lab-sessions-26a

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In [17]:

# data for the OLS
import numpy as np
import matplotlib.pyplot as plt

theta0 = 2.6486
theta1 = 4.98
m = 500
disp = 1
x = np.linspace(0,5,m)
y = theta0+theta1*x+disp*np.random.randn(m)

In [18]:

plt.plot(x,y, '.k')
plt.show()

No description has been provided for this image

In [20]:

xMean = x.mean()
yMean = y.mean()

Sxx = np.sum((x - xMean)**2)
Syy = np.sum((y - yMean)**2)
Sxy = np.sum((x - xMean) * (y - yMean))

r = Sxy / np.sqrt(Sxx * Syy)
r

Out[20]:

np.float64(0.990800615658485)

In [22]:

t1 = Sxy / Sxx
t0 = yMean - t1 * xMean

yHat = t0 + t1 * x
residuals = y - yHat
residuals

Out[22]:

array([-0.95082413, -0.64573564,  0.87130595, -0.53988214, -0.56134601,
        0.718763  , -1.57062343,  0.39547852,  0.28642336,  0.67589762,
       -0.05256052,  0.52740588, -0.2622087 ,  0.95248882,  0.77106812,
        1.04923914, -0.56930707,  0.0336549 , -1.23845916,  1.34279223,
       -1.28317243,  0.23367606,  1.5810006 , -1.28925369, -0.91503262,
       -0.47238037, -0.51189083,  0.22029059,  0.1362538 ,  0.10890607,
        1.45135919,  1.07239237, -0.9002686 ,  1.64189709, -0.28342477,
       -1.09095379, -0.95598686, -0.40362419, -1.11918067,  0.20567161,
       -0.87438623, -1.84805568, -0.00444769, -0.11272938,  0.50923524,
       -0.89881567,  1.16206936, -1.9768896 , -0.05677879,  0.38329957,
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       -0.22397765,  0.09222097,  0.32542925, -0.64405536,  0.69482021,
        0.39390481, -0.42203615,  1.46687941, -0.54187665, -1.21430829,
        0.27910103,  1.39819159, -1.17537755, -0.65009484,  0.99475956,
        1.0573686 , -1.03661444,  0.1850461 ,  0.47221162,  0.30483808,
        0.44341025,  1.80746389, -0.08666078,  0.38689218, -0.79105298,
       -0.46934343, -2.98387474,  0.49558085, -0.82921583,  0.30384201,
       -0.41699037,  0.71682262,  0.08100853,  0.95911846,  0.1805543 ,
        1.24498695,  2.13075034, -0.79247779, -1.46850503, -1.45895857,
       -0.10525365, -0.24755927, -0.4308391 , -0.14923592,  0.65655332,
        0.4330343 , -1.14340091, -0.19470556, -1.27218357,  0.1085798 ,
        1.00076225,  1.064037  , -0.69196389, -2.42524639,  0.27019292,
       -0.73286207,  1.14572395,  0.51588267,  1.5363161 ,  0.85035987,
       -0.58268652,  0.15414688, -1.88700757, -0.09797611, -0.86406005,
        0.21557541,  0.29166805,  2.06165041,  0.54580793,  2.23401705,
        1.04566736, -1.40389533,  1.76667624, -0.83624078, -0.51967116,
        0.46900263, -0.28565591,  0.0056141 ,  1.7469312 , -0.02517737,
        0.10641521,  0.6377089 , -0.92879003, -1.98053636, -0.38344278,
       -0.80088618,  0.77383712,  0.9961228 ,  0.43158633,  2.09416231,
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       -0.38056226,  0.62933008,  1.69891605,  0.85233287, -0.64892305,
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        1.19596399, -0.46304232, -0.54587135,  0.58229878,  1.03146887,
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        1.5261428 , -0.86068793,  0.26628651,  0.8725808 ,  0.10768281,
       -0.02945102,  1.08162492,  0.96757504,  0.8487286 ,  0.40946741,
       -0.02082662, -2.8380769 ,  1.00451544,  0.33697275, -0.99442708,
        1.06901056,  1.17072805, -0.5387951 ,  0.48246067, -0.39052061,
       -1.28477645, -1.04858003, -0.88628018, -0.47541591,  0.50383958,
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       -2.83929432, -0.09720386, -0.07391387, -1.31838473,  0.75623635,
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        1.40348052, -0.32213203,  0.85653684,  0.54061916, -0.06615145,
       -0.06541276, -0.1874332 ,  1.64653204,  0.27782626, -1.56029627,
        1.65082298,  0.67656205, -1.58084623, -0.43152429,  1.2077006 ,
       -0.01064432,  0.39992249,  0.7204933 , -0.97893129,  0.3783936 ,
        1.42397314,  2.2570722 ,  0.73192937, -0.6796615 ,  0.79733617,
       -1.22591839, -0.3439277 ,  0.04718119,  1.01248675, -0.25599286,
       -0.78884594,  0.40217221,  0.54593394,  0.77981857,  1.6034912 ,
       -0.58348904,  0.23082941, -0.84446779,  0.0897946 ,  0.32875383,
        0.5969846 ,  0.2950068 , -1.85252387,  0.20742835, -1.99840818,
        1.13742464,  0.32724843,  0.80450174,  0.25135255,  0.6623171 ,
       -0.92213262, -0.07474869, -1.32358274, -1.36633224, -0.17526807,
       -1.49026666, -0.04415546,  0.51904066, -0.10048653,  0.10796533])

In [23]:

R2 = r**2
R2

Out[23]:

np.float64(0.9816858599892329)

In [25]:

print("Correlation r:", r)
print("Intercept t0:", t0)
print("Slope t1:", t1)
print("R2:", R2)

Correlation r: 0.990800615658485
Intercept t0: 2.715728336655882
Slope t1: 4.987467248897177
R2: 0.9816858599892329

In [30]:

plt.scatter(x, y, color='black', alpha=0.3)
plt.plot(x, yHat, ':r')
plt.xlabel("x")
plt.ylabel("y")
plt.title("Simple Linear Regression")
plt.show()

In [38]:

plt.scatter(x, residuals, color="red", alpha=0.3)
plt.axhline(0, color='black')
plt.xlabel("x")
plt.ylabel("Residuals")
plt.title("Residual Plot")
plt.show()

In [1]:

import numpy as np
import matplotlib.pyplot as plt

x = np.array([1, 2, 3, 4, 5, 6, 7, 8], dtype=float)
y = np.array([2.2, 4.1, 5.9, 8.3, 10.1, 12.0, 13.8, 16.2], dtype=float)

x_mean = x.mean()
y_mean = y.mean()

Sxx = np.sum((x - x_mean)**2)
Syy = np.sum((y - y_mean)**2)
Sxy = np.sum((x - x_mean) * (y - y_mean))

r = Sxy / np.sqrt(Sxx * Syy)

b1 = Sxy / Sxx
b0 = y_mean - b1 * x_mean

y_hat = b0 + b1 * x
residuals = y - y_hat

R2 = r**2

print("Correlation r:", r)
print("Intercept b0:", b0)
print("Slope b1:", b1)
print("R²:", R2)

plt.scatter(x, y)
plt.plot(x, y_hat)
plt.xlabel("x")
plt.ylabel("y")
plt.title("Simple Linear Regression")
plt.show()

plt.scatter(x, residuals)
plt.axhline(0)
plt.xlabel("x")
plt.ylabel("Residuals")
plt.title("Residual Plot")
plt.show()

Correlation r: 0.999445617564577
Intercept b0: 0.15000000000000036
Slope b1: 1.9833333333333332
R²: 0.9988915424690386

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196 KiB

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