b = st.slider('b',-2.0, 2.0, step = 0.1, value = 0.0)
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c = st.slider('c',-2.0, 2.0, step = 0.1, value = 0.0)
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d = st.slider('d',-2.0, 2.0, step = 0.1, value = 1.0)
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#%%
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x1_ = np.linspace(-1, 1, 11)
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x2_ = np.linspace(-1, 1, 11)
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xx1,xx2 = np.meshgrid(x1_, x2_)
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X = np.column_stack((xx1.flatten(), xx2.flatten()))
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# st.write(X)
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A = np.array([[a, b],
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[c, d]])
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X = X@A
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# st.write(len(X))
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#%%
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color_array = np.linspace(0,1,len(X))
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# st.write(color_array)
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X = np.column_stack((X, color_array))
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df = pd.DataFrame(X, columns=['z1','z2', 'color'])
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#%% Scatter
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st.latex('A = ' + bmatrix(A))
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fig = px.scatter(df,
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x="z1",
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y="z2",
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color='color',
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color_continuous_scale = 'rainbow')
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fig.update_layout(
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autosize=False,
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width=500,
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height=500)
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fig.add_hline(y=0, line_color = 'black')
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fig.add_vline(x=0, line_color = 'black')
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fig.update_xaxes(range=[-3, 3])
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fig.update_yaxes(range=[-3, 3])
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fig.update_coloraxes(showscale=False)
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st.plotly_chart(fig)
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