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https://github.com/Visualize-ML/Book4_Power-of-Matrix.git
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107 lines
2.3 KiB
Python
107 lines
2.3 KiB
Python
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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_Ch14_03.py
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import sympy
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import numpy as np
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import matplotlib.pyplot as plt
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from numpy import linalg as L
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def mesh_circ(c1, c2, r, num):
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theta = np.linspace(0, 2*np.pi, num)
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r = np.linspace(0,r, num)
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theta,r = np.meshgrid(theta,r)
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xx1 = np.cos(theta)*r + c1
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xx2 = np.sin(theta)*r + c2
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return xx1, xx2
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#define symbolic vars, function
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x1,x2 = sympy.symbols('x1 x2')
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A = np.array([[0.5, -0.5],
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[-0.5, 0.5]])
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Lambda, V = L.eig(A)
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x = np.array([[x1,x2]]).T
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f_x = x.T@A@x
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f_x = f_x[0][0]
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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xx1, xx2 = mesh_circ(0, 0, 1, 50)
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ff_x = f_x_fcn(xx1,xx2)
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if Lambda[1] > 0:
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levels = np.linspace(0,Lambda[0],21)
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else:
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levels = np.linspace(Lambda[1],Lambda[0],21)
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t = np.linspace(0,np.pi*2,100)
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# 2D visualization
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fig, ax = plt.subplots()
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ax.plot(np.cos(t), np.sin(t), color = 'k')
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cs = plt.contourf(xx1, xx2, ff_x,
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levels=levels, cmap = 'RdYlBu_r')
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plt.show()
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ax.set_aspect('equal')
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ax.xaxis.set_ticks([])
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ax.yaxis.set_ticks([])
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ax.set_xlabel('$x_1$')
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ax.set_ylabel('$x_2$')
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ax.set_xlim(-1,1)
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ax.set_ylim(-1,1)
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clb = fig.colorbar(cs, ax=ax)
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clb.set_ticks(levels)
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#%% 3D surface of f(x1,x2)
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x1_ = np.linspace(-1.2,1.2,31)
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x2_ = np.linspace(-1.2,1.2,31)
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xx1_fine, xx2_fine = np.meshgrid(x1_,x2_)
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ff_x_fine = f_x_fcn(xx1_fine,xx2_fine)
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f_circle = f_x_fcn(np.cos(t), np.sin(t))
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# 3D visualization
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fig, ax = plt.subplots()
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ax = plt.axes(projection='3d')
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ax.plot(np.cos(t), np.sin(t), f_circle, color = 'k')
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# circle projected to f(x1,x2)
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ax.plot_wireframe(xx1_fine,xx2_fine,ff_x_fine,
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color = [0.8,0.8,0.8],
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linewidth = 0.25)
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ax.contour3D(xx1_fine,xx2_fine,ff_x_fine,15,
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cmap = 'RdYlBu_r')
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ax.view_init(elev=30, azim=60)
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ax.xaxis.set_ticks([])
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ax.yaxis.set_ticks([])
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ax.zaxis.set_ticks([])
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ax.set_xlim(xx1_fine.min(),xx1_fine.max())
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ax.set_ylim(xx2_fine.min(),xx2_fine.max())
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plt.tight_layout()
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ax.set_proj_type('ortho')
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plt.show()
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