Linear regression
A very simple approach to perform a linear regression with a single neuron using Keras.
Package import
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('ggplot')
Random data generation
We create our samples.
x = np.linspace(0, 50, 51)
x
array([ 0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12.,
13., 14., 15., 16., 17., 18., 19., 20., 21., 22., 23., 24., 25.,
26., 27., 28., 29., 30., 31., 32., 33., 34., 35., 36., 37., 38.,
39., 40., 41., 42., 43., 44., 45., 46., 47., 48., 49., 50.])
And for each sample add some random noise for the $y$ value.
y = x + 10 * np.random.random((len(x)))
y
array([ 4.59333333, 10.1888939 , 3.10342612, 3.78295369, 10.57957367,
8.31186454, 11.97466834, 8.38350117, 14.58165746, 12.7179244 ,
16.04800824, 16.04253531, 17.7570093 , 15.36504093, 23.32077078,
19.38063338, 25.72030949, 20.81364232, 18.55875267, 25.1340618 ,
28.48036 , 21.73467374, 30.81790828, 28.56736033, 28.83225669,
28.18684725, 34.95836113, 29.90731219, 30.90521404, 38.67280311,
33.28501437, 40.01292045, 33.16216509, 34.99748693, 35.87077378,
35.66317699, 36.37898628, 42.26194454, 47.36216501, 45.62434907,
46.47169133, 48.05329522, 45.83970327, 50.68483177, 53.01284414,
54.96997999, 46.42993498, 50.41667051, 53.26256812, 58.97071734,
55.3635401 ])
The generated data looks like this:
plt.scatter(x, y, label='Generated data')
plt.legend()
plt.show()
Modelling
The model has just a single neuron that will model the linear equation $y = mx + b$.
The trained weight will correspond to the slope $m$ of the equation and the bias to the intersection value $b$.
model = tf.keras.Sequential()
model.add(tf.keras.layers.Dense(1, input_shape=[1]))
model.compile(loss='mean_squared_error', optimizer=tf.keras.optimizers.Adam(0.1))
model.summary()
Model: "sequential"
_________________________________________________________________
Layer (type) Output Shape Param #
=================================================================
dense (Dense) (None, 1) 2
=================================================================
Total params: 2
Trainable params: 2
Non-trainable params: 0
_________________________________________________________________
We proceed to fit the model.
history = model.fit(x, y, epochs=200)
Epoch 1/200
2/2 [==============================] - 0s 4ms/step - loss: 653.8051
Epoch 2/200
2/2 [==============================] - 0s 2ms/step - loss: 399.1392
Epoch 3/200
2/2 [==============================] - 0s 2ms/step - loss: 205.5257
Epoch 4/200
2/2 [==============================] - 0s 2ms/step - loss: 83.4503
Epoch 5/200
2/2 [==============================] - 0s 2ms/step - loss: 23.8983
Epoch 6/200
2/2 [==============================] - 0s 2ms/step - loss: 14.8980
Epoch 7/200
2/2 [==============================] - 0s 2ms/step - loss: 30.8598
Epoch 8/200
2/2 [==============================] - 0s 2ms/step - loss: 53.7728
Epoch 9/200
2/2 [==============================] - 0s 2ms/step - loss: 69.0219
Epoch 10/200
2/2 [==============================] - 0s 2ms/step - loss: 70.0617
Epoch 11/200
2/2 [==============================] - 0s 2ms/step - loss: 58.6065
Epoch 12/200
2/2 [==============================] - 0s 2ms/step - loss: 43.6842
Epoch 13/200
2/2 [==============================] - 0s 2ms/step - loss: 26.3015
Epoch 14/200
2/2 [==============================] - 0s 2ms/step - loss: 15.8109
Epoch 15/200
2/2 [==============================] - 0s 1ms/step - loss: 12.0864
Epoch 16/200
2/2 [==============================] - 0s 2ms/step - loss: 13.2904
Epoch 17/200
2/2 [==============================] - 0s 2ms/step - loss: 17.2666
Epoch 18/200
2/2 [==============================] - 0s 2ms/step - loss: 20.2788
Epoch 19/200
2/2 [==============================] - 0s 1ms/step - loss: 21.0044
Epoch 20/200
2/2 [==============================] - 0s 2ms/step - loss: 19.3135
Epoch 21/200
2/2 [==============================] - 0s 2ms/step - loss: 16.5099
Epoch 22/200
2/2 [==============================] - 0s 1ms/step - loss: 13.8078
Epoch 23/200
2/2 [==============================] - 0s 1ms/step - loss: 12.4786
Epoch 24/200
2/2 [==============================] - 0s 1ms/step - loss: 11.5450
Epoch 25/200
2/2 [==============================] - 0s 2ms/step - loss: 12.2430
Epoch 26/200
2/2 [==============================] - 0s 2ms/step - loss: 12.6910
Epoch 27/200
2/2 [==============================] - 0s 2ms/step - loss: 13.0053
Epoch 28/200
2/2 [==============================] - 0s 1ms/step - loss: 12.8281
Epoch 29/200
2/2 [==============================] - 0s 1ms/step - loss: 12.4289
Epoch 30/200
2/2 [==============================] - 0s 1ms/step - loss: 11.6947
Epoch 31/200
2/2 [==============================] - 0s 1ms/step - loss: 11.3287
Epoch 32/200
2/2 [==============================] - 0s 2ms/step - loss: 11.3578
Epoch 33/200
2/2 [==============================] - 0s 1ms/step - loss: 11.3632
Epoch 34/200
2/2 [==============================] - 0s 1ms/step - loss: 11.5163
Epoch 35/200
2/2 [==============================] - 0s 1ms/step - loss: 11.5517
Epoch 36/200
2/2 [==============================] - 0s 2ms/step - loss: 11.4354
Epoch 37/200
2/2 [==============================] - 0s 2ms/step - loss: 11.2493
Epoch 38/200
2/2 [==============================] - 0s 2ms/step - loss: 11.0766
Epoch 39/200
2/2 [==============================] - 0s 2ms/step - loss: 11.0697
Epoch 40/200
2/2 [==============================] - 0s 2ms/step - loss: 10.9828
Epoch 41/200
2/2 [==============================] - 0s 1ms/step - loss: 11.0126
Epoch 42/200
2/2 [==============================] - 0s 1ms/step - loss: 11.0186
Epoch 43/200
2/2 [==============================] - 0s 1ms/step - loss: 10.9801
Epoch 44/200
2/2 [==============================] - 0s 1ms/step - loss: 10.9340
Epoch 45/200
2/2 [==============================] - 0s 1ms/step - loss: 10.7914
Epoch 46/200
2/2 [==============================] - 0s 2ms/step - loss: 10.7687
Epoch 47/200
2/2 [==============================] - 0s 2ms/step - loss: 10.6795
Epoch 48/200
2/2 [==============================] - 0s 2ms/step - loss: 10.6936
Epoch 49/200
2/2 [==============================] - 0s 1ms/step - loss: 10.6416
Epoch 50/200
2/2 [==============================] - 0s 1ms/step - loss: 10.6058
Epoch 51/200
2/2 [==============================] - 0s 1ms/step - loss: 10.5609
Epoch 52/200
2/2 [==============================] - 0s 1ms/step - loss: 10.5371
Epoch 53/200
2/2 [==============================] - 0s 1ms/step - loss: 10.4772
Epoch 54/200
2/2 [==============================] - 0s 3ms/step - loss: 10.4357
Epoch 55/200
2/2 [==============================] - 0s 1ms/step - loss: 10.4067
Epoch 56/200
2/2 [==============================] - 0s 2ms/step - loss: 10.3866
Epoch 57/200
2/2 [==============================] - 0s 2ms/step - loss: 10.3872
Epoch 58/200
2/2 [==============================] - 0s 1ms/step - loss: 10.3384
Epoch 59/200
2/2 [==============================] - 0s 2ms/step - loss: 10.3408
Epoch 60/200
2/2 [==============================] - 0s 1ms/step - loss: 10.3051
Epoch 61/200
2/2 [==============================] - 0s 1ms/step - loss: 10.2206
Epoch 62/200
2/2 [==============================] - 0s 1ms/step - loss: 10.1830
Epoch 63/200
2/2 [==============================] - 0s 2ms/step - loss: 10.2140
Epoch 64/200
2/2 [==============================] - 0s 2ms/step - loss: 10.1338
Epoch 65/200
2/2 [==============================] - 0s 1ms/step - loss: 10.1010
Epoch 66/200
2/2 [==============================] - 0s 1ms/step - loss: 10.0534
Epoch 67/200
2/2 [==============================] - 0s 2ms/step - loss: 10.0265
Epoch 68/200
2/2 [==============================] - 0s 1ms/step - loss: 9.9996
Epoch 69/200
2/2 [==============================] - 0s 1ms/step - loss: 9.9761
Epoch 70/200
2/2 [==============================] - 0s 1ms/step - loss: 9.9521
Epoch 71/200
2/2 [==============================] - 0s 2ms/step - loss: 9.9255
Epoch 72/200
2/2 [==============================] - 0s 1ms/step - loss: 9.8958
Epoch 73/200
2/2 [==============================] - 0s 4ms/step - loss: 9.8855
Epoch 74/200
2/2 [==============================] - 0s 2ms/step - loss: 9.8750
Epoch 75/200
2/2 [==============================] - 0s 2ms/step - loss: 9.8222
Epoch 76/200
2/2 [==============================] - 0s 2ms/step - loss: 9.7902
Epoch 77/200
2/2 [==============================] - 0s 2ms/step - loss: 9.7729
Epoch 78/200
2/2 [==============================] - 0s 2ms/step - loss: 9.7519
Epoch 79/200
2/2 [==============================] - 0s 2ms/step - loss: 9.7254
Epoch 80/200
2/2 [==============================] - 0s 2ms/step - loss: 9.7084
Epoch 81/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6988
Epoch 82/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6648
Epoch 83/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6532
Epoch 84/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6367
Epoch 85/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6036
Epoch 86/200
2/2 [==============================] - 0s 2ms/step - loss: 9.6125
Epoch 87/200
2/2 [==============================] - 0s 2ms/step - loss: 9.5628
Epoch 88/200
2/2 [==============================] - 0s 2ms/step - loss: 9.5510
Epoch 89/200
2/2 [==============================] - 0s 2ms/step - loss: 9.5662
Epoch 90/200
2/2 [==============================] - 0s 2ms/step - loss: 9.5147
Epoch 91/200
2/2 [==============================] - 0s 2ms/step - loss: 9.5191
Epoch 92/200
2/2 [==============================] - 0s 1ms/step - loss: 9.5031
Epoch 93/200
2/2 [==============================] - 0s 1ms/step - loss: 9.4601
Epoch 94/200
2/2 [==============================] - 0s 2ms/step - loss: 9.4431
Epoch 95/200
2/2 [==============================] - 0s 2ms/step - loss: 9.4269
Epoch 96/200
2/2 [==============================] - 0s 2ms/step - loss: 9.4388
Epoch 97/200
2/2 [==============================] - 0s 2ms/step - loss: 9.3951
Epoch 98/200
2/2 [==============================] - 0s 2ms/step - loss: 9.4153
Epoch 99/200
2/2 [==============================] - 0s 1ms/step - loss: 9.3706
Epoch 100/200
2/2 [==============================] - 0s 1ms/step - loss: 9.3629
Epoch 101/200
2/2 [==============================] - 0s 4ms/step - loss: 9.3394
Epoch 102/200
2/2 [==============================] - 0s 2ms/step - loss: 9.3237
Epoch 103/200
2/2 [==============================] - 0s 2ms/step - loss: 9.3269
Epoch 104/200
2/2 [==============================] - 0s 2ms/step - loss: 9.3081
Epoch 105/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2812
Epoch 106/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2657
Epoch 107/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2867
Epoch 108/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2592
Epoch 109/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2355
Epoch 110/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2726
Epoch 111/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2075
Epoch 112/200
2/2 [==============================] - 0s 2ms/step - loss: 9.2053
Epoch 113/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1904
Epoch 114/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1810
Epoch 115/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1972
Epoch 116/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1841
Epoch 117/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1773
Epoch 118/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1964
Epoch 119/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1943
Epoch 120/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1598
Epoch 121/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1316
Epoch 122/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1139
Epoch 123/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1245
Epoch 124/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1076
Epoch 125/200
2/2 [==============================] - 0s 1ms/step - loss: 9.0872
Epoch 126/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1018
Epoch 127/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0746
Epoch 128/200
2/2 [==============================] - 0s 4ms/step - loss: 9.0583
Epoch 129/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1542
Epoch 130/200
2/2 [==============================] - 0s 2ms/step - loss: 9.1178
Epoch 131/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0470
Epoch 132/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0456
Epoch 133/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0353
Epoch 134/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0327
Epoch 135/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0275
Epoch 136/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0258
Epoch 137/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0356
Epoch 138/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0048
Epoch 139/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0333
Epoch 140/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0226
Epoch 141/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0144
Epoch 142/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0049
Epoch 143/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9832
Epoch 144/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9752
Epoch 145/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0019
Epoch 146/200
2/2 [==============================] - 0s 1ms/step - loss: 9.0445
Epoch 147/200
2/2 [==============================] - 0s 4ms/step - loss: 8.9637
Epoch 148/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9682
Epoch 149/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9812
Epoch 150/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9747
Epoch 151/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9863
Epoch 152/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9593
Epoch 153/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9487
Epoch 154/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9373
Epoch 155/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9841
Epoch 156/200
2/2 [==============================] - 0s 1ms/step - loss: 9.1398
Epoch 157/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0232
Epoch 158/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9637
Epoch 159/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9288
Epoch 160/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9415
Epoch 161/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9729
Epoch 162/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9534
Epoch 163/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9379
Epoch 164/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9516
Epoch 165/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9502
Epoch 166/200
2/2 [==============================] - 0s 2ms/step - loss: 9.0555
Epoch 167/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9306
Epoch 168/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9888
Epoch 169/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9640
Epoch 170/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9570
Epoch 171/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9267
Epoch 172/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9134
Epoch 173/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9231
Epoch 174/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9520
Epoch 175/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9192
Epoch 176/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9162
Epoch 177/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9310
Epoch 178/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9030
Epoch 179/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9056
Epoch 180/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9027
Epoch 181/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9037
Epoch 182/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8955
Epoch 183/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9029
Epoch 184/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9420
Epoch 185/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9328
Epoch 186/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9198
Epoch 187/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9586
Epoch 188/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8910
Epoch 189/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8981
Epoch 190/200
2/2 [==============================] - 0s 1ms/step - loss: 8.9199
Epoch 191/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9941
Epoch 192/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9090
Epoch 193/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8938
Epoch 194/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9167
Epoch 195/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9029
Epoch 196/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8897
Epoch 197/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8889
Epoch 198/200
2/2 [==============================] - 0s 1ms/step - loss: 8.8904
Epoch 199/200
2/2 [==============================] - 0s 2ms/step - loss: 8.8932
Epoch 200/200
2/2 [==============================] - 0s 2ms/step - loss: 8.9216
And we can plot the loss during the training.
plt.plot(history.history['loss'])
plt.show()
Model prediction
There are two ways to generate the adjusted model. The first one will be simlpy to use the .predict() method directly over the $x$ samples:
y_pred_model = model.predict(x)
plt.scatter(x, y, label='Generated data')
plt.plot(x, y_pred_model, label='Predicted with model', color='c')
plt.legend()
plt.show()
The second (and my favorite) way is to understand the guts inside the network and access the information to literally replicate the model.
In this case we acces the first (and only) layer:
layer = model.get_layer(index=0)
layer
<tensorflow.python.keras.layers.core.Dense at 0x7fe81910e828>
Then, we get and print the weights and biases:
weights = layer.get_weights()
weights
[array([[1.014319]], dtype=float32), array([4.2396894], dtype=float32)]
As we previously mentioned, the only weight will correspond to the slope and the bias to the intersection point. In order to replicate the linear equation we simply do:
m, b = weights[0][0], weights[1]
print(m)
print(b)
[1.014319]
[4.2396894]
y_pred_params = m * x + b
y_pred_params
array([ 4.23968935, 5.25400829, 6.26832724, 7.28264618, 8.29696512,
9.31128407, 10.32560301, 11.33992195, 12.35424089, 13.36855984,
14.38287878, 15.39719772, 16.41151667, 17.42583561, 18.44015455,
19.4544735 , 20.46879244, 21.48311138, 22.49743032, 23.51174927,
24.52606821, 25.54038715, 26.5547061 , 27.56902504, 28.58334398,
29.59766293, 30.61198187, 31.62630081, 32.64061975, 33.6549387 ,
34.66925764, 35.68357658, 36.69789553, 37.71221447, 38.72653341,
39.74085236, 40.7551713 , 41.76949024, 42.78380919, 43.79812813,
44.81244707, 45.82676601, 46.84108496, 47.8554039 , 48.86972284,
49.88404179, 50.89836073, 51.91267967, 52.92699862, 53.94131756,
54.9556365 ])
plt.scatter(x, y, label='Generated data')
plt.plot(x, y_pred_params, label='Line fitted using parameter values', color='c')
plt.legend()
plt.show()
If you want to run the code above directly on Google Colab, please follow this link.
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