mirror of
https://github.com/Visualize-ML/Book4_Power-of-Matrix.git
synced 2026-02-03 02:24:03 +08:00
Delete Book4_Ch14_Python_Codes directory
This commit is contained in:
Iris Series: Visualize Math -- From Arithmetic Basics to Machine Learning
committed by
GitHub
GitHub
parent
a722f54863
commit
d9ca162c8e
@@ -1,22 +0,0 @@
|
||||
|
||||
###############
|
||||
# Authored by Weisheng Jiang
|
||||
# Book 4 | From Basic Arithmetic to Machine Learning
|
||||
# Published and copyrighted by Tsinghua University Press
|
||||
# Beijing, China, 2022
|
||||
###############
|
||||
|
||||
# Bk4_Ch14_01.py
|
||||
|
||||
import numpy as np
|
||||
|
||||
A = np.matrix([[1.25, -0.75],
|
||||
[-0.75, 1.25]])
|
||||
|
||||
LAMBDA, V = np.linalg.eig(A)
|
||||
|
||||
B = V@np.diag(np.sqrt(LAMBDA))@np.linalg.inv(V)
|
||||
|
||||
A_reproduced = B@B
|
||||
|
||||
print(A_reproduced)
|
||||
@@ -1,53 +0,0 @@
|
||||
|
||||
###############
|
||||
# Authored by Weisheng Jiang
|
||||
# Book 4 | From Basic Arithmetic to Machine Learning
|
||||
# Published and copyrighted by Tsinghua University Press
|
||||
# Beijing, China, 2022
|
||||
###############
|
||||
|
||||
# Bk4_Ch14_02.py
|
||||
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
# transition matrix
|
||||
T = np.matrix([[0.7, 0.2],
|
||||
[0.3, 0.8]])
|
||||
|
||||
# steady state
|
||||
sstate = np.linalg.eig(T)[1][:,1]
|
||||
sstate = sstate/sstate.sum()
|
||||
print(sstate)
|
||||
|
||||
# initial states
|
||||
initial_x_array = np.array([[1, 0, 0.5, 0.4], # Chicken
|
||||
[0, 1, 0.5, 0.6]]) # Rabbit
|
||||
|
||||
num_iterations = 10;
|
||||
|
||||
for i in np.arange(0,4):
|
||||
|
||||
initial_x = initial_x_array[:,i][:, None]
|
||||
|
||||
x_i = np.zeros_like(initial_x)
|
||||
x_i = initial_x
|
||||
X = initial_x.T;
|
||||
|
||||
# matrix power through iterations
|
||||
|
||||
for x in np.arange(0,num_iterations):
|
||||
x_i = T@x_i;
|
||||
X = np.concatenate([X, x_i.T],axis = 0)
|
||||
|
||||
fig, ax = plt.subplots()
|
||||
|
||||
itr = np.arange(0,num_iterations+1);
|
||||
plt.plot(itr,X[:,0],marker = 'x',color = (1,0,0))
|
||||
plt.plot(itr,X[:,1],marker = 'x',color = (0,0.6,1))
|
||||
|
||||
ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
|
||||
ax.set_xlim(0, num_iterations)
|
||||
ax.set_ylim(0, 1)
|
||||
ax.set_xlabel('Iteration, k')
|
||||
ax.set_ylabel('State')
|
||||
@@ -1,106 +0,0 @@
|
||||
|
||||
###############
|
||||
# Authored by Weisheng Jiang
|
||||
# Book 4 | From Basic Arithmetic to Machine Learning
|
||||
# Published and copyrighted by Tsinghua University Press
|
||||
# Beijing, China, 2022
|
||||
###############
|
||||
|
||||
# Bk4_Ch14_03.py
|
||||
|
||||
import sympy
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from numpy import linalg as L
|
||||
|
||||
|
||||
def mesh_circ(c1, c2, r, num):
|
||||
|
||||
theta = np.linspace(0, 2*np.pi, num)
|
||||
r = np.linspace(0,r, num)
|
||||
theta,r = np.meshgrid(theta,r)
|
||||
xx1 = np.cos(theta)*r + c1
|
||||
xx2 = np.sin(theta)*r + c2
|
||||
|
||||
return xx1, xx2
|
||||
|
||||
|
||||
#define symbolic vars, function
|
||||
x1,x2 = sympy.symbols('x1 x2')
|
||||
|
||||
A = np.array([[0.5, -0.5],
|
||||
[-0.5, 0.5]])
|
||||
|
||||
Lambda, V = L.eig(A)
|
||||
|
||||
x = np.array([[x1,x2]]).T
|
||||
|
||||
f_x = x.T@A@x
|
||||
f_x = f_x[0][0]
|
||||
|
||||
f_x_fcn = sympy.lambdify([x1,x2],f_x)
|
||||
|
||||
xx1, xx2 = mesh_circ(0, 0, 1, 50)
|
||||
|
||||
ff_x = f_x_fcn(xx1,xx2)
|
||||
|
||||
if Lambda[1] > 0:
|
||||
levels = np.linspace(0,Lambda[0],21)
|
||||
else:
|
||||
levels = np.linspace(Lambda[1],Lambda[0],21)
|
||||
|
||||
t = np.linspace(0,np.pi*2,100)
|
||||
|
||||
# 2D visualization
|
||||
fig, ax = plt.subplots()
|
||||
|
||||
ax.plot(np.cos(t), np.sin(t), 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()
|
||||
@@ -1,54 +0,0 @@
|
||||
|
||||
###############
|
||||
# Authored by Weisheng Jiang
|
||||
# Book 4 | From Basic Arithmetic to Machine Learning
|
||||
# Published and copyrighted by Tsinghua University Press
|
||||
# Beijing, China, 2022
|
||||
###############
|
||||
|
||||
|
||||
import numpy as np
|
||||
import streamlit as st
|
||||
import time
|
||||
|
||||
# transition matrix
|
||||
A = np.matrix([[0.7, 0.2],
|
||||
[0.3, 0.8]])
|
||||
|
||||
|
||||
with st.sidebar:
|
||||
pi_0_chicken = st.slider('Ratio of chicken:',
|
||||
0.0, 1.0, step = 0.1)
|
||||
pi_0_rabbit = 1 - pi_0_chicken
|
||||
st.write('Ratio of rabbit: ' + str(round(pi_0_rabbit,1)))
|
||||
|
||||
num_iterations = st.slider('Number of nights:',
|
||||
20,100,step = 5)
|
||||
|
||||
|
||||
progress_bar = st.sidebar.progress(0)
|
||||
status_text = st.sidebar.empty()
|
||||
last_rows = np.array([[pi_0_chicken, pi_0_rabbit]])
|
||||
# st.write(last_rows) # row vector
|
||||
|
||||
chart = st.line_chart(last_rows)
|
||||
|
||||
for i in range(1, num_iterations):
|
||||
|
||||
last_status = last_rows[-1,:]
|
||||
|
||||
# st.write(last_status)
|
||||
|
||||
new_rows = last_status@A.T
|
||||
|
||||
percent = (i + 1)*100/num_iterations
|
||||
|
||||
status_text.text("%i%% Complete" % percent)
|
||||
|
||||
chart.add_rows(new_rows)
|
||||
progress_bar.progress(i)
|
||||
last_rows = new_rows
|
||||
time.sleep(0.1)
|
||||
|
||||
progress_bar.empty()
|
||||
|
||||
@@ -1,119 +0,0 @@
|
||||
|
||||
###############
|
||||
# Authored by Weisheng Jiang
|
||||
# Book 4 | From Basic Arithmetic to Machine Learning
|
||||
# Published and copyrighted by Tsinghua University Press
|
||||
# Beijing, China, 2022
|
||||
###############
|
||||
|
||||
import plotly.graph_objects as go
|
||||
import streamlit as st
|
||||
import numpy as np
|
||||
import plotly.express as px
|
||||
import pandas as pd
|
||||
import sympy
|
||||
|
||||
def bmatrix(a):
|
||||
"""Returns a LaTeX bmatrix
|
||||
|
||||
:a: numpy array
|
||||
:returns: LaTeX bmatrix as a string
|
||||
"""
|
||||
if len(a.shape) > 2:
|
||||
raise ValueError('bmatrix can at most display two dimensions')
|
||||
lines = str(a).replace('[', '').replace(']', '').splitlines()
|
||||
rv = [r'\begin{bmatrix}']
|
||||
rv += [' ' + ' & '.join(l.split()) + r'\\' for l in lines]
|
||||
rv += [r'\end{bmatrix}']
|
||||
return '\n'.join(rv)
|
||||
|
||||
with st.sidebar:
|
||||
|
||||
st.latex(r'''
|
||||
A = \begin{bmatrix}
|
||||
a & b\\
|
||||
b & c
|
||||
\end{bmatrix}''')
|
||||
|
||||
a = st.slider('a',-2.0, 2.0, step = 0.05, value = 1.0)
|
||||
b = st.slider('b',-2.0, 2.0, step = 0.05, value = 0.0)
|
||||
c = st.slider('c',-2.0, 2.0, step = 0.05, value = 1.0)
|
||||
|
||||
#%%
|
||||
|
||||
theta_array = np.linspace(0, 2*np.pi, 36)
|
||||
|
||||
X = np.column_stack((np.cos(theta_array),
|
||||
np.sin(theta_array)))
|
||||
|
||||
# st.write(X)
|
||||
A = np.array([[a, b],
|
||||
[b, c]])
|
||||
|
||||
st.latex(r'''z^Tz = 1''')
|
||||
st.latex(r'''x = Az''')
|
||||
|
||||
st.latex('A =' + bmatrix(A))
|
||||
|
||||
X_ = X@A
|
||||
|
||||
#define symbolic vars, function
|
||||
x1,x2 = sympy.symbols('x1 x2')
|
||||
y1,y2 = sympy.symbols('y1 y2')
|
||||
x = np.array([[x1,x2]]).T
|
||||
y = np.array([[y1,y2]]).T
|
||||
|
||||
Q = np.linalg.inv(A@A.T)
|
||||
D,V = np.linalg.eig(Q)
|
||||
D = np.diag(D)
|
||||
|
||||
st.latex(r'Q = \left( AA^T\right)^{-1} = ' + bmatrix(np.round(Q, 3)))
|
||||
|
||||
st.latex(r'''Q = V \Lambda V^{T}''')
|
||||
st.latex(bmatrix(np.around(Q, decimals=3)) + '=' +
|
||||
bmatrix(np.around(V, decimals=3)) + '@' +
|
||||
bmatrix(np.around(D, decimals=3)) + '@' +
|
||||
bmatrix(np.around(V.T, decimals=3)))
|
||||
|
||||
f_x = x.T@np.round(Q, 3)@x
|
||||
f_y = y.T@np.round(D, 3)@y
|
||||
|
||||
from sympy import *
|
||||
st.write('The formula of the ellipse:')
|
||||
st.latex(latex(simplify(f_x[0][0])) + ' = 1')
|
||||
|
||||
st.write('The formula of the transformed ellipse:')
|
||||
st.latex(latex(simplify(f_y[0][0])) + ' = 1')
|
||||
|
||||
#%%
|
||||
color_array = np.linspace(0,1,len(X))
|
||||
# st.write(color_array)
|
||||
X_c = np.column_stack((X_, color_array))
|
||||
df = pd.DataFrame(X_c, columns=['x1','x2', 'color'])
|
||||
|
||||
#%% Scatter
|
||||
|
||||
fig = px.scatter(df,
|
||||
x="x1",
|
||||
y="x2",
|
||||
color='color',
|
||||
color_continuous_scale=px.colors.sequential.Rainbow)
|
||||
|
||||
|
||||
|
||||
fig.update_layout(
|
||||
autosize=False,
|
||||
width=500,
|
||||
height=500)
|
||||
|
||||
|
||||
fig.add_hline(y=0, line_color = 'black')
|
||||
fig.add_vline(x=0, line_color = 'black')
|
||||
fig.update_layout(coloraxis_showscale=False)
|
||||
fig.update_xaxes(range=[-3, 3])
|
||||
fig.update_yaxes(range=[-3, 3])
|
||||
|
||||
st.plotly_chart(fig)
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user