Regression Analysiss
Polynomial Regression
Learning objectives:
What is Polynomial Regression?
Polynomial regression is a regression analysis where we model the relationship between the independent x and the dependent variable y as an nth-degree polynomial.
Polynomial regression can be used to model relationships between variables that are not linear. In contrast, multiple linear regression models the relationship between more than a single independent variable and a single dependent variable using a straight line.
The polynomial regression model is a more general form of linear regression, which is formed by adding powers of the original features as new features. When we have multiple features, polynomial regression is extended by adding interaction terms and polynomial terms of the original features.
In this tutorial, we will explore the various techniques for fitting a polynomial regression model, including linear regression, ridge regression, and lasso regression.
We will also examine how to select the appropriate degree of the polynomial and assess the model's goodness of fit. Finally, we will look at examples of polynomial regression in action and how we can use it in real-world applications.
The Mathematical Notation
To start, let's first look at the mathematical notation of a polynomial regression model. Given a set of n pairs of data \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\), the polynomial regression model takes the form:
\[y = b_0 + b_1x + b_2x^2 + ... + b_nx^n\]
where \(b_0, b_1, b_2, ..., b_n\) are the coefficients of the polynomial and \(x\) is the independent variable.
The degree of the polynomial, \(n\), determines the complexity of the model. A model with a high degree of \(n\) will have higher flexibility and can potentially fit the data better, but it also runs the risk of overfitting.
Now, let's consider the case of multiple independent variables. The polynomial regression model can be extended to include multiple features as follows:
\[y = b_0 + b_1x_1 + b_2x_2 + ... + b_nx_n + b_n+1x_1x_2 + b_n+2x_1x_3 + ... + b_mx_n-1x_n\]
where \(x_1, x_2, ..., x_n\) are the independent variables and \(b_0, b_1, b_2, ..., b_m\) are the coefficients.
Fitting The Model
Several techniques for fitting a polynomial regression model include linear regression, ridge regression, and lasso regression. Let's go through each of these techniques in turn.
Linear regression is the most straightforward and most commonly used technique for fitting a polynomial regression model. It seeks to reduce the residual sum of squares between the observed answers in the dataset and those that the linear approximation predicted.
The polynomial coefficients are estimated using the least squares method, which minimizes the sum of the squared residuals.
A variant of linear regression, Ridge regression, adds a regularization term to the loss function. The regularization term is a penalty on the size of the coefficients, which helps to prevent overfitting by keeping the coefficients small. Ridge regression is well-suited for polynomial regression with multiple features, as it can help reduce the model's complexity.
Lasso regression is another variant of linear regression that adds a regularization term to the loss function. However, unlike ridge regression, which uses the L2 norm of the coefficients as the regularization term, lasso regression uses the L1 norm. This results in a sparse model, where some coefficients are precisely equal to zero. Lasso regression is particularly useful for feature selection, as it automatically selects a subset of the essential features.
Now that we have an overview of the different techniques for fitting a polynomial regression model let's discuss how to select the appropriate degree of the polynomial.
Selecting The Degree
The polynomial degree should be chosen based on the complexity of the relationship between the independent and dependent variables. Plus, the number of observations in the dataset.
A model with a high degree of the polynomial will have higher flexibility and can potentially fit the data better, but it also runs the risk of overfitting.
One way to select the appropriate degree of the polynomial is to use cross-validation to evaluate the model's performance for different degrees and choose the one that performs the best.
Once the polynomial regression model has been fit, it is essential to assess its goodness of fit. One way to do this is to use the coefficient of determination, \(R^2\), which measures the proportion of the variance in the dependent variable that the model explains. \(R^2\) is calculated as follows:
\[R^2 = 1 - {sum \ of \ squared \ residuals \over sum \ of \ squared \ total}\]
A model with a high \(R^2\) value is considered a good fit for the data. However, it is important to note that \(R^2\) is sensitive to the number of observations in the dataset, so it is not always a reliable measure of model performance.
Another way to assess the goodness of fit of a polynomial regression model is to plot the observed data and the predicted values. This kind of plot can help visualize the model's fit and identify any patterns or discrepancies in the data.
Now that we have a general understanding of polynomial regression and how it can be used, let's look at some examples of polynomial regression in action.
Common Applications
One common application of polynomial regression is finance, where it can be used to model the relationship between financial variables such as stock prices or exchange rates. For example, a polynomial regression model could predict stock prices based on historical data and other factors such as economic indicators.
Polynomial regression can also be used in engineering, where it can be used to model the relationship between variables such as temperature and pressure. For example, a polynomial regression model could be used to predict the pressure of a gas based on its temperature and volume.
Finally, polynomial regression can be used in the field of biology it can be used to model the relationship between variables—for example, the concentration of a drug in the blood and the time since it was administered.
For example, a polynomial regression model could be used to predict the concentration of a drug in the blood based on the dosage and the time since it was administered.
In summary, polynomial regression is a powerful tool for modeling the relationship between variables that are not linearly related. It can be used in various fields, including finance, engineering, and biology, to make predictions and understand complex relationships in data.