18.4: multiple linear regression
- Page ID
- 25000
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Multiple linear regression is a statistical method used to understand the relationship between one dependent variable and two or more independent variables. By creating an equation, this method predicts the value of the dependent variable based on the values of the independents. In practice, it helps to answer questions like how changes in factors like price and promotion affect sales, or how different health indicators can predict a patient's blood pressure. It's a step beyond simple linear regression, which only considers one independent variable, allowing for more complex analyses and more accurate predictions when several factors are at play simultaneously.
In the provided multiple linear regression example using scikit-learn, we have a dataset with two independent variables: hours studied and hours slept, and one dependent variable, which is the exam score.
from sklearn.linear_model import LinearRegression
import numpy as np
# Sample data with two independent variables (features) and one dependent variable (target)
# Let's assume X1 represents hours studied and X2 represents hours slept
# Y represents the exam score
X = np.array([[1, 8], [2, 7], [3, 6], [4, 5], [5, 4]]) # Features: Hours Studied, Hours Slept
y = np.array([5, 12, 14, 22, 32]) # Target: Exam Score
# Create an instance of the model
model = LinearRegression()
# Fit the model
model.fit(X, y)
# Coefficients for each independent variable
coefficients = model.coef_
# Intercept of the model
intercept = model.intercept_
# Let's predict the exam score for someone who studied for 6 hours and slept for 3 hours
new_data = np.array([[6, 3]])
predicted_score = model.predict(new_data)
coefficients, intercept, predicted_score
After fitting the model with the sample data, we obtain the following output:
(array([ 3.2, -3.2]), 26.599999999999998, array([36.2]))
- The coefficients for the independent variables are approximately 3.2 for hours studied and -3.2 for hours slept. This suggests that for each additional hour studied, the exam score is expected to increase by 3.2 points, and for each additional hour slept, the score is expected to decrease by 3.2 points, according to the model.
- The intercept of the model is approximately 26.6, which is the expected exam score if both independent variables were zero (which isn't practically possible, but it is a feature of the linear model).
Using this model, we predict that someone who studied for 6 hours and slept for 3 hours would have an exam score of approximately 36.2 points. This prediction is based on the linear relationship learned from the provided sample data.