SSJ
3.2.1
Stochastic Simulation in Java
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This class implements a density derivative estimator (DDE) with a Gaussian ( i.e., standard normal) kernel function. More...
Public Member Functions | |
DEDerivativeGaussian (int order) | |
Constructs a DDE with a Gaussian kernel function of order order. More... | |
DEDerivativeGaussian (int order, double h) | |
Constructs a DDE with a Gaussian kernel function of order order and bandwidth h. More... | |
double | evalDensity (double x, double[] data, double a, double b) |
double [] | evalDensity (double[] evalPoints, double[] data, double a, double b) |
Constructs a DDE with a Gaussian kernel function from the observations data and the bandwith \(h\) set to getH(). More... | |
String | toString () |
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int | getOrder () |
Gives the order \(r\) of the DDE. More... | |
void | setOrder (int order) |
Sets the order \(r\) of the DDE to order. More... | |
double | getH () |
Gives the current bandwidth \(h\). More... | |
void | setH (double h) |
Sets the bandwidth \(h\) to h. More... | |
String | toString () |
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abstract double | evalDensity (double x, double data[], double a, double b) |
Constructs a density estimator over the interval \([a,b]\) based on the observations data if necessary, and evaluates it at x. More... | |
double [] | evalDensity (double[] evalPoints, double[] data, double a, double b) |
Constructs a density estimator over the interval \([a,b]\) based on the observations data if necessary, and evaluates it at the points in evalPoints. More... | |
double [][] | evalDensity (double[] evalPoints, double[][] data, double a, double b) |
This method is particularly designed to evaluate the density estimator in such a way that the result can be easily used to estimate the empirical IV and other convergence-related quantities. More... | |
abstract String | toString () |
Gives a short description of the estimator. More... | |
Static Public Member Functions | |
static double | hermitePoly (int r, double x) |
Computes the probabilist's Hermite polynomial of order r at x, which is defined by the recursion \[H_{r+1}(x)= x H_r(x) - r H_{r-1}(x) \] with initial values \(H_0(x) = 0\), \(H_1(x) = x\). More... | |
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static double | hAmiseR (int r, double mu2, double mu2Derivative, double init, int n) |
Given a value init for the roughness functional of \( f^{(r+2)}\), mu2 the second moment of the kernel function \(K\), and mu2Derivative the second moment of \(K^{(r)}\), this method computes the asymptotically optimal bandwidth for the DDE based on \(n\) observations simulated by Monte Carlo for the \(r\)-th derivative of the sought density \(f\). More... | |
static double | hAmiseR (int r, int t, double mu2, double[] mu2Derivative, double init, DensityDerivativeEstimator dde, double[] data, double[] evalPoints, double a, double b) |
Given an estimate of \(R(f^{(r+2t)})\) via init as initial value, this function iterates over (hopt) \(t\) times to obtain the asymptotically optimal bandwidth for the DDE based on \(n\) observations simulated by Monte Carlo for the \(r\)-th derivative of the sought density \(f\). More... | |
static double | densityFunctionalGaussian (int r, double sigma) |
Computes \(\Phi_{2r}(p)\), i.e. More... | |
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static void | evalDensity (ArrayList< DensityEstimator > listDE, double[] evalPoints, double[][] data, double a, double b, ArrayList< double[][]> listDensity) |
This function is particularly designed for experiments with many different types of density estimators, as it evaluates all of these estimators at the points in evalPoints. More... | |
static double [] | computeVariance (double[][] density) |
This method computes the empirical variance based on the values given in data. More... | |
static double | computeIV (double[][] density, double a, double b, double[] variance) |
This method estimates the empirical IV over the interval \([a,b]\). More... | |
static void | computeIV (ArrayList< double[][]> listDensity, double a, double b, ArrayList< Double > listIV) |
This method estimates the empirical IV over the interval \([a,b]\) for a collection of different estimators. More... | |
static double [] | computeMISE (ContinuousDistribution dist, double[] evalPoints, double[][] density, double a, double b, double[] variance, double[] sqBias, double[] mse) |
In situations where the true density is known this method can estimate the empirical MISE over the interval \([a,b]\). More... | |
static void | computeMISE (ContinuousDistribution dist, double[] evalPoints, ArrayList< double[][]> listDensity, double a, double b, ArrayList< double[]> listMISE) |
This method estimates the empirical MISE over the interval \([a,b]\) for a collection of different estimators. More... | |
static String | plotDensity (double[] evalPoints, double[] density, String plotTitle, String[] axisTitles) |
Gives a plot of the estimated density. More... | |
static double | roughnessFunctional (double[] density, double a, double b) |
Estimates the roughness functional. More... | |
Additional Inherited Members | |
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static double | coefficientOfDetermination (double[] data, double[] dataEstimated) |
Computes the Coefficient of determination \(R^2\) of the observed data data and the estimated data dataEstimated. More... | |
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double [] | data |
The data associated with this DensityEstimator object, if any. | |
This class implements a density derivative estimator (DDE) with a Gaussian ( i.e., standard normal) kernel function.
While many general methods are already handled by its superclass, this class is mainly concerned with the construction and evaluation of this kind of DDE. The formula specific to a Gaussian DDE is given by
\[ \hat{f}^{(r)}_{n,h}(x) = \frac{(-1)^r}{n h^{r+1}}\sum_{i=0}^{n-1} \phi \left( \frac{x-X_i}{h}\right) H_r\left( \frac{x-X_i}{h}\right), \]
where \(H_r\) denotes the probabilist's Hermite polynomial of order \(r\) and \(\phi\) denotes the standard normal density.
DEDerivativeGaussian | ( | int | order | ) |
Constructs a DDE with a Gaussian kernel function of order order.
order | the order of the derivative considered. |
DEDerivativeGaussian | ( | int | order, |
double | h | ||
) |
Constructs a DDE with a Gaussian kernel function of order order and bandwidth h.
h | the bandwidth. |
order | the order of the derivative considered. |
double evalDensity | ( | double | x, |
double [] | data, | ||
double | a, | ||
double | b | ||
) |
double [] evalDensity | ( | double [] | evalPoints, |
double [] | data, | ||
double | a, | ||
double | b | ||
) |
Constructs a DDE with a Gaussian kernel function from the observations data and the bandwith \(h\) set to getH().
Furthermore, the DDE is evaluated at the points in evalPoints and the resulting values are returned in an array.
evalPoints | the evaluation points. |
data | the observations for constructing the density estimator. |
a | the left boundary of the interval. |
b | the right boundary of the interval |
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static |
Computes the probabilist's Hermite polynomial of order r at x, which is defined by the recursion
\[H_{r+1}(x)= x H_r(x) - r H_{r-1}(x) \]
with initial values \(H_0(x) = 0\), \(H_1(x) = x\).
r | the order of the Hermite polynomial. |
x | the evaluation point. |