ConditionalSumOfSquares (NMDEV 1.2.9 API)

java.lang.Object
- dev.nm.stat.timeseries.linear.univariate.stationaryprocess.arma.ConditionalSumOfSquares

All Implemented Interfaces:

ARMAFit
```
public class ConditionalSumOfSquares
extends Object
implements ARMAFit
```
The method Conditional Sum of Squares (CSS) fits an ARIMA model by minimizing the conditional sum of squares. The CSS estimates are conditional on the assumption that the past unobserved errors are 0s. The estimation produced by CSS can be used as a starting point for a better algorithm, e.g., the maximum likelihood. Note that the order of integration is taken as an input, not estimated. The R equivalent function is arima.
Fit, inspect, then forecast

Supply finite observations in chronological order and nonnegative p, d, q. Choose d before fitting and retain enough observations after differencing to identify the requested model. The constructor fits immediately using a numerical optimizer; the iteration limit is not a convergence certificate.

getModel() returns the fitted ARIMA model, including its supplied differencing order. getARMAModel() extracts the stationary ARMA part: when d is nonzero it must not be applied directly to the original undifferenced levels. Use ARIMAForecast with the original history and getModel() in that case. For d = 0, ARMAForecast is sufficient.

Inspect fitted coefficients and var() for finite values, check stationarity/invertibility and residual behavior independently, and compare forecasts on held-out observations. No automatic data cleaning or order search is performed by this fitter.

Fitting a reproducible sample

The synthetic observations below demonstrate the calling sequence. They are not market data, and the assertions check usable output rather than statistical model adequacy.
```
 import dev.nm.stat.timeseries.datastructure.univariate.realtime.inttime.SimpleTimeSeries;
 import dev.nm.stat.timeseries.linear.univariate.stationaryprocess.arma.ConditionalSumOfSquares;
 import dev.nm.stat.timeseries.linear.univariate.stationaryprocess.arma.ARMAForecast;
 import dev.nm.stat.timeseries.linear.univariate.arima.ARIMAForecast.Forecast;
 double[] observed = new double[40];
 for (int i = 1; i < observed.length; ++i) {
     observed[i] = 0.6 * observed[i - 1] + Math.sin(1.7 * i) + 0.2 * Math.cos(0.3 * i);
 }
 ConditionalSumOfSquares fit = new ConditionalSumOfSquares(observed, 1, 0, 0, 500);
 assert fit.getARMAModel().p() == 1;
 assert fit.getModel().d() == 0;
 assert Double.isFinite(fit.var()) && fit.var() > 0;
 ARMAForecast predictor = new ARMAForecast(new SimpleTimeSeries(observed), fit.getARMAModel());
 Forecast next = predictor.next();
 assert next.time() == observed.length + 1;
 assert Double.isFinite(next.xHat());
 
```
See Also:

"P. J. Brockwell and R. A. Davis, "Chapter 8.7, Model Building and Forecasting with ARIMA Processes," Time Series: Theory and Methods, Springer, 2006."

Constructor Summary

Constructors
Constructor and Description
`ConditionalSumOfSquares(double[] x, int p, int d, int q)` Fit an ARIMA model for the observations using CSS.
`ConditionalSumOfSquares(double[] x, int p, int d, int q, int maxIterations)` Fit an ARIMA model for the observations using CSS.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`AIC()` Compute the AIC, a model selection criterion.
`double`	`AICC()` Compute the AICC, a model selection criterion.
`Matrix`	`covariance()` Get the asymptotic covariance matrix of the estimated parameters, φ and θ.
`ARMAModel`	`getARMAModel()` Get the fitted ARMA model.
`ARIMAModel`	`getModel()` Get the fitted ARIMA model.
`int`	`nParams()` Get the number of parameters for the estimation/fitting.
`ImmutableVector`	`stderr()` Get the asymptotic standard errors of the estimated parameters, φ and θ.
`String`	`toString()`
`double`	`var()` Get the variance of the white noise.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - ConditionalSumOfSquares
```
public ConditionalSumOfSquares(double[] x,
                               int p,
                               int d,
                               int q,
                               int maxIterations)
```
    Fit an ARIMA model for the observations using CSS. Note that the algorithm fits only an ARMA model. d is taken as an input. If the differenced input time series is not zero-mean, it is first de-mean-ed before running the algorithm as in Brockwell and Davis. When reporting the model, we compute the intercept to match the mean.
    
    Parameters:
    
    x - the time series of observations
    
    p - the number of AR terms
    
    d - the order of integration
    
    q - the number of MA terms
    
    maxIterations - the maximum number of iterations
  - ConditionalSumOfSquares
```
public ConditionalSumOfSquares(double[] x,
                               int p,
                               int d,
                               int q)
```
    Fit an ARIMA model for the observations using CSS. Note that the algorithm fits only an ARMA model. d is taken as an input. If the differenced input time series is not zero-mean, it is first de-mean-ed before running the algorithm as in Brockwell and Davis. When reporting the model, we compute the intercept to match the mean.
    
    Parameters:
    
    x - the time series of observations
    
    p - the number of AR terms
    
    d - the order of integration
    
    q - the number of MA terms
- Method Detail
  - nParams
```
public int nParams()
```
    Get the number of parameters for the estimation/fitting. They are the AR terms, MA terms, and variance (sigma^2).
    
    Returns:
    
    the number of parameters
  - getModel
```
public ARIMAModel getModel()
```
    Get the fitted ARIMA model.
    
    Specified by:
    
    getModel in interface ARMAFit
    
    Returns:
    
    the fitted ARIMA model
  - getARMAModel
```
public ARMAModel getARMAModel()
```
    Get the fitted ARMA model.
    
    Returns:
    
    the fitted ARMA model
  - var
```
public double var()
```
    Description copied from interface: ARMAFit
    
    Get the variance of the white noise.
    
    Specified by:
    
    var in interface ARMAFit
    
    Returns:
    
    \(\sigma ^{2}\)
  - covariance
```
public Matrix covariance()
```
    Get the asymptotic covariance matrix of the estimated parameters, φ and θ. The estimators are asymptotically normal.
    
    Specified by:
    
    covariance in interface ARMAFit
    
    Returns:
    
    the asymptotic covariance matrix
    
    See Also:
    
    "P. J. Brockwell and R. A. Davis, "Eq. 10.8.27, Thm. 10.8.2, Chapter 10.8, Model Building and Forecasting with ARIMA Processes," Time Series: Theory and Methods, Springer, 2006."
  - stderr
```
public ImmutableVector stderr()
```
    Get the asymptotic standard errors of the estimated parameters, φ and θ. The estimators are asymptotically normal.
    
    Specified by:
    
    stderr in interface ARMAFit
    
    Returns:
    
    the asymptotic errors
    
    See Also:
    
    "P. J. Brockwell and R. A. Davis, "Eq. 10.8.27, Thm. 10.8.2, Chapter 10.8, Model Building and Forecasting with ARIMA Processes," Time Series: Theory and Methods, Springer, 2006."
  - AIC
```
public double AIC()
```
    Compute the AIC, a model selection criterion.
    
    Specified by:
    
    AIC in interface ARMAFit
    
    Returns:
    
    the AIC
    
    See Also:
    
    Wikipedia: Akaike information criterion
  - AICC
```
public double AICC()
```
    Compute the AICC, a model selection criterion.
    
    Specified by:
    
    AICC in interface ARMAFit
    
    Returns:
    
    the AICC
    
    See Also:
    
    "P. J. Brockwell and R. A. Davis, "Eq. 9.2.1, Chapter 9.2, Model Building and Forecasting with ARIMA Processes," Time Series: Theory and Methods, Springer, 2006."
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object

Class ConditionalSumOfSquares

Fit, inspect, then forecast

Fitting a reproducible sample

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

ConditionalSumOfSquares

ConditionalSumOfSquares

Method Detail

nParams

getModel

getARMAModel

var

covariance

stderr

AIC

AICC

toString