Processing math: 100%

import pandas as pd
import numpy as np
import plotly.express as px
import plotly.graph_objects as go
import seaborn as sns

data = sns.load_dataset('mpg')
data.head()

	mpg	cylinders	displacement	horsepower	weight	acceleration	model_year	origin	name
0	18.0	8	307.0	130.0	3504	12.0	70	usa	chevrolet chevelle malibu
1	15.0	8	350.0	165.0	3693	11.5	70	usa	buick skylark 320
2	18.0	8	318.0	150.0	3436	11.0	70	usa	plymouth satellite
3	16.0	8	304.0	150.0	3433	12.0	70	usa	amc rebel sst
4	17.0	8	302.0	140.0	3449	10.5	70	usa	ford torino

fig = px.scatter(data, x="displacement", y="mpg", trendline="ols")
fig

model = px.get_trendline_results(fig).iloc[0][0]
model.summary()

OLS Regression Results

Dep. Variable:	y	R-squared:	0.647
Model:	OLS	Adj. R-squared:	0.646
Method:	Least Squares	F-statistic:	725.0
Date:	Tue, 02 Mar 2021	Prob (F-statistic):	1.66e-91
Time:	14:50:54	Log-Likelihood:	-1175.5
No. Observations:	398	AIC:	2355.
Df Residuals:	396	BIC:	2363.
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	35.1748	0.492	71.519	0.000	34.208	36.142
x1	-0.0603	0.002	-26.926	0.000	-0.065	-0.056

Omnibus:	41.373	Durbin-Watson:	1.394
Prob(Omnibus):	0.000	Jarque-Bera (JB):	60.024
Skew:	0.711	Prob(JB):	9.24e-14
Kurtosis:	4.264	Cond. No.	463.

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Making the Design Matrix

X = data[ ['displacement'] ]
y = data[ ['mpg'] ]

X.head()

	displacement
0	307.0
1	350.0
2	318.0
3	304.0
4	302.0

Adding the constant term:

data["constant"] = 1
X = data[['displacement', "constant"]]

X.head()

	displacement	constant
0	307.0	1
1	350.0	1
2	318.0	1
3	304.0	1
4	302.0	1

X.head()

	displacement	constant
0	307.0	1
1	350.0	1
2	318.0	1
3	304.0	1
4	302.0	1

$$(X^T X)^{-1} X^T y$$

X.T @ X

	displacement	constant
displacement	19206864.25	76983.5
constant	76983.50	398.0

X.T @ y

	mpg
displacement	1550039.4
constant	9358.8

Solving the Linear System

from numpy.linalg import inv, solve

The exact math from lecture.

$$(X^T X)^{-1} X^T y$$

inv(X.T @ X) @ (X.T @ y)

	mpg
0	-0.060282
1	35.174750

More numerically stable and computationally efficient.

$$A\theta = b$$ $$X^T X \theta = X^T y$$

solve(X.T @ X, X.T @ y)

array([[-0.06028241],
       [35.17475015]])

Using a software package:

from sklearn.linear_model import LinearRegression

model = LinearRegression(fit_intercept=False)
model.fit(X, y)
print(model.coef_)

[[-0.06028241 35.17475015]]

Making Predictions

theta = solve(X.T @ X, X.T @ y)

theta

array([[-0.06028241],
       [35.17475015]])

data['yhat'] = (X @ theta)
data.head()

	mpg	cylinders	displacement	horsepower	weight	acceleration	model_year	origin	name	constant	yhat
0	18.0	8	307.0	130.0	3504	12.0	70	usa	chevrolet chevelle malibu	1	16.668052
1	15.0	8	350.0	165.0	3693	11.5	70	usa	buick skylark 320	1	14.075908
2	18.0	8	318.0	150.0	3436	11.0	70	usa	plymouth satellite	1	16.004945
3	16.0	8	304.0	150.0	3433	12.0	70	usa	amc rebel sst	1	16.848899
4	17.0	8	302.0	140.0	3449	10.5	70	usa	ford torino	1	16.969464

data['yhat'] = model.predict(X)
data.head()

	mpg	cylinders	displacement	horsepower	weight	acceleration	model_year	origin	name	constant	yhat
0	18.0	8	307.0	130.0	3504	12.0	70	usa	chevrolet chevelle malibu	1	16.668052
1	15.0	8	350.0	165.0	3693	11.5	70	usa	buick skylark 320	1	14.075908
2	18.0	8	318.0	150.0	3436	11.0	70	usa	plymouth satellite	1	16.004945
3	16.0	8	304.0	150.0	3433	12.0	70	usa	amc rebel sst	1	16.848899
4	17.0	8	302.0	140.0	3449	10.5	70	usa	ford torino	1	16.969464

Examining the Residuals

data["residual"] = data["mpg"] - data["yhat"] 

fig = px.scatter(data, x="displacement", y="residual")
fig.add_trace(go.Scatter(x=[50, 475], y=[0,0], name = "Model"))

data['residual'].sum()

-1.7053025658242404e-13

Root Mean Squared Error

$$\sqrt{\frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2 }$$

np.sqrt(np.mean(data['residual']**2))

4.6396293441245815

data["residual"].abs().mean()

3.526327374875378

Improving the Model

data.head()

	mpg	cylinders	displacement	horsepower	weight	acceleration	model_year	origin	name	constant	yhat	residual
0	18.0	8	307.0	130.0	3504	12.0	70	usa	chevrolet chevelle malibu	1	16.668052	1.331948
1	15.0	8	350.0	165.0	3693	11.5	70	usa	buick skylark 320	1	14.075908	0.924092
2	18.0	8	318.0	150.0	3436	11.0	70	usa	plymouth satellite	1	16.004945	1.995055
3	16.0	8	304.0	150.0	3433	12.0	70	usa	amc rebel sst	1	16.848899	-0.848899
4	17.0	8	302.0	140.0	3449	10.5	70	usa	ford torino	1	16.969464	0.030536

model2 = LinearRegression(fit_intercept=True)
features = ["cylinders", "displacement", "weight", "model_year", "acceleration"]

model2.fit(data[features], data[['mpg']])
print(model2.coef_)

[[-0.25858516  0.00726771 -0.00692571  0.75530084  0.08034746]]

data['yhat2'] = model2.predict(data[features])

What can we say about the magnitudes of the weights?

px.histogram(data, 
             x = ["cylinders", "displacement", "weight"],
             barmode="overlay",
             marginal="box")

Rescaling the features

from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
Z = scaler.fit_transform(data[features])

Z

array([[ 1.49819126,  1.0906037 ,  0.63086987, -1.62742629, -1.29549834],
       [ 1.49819126,  1.5035143 ,  0.85433297, -1.62742629, -1.47703779],
       [ 1.49819126,  1.19623199,  0.55047045, -1.62742629, -1.65857724],
       ...,
       [-0.85632057, -0.56103873, -0.79858454,  1.62198339, -1.4407299 ],
       [-0.85632057, -0.70507731, -0.40841088,  1.62198339,  1.10082237],
       [-0.85632057, -0.71467988, -0.29608816,  1.62198339,  1.39128549]])

px.histogram(x = [Z[:,0], Z[:,1], Z[:,2]],
             barmode="overlay")

model3 = LinearRegression(fit_intercept=True)
model3.fit(Z, data[['mpg']])
print(model3.coef_)
data['yhat3'] = model3.predict(Z)

[[-0.43930153  0.75684991 -5.85760433  2.78930976  0.22129477]]

features

['cylinders', 'displacement', 'weight', 'model_year', 'acceleration']

data

	mpg	cylinders	displacement	horsepower	weight	acceleration	model_year	origin	name	constant	yhat	residual	yhat2	yhat3
0	18.0	8	307.0	130.0	3504	12.0	70	usa	chevrolet chevelle malibu	1	16.668052	1.331948	15.160369	15.160369
1	15.0	8	350.0	165.0	3693	11.5	70	usa	buick skylark 320	1	14.075908	0.924092	14.123749	14.123749
2	18.0	8	318.0	150.0	3436	11.0	70	usa	plymouth satellite	1	16.004945	1.995055	15.630915	15.630915
3	16.0	8	304.0	150.0	3433	12.0	70	usa	amc rebel sst	1	16.848899	-0.848899	15.630291	15.630291
4	17.0	8	302.0	140.0	3449	10.5	70	usa	ford torino	1	16.969464	0.030536	15.384423	15.384423
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
393	27.0	4	140.0	86.0	2790	15.6	82	usa	ford mustang gl	1	26.735213	0.264787	29.278818	29.278818
394	44.0	4	97.0	52.0	2130	24.6	82	europe	vw pickup	1	29.327357	14.672643	34.260400	34.260400
395	32.0	4	135.0	84.0	2295	11.6	82	usa	dodge rampage	1	27.036625	4.963375	32.349314	32.349314
396	28.0	4	120.0	79.0	2625	18.6	82	usa	ford ranger	1	27.940861	0.059139	30.517248	30.517248
397	31.0	4	119.0	82.0	2720	19.4	82	usa	chevy s-10	1	28.001144	2.998856	29.916316	29.916316

398 rows × 14 columns

Residual Analysis

data['residual2'] = data['mpg'] - data['yhat2']

np.sqrt(np.mean(data['residual2']**2))

3.4143567605602265

Computing $R^2$

from sklearn.metrics import r2_score

r2_score(data["mpg"], data["yhat"])

0.646742183425786

r2_score(data["mpg"], data["yhat2"])

0.8086876515237436