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Linear Regression

Introduction

A data model explicitly describes a relationship between predictor and response variables. Linear regression fits a data model that is linear in the model coefficients. The most common type of linear regression is a least-squares fit, which can fit both lines and polynomials, among other linear models.

Before you model the relationship between pairs of quantities, it is a good idea to perform correlation analysis to establish if a linear relationship exists between these quantities. Be aware that variables can have nonlinear relationships, which correlation analysis cannot detect. For more information, see Linear Correlation.

The MATLAB® Basic Fitting GUI helps you to fit your data, so you can calculate model coefficients and plot the model on top of the data. For an example, see Example: Using Basic Fitting GUI. You also can use the MATLAB polyfit and polyval functions to fit your data to a model that is linear in the coefficients. For an example, see Programmatic Fitting.

If you need to fit data with a nonlinear model, transforming the variables to make the relationship linear. Alternatively, try to fit a nonlinear function directly using either the Statistics Toolbox™ nlinfit function, the Optimization Toolbox™ lsqcurvefit function, or by applying functions in the Curve Fitting Toolbox™.

This topic explains how to:

  • Use correlation analysis to determine whether two quantities are related to justify fitting the data.

  • Fit a linear model to the data.

  • Evaluate the goodness of fit by plotting residuals and looking for patterns.

  • Calculate measures of goodness of fit R2 and adjusted R2

Residuals and Goodness of Fit

Residuals are the difference between the observed values of the response (dependent) variable and the values that a model predicts. When you fit a model that is appropriate for your data, the residuals approximate independent random errors. That is, the distribution of residuals ought not to exhibit a discernible pattern.

Producing a fit using a linear model requires minimizing the sum of the squares of the residuals. This minimization yields what is called a least-squares fit. You can gain insight into the "goodness" of a fit by visually examining a plot of the residuals. If the residual plot has a pattern (that is, residual data points do not appear to have a random scatter), the randomness indicates that the model does not properly fit the data.

Evaluate each fit you make in the context of your data. For example, if your goal of fitting the data is to extract coefficients that have physical meaning, then it is important that your model reflect the physics of the data. Understanding what your data represents, how it was measured, and how it is modeled is important when evaluating the goodness of fit.

One measure of goodness of fit is the coefficient of determination, or R2 (pronounced r-square). This statistic indicates how closely values you obtain from fitting a model match the dependent variable the model is intended to predict. Statisticians often define R2 using the residual variance from a fitted model:

R2 = 1 – SSresid / SStotal

SSresid is the sum of the squared residuals from the regression. SStotal is the sum of the squared differences from the mean of the dependent variable (total sum of squares). Both are positive scalars.

To learn how to compute R2 when you use the Basic Fitting tool, see Derive R2, the Coefficient of Determination. To learn more about calculating the R2 statistic and its multivariate generalization, continue reading here.

Example: Computing R2 from Polynomial Fits

You can derive R2 from the coefficients of a polynomial regression to determine how much variance in y a linear model explains, as the following example describes:

  1. Create two variables, x and y from the first two columns of the count variable in the data file count.dat:

    load count.dat
    x = count(:,1);
    y = count(:,2);
  2. Use polyfit to compute a linear regression that predicts y from x:

    p = polyfit(x,y,1)
    
    p =
        1.5229   -2.1911
    

    p(1) is the slope and p(2) is the intercept of the linear predictor. You can also obtain regression coefficients using the Basic Fitting GUI.

  3. Call polyval to use p to predict y, calling the result yfit:

    yfit = polyval(p,x);

    Using polyval saves you from typing the fit equation yourself, which in this case looks like:

    yfit =  p(1) * x + p(2);
  4. Compute the residual values as a vector signed numbers:

    yresid = y - yfit;
  5. Square the residuals and total them obtain the residual sum of squares:

    SSresid = sum(yresid.^2);
  6. Compute the total sum of squares of y by multiplying the variance of y by the number of observations minus 1:

    SStotal = (length(y)-1) * var(y);
    
  7. Compute R2 using the formula given in the introduction of this topic:

    rsq = 1 - SSresid/SStotal
    
    rsq =
        0.8707

    This demonstrates that the linear equation 1.5229 * x -2.1911 predicts 87% of the variance in the variable y.

Computing Adjusted R2 for Polynomial Regressions

You can usually reduce the residuals in a model by fitting a higher degree polynomial. When you add more terms, you increase the coefficient of determination, R2. You get a closer fit to the data, but at the expense of a more complex model, for which R2 cannot account. However, a refinement of this statistic, adjusted R2, does include a penalty for the number of terms in a model. Adjusted R2, therefore, is more appropriate for comparing how different models fit to the same data. The adjusted R2 is defined as:

R2adjusted = 1 - (SSresid / SStotal)*((n-1)/(n-d-1))

where n is the number of observations in your data, and d is the degree of the polynomial. (A linear fit has a degree of 1, a quadratic fit 2, a cubic fit 3, and so on.)

The following example repeats the steps of the previous example, Example: Computing R2 from Polynomial Fits, but performs a cubic (degree 3) fit instead of a linear (degree 1) fit. From the cubic fit, you compute both simple and adjusted R2 values to evaluate whether the extra terms improve predictive power:

  1. Create two variables, x and y from the first two columns of the count variable in the data file count.dat:

    load count.dat
    x = count(:,1);
    y = count(:,2);
  2. Call polyfit to generate a cubic fit to predict y from x::

    p = polyfit(x,y,3)
    
    p =
       -0.0003    0.0390    0.2233    6.2779

    p(1) is the slope and p(2) is the intercept of the linear predictor. You can also obtain regression coefficients using the Basic Fitting GUI.

  3. Call polyval to use the coefficients in p to predict y, naming the result yfit:

    yfit = polyval(p,x);

    polyval evaluates the explicit equation you could manually enter as:

    yfit =  p(1) * x.^3 + p(2) * x.^2 + p(3) * x + p(4);
  4. Compute the residual values as a vector signed numbers:

    yresid = y - yfit;
  5. Square the residuals and total them obtain the residual sum of squares:

    SSresid = sum(yresid.^2);
  6. Compute the total sum of squares of y by multiplying the variance of y by the number of observations minus 1:

    SStotal = (length(y)-1) * var(y);
    
  7. Compute simple R2 for the cubic fit using the formula given in the introduction of this topic:

    rsq = 1 - SSresid/SStotal
    
    rsq =
        0.9083
  8. Finally, compute adjusted R2 to account for degrees of freedom:

    rsq_adj = 1 - SSresid/SStotal * (length(y)-1)/(length(y)-length(p))
    
    rsq_adj =
        0.8945

    The adjusted R2, 0.8945, is smaller than simple R2, .9083. It provides a more reliable estimate of the power of your polynomial model to predict.

In many polynomial regression models, adding terms to the equation increases both R2 and adjusted R2. In the preceding example, using a cubic fit increased both statistics compared to a linear fit. (You can compute adjusted R2 for the linear fit for yourself to demonstrate that it has a lower value.) However, it is not always true that a linear fit is worse than a higher-order fit: a more complicated fit can have a lower adjusted R2 than a simpler fit, indicating that the increased complexity is not justified. Also, while R2 always varies between 0 and 1 for the polynomial regression models that the Basic Fitting tool generates, adjusted R2 for some models can be negative, indicating that a model that has too many terms.

Correlation does not imply causality. Always interpret coefficients of correlation and determination cautiously. The coefficients only quantify how much variance in a dependent variable a fitted model removes. Such measures do not describe how appropriate your model—or the independent variables you select—are for explaining the behavior of the variable the model predicts.

Fitting Data with Curve Fitting Toolbox Functions

The Curve Fitting Toolbox software extends core MATLAB functionality by enabling the following data-fitting capabilities:

  • Linear and nonlinear parametric fitting, including standard linear least squares, nonlinear least squares, weighted least squares, constrained least squares, and robust fitting procedures

  • Nonparametric fitting

  • Statistics for determining the goodness of fit

  • Extrapolation, differentiation, and integration

  • GUI that facilitates data sectioning and smoothing

  • Saving fit results in various formats, including MATLAB code files, MAT-files, and workspace variables

For more information, see the Curve Fitting Toolbox documentation.

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