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%% Linear Regression (retired, for future use) % Linear regression models the relation between a dependent, or response, % variable $y$ and one or more independent, or predictor, variables $x_1, ..., x_n$. % Simple linear regression considers only one independent variable using the % relation % % $$y = \beta_0 + \beta_1x + \epsilon,$$ % % where $\beta_0$ is the y-intercept, $\beta_1$ is the slope (or regression % coefficient), and $\epsilon$ is the error term. % % Multiple (or general) linear regression considers multiple independent variables, where % % $$y = \beta_0 + \beta_1x_1 + ... + \beta_nx_n + \epsilon.$$ % % You can model a curve by using higher powers of $x$ in place of $x_1, % ..., x_n$. The relation is now % % $$y = \beta_0 + \beta_1x + \beta_2x^2 + ... + \beta_nx^n + \epsilon.$$ % % Linear regression is defined as being linear in the coefficients % $\beta_0, ..., \beta_n$. The independent variables $x_1, ..., x_n$ can be % nonlinear. %% Preparing Data % Before running a linear regression, you might need to import your data % into MATLAB(TM). % % To import data from a *text file*, see <docid:import_export.f5-35378>. % % To import data from an *Excel(R)* spreadsheet, see % <docid:import_export.bs5bkj_>. % %% Simple Linear Regression Using the \ Operator % You can find the linear regression relation between dataset $X$ for the % independent variable and dataset $Y$ for % the dependent variable using the <docid:matlab_ref.btg5qam> operator. % % The dataset |accidents| contains data for fatal traffic accidents in U.S. % states. Load accident data in |y| and state population data in |x|. % Find the linear regression relation $y = \beta_1x$ between the accidents in a state and % a state's population using the |\| operator. The |\| operator performs % a least-squares regression. % Copyright 2015 The MathWorks, Inc. load accidents x = hwydata(:,14); %Population of states y = hwydata(:,4); %Accidents per state b1 = x\y %% % |b1| is the slope or regression coefficient. The linear relation is |y = b1*x = 1.4802*x|. % % Calculate the accidents per state |yCalc| from |x| using % the relation. Visualize the regression by plotting the actual values % |y| and the calculated values |yCalc|. yCalc = b1*x; h1 = figure(1); plot(x,y,x,yCalc) xlabel('Population of state') ylabel('Fatal traffic accidents per state') title('Linear Regression Relation Between Accidents & Population') grid on %% % You can improve the fit by including a y-intercept $\beta_0$ in your model as $y = % \beta_0 + \beta_1x$. Calculate the y-intercept by padding |x| with a % column of ones and using the |\| operator. X = [ones(length(x),1) x]; b = X\y %% % This result represents the relation |y = b0 + b1*x = 142.7120 + % 1.2564e-04x|. % % Visualize the relation by plotting it on the same figure. The plot shows that including a % y-intercept in the relation improves the fit. yCalc = (b'*X')'; hold on h = plot(x,yCalc,'--'); legend('Data','Slope','Slope & Intercept','Location','best'); %% Evaluate Model Using Residues, $R^2$ (R-Squared), and adjusted $R^2$ % Residues are the difference between the predicted values and the actual % values. A pattern in the residues means your model can be improved. % % Calculate and plot the residues for simple linear regression. yResi = yCalc - y; h2 = figure(2); scatter(x,yResi,'+'); %% % The residues show that the model is a good fit at the extremes but loses % accurate between the extremes. Thus, the model can be improved. % % The coefficient of determination, $R^2$, is another measure of how useful the % model is in predicting the data. $R^2$ falls between $0$ to $1$. The % higher the value, the better the model is at predicting the data. % % For simple linear regression, R-squared is defined as the square of the oorrelation % coefficient R between $Y$ and $X$. Repeat the simple linear regression example, % calculate |R| using the <docid:matlab_ref.f80-999628> function, and find R-squared. b1 = x\y; yCalc = b1*x; R = corrcoef(x,y); Rsq = R(1,2)^2 %% % Adjusted $R^2$ is defined as % $$ R^2_{adj} = 1 - % \frac{SS_{resid}}{SS_{total}}\bigg(\frac{n_{obs}-1}{n_{obs} - d - % 1}\biggr),$$ % where $SS_{resid}$ is the sum of the residuals and $SS_{total}$ is the % sum of the differences of the data from mean. The residuals are the % differences between the calculated values and the data. % % Find the adjusted $R^2$. SSresid = sum((yCalc - y).^2); SStotal = (length(y)-1)*var(y); deg = length(b1); Radj = 1 - SSresid/SStotal*((length(y)-1)/(length(y)-deg-1)) %% Linear Regression by Fitting a Curve % To improve your fit, you can use a higher-order polynomial. This section % shows how to fit polynomials models using the |\| operator. If you have % only one independent variable, use |polyfit| instead. For details, see <docid:matlab_math.f1-8450>. % % A polynomial regression is linear because the coefficients are linear, % while the terms are nonlinear. Choose a fourth-order polynomial model % % $$y = \beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3 + \beta_4x^4$$ % % Scale down |x| to avoid a rank deficient matrix. If you do not scale |x|, the |\| operator % returns a warning stating that your matrix is rank deficient. xSc = x/10000; %% % Form the design matrix |X|. X = [ones(length(xSc),1) xSc xSc.^2 xSc.^3 xSc.^4]; %% % Perform linear regression using the |\| operator. B = X\y; %% % Plot the result on the first figure. yCalc = (B'*X')'; figure(h1) plot(x,yCalc,'-.') legend('Data','Slope','Slope & intercept','Polynomial fit','Location','best') %% Linear Regression Using Custom Equation % To fit the data, you can use a custom equation. Use the equation % % $$y = \beta_0 + \beta_1x\sin(x) + \beta_2\log(x)$$ % % Construct the design matrix |X|. X = [ones(length(x),1) x.*sin(x) log(x)]; %% % Perform linear regression using the |\| operator and plot the result. B = X\y; yCalc = (B'*X')'; plot(x,yCalc,':') legend('Data','Slope','Slope & intercept','Polynomial fit','Custom Equation','Location','best') %% Multiple Linear Regression Using the |\| Operator % Multiple linear regression relates a dependent variable $Y$ to % multiple independent variables $x_1, ..., x_n$ as % % $$y = \beta_0 + \beta_1x_1 + ... + \beta_nx_n + \epsilon.$$ % % Model accidents per state using the independent variables of population % per state and vehicles per state. First, load data from |hwydata|. x = hwydata(:,14); % population y = hwydata(:,6); % vehicles z = hwydata(:,4); % accidents %% % Concatenate |x| and |y| into a matrix. Pad the matrix with ones to % include a y-intercept in the model. Calcuate the coefficients of the linear model using % the |\| operator. Use |format long| to view the result. M = [ones(length(x),1) x y]; format long v = M\z %% % Reset |format| to the default. format short %% % Calculate accident values from the model. Compare the actual values and % calculated values by plotting them on the same figure. zCalc = (v'*M')'; clf plot3(x,y,z,'LineWidth',1) hold on plot3(x,y,zCalc,'--','LineWidth',1) view(-11,38) grid on xlabel('Population of State') ylabel('Vehicles in State') zlabel('Accidents') title('Linear Regression of Accidents in States from Population and Vehicles') %% Evaluate Model Using Residues, $R^2$ (R-squared), and adjusted $R^2$ % Calculate and plot the residues for the multiple linear regression. zResi = zCalc - z; figure(h2) subplot(2,1,1) scatter(x,zResi) subplot(2,1,2) scatter(y,zResi) %% % The residues show that the model is a good fit at the extremes but loses % accurate between the extremes. Thus, the model can be improved. % % Calculate $R^2$ for Multiple Linear Regression by squaring the % correlation coefficient between the actual values and calculated values. % % Find the correlation coefficient using |corrcoef|. Calculate $R^2$ % for the multiple linear regression previously performed. R = corrcoef(z,zCalc); Rsq = R(1,2)^2 %% % As previously defined, adjusted-$R^2$ is % % $$ R^2_{adj} = 1 - % \frac{SS_{resid}}{SS_{total}}\bigg(\frac{n_{obs}-1}{n_{obs} - d - % 1}\biggr).$$ % % Calculate adjusted $R^2$ for multivariate regression. SSresid = sum((zCalc - z).^2); SStotal = (length(z) - 1)*var(z); deg = length(v) - 1; Radj = 1 - SSresid/SStotal*((length(z)-1)/(length(z)-deg-1)) %% % This value of adjusted $R^2$ is higher than the adjusted $R^2$ of % |0.8186| with one independent variable. The higher value of % adjusted $R^2$ indicates that the two-variable model is a better % predictor than the one-variable model. However, you should use % adjusted $R^2$ in conjunction with other measures to determine how good % your model is. %% Linear Regression Using the Basic Fitting Tool % The MATLAB Basic Fitting GUI allows you to interactively: %% % * Model data using a spline interpolant, a shape-preserving interpolant, or a polynomial up to the tenth degree % * Plot one or more fits together with data % * Plot the residuals of the fits % * Compute model coefficients % * Compute the norm of the residuals (a statistic you can use to analyze how well a model fits your data) % * Use the model to interpolate or extrapolate outside of the data % * Save coefficients and computed values to the MATLAB workspace for use outside of the GUI % * Generate MATLAB code to recompute fits and reproduce plots with new data % % For details, see <docid:data_analysis.f1-15377>. %% Advanced Functionality Using Statistics or Curve Fitting Toolboxes % The Statistics and Machine Learning Toolbox(TM) and Curve Fitting Toolbox(TM) toolboxes provide advanced functionality % % The Curve Fitting Toolbox The toolbox supports over 100 regression % models. All of these standard regression models include optimized solver % parameters and starting conditions to improve fit quality. Supported models include: %% % * Lines and planes % * High order polynomials (up to ninth degree for curves and fifth degree for surfaces) % * Fourier and power series % * Gaussians % * Weibull functions % * Exponentials % * Rational functions % * Sum of sines % * Custom Equation option to specify your own regression model % % For information on linear regression in the Curve Fitting Toolbox, please % see <docid:curvefit_doc_center.linear-and-nonlinear-regression-103> and % <docid:curvefit_ug.bszh0l1>. % % The Statistics and Machine Learning Toolbox(TM) The toolbox offers % several types of linear regression models and fitting methods, including: %% % * Simple: model with only one predictor % * Multiple: model with multiple predictors % * Multivariate: model with multiple response variables % * Robust: model in the presence of outliers % * Stepwise: model with automatic variable selection % * Regularized: model that can deal with redundant predictors and prevent overfitting using ridge, lasso, and elastic net algorithms % % For information on linear regression in the Statistics Toolbox, please % see <docid:stats_ug.bs9kc51>.