Java程序辅导

Michael Thomas Flanagan's Java Scientific Library: Linear and non-linear regression       Michael Thomas Flanagan's Java Scientific Library Regression Class:     Linear and Non-linear Regression       Last update: 13 August 2016                                                                                                                             PERMISSION TO USE Home Page of Michael Thomas Flanagan's Java Scientific Library. This class contains the methods for performing the following regressions: Fitting to a constant; yi = a0; Linear regression with intercept; yi = a0 + a1.x0,i + a2.x1,i + . . . ; General linear regression; yi = a0.f1(x0, x1 . .) + a1.f2(x0, x1 . .) + . . . ; Polynomial fitting; yi = a0 + a1.xi + a2.xi2 + a3.xi3. . .;                            yi = a0 + a1.xian+1 + a2.xian+2 + . . . + an.xia2n+1 [non-linear] Non linear regression; Nelder and Mead Simplex. [Unconstrained and constrained] y = f1(a0,a1,a2..., x0,x1,x2...) y0 = f1(a0,a1,a2..., x0,x1,x2...) y1 = f2(a0,a1,a2..., x0,x1,x2...) . . . yn = fn(a0,a1,a2..., x0,x1,x2...) Fitting data to exponentials Fitting to a single exponential, yi = a.exp(b.xi) Fitting to a multiple exponentials Fitting to yi = a.(1 - exp(b.xi)) See also Fitting to an Exponential Distribution Fitting data to a Rectangular Hyperbola Fitting data to a Sigmoidal Function Sigmoid threshold function: y = Ao/(1 + e-α(x-θ)) Hill/Sips Sigmoid: y = xnAo/(θn + xn) Four parameter logistic (EC50 dose response curve): y = top + (bottom - top)/(1 + (x/EC50)Hill Slope) Five parameter logistic: y = top + (bottom - top)/(1 + (x/C50)Hill Slope)asymm Fitting data to a Scaled Heaviside Step Function Fitting data to a Gaussian Distribution [Normal Distribution]; See also ProbabilityPlot Fitting data to a Log-normal Distribution Two Parameter Statistic ; Three Parameter Statistic ; Fitting data to a Logistic Distribution [sech squared Distribution]; See also ProbabilityPlot Fitting data to a Lorentzian Distribution [Cauchy Distribution]; See also ProbabilityPlot Fitting data to a Poisson Distribution; Fitting data to a Beta Distribution Beta Distribution on the interval [0,1]; Beta Distribution on the interval [min, max]; Fitting data to a Gamma Distribution Three Parameter Gamma Distribution; Standard Gamma Distribution; Fitting data to an Erlang Distribution Fitting data to a Type 1 Extreme Value Distribution: Minimum order statistic General Gumbel Distribution One Parameter Gumbel Distribution Standard Gumbel Distribution See also ProbabilityPlot Maximum order statistic General Gumbel Distribution One Parameter Gumbel Distribution Standard Gumbel Distribution See also ProbabilityPlot Fitting data to a Type 2 Extreme Value Distribution: General Fréchet Distribution Two Parameter Fréchet Distribution Standard Fréchet Distribution See also ProbabilityPlot Fitting data to a Type 3 Extreme Value Distribution: Weibull Distribution General Weibull Distribution Two Parameter Weibull Distribution Standard Weibull Distribution See also ProbabilityPlot Exponential Distribution General Exponential Distribution One Parameter Exponential Distribution Standard Exponential Distribution See also ProbabilityPlot Rayleigh Distribution See also ProbabilityPlot Fitting data to a Pareto Distribution Shifted Pareto Distribution Two Parameter Pareto Distribution One Parameter Pareto Distribution See also ProbabilityPlot Fitting data to a F-Distribution See ProbabilityPlot   Check best fit model: Extra sum of squares analysis Check normality of the data: See separate class Normality Check normality of the residuals Durbin-Watson d Statistic Fitting Immunoassay data to model assay equations See separate subclass ImmunoAssay Fitting Impedance and Electrochemical Impedance Spectra data to a circuit model See separate subclass ImpedSpecRegression Smoothing 1D data, i.e. a curve See a separate class CurveSmooth Smoothing 2D data, i.e. a surface See a separate class SurfaceSmooth Smoothing 3D data See a separate class ThreeDimensionalSmooth The regressions may be performed on sets of x - y data or on a single set of data which is first binned into an histogram. Weighted regression using both dependent and independent variable errors The above listed regressions may all be performed as unweighted regressions or as classically weighted regressions, i.e. regressions in which the weighting factors are equated to the measurement errors in the dependent variable (y) and the dependent variables (x) are assumed to be error free. The following regressions may also be performed as weighted regressions in which the weighting factors are derived from the measurement errors in both the dependent and independent variables: Linear regression with intercept: yi = a0 + a1.x0,i + a2.x1,i + . . . General linear regression: yi = a0.f1(x0, x1 . .) + a1.f2(x0, x1 . .) + . . . Polynomial fitting: yi = a0 + a1.xi + a2.xi2 + a3.xi3. . .         and         yi = a0 + a1.xian+1 + a2.xian+2 + . . . + an.xia2n+1 Rectangular Hyperbola: yi = A0.xi/(θ + xi)         and         yi = A0.xi/(θ + xi) + α Hill/Sips Sigmoid: y = xnAo/(θn + xn) Sigmoid threshold function: y = Ao/(1 + e-α(x-θ)) Five parameter logistic: y = top + (bottom - top)/(1 + (x/C50)Hill Slope)asymm Four parameter logistic (EC50 dose response curve): y = top + (bottom - top)/(1 + (x/EC50)Hill Slope) These methods calculate the combined weight as           i.e. assuming all errors are uncorrelated. See ErrorProp for the equations needed if correlated errors are to handled in a user supplied function. Both the data and any measurement errors (estimates of σ) are entered via a constructor. Non-linear regression with a user supplied function: The user may code the calculation of a value for the supplied function and a value of the corresponding square of the weight using both dependent and independent variable measurement errors, as described above, in a method that overrides the function method in the interface RegressionFunction3 and then call either the simplex or simplexPlot method. Coding a user supplied function The sum of squares function required by the non-linear regression procedure is supplied via one of the following interfaces: RegressionFunction, typically used for functions with a one-dimensional y array and either no weighting or weighting derived only from the measurement errors in the dependent (y) variable RegressionFunction2 typically used for functions with a two-dimensional y array and either no weighting or weighting derived only from the measurement errors in the dependent (y) variable RegressionFunction3 typically used for functions with weighting derived from the measurement errors in both the dependent (y) variable and the independent (x) variables, irrespective of the dimensionality of the dependent (y) array. Covariance matrix options The covariance matrix used in the statistical analysis of the best estimates may, in the case of non-linear regression, be calculated either numerically or analytically. If the latter option is chosen the first and second derivatives of the fitted function for pairs of parameters are supplied via an interface, RegressionDerivativeFunction or RegressionDerivativeFunction2 (more than one dependent variable). Deprecated and renamed methods Method names are occasionly changed to maintain naming consistency and some methods are occasionally deprecated. The older named methods are always retained in the flanagan.jar file but may not be listed on this page. Warning before attempting a non-linear regression: Click here. Fitting to statistical distributions and unbinned data: the ProbabilityPlot class may be as or more useful in fitting data to statistical distributions if the data is unbinned. Outlier removal methods may be found in the Outliers Class. import statement/s: For all regressions      import flanagan.analysis.Regression; In addition for relevant non-linear regressions      import flanagan.analysis.RegressionFunction;      import flanagan.analysis.RegressionFunction2;      import flanagan.analysis.RegressionFunction3; Summary table of constructors and methods Details of constuctors and methods Example programs demonstrating the use of non-linear regression. Last update of the source code for Regression.java: 13 August 2016. Last update of the source code for RegressionFunction.java: 14 April 2004. Last update of the source code for RegressionFunction2.java: 14 April 2004. Last update of the source code for RegressionFunction3.java: 25 March 2012. Last update of the source code for RegressionDerivativeFunction.java: 16 January 2012. Last update of the source code for RegressionDerivativeFunction2.java: 16 January 2012. Home Page of Michael Thomas Flanagan's Java Scientific Library. Other classes, from this library, used by this class. Related classes ProbabilityPlot Probability plots Stat - statistical methods. ErrorProp - error propagation SUMMARY OF CONSTRUCTORS AND METHODS Constructors   public Regression(double[ ][ ] xdata, double[ ] ydata, double[ ] yErrors) public Regression(double[ ][ ] xdata, double[ ] ydata, double[ ][] xerrors, double[ ] yerrors) public Regression(double[ ][ ] xdata, double[ ][] ydata, double[ ][ ] yerrors) public Regression(double[ ][ ] xdata, double[ ][] ydata, double[ ][ ] xerrors, double[ ][ ] yerrors) public Regression(double[ ] xdata, double[ ] ydata, double[ ] yErrors) public Regression(double[ ] xdata, double[ ] ydata, double[ ] xerrors, double[ ] yerrors) public Regression(double[ ] xdata, double[ ][] ydata, double[ ][ ] yerrors) public Regression(double[ ] xdata, double[ ][] ydata, double[ ] xerrors, double[ ][ ] yerrors) public Regression(double[ ][ ] xdata, double[ ] ydata) public Regression(double[ ][ ] xdata, double[ ][] ydata) public Regression(double[ ] xdata, double[ ] ydata) public Regression(double[ ] xdata, double[ ][] ydata) public Regression(double[ ] xdata, double binWidth, double binZero) public Regression(double[ ] xdata, double binWidth) Linear Regression Fitting to a constant yi = a0 public void constant() public void constantPlot() public void constantPlot(String xLegend, String yLegend) Linear with intercept yi = a0+a1.x0,i+a2.x1,i+... public void linear() public void linear(double fixedIntercept) public void linearPlot() public void linearPlot(double fixedIntercept) public void linearPlot(String xLegend, String yLegend) public void linearPlot(double fixedIntercept, String xLegend, String yLegend) General linear yi = a0.f1(x0,x1..)+a1.f2(x0,x1+...) public void linearGeneral() public void linearGeneralPlot() public void linearGeneralPlot(String xLegend, String yLegend) Polynomial yi = a0+a1.x+a2.x2+a3x3 ... public void polynomial(int n) public void polynomial(int n, double fixedIntercept) public void polynomialPlot(int n) public void polynomialPlot(int n, double fixedIntercept) public void polynomialPlot(int n, String xLegend, String yLegend) public void polynomialPlot(int n, double fixedIntercept, String xLegend, String yLegend) public ArrayList