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SSJ
3.3.1
Stochastic Simulation in Java
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This class implements different linear regression models, using the least squares method to estimate the regression coefficients. More...
Static Public Member Functions | |
| static double [] | calcCoefficients (double[] X, double[] Y) |
| Computes the regression coefficients using the least squares method. More... | |
| static double [] | calcCoefficients (double[] X, double[] Y, int deg) |
| Computes the regression coefficients using the least squares method. More... | |
| static double [] | calcCoefficients0 (double[][] X, double[] Y) |
| Computes the regression coefficients using the least squares method. More... | |
| static double [] | calcCoefficients (double[][] X, double[] Y) |
| Computes the regression coefficients using the least squares method. More... | |
This class implements different linear regression models, using the least squares method to estimate the regression coefficients.
Given input data \(x_{ij}\) and response \(y_i\), one want to find the coefficients \(\beta_j\) that minimize the residuals of the form (using matrix notation)
\[ r = \min_{\beta}\| Y - X\beta\|_2, \]
where the \(L_2\) norm is used. Particular cases are
\[ r = \min_{\beta}\sum_i \left(y_i - \beta_0 - \sum_{j=1}^k \beta_j x_{ij}\right)^2. \]
for \(k\) regressor variables \(x_j\). The well-known case of the single variable \(x\) is
\[ r = \min_{\alpha,\beta} \sum_i \left(y_i - \alpha- \beta x_i\right)^2. \]
Sometimes, one wants to use a basis of general functions \(\psi_j(t)\) with a minimization of the form
\[ r = \min_{\beta}\sum_i \left(y_i - \sum_{j=1}^k \beta_j\psi_j(t_i)\right)^2. \]
For example, we could have \(\psi_j(t) = e^{-\lambda_j t}\) or some other functions. In that case, one has to choose the points \(t_i\) at which to compute the basis functions, and use a method below with \(x_{ij} = \psi_j(t_i)\).
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Computes the regression coefficients using the least squares method.
This is a simple linear regression with 2 regression coefficients, \(\alpha\) and \(\beta\). The model is
\[ y = \alpha+ \beta x. \]
Given the \(n\) data points \((X_i, Y_i)\), \(i=0,1,…,(n-1)\), the method computes and returns the array \([\alpha, \beta]\).
| X | the regressor variables |
| Y | the response |
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Computes the regression coefficients using the least squares method.
This is a linear regression with a polynomial of degree deg \(= k\) and \(k+1\) regression coefficients \(\beta_j\). The model is
\[ y = \beta_0 + \sum_{j=1}^k \beta_j x^j. \]
Given the \(n\) data points \((X_i, Y_i)\), \(i=0,1,…,(n-1)\), the method computes and returns the array \([\beta_0, \beta_1, …, \beta_k]\). Restriction: \(n > k\).
| X | the regressor variables |
| Y | the response |
| deg | degree of the function |
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Computes the regression coefficients using the least squares method.
This is a model for multiple linear regression. There are \(k\) regression coefficients \(\beta_j\), \(j=0,1,…,(k-1)\) and \(k\) regressors variables \(x_j\). The model is
\[ y = \sum_{j=0}^{k-1} \beta_j x_j. \]
There are \(n\) data points \(Y_i\), \(X_{ij}\), \(i=0,1,…,(n-1)\), and each \(X_i\) is a \(k\)-dimensional point. Given the response Y[i] and the regressor variables X[i][j], \(\mathtt{i} =0,1,…,(n-1)\), \(\mathtt{j} =0,1,…,(k-1)\), the method computes and returns the array \([\beta_0, \beta_1, …, \beta_{k-1}]\). Restriction: \(n > k\).
| X | the regressor variables |
| Y | the response |
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static |
Computes the regression coefficients using the least squares method.
This is a model for multiple linear regression. There are \(k+1\) regression coefficients \(\beta_j\), and \(k\) regressors variables \(x_j\). The model is
\[ y = \beta_0 + \sum_{j=1}^k \beta_j x_j. \]
There are \(n\) data points \(Y_i\), \(X_{ij}\), \(i=0,1,…,(n-1)\), and each \(X_i\) is a \(k\)-dimensional point. Given the response Y[i] and the regressor variables X[i][j], \(\mathtt{i} =0,1,…,(n-1)\), \(\mathtt{j} =0,1,…,(k-1)\), the method computes and returns the array \([\beta_0, \beta_1, …, \beta_k]\). Restriction: \(n > k+1\).
| X | the regressor variables |
| Y | the response |
1.8.14