Polynomial Regression

Polynomial regression is a type of regression analysis where the relationship between the independent variable(s) and the dependent variable is modeled as an nth-degree polynomial. Unlike simple or multiple linear regression, which assume a linear relationship between variables, polynomial regression can capture non-linear relationships between variables.

y=β₀ ​+β₁ ​x + β₂ x² +…+ β_{n ​} xⁿ + ε

Where:

• y is the dependent variable.
• x is the independent variable.
• x², x³, …, xⁿ are the polynomial terms, representing the square, cube, and higher-order terms of x.
• β₀ , β₁ ​, … , β_n are the coefficients, representing the effect of each polynomial term on y.
• ε is the error term, representing the difference between the observed and predicted values of y.

Polynomial regression extends linear regression by considering polynomial terms of the independent variable.

• Use the poly() function in R to generate polynomial terms.
• Fit the model using lm() with polynomial terms included.

Output:

Call:
lm(formula = y ~ poly(x, degree = 2))

Residuals:
      1       2       3       4       5 
-0.3143  0.4571  0.5143 -1.1429  0.4857 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)  
(Intercept)            4.2000     0.4598   9.134   0.0118 *
poly(x, degree = 2)1   2.5298     1.0282   2.460   0.1330  
poly(x, degree = 2)2  -0.5345     1.0282  -0.520   0.6550  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.028 on 2 degrees of freedom
Multiple R-squared:  0.7597,    Adjusted R-squared:  0.5195 
F-statistic: 3.162 on 2 and 2 DF,  p-value: 0.2403

This output is from a polynomial regression model fitted to data with one independent variable (x) and one dependent variable (y).

• Residuals: Differences between observed and predicted values of y.
• Coefficients: Estimates of intercept and polynomial terms (poly(x, degree = 2)).
  • Estimate: Coefficient values.
  • Std. Error: Standard errors.
  • t value: T-statistics for significance.
  • Pr(>|t|): P-values indicating significance.
• Model Fit:
  • Residual standard error: Standard deviation of residuals.
  • Multiple R-squared: Approximately 75.97% of variance explained.
  • Adjusted R-squared: Adjusted for predictors.
  • F-statistic: Tests overall model significance.

Regression analysis allows us to understand how one or more independent variables relate to a dependent variable. Simple linear regression, which explores the relationship between two variables. Multiple linear regression extends this to include several predictors simultaneously. Finally, polynomial regression introduces flexibility by accommodating non-linear relationships in the R Programming Language.