From 6f64557eb66be91cf3680b353fbb2588752dc29a Mon Sep 17 00:00:00 2001 From: Visualize-ML <105787223+Visualize-ML@users.noreply.github.com> Date: Sat, 16 Jul 2022 22:00:56 -0400 Subject: [PATCH] Add files via upload --- Book4_Ch13_Python_Codes/Bk4_Ch13_01.py | 89 +++++++++++++++ Book4_Ch13_Python_Codes/Bk4_Ch13_02.py | 96 ++++++++++++++++ ...值分解__数学要素__从加减乘除到机器学习.pdf | Bin 0 -> 886511 bytes Book4_Ch14_Python_Codes/Bk4_Ch14_01.py | 22 ++++ Book4_Ch14_Python_Codes/Bk4_Ch14_02.py | 53 +++++++++ Book4_Ch14_Python_Codes/Bk4_Ch14_03.py | 106 ++++++++++++++++++ Book4_Ch14_Python_Codes/Bk4_Ch14_04.py | 70 ++++++++++++ ...征值分解__数学要素__从加减乘除到机器学习.pdf | Bin 0 -> 1531512 bytes 8 files changed, 436 insertions(+) create mode 100644 Book4_Ch13_Python_Codes/Bk4_Ch13_01.py create mode 100644 Book4_Ch13_Python_Codes/Bk4_Ch13_02.py create mode 100644 Book4_Ch13_特征值分解__数学要素__从加减乘除到机器学习.pdf create mode 100644 Book4_Ch14_Python_Codes/Bk4_Ch14_01.py create mode 100644 Book4_Ch14_Python_Codes/Bk4_Ch14_02.py create mode 100644 Book4_Ch14_Python_Codes/Bk4_Ch14_03.py create mode 100644 Book4_Ch14_Python_Codes/Bk4_Ch14_04.py create mode 100644 Book4_Ch14_深入特征值分解__数学要素__从加减乘除到机器学习.pdf diff --git a/Book4_Ch13_Python_Codes/Bk4_Ch13_01.py b/Book4_Ch13_Python_Codes/Bk4_Ch13_01.py new file mode 100644 index 0000000..fb601ea --- /dev/null +++ b/Book4_Ch13_Python_Codes/Bk4_Ch13_01.py @@ -0,0 +1,89 @@ + +############### +# Authored by Weisheng Jiang +# Book 4 | From Basic Arithmetic to Machine Learning +# Published and copyrighted by Tsinghua University Press +# Beijing, China, 2022 +############### + +# Bk4_Ch13_01.py + +import numpy as np +import matplotlib.pyplot as plt + +A = np.array([[1.25, -0.75], + [-0.75, 1.25]]) + +xx1, xx2 = np.meshgrid(np.linspace(-8, 8, 9), np.linspace(-8, 8, 9)) +num_vecs = np.prod(xx1.shape); + +thetas = np.linspace(0, 2*np.pi, num_vecs) + +thetas = np.reshape(thetas, (-1, 9)) +thetas = np.flipud(thetas); + +uu = np.cos(thetas); +vv = np.sin(thetas); + +fig, ax = plt.subplots() + +ax.quiver(xx1,xx2,uu,vv, + angles='xy', scale_units='xy',scale=1, + edgecolor='none', facecolor= 'b') + +plt.ylabel('$x_2$') +plt.xlabel('$x_1$') +plt.axis('scaled') +ax.set_xlim([-10, 10]) +ax.set_ylim([-10, 10]) +ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5]) +ax.set_xticks(np.linspace(-10,10,11)); +ax.set_yticks(np.linspace(-10,10,11)); +plt.show() + +# Matrix multiplication +V = np.array([uu.flatten(),vv.flatten()]).T; +W = V@A; + +uu_new = np.reshape(W[:,0],(-1, 9)); +vv_new = np.reshape(W[:,1],(-1, 9)); + +fig, ax = plt.subplots() + +ax.quiver(xx1,xx2,uu,vv, + angles='xy', scale_units='xy',scale=1, + edgecolor='none', facecolor= 'b') + +ax.quiver(xx1,xx2,uu_new,vv_new, + angles='xy', scale_units='xy',scale=1, + edgecolor='none', facecolor= 'r') + +plt.ylabel('$x_2$') +plt.xlabel('$x_1$') +plt.axis('scaled') +ax.set_xlim([-10, 10]) +ax.set_ylim([-10, 10]) +ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5]) +ax.set_xticks(np.linspace(-10,10,11)); +ax.set_yticks(np.linspace(-10,10,11)); +plt.show() + + +fig, ax = plt.subplots() +ax.quiver(xx1*0,xx2*0,uu,vv, + angles='xy', scale_units='xy',scale=1, + edgecolor='none', facecolor= 'b') + +ax.quiver(xx1*0,xx2*0,uu_new,vv_new, + angles='xy', scale_units='xy',scale=1, + edgecolor='none', facecolor= 'r') + +plt.ylabel('$x_2$') +plt.xlabel('$x_1$') +plt.axis('scaled') +ax.set_xlim([-2, 2]) +ax.set_ylim([-2, 2]) +ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5]) +ax.set_xticks(np.linspace(-2,2,5)); +ax.set_yticks(np.linspace(-2,2,5)); +plt.show() diff --git a/Book4_Ch13_Python_Codes/Bk4_Ch13_02.py b/Book4_Ch13_Python_Codes/Bk4_Ch13_02.py new file mode 100644 index 0000000..c18e6e1 --- /dev/null +++ b/Book4_Ch13_Python_Codes/Bk4_Ch13_02.py @@ -0,0 +1,96 @@ + +############### +# Authored by Weisheng Jiang +# Book 4 | From Basic Arithmetic to Machine Learning +# Published and copyrighted by Tsinghua University Press +# Beijing, China, 2022 +############### + +# Bk4_Ch13_02.py + +import numpy as np +import matplotlib.pyplot as plt + +def visualize(X_circle,X_vec,title_txt): + + fig, ax = plt.subplots() + + plt.plot(X_circle[0,:], X_circle[1,:],'k', + linestyle = '--', + linewidth = 0.5) + + plt.quiver(0,0,X_vec[0,0],X_vec[1,0], + angles='xy', scale_units='xy',scale=1, + color = [0, 0.4392, 0.7529]) + + plt.quiver(0,0,X_vec[0,1],X_vec[1,1], + angles='xy', scale_units='xy',scale=1, + color = [1,0,0]) + + plt.axvline(x=0, color= 'k', zorder=0) + plt.axhline(y=0, color= 'k', zorder=0) + + plt.ylabel('$x_2$') + plt.xlabel('$x_1$') + + ax.set_aspect(1) + ax.set_xlim([-2.5, 2.5]) + ax.set_ylim([-2.5, 2.5]) + ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5]) + ax.set_xticks(np.linspace(-2,2,5)); + ax.set_yticks(np.linspace(-2,2,5)); + plt.title(title_txt) + plt.show() + + +theta = np.linspace(0, 2*np.pi, 100) + +circle_x1 = np.cos(theta) +circle_x2 = np.sin(theta) + +V_vec = np.array([[np.sqrt(2)/2, -np.sqrt(2)/2], + [np.sqrt(2)/2, np.sqrt(2)/2]]) + +X_circle = np.array([circle_x1, circle_x2]) + +# plot original circle and two vectors +visualize(X_circle,V_vec,'Original') + +A = np.array([[1.25, -0.75], + [-0.75, 1.25]]) + +# plot the transformation of A + +visualize(A@X_circle, A@V_vec,'$A$') + + +#%% Eigen deomposition + +# A = V @ D @ V.T + +lambdas, V = np.linalg.eig(A) + +D = np.diag(np.flip(lambdas)) +V = V.T # reverse the order + +print('=== LAMBDA ===') +print(D) +print('=== V ===') +print(V) + +# plot the transformation of V.T + +visualize(V.T@X_circle, V.T@V_vec,'$V^T$') + +# plot the transformation of D @ V.T + +visualize(D@V.T@X_circle, D@V.T@V_vec,'$\u039BV^T$') + +# plot the transformation of V @ D @ V.T + +visualize(V@D@V.T@X_circle, V@D@V.T@V_vec,'$V\u039BV^T$') + +# plot the transformation of A + +visualize(A@X_circle, A@V_vec,'$A$') + diff --git a/Book4_Ch13_特征值分解__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch13_特征值分解__数学要素__从加减乘除到机器学习.pdf new file mode 100644 index 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color = 'k') + +cs = plt.contourf(xx1, xx2, ff_x, + levels=levels, cmap = 'RdYlBu_r') +plt.show() +ax.set_aspect('equal') +ax.xaxis.set_ticks([]) +ax.yaxis.set_ticks([]) +ax.set_xlabel('$x_1$') +ax.set_ylabel('$x_2$') +ax.set_xlim(-1,1) +ax.set_ylim(-1,1) +clb = fig.colorbar(cs, ax=ax) +clb.set_ticks(levels) + +#%% 3D surface of f(x1,x2) + +x1_ = np.linspace(-1.2,1.2,31) +x2_ = np.linspace(-1.2,1.2,31) + +xx1_fine, xx2_fine = np.meshgrid(x1_,x2_) + +ff_x_fine = f_x_fcn(xx1_fine,xx2_fine) + +f_circle = f_x_fcn(np.cos(t), np.sin(t)) + +# 3D visualization + +fig, ax = plt.subplots() +ax = plt.axes(projection='3d') + +ax.plot(np.cos(t), np.sin(t), f_circle, color = 'k') +# circle projected to f(x1,x2) + +ax.plot_wireframe(xx1_fine,xx2_fine,ff_x_fine, + color = [0.8,0.8,0.8], + linewidth = 0.25) + +ax.contour3D(xx1_fine,xx2_fine,ff_x_fine,15, + cmap = 'RdYlBu_r') + +ax.view_init(elev=30, azim=60) +ax.xaxis.set_ticks([]) +ax.yaxis.set_ticks([]) +ax.zaxis.set_ticks([]) +ax.set_xlim(xx1_fine.min(),xx1_fine.max()) +ax.set_ylim(xx2_fine.min(),xx2_fine.max()) +plt.tight_layout() +ax.set_proj_type('ortho') +plt.show() diff --git a/Book4_Ch14_Python_Codes/Bk4_Ch14_04.py b/Book4_Ch14_Python_Codes/Bk4_Ch14_04.py new file mode 100644 index 0000000..a53c9db --- /dev/null +++ b/Book4_Ch14_Python_Codes/Bk4_Ch14_04.py @@ -0,0 +1,70 @@ + +############### +# Authored by Weisheng Jiang +# Book 4 | From Basic Arithmetic to Machine Learning +# Published and copyrighted by Tsinghua University Press +# Beijing, China, 2022 +############### + +# Bk4_Ch14_04.py + +import numpy as np +import matplotlib.pyplot as plt + +theta = np.deg2rad(30) + +r = 0.8 # 1.2, scaling factor + +R = np.array([[np.cos(theta), -np.sin(theta)], + [np.sin(theta), np.cos(theta)]]) + +S = np.array([[r, 0], + [0, r]]) + +A = R@S + +# A = np.array([[1, -1], +# [1, 1]]) + +Lamb, V = np.linalg.eig(A) + +theta_array = np.arange(0,np.pi*2,np.pi*2/18) + + +colors = plt.cm.rainbow(np.linspace(0,1,len(theta_array))) + + +fig, ax = plt.subplots() + +for j, theat_i in enumerate(theta_array): + + # initial point + x = np.array([[5*np.cos(theat_i)], + [5*np.sin(theat_i)]]) + + plt.plot(x[0],x[1], + marker = 'x',color = colors_j, + markersize = 15) + # plot the initial point + + x_array = x + + for i in np.arange(20): + + x = A@x + x_array = np.column_stack((x_array,x)) + + + colors_j = colors[j,:] + plt.plot(x_array[0,:],x_array[1,:], + marker = '.',color = colors_j) + + +plt.axis('scaled') + +ax.spines['top'].set_visible(False) +ax.spines['right'].set_visible(False) +ax.spines['bottom'].set_visible(False) +ax.spines['left'].set_visible(False) +ax.axvline(x=0,color = 'k') +ax.axhline(y=0,color = 'k') diff --git a/Book4_Ch14_深入特征值分解__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch14_深入特征值分解__数学要素__从加减乘除到机器学习.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7a5314344f14ab58a56d2491fa7adb5e139e4e8d GIT binary patch literal 1531512 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