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Delete Book4_Ch17_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_Ch17_01.py
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import sympy
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import numpy as np
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from sympy.functions import exp
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#define symbolic vars, function
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x1,x2 = sympy.symbols('x1 x2')
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f_x = x1*exp(-(x1**2 + x2**2))
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print(f_x)
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#take the gradient symbolically
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grad_f = [sympy.diff(f_x,var) for var in (x1,x2)]
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print(grad_f)
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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#turn into a bivariate lambda for numpy
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grad_fcn = sympy.lambdify([x1,x2],grad_f)
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import matplotlib.pyplot as plt
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xx1, xx2 = np.meshgrid(np.linspace(-2,2,40),np.linspace(-2,2,40))
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# coarse mesh
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xx1_, xx2_ = np.meshgrid(np.linspace(-2,2,20),np.linspace(-2,2,20))
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V = grad_fcn(xx1_,xx2_)
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ff_x = f_x_fcn(xx1,xx2)
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color_array = np.sqrt(V[0]**2 + V[1]**2)
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# 3D visualization
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ax = plt.figure().add_subplot(projection='3d')
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ax.plot_wireframe(xx1, xx2, ff_x, rstride=1,
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cstride=1, color = [0.5,0.5,0.5],
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linewidth = 0.2)
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ax.contour3D(xx1, xx2, ff_x, 20, cmap = 'RdBu_r')
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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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plt.xlim(-2,2)
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plt.ylim(-2,2)
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ax.view_init(30, -125)
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ax.set_xlabel('$x_1$')
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ax.set_ylabel('$x_2$')
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ax.set_zlabel('$f(x_1,x_2)$')
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plt.tight_layout()
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# 2D visualization
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fig, ax = plt.subplots()
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plt.contourf(xx1, xx2, ff_x,20, cmap = 'RdBu_r')
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plt.quiver (xx1_, xx2_, V[0], V[1],
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angles='xy', scale_units='xy',
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edgecolor='none', facecolor= 'k')
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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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plt.tight_layout()
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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_Ch17_02.py
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import sympy
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import numpy as np
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from sympy.functions import exp
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#define symbolic vars, function
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x1,x2 = sympy.symbols('x1 x2')
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f_x = x1*exp(-(x1**2 + x2**2))
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print(f_x)
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#take the gradient symbolically
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grad_f = [sympy.diff(f_x,var) for var in (x1,x2)]
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print(grad_f)
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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#turn into a bivariate lambda for numpy
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grad_fcn = sympy.lambdify([x1,x2],grad_f)
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import matplotlib.pyplot as plt
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xx1, xx2 = np.meshgrid(np.linspace(-2,2,40),np.linspace(-2,2,40))
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# coarse mesh
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xx1_, xx2_ = np.meshgrid(np.linspace(-2,2,15),np.linspace(-2,2,15))
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V = grad_fcn(xx1_,xx2_)
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ff_x = f_x_fcn(xx1,xx2)
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ff_x_ = f_x_fcn(xx1_,xx2_)
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color_array = np.sqrt(V[0]**2 + V[1]**2)
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# 3D visualization + vectors
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ax = plt.figure().add_subplot(projection='3d')
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ax.plot_wireframe(xx1, xx2, ff_x, rstride=1,
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cstride=1, color = [0.5,0.5,0.5],
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linewidth = 0.2)
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ax.contour3D(xx1, xx2, ff_x, 20, cmap = 'RdBu_r')
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lengths = np.sqrt(V[0]**2+V[1]**2+1**2)
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for x1,y1,z1,u1,v1,w1,l in zip(xx1_.flatten(),
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xx2_.flatten(),
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ff_x_.flatten(),
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V[0].flatten(),
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V[1].flatten(),
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V[0].flatten()*0 - 1,
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lengths.flatten()):
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ax.quiver(x1, y1, z1, u1, v1, w1, length=l*0.2,
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color = 'k')
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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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plt.xlim(-2,2)
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plt.ylim(-2,2)
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ax.view_init(30, -125)
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ax.set_xlabel('$x_1$')
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ax.set_ylabel('$x_2$')
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ax.set_zlabel('$f(x_1,x_2)$')
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plt.tight_layout()
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@@ -1,89 +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_Ch17_03.py
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import sympy
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from sympy import Matrix, Transpose
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import numpy as np
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from sympy.functions import exp
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import matplotlib.pyplot as plt
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def mesh_circ(c1, c2, r, num):
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theta = np.arange(0,2*np.pi+np.pi/num,np.pi/num)
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r = np.arange(0,r,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,p1,p2 = sympy.symbols('x1 x2 p1 p2')
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f_x = -4*x1**2 -4*x2**2
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f_p = -4*p1**2 -4*p2**2
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print(f_x)
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#take the gradient symbolically
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grad_f = [sympy.diff(f_p,var) for var in (p1,p2)]
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print(grad_f)
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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t_x = Matrix(grad_f).T*Matrix([[x1 - p1], [x2 - p2]]) + Matrix([f_p])
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print(t_x)
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t_x_fcn = sympy.lambdify([x1,x2,p1,p2],t_x)
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#turn into a bivariate lambda for numpy
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grad_fcn = sympy.lambdify([x1,x2],grad_f)
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xx1, xx2 = mesh_circ(0, 0, 3, 20)
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# expansion point
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p1 = -1.5
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p2 = 0
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py = f_x_fcn(p1,p2)
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# coarse mesh
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xx1_, xx2_ = np.meshgrid(np.linspace(p1-1, p1+1, 10),
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np.linspace(p2-1, p2+1, 10))
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# quadratic surface
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ff_x = f_x_fcn(xx1,xx2)
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# tangent plane
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tt_x = t_x_fcn(xx1_, xx2_, p1, p2)
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# 3D visualization
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fig = plt.figure()
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ax = fig.gca(projection='3d')
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ax.plot_wireframe(xx1, xx2, ff_x, rstride=1,
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cstride=1, color = [0.5,0.5,0.5],
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linewidth = 0.2)
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ax.plot_wireframe(xx1_, xx2_, np.squeeze(tt_x), rstride=1,
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cstride=1, color = [1,0,0],
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linewidth = 0.2)
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ax.plot(p1,p2,py,'x')
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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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plt.xlim(-3,3)
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plt.ylim(-3,3)
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ax.view_init(30, -125)
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ax.set_xlabel('$x_1$')
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ax.set_ylabel('$x_2$')
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ax.set_zlabel('$f(x_1,x_2)$')
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@@ -1,85 +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 sympy
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import numpy as np
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from sympy.functions import exp
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import streamlit as st
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import plotly.figure_factory as ff
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import plotly.graph_objects as go
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#define symbolic vars, function
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x1,x2 = sympy.symbols('x1 x2')
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f_x = x1*exp(-(x1**2 + x2**2))
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st.latex('f(x_1, x_2) = ' + sympy.latex(f_x))
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#take the gradient symbolically
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grad_f = [sympy.diff(f_x,var) for var in (x1,x2)]
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st.latex(r'\nabla f = ' + sympy.latex(grad_f) + '^T')
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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#turn into a bivariate lambda for numpy
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grad_fcn = sympy.lambdify([x1,x2],grad_f)
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x1_array = np.linspace(-2,2,100)
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x2_array = np.linspace(-2,2,100)
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xx1, xx2 = np.meshgrid(x1_array,x2_array)
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# coarse mesh
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xx1_, xx2_ = np.meshgrid(np.linspace(-2,2,20),np.linspace(-2,2,20))
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V = grad_fcn(xx1_,xx2_)
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ff_x = f_x_fcn(xx1,xx2)
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#%% visualizations
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fig_surface = go.Figure(go.Surface(
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x = x1_array,
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y = x2_array,
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z = ff_x,
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showscale=False))
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fig_surface.update_layout(
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autosize=False,
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width =800,
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height=800)
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st.plotly_chart(fig_surface)
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f = ff.create_quiver(xx1_, xx2_, V[0], V[1])
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trace1 = f.data[0]
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trace2 = go.Contour(
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x = x1_array,
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y = x2_array,
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z = ff_x,
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showscale=False)
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data=[trace1,trace2]
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fig = go.FigureWidget(data)
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fig.update_layout(
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autosize=False,
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width =800,
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height=800)
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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=[-2, 2])
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fig.update_yaxes(range=[-2, 2])
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fig.update_coloraxes(showscale=False)
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st.plotly_chart(fig)
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