AR

java.lang.Object
- smile.timeseries.AR

All Implemented Interfaces:

java.io.Serializable
```
public class AR
extends java.lang.Object
implements java.io.Serializable
```
Autoregressive model. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term; thus the model is in the form of a stochastic difference equation. Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
Contrary to the moving-average (MA) model, the autoregressive model is not always stationary as it may contain a unit root.
The notation AR(p) indicates an autoregressive model of order p. For an AR(p) model to be weakly stationary, the inverses of the roots of the characteristic polynomial must be less than 1 in modulus.
Two general approaches are available for determining the order p. The first approach is to use the partial autocorrelation function, and the second approach is to use some information criteria.
Autoregression is a good start point for more complicated models. They often fit quite well (don’t need the MA terms). And the fitting process is fast (MLEs require some iterations). In applications, easily fitting autoregressions is important for obtaining initial values of parameters and in getting estimates of the error process. Least squares is a popular choice, as is the Yule-Walker procedure. Unlike the Yule-Walker procedure, least squares can produce a non-stationary fitted model. Both the Yule-Walker and least squares estimators are non-iterative and consistent, so they can be used as starting values for iterative methods like MLE.

See Also:

Serialized Form

Nested Class Summary

Nested Classes
Modifier and Type Class and Description

static class AR.Method
The fitting method.

Nested Classes
Modifier and Type	Class and Description
`static class`	`AR.Method` The fitting method.

Constructor Summary

Constructors
Constructor and Description

AR(double[] x, double[] ar, double b, AR.Method method)
Constructor.

Constructors
Constructor and Description
`AR(double[] x, double[] ar, double b, AR.Method method)` Constructor.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`adjustedRSquared()` Returns adjusted R² statistic.
`double[]`	`ar()` Returns the linear coefficients of AR (without intercept).
`int`	`df()` Returns the degree-of-freedom of residual standard error.
`static AR`	`fit(double[] x, int p)` Fits an autoregressive model with Yule-Walker procedure.
`double[]`	`fittedValues()` Returns the fitted values.
`double`	`forecast()` Returns 1-step ahead forecast.
`double[]`	`forecast(int l)` Returns l-step ahead forecast.
`double`	`intercept()` Returns the intercept.
`double`	`mean()` Returns the mean of time series.
`static AR`	`ols(double[] x, int p)` Fits an autoregressive model with least squares method.
`static AR`	`ols(double[] x, int p, boolean stderr)` Fits an autoregressive model with least squares method.
`int`	`p()` Returns the order of AR.
`double[]`	`residuals()` Returns the residuals, that is response minus fitted values.
`double`	`RSquared()` Returns R² statistic.
`double`	`RSS()` Returns the residual sum of squares.
`java.lang.String`	`toString()`
`double[][]`	`ttest()` Returns the t-test of the coefficients (including intercept).
`double`	`variance()` Returns the residual variance.
`double[]`	`x()` Returns the time series.

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

- Constructor Detail
  - AR
```
public AR(double[] x,
          double[] ar,
          double b,
          AR.Method method)
```
    Constructor.
    
    Parameters:
    
    x - the time series
    
    ar - the estimated weight parameters of AR(p).
    
    b - the intercept.
    
    method - the fitting method.
- Method Detail
  - x
```
public double[] x()
```
    Returns the time series.
  - mean
```
public double mean()
```
    Returns the mean of time series.
  - p
```
public int p()
```
    Returns the order of AR.
  - ttest
```
public double[][] ttest()
```
    Returns the t-test of the coefficients (including intercept). The first column is the coefficients, the second column is the standard error of coefficients, the third column is the t-score of the hypothesis test if the coefficient is zero, the fourth column is the p-values of test. The last row is of intercept.
  - ar
```
public double[] ar()
```
    Returns the linear coefficients of AR (without intercept).
  - intercept
```
public double intercept()
```
    Returns the intercept.
  - residuals
```
public double[] residuals()
```
    Returns the residuals, that is response minus fitted values.
  - fittedValues
```
public double[] fittedValues()
```
    Returns the fitted values.
  - RSS
```
public double RSS()
```
    Returns the residual sum of squares.
  - variance
```
public double variance()
```
    Returns the residual variance.
  - df
```
public int df()
```
    Returns the degree-of-freedom of residual standard error.
  - RSquared
```
public double RSquared()
```
    Returns R² statistic. In regression, the R² coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R² of 1.0 indicates that the regression line perfectly fits the data.
    In the case of ordinary least-squares regression, R² increases as we increase the number of variables in the model (R² will not decrease). This illustrates a drawback to one possible use of R², where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R².
  - adjustedRSquared
```
public double adjustedRSquared()
```
    Returns adjusted R² statistic. The adjusted R² has almost same explanation as R² but it penalizes the statistic as extra variables are included in the model.
  - fit
```
public static AR fit(double[] x,
                     int p)
```
    Fits an autoregressive model with Yule-Walker procedure.
    
    Parameters:
    
    x - the time series.
    
    p - the order.
  - ols
```
public static AR ols(double[] x,
                     int p)
```
    Fits an autoregressive model with least squares method.
    
    Parameters:
    
    x - the time series.
    
    p - the order.
  - ols
```
public static AR ols(double[] x,
                     int p,
                     boolean stderr)
```
    Fits an autoregressive model with least squares method.
    
    Parameters:
    
    x - the time series.
    
    p - the order.
    
    stderr - the flag if estimate the standard errors of parameters.
  - forecast
```
public double forecast()
```
    Returns 1-step ahead forecast.
  - forecast
```
public double[] forecast(int l)
```
    Returns l-step ahead forecast.
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class AR

Nested Class Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

AR

Method Detail

x

mean

p

ttest

ar

intercept

residuals

fittedValues

RSS

variance

df

RSquared

adjustedRSquared

fit

ols

ols

forecast

forecast

toString