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矩阵转置法

import tensorflow.compat.v1 as tf
tf.disable_eager_execution()
tf.disable_v2_behavior()
sess = tf.Session()
/opt/anaconda3/lib/python3.8/site-packages/requests/__init__.py:89: RequestsDependencyWarning: urllib3 (1.26.8) or chardet (3.0.4) doesn't match a supported version!

warnings.warn("urllib3 ({}) or chardet ({}) doesn't match a supported "

WARNING:tensorflow:From /opt/anaconda3/lib/python3.8/site-packages/tensorflow/python/compat/v2_compat.py:111: disable_resource_variables (from tensorflow.python.ops.variable_scope) is deprecated and will be removed in a future version. Instructions for updating: non-resource variables are not supported in the long term

import matplotlib.pyplot as plt
import numpy as np
# numpy linspace (0, 0.01, 0.02,..., 1)
x_vals = np.linspace(0, 10, 100)
# floating numbers, fluctuation between x_vals
y_vals = x_vals + np.random.normal(0, 1, 100)
plt.plot(x_vals, y_vals, 'rx')

[<matplotlib.lines.Line2D at 0x7fc195a22400>]

image

# Change x_vals into matrix and then transpose it
x_vals_transpose = np.transpose(np.matrix(x_vals))
# Turn one hundred 1s into matrix and then transpose it
ones_column = np.transpose(np.matrix(np.repeat(1, 100)))
# Add the two previous matrix together
A = np.column_stack((x_vals_transpose, ones_column))
# change y_vals into matrix and then transpose it
b = np.transpose(np.matrix(y_vals))
np.matrix.flatten(A)
matrix([[ 0. , 1. , 0.1010101 , 1. , 0.2020202 ,

1. , 0.3030303 , 1. , 0.4040404 , 1. , 0.50505051, 1. , 0.60606061, 1. , 0.70707071, 1. , 0.80808081, 1. , 0.90909091, 1. , 1.01010101, 1. , 1.11111111, 1. , 1.21212121, 1. , 1.31313131, 1. , 1.41414141, 1. , 1.51515152, 1. , 1.61616162, 1. , 1.71717172, 1. , 1.81818182, 1. , 1.91919192, 1. , 2.02020202, 1. , 2.12121212, 1. , 2.22222222, 1. , 2.32323232, 1. , 2.42424242, 1. , 2.52525253, 1. , 2.62626263, 1. , 2.72727273, 1. , 2.82828283, 1. , 2.92929293, 1. , 3.03030303, 1. , 3.13131313, 1. , 3.23232323, 1. , 3.33333333, 1. , 3.43434343, 1. , 3.53535354, 1. , 3.63636364, 1. , 3.73737374, 1. , 3.83838384, 1. , 3.93939394, 1. , 4.04040404, 1. , 4.14141414, 1. , 4.24242424, 1. , 4.34343434, 1. , 4.44444444, 1. , 4.54545455, 1. , 4.64646465, 1. , 4.74747475, 1. , 4.84848485, 1. , 4.94949495, 1. , 5.05050505, 1. , 5.15151515, 1. , 5.25252525, 1. , 5.35353535, 1. , 5.45454545, 1. , 5.55555556, 1. , 5.65656566, 1. , 5.75757576, 1. , 5.85858586, 1. , 5.95959596, 1. , 6.06060606, 1. , 6.16161616, 1. , 6.26262626, 1. , 6.36363636, 1. , 6.46464646, 1. , 6.56565657, 1. , 6.66666667, 1. , 6.76767677, 1. , 6.86868687, 1. , 6.96969697, 1. , 7.07070707, 1. , 7.17171717, 1. , 7.27272727, 1. , 7.37373737, 1. , 7.47474747, 1. , 7.57575758, 1. , 7.67676768, 1. , 7.77777778, 1. , 7.87878788, 1. , 7.97979798, 1. , 8.08080808, 1. , 8.18181818, 1. , 8.28282828, 1. , 8.38383838, 1. , 8.48484848, 1. , 8.58585859, 1. , 8.68686869, 1. , 8.78787879, 1. , 8.88888889, 1. , 8.98989899, 1. , 9.09090909, 1. , 9.19191919, 1. , 9.29292929, 1. , 9.39393939, 1. , 9.49494949, 1. , 9.5959596 , 1. , 9.6969697 , 1. , 9.7979798 , 1. , 9.8989899 , 1. , 10. , 1. ]])

np.matrix.flatten(b)
matrix([[-1.32657576, 0.56506495, -0.45773644, 2.05872291, -0.76011994,

2.11881469, 2.02012218, 1.98189628, 1.13606984, -0.64178737, 0.63951158, 2.13227367, 2.14387046, 1.63050021, 0.14826131, 0.79041461, 3.38163066, 1.81012214, 1.61740339, 1.8660018 , 1.06663904, 2.57995005, 2.51257911, 2.51397268, 2.30026928, 1.66643123, 2.74435042, 2.88618636, 3.21936111, 2.28648487, 1.56108793, 3.27443017, 0.95188115, 3.25380978, 4.4599258 , 1.7161918 , 2.57034704, 3.77175977, 3.37298323, 4.93334055, 6.24041238, 3.79034625, 5.11975499, 4.69633511, 2.37338909, 3.86932804, 7.15341682, 6.55787247, 6.44285216, 5.787033 , 3.81065563, 5.47691796, 6.7668701 , 4.32195582, 5.88448108, 5.84259411, 7.25223478, 6.17904533, 4.91669862, 4.66554841, 5.94659988, 5.56600608, 7.42489335, 5.756941 , 6.57996822, 9.17010871, 4.99166266, 8.60093335, 6.55241788, 8.47318977, 7.82032544, 7.45544871, 7.646155 , 8.11039097, 7.48708838, 8.03282116, 7.45815892, 8.65651986, 7.16416238, 8.27322034, 8.29417968, 8.54683231, 6.4814741 , 9.13929264, 8.84793761, 9.84207706, 8.30548891, 9.06064814, 10.40503202, 10.25362638,

11.47564933, 7.4315958 , 9.12460191, 9.99417217, 9.22592671,

10.13666882, 10.58905165, 8.53342043, 9.85186347, 12.82439669]])

A_tensor = tf.constant(A)
b_tensor = tf.constant(b)
tA_A = tf.matmul(tf.transpose(A_tensor), A_tensor)
sess.run(A_tensor)
array([[ 0. , 1. ],

[ 0.1010101 , 1. ], [ 0.2020202 , 1. ], [ 0.3030303 , 1. ], [ 0.4040404 , 1. ], [ 0.50505051, 1. ], [ 0.60606061, 1. ], [ 0.70707071, 1. ], [ 0.80808081, 1. ], [ 0.90909091, 1. ], [ 1.01010101, 1. ], [ 1.11111111, 1. ], [ 1.21212121, 1. ], [ 1.31313131, 1. ], [ 1.41414141, 1. ], [ 1.51515152, 1. ], [ 1.61616162, 1. ], [ 1.71717172, 1. ], [ 1.81818182, 1. ], [ 1.91919192, 1. ], [ 2.02020202, 1. ], [ 2.12121212, 1. ], [ 2.22222222, 1. ], [ 2.32323232, 1. ], [ 2.42424242, 1. ], [ 2.52525253, 1. ], [ 2.62626263, 1. ], [ 2.72727273, 1. ], [ 2.82828283, 1. ], [ 2.92929293, 1. ], [ 3.03030303, 1. ], [ 3.13131313, 1. ], [ 3.23232323, 1. ], [ 3.33333333, 1. ], [ 3.43434343, 1. ], [ 3.53535354, 1. ], [ 3.63636364, 1. ], [ 3.73737374, 1. ], [ 3.83838384, 1. ], [ 3.93939394, 1. ], [ 4.04040404, 1. ], [ 4.14141414, 1. ], [ 4.24242424, 1. ], [ 4.34343434, 1. ], [ 4.44444444, 1. ], [ 4.54545455, 1. ], [ 4.64646465, 1. ], [ 4.74747475, 1. ], [ 4.84848485, 1. ], [ 4.94949495, 1. ], [ 5.05050505, 1. ], [ 5.15151515, 1. ], [ 5.25252525, 1. ], [ 5.35353535, 1. ], [ 5.45454545, 1. ], [ 5.55555556, 1. ], [ 5.65656566, 1. ], [ 5.75757576, 1. ], [ 5.85858586, 1. ], [ 5.95959596, 1. ], [ 6.06060606, 1. ], [ 6.16161616, 1. ], [ 6.26262626, 1. ], [ 6.36363636, 1. ], [ 6.46464646, 1. ], [ 6.56565657, 1. ], [ 6.66666667, 1. ], [ 6.76767677, 1. ], [ 6.86868687, 1. ], [ 6.96969697, 1. ], [ 7.07070707, 1. ], [ 7.17171717, 1. ], [ 7.27272727, 1. ], [ 7.37373737, 1. ], [ 7.47474747, 1. ], [ 7.57575758, 1. ], [ 7.67676768, 1. ], [ 7.77777778, 1. ], [ 7.87878788, 1. ], [ 7.97979798, 1. ], [ 8.08080808, 1. ], [ 8.18181818, 1. ], [ 8.28282828, 1. ], [ 8.38383838, 1. ], [ 8.48484848, 1. ], [ 8.58585859, 1. ], [ 8.68686869, 1. ], [ 8.78787879, 1. ], [ 8.88888889, 1. ], [ 8.98989899, 1. ], [ 9.09090909, 1. ], [ 9.19191919, 1. ], [ 9.29292929, 1. ], [ 9.39393939, 1. ], [ 9.49494949, 1. ], [ 9.5959596 , 1. ], [ 9.6969697 , 1. ], [ 9.7979798 , 1. ], [ 9.8989899 , 1. ], [10. , 1. ]])

sess.run(tf.transpose(A_tensor))
array([[ 0. , 0.1010101 , 0.2020202 , 0.3030303 , 0.4040404 ,

0.50505051, 0.60606061, 0.70707071, 0.80808081, 0.90909091, 1.01010101, 1.11111111, 1.21212121, 1.31313131, 1.41414141, 1.51515152, 1.61616162, 1.71717172, 1.81818182, 1.91919192, 2.02020202, 2.12121212, 2.22222222, 2.32323232, 2.42424242, 2.52525253, 2.62626263, 2.72727273, 2.82828283, 2.92929293, 3.03030303, 3.13131313, 3.23232323, 3.33333333, 3.43434343, 3.53535354, 3.63636364, 3.73737374, 3.83838384, 3.93939394, 4.04040404, 4.14141414, 4.24242424, 4.34343434, 4.44444444, 4.54545455, 4.64646465, 4.74747475, 4.84848485, 4.94949495, 5.05050505, 5.15151515, 5.25252525, 5.35353535, 5.45454545, 5.55555556, 5.65656566, 5.75757576, 5.85858586, 5.95959596, 6.06060606, 6.16161616, 6.26262626, 6.36363636, 6.46464646, 6.56565657, 6.66666667, 6.76767677, 6.86868687, 6.96969697, 7.07070707, 7.17171717, 7.27272727, 7.37373737, 7.47474747, 7.57575758, 7.67676768, 7.77777778, 7.87878788, 7.97979798, 8.08080808, 8.18181818, 8.28282828, 8.38383838, 8.48484848, 8.58585859, 8.68686869, 8.78787879, 8.88888889, 8.98989899, 9.09090909, 9.19191919, 9.29292929, 9.39393939, 9.49494949, 9.5959596 , 9.6969697 , 9.7979798 , 9.8989899 , 10. ],

[ 1. , 1. , 1. , 1. , 1. ,
  1. , 1. , 1. , 1. , 1. ,
  2. , 1. , 1. , 1. , 1. ,
  3. , 1. , 1. , 1. , 1. ,
  4. , 1. , 1. , 1. , 1. ,
  5. , 1. , 1. , 1. , 1. ,
  6. , 1. , 1. , 1. , 1. ,
  7. , 1. , 1. , 1. , 1. ,
  8. , 1. , 1. , 1. , 1. ,
  9. , 1. , 1. , 1. , 1. ,
  10. , 1. , 1. , 1. , 1. ,
  11. , 1. , 1. , 1. , 1. ,
  12. , 1. , 1. , 1. , 1. ,
  13. , 1. , 1. , 1. , 1. ,
  14. , 1. , 1. , 1. , 1. ,
  15. , 1. , 1. , 1. , 1. ,
  16. , 1. , 1. , 1. , 1. ,
  17. , 1. , 1. , 1. , 1. ,
  18. , 1. , 1. , 1. , 1. ,
  19. , 1. , 1. , 1. , 1. ]])
tA_A_inv = tf.matrix_inverse(tA_A)
sess.run(tA_A_inv), sess.run(tA_A)
(array([[ 0.00117624, -0.00588119],

[-0.00588119, 0.03940594]]),

array([[3350.16835017, 500. ],

[ 500. , 100. ]]))

component_1_1 = (sess.run(tA_A_inv))[:, 0] * (sess.run(tA_A))[0,:]
sess.run(tf.reduce_sum(component_1_1))

1.0

component_1_2 = (sess.run(tA_A_inv))[:,0] * (sess.run(tA_A))[1,:]
sess.run(tf.reduce_sum(component_1_2))

0.0

component_2_1 = (sess.run(tA_A_inv))[:,1] * (sess.run(tA_A))[0,:]
sess.run(tf.reduce_sum(component_2_1))

-3.552713678800501e-15

component_2_2 = (sess.run(tA_A_inv))[:,1] * (sess.run(tA_A))[1,:]
sess.run(tf.reduce_sum(component_2_2))

0.9999999999999996

product = tf.matmul(tA_A_inv, tf.transpose(A_tensor))
solution = tf.matmul(product, b_tensor)
solution_eval = sess.run(solution)
solution_eval
array([[ 1.04588188],

[-0.03765808]])

sess.run(product)
array([[-5.88118812e-03, -5.76237624e-03, -5.64356436e-03,

-5.52475248e-03, -5.40594059e-03, -5.28712871e-03, -5.16831683e-03, -5.04950495e-03, -4.93069307e-03, -4.81188119e-03, -4.69306931e-03, -4.57425743e-03, -4.45544554e-03, -4.33663366e-03, -4.21782178e-03, -4.09900990e-03, -3.98019802e-03, -3.86138614e-03, -3.74257426e-03, -3.62376238e-03, -3.50495050e-03, -3.38613861e-03, -3.26732673e-03, -3.14851485e-03, -3.02970297e-03, -2.91089109e-03, -2.79207921e-03, -2.67326733e-03, -2.55445545e-03, -2.43564356e-03, -2.31683168e-03, -2.19801980e-03, -2.07920792e-03, -1.96039604e-03, -1.84158416e-03, -1.72277228e-03, -1.60396040e-03, -1.48514851e-03, -1.36633663e-03, -1.24752475e-03, -1.12871287e-03, -1.00990099e-03, -8.91089109e-04, -7.72277228e-04, -6.53465347e-04, -5.34653465e-04, -4.15841584e-04, -2.97029703e-04, -1.78217822e-04, -5.94059406e-05, 5.94059406e-05, 1.78217822e-04, 2.97029703e-04, 4.15841584e-04, 5.34653465e-04, 6.53465347e-04, 7.72277228e-04, 8.91089109e-04, 1.00990099e-03, 1.12871287e-03, 1.24752475e-03, 1.36633663e-03, 1.48514851e-03, 1.60396040e-03, 1.72277228e-03, 1.84158416e-03, 1.96039604e-03, 2.07920792e-03, 2.19801980e-03, 2.31683168e-03, 2.43564356e-03, 2.55445545e-03, 2.67326733e-03, 2.79207921e-03, 2.91089109e-03, 3.02970297e-03, 3.14851485e-03, 3.26732673e-03, 3.38613861e-03, 3.50495050e-03, 3.62376238e-03, 3.74257426e-03, 3.86138614e-03, 3.98019802e-03, 4.09900990e-03, 4.21782178e-03, 4.33663366e-03, 4.45544554e-03, 4.57425743e-03, 4.69306931e-03, 4.81188119e-03, 4.93069307e-03, 5.04950495e-03, 5.16831683e-03, 5.28712871e-03, 5.40594059e-03, 5.52475248e-03, 5.64356436e-03, 5.76237624e-03, 5.88118812e-03],

[ 3.94059406e-02, 3.88118812e-02, 3.82178218e-02,

3.76237624e-02, 3.70297030e-02, 3.64356436e-02, 3.58415842e-02, 3.52475248e-02, 3.46534653e-02, 3.40594059e-02, 3.34653465e-02, 3.28712871e-02, 3.22772277e-02, 3.16831683e-02, 3.10891089e-02, 3.04950495e-02, 2.99009901e-02, 2.93069307e-02, 2.87128713e-02, 2.81188119e-02, 2.75247525e-02, 2.69306931e-02, 2.63366337e-02, 2.57425743e-02, 2.51485149e-02, 2.45544554e-02, 2.39603960e-02, 2.33663366e-02, 2.27722772e-02, 2.21782178e-02, 2.15841584e-02, 2.09900990e-02, 2.03960396e-02, 1.98019802e-02, 1.92079208e-02, 1.86138614e-02, 1.80198020e-02, 1.74257426e-02, 1.68316832e-02, 1.62376238e-02, 1.56435644e-02, 1.50495050e-02, 1.44554455e-02, 1.38613861e-02, 1.32673267e-02, 1.26732673e-02, 1.20792079e-02, 1.14851485e-02, 1.08910891e-02, 1.02970297e-02, 9.70297030e-03, 9.10891089e-03, 8.51485149e-03, 7.92079208e-03, 7.32673267e-03, 6.73267327e-03, 6.13861386e-03, 5.54455446e-03, 4.95049505e-03, 4.35643564e-03, 3.76237624e-03, 3.16831683e-03, 2.57425743e-03, 1.98019802e-03, 1.38613861e-03, 7.92079208e-04, 1.98019802e-04, -3.96039604e-04, -9.90099010e-04,

-1.58415842e-03, -2.17821782e-03, -2.77227723e-03,

-3.36633663e-03, -3.96039604e-03, -4.55445545e-03, -5.14851485e-03, -5.74257426e-03, -6.33663366e-03, -6.93069307e-03, -7.52475248e-03, -8.11881188e-03, -8.71287129e-03, -9.30693069e-03, -9.90099010e-03, -1.04950495e-02, -1.10891089e-02, -1.16831683e-02, -1.22772277e-02, -1.28712871e-02, -1.34653465e-02, -1.40594059e-02, -1.46534653e-02, -1.52475248e-02, -1.58415842e-02, -1.64356436e-02, -1.70297030e-02, -1.76237624e-02, -1.82178218e-02, -1.88118812e-02, -1.94059406e-02]])

slope = solution_eval[0][0]
y_intercept = solution_eval[1][0]
print('slope: ' + str(slope))
print('y_intercept: ' + str(y_intercept))

slope: 1.045881880470386 y_intercept: -0.03765807642003677

best_fit = []
for i in x_vals:
    best_fit.append(slope*i+y_intercept)
plt.plot(x_vals, y_vals, 'o', label='Original Data')
plt.plot(x_vals, best_fit, 'r-', label= 'Best fit line', linewidth=3)
plt.legend(loc='upper left')
plt.xlim(0,10)
plt.ylim(0,10)
plt.show()

image