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Delete Book4_Ch21_Python_Codes directory
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Iris Series: Visualize Math -- From Arithmetic Basics to Machine Learning
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@@ -1,26 +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_Ch21_01.py
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import numpy as np
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def is_pos_def(A):
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if np.array_equal(A, A.T):
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try:
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np.linalg.cholesky(A)
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return True
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except np.linalg.LinAlgError:
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return False
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else:
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return False
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A = np.array([[1,0],
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[0,0]])
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print(is_pos_def(A))
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@@ -1,100 +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_Ch21_02.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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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 = sympy.symbols('x1 x2')
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A = np.array([[1.5, 0.5],
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[0.5, 1.5]])
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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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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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xx1, xx2 = mesh_circ(0, 0, 4, 20)
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# coarse mesh
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xx1_, xx2_ = mesh_circ(0, 0, 4, 10)
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V = grad_fcn(xx1_,xx2_)
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V_z = np.ones_like(V[1]);
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if isinstance(V[1], int):
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V[1] = np.zeros_like(V[0])
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elif isinstance(V[0], int):
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V[0] = np.zeros_like(V[1])
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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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l_3D_vectors = np.sqrt(V[0]**2 + V[1]**2 + V_z**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 = 'RdYlBu_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(xx1.min(),xx1.max())
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plt.ylim(xx2.min(),xx2.max())
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ax.set_proj_type('ortho')
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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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color_array = np.sqrt(V[0]**2 + V[1]**2)
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# 2D visualization
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fig, ax = plt.subplots()
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plt.quiver (xx1_, xx2_, -V[0], -V[1],color_array,
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angles='xy', scale_units='xy',
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edgecolor='none', alpha=0.8,cmap = 'RdYlBu_r')
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plt.contour(xx1, xx2, ff_x,20, 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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plt.tight_layout()
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@@ -1,110 +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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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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b & c
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\end{bmatrix}''')
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st.latex(r'''
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f(x_1,x_2) = ax_1^2 + 2bx_1x_2 + cx_2^2
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''')
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a = st.slider('a',-2.0, 2.0, step = 0.1)
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b = st.slider('b',-2.0, 2.0, step = 0.1)
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c = st.slider('c',-2.0, 2.0, step = 0.1)
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#%%
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x1_ = np.linspace(-2, 2, 101)
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x2_ = np.linspace(-2, 2, 101)
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xx1,xx2 = np.meshgrid(x1_, x2_)
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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([[a, b],
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[b, c]])
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D,V = np.linalg.eig(A)
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D = np.diag(D)
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st.latex(r'''A = \begin{bmatrix}%s & %s\\%s & %s\end{bmatrix}''' %(a, b, b, c))
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st.latex(r'''A = V \Lambda V^{T}''')
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st.latex(bmatrix(A) + '=' +
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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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x = np.array([[x1,x2]]).T
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f_x = a*x1**2 + 2*b*x1*x2 + c*x2**2
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st.latex(r'''f(x_1,x_2) = ''')
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st.write(f_x)
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f_x_fcn = sympy.lambdify([x1,x2],f_x)
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ff_x = f_x_fcn(xx1,xx2)
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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 = ff_x,
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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=ff_x,
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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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@@ -1,117 +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 numpy as np
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from sympy import lambdify, diff, exp, latex, simplify, symbols
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import numpy as np
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import plotly.figure_factory as ff
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import plotly.graph_objects as go
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import streamlit as st
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x1,x2 = symbols('x1 x2')
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num = 301; # number of mesh grids
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x1_array = np.linspace(-3,3,num)
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x2_array = np.linspace(-3,3,num)
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xx1,xx2 = np.meshgrid(x1_array,x2_array)
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# f_xy = x*exp(- x**2 - y**2);
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f_x = 3*(1-x1)**2*exp(-(x1**2) - (x2+1)**2)\
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- 10*(x1/5 - x1**3 - x2**5)*exp(-x1**2-x2**2)\
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- 1/3*exp(-(x1+1)**2 - x2**2)
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f_x_fcn = lambdify([x1,x2],f_x)
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f_zz = f_x_fcn(xx1,xx2)
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st.latex('f(x_1, x_2) = ' + latex(f_x))
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#%% gradient
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#take the gradient symbolically
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grad_f = [diff(f_x,var) for var in (x1,x2)]
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#turn into a bivariate lambda for numpy
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grad_fcn = lambdify([x1,x2],grad_f)
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x1__ = np.linspace(-3,3,40)
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x2__ = np.linspace(-3,3,40)
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# coarse mesh
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xx1_, xx2_ = np.meshgrid(x1__,x2__)
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V = grad_fcn(xx1_,xx2_)
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#%%
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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 = f_zz,
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showscale=False,
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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 =800,
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height=600)
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st.plotly_chart(fig_surface)
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#%% gradient vector plot
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f = ff.create_quiver(xx1_, xx2_,
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V[0], V[1],
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arrow_scale=.1,
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scale = 0.03)
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f_stream = ff.create_streamline(x1__,x2__,
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V[0], V[1],
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arrow_scale=.1)
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trace1 = f.data[0]
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trace3 = f_stream.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 = f_zz,
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showscale=False,
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colorscale = 'RdYlBu_r')
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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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#%% streamlit plot
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data2=[trace3,trace2]
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fig2 = go.FigureWidget(data2)
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fig2.update_layout(
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autosize=False,
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width =800,
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height=800)
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fig2.add_hline(y=0, line_color = 'black')
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fig2.add_vline(x=0, line_color = 'black')
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fig2.update_xaxes(range=[-2, 2])
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fig2.update_yaxes(range=[-2, 2])
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fig2.update_coloraxes(showscale=False)
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st.plotly_chart(fig2)
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