SARIMA Model

Seasonal Autoregressive Integrated Moving Average (SARIMA) Model. More...

Functions
int __stdcall	NDK_SARIMA_GOF (double *pData, size_t nSize, double mean, double sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ, GOODNESS_OF_FIT_FUNC retType, double *retVal)
int __stdcall	NDK_SARIMA_PARAM (double *pData, size_t nSize, double *mean, double *sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ, MODEL_RETVAL_FUNC retType, size_t maxIter)
int __stdcall	NDK_SARIMA_SIM (double mean, double sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ, double *pData, size_t nSize, size_t nSeed, double *retVal, size_t nStep)
int __stdcall	NDK_SARIMA_FORE (double *pData, size_t nSize, double mean, double sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ, size_t nStep, FORECAST_RETVAL_FUNC retType, double alpha, double *retVal)
int __stdcall	NDK_SARIMA_FITTED (double *pData, size_t nSize, double mean, double sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ, FIT_RETVAL_FUNC retType)
int __stdcall	NDK_SARIMA_VALIDATE (double mean, double sigma, WORD nIntegral, double *phis, size_t p, double *thetas, size_t q, WORD nSIntegral, WORD nSPeriod, double *sPhis, size_t sP, double *sThetas, size_t sQ)

Detailed Description

The SARIMA model is an extension of the ARIMA model, often used when we suspect a model may have a seasonal effect.

By definition, the seasonal auto-regressive integrated moving average - SARIMA(p, d, q)(P, D, Q)s - process is a multiplicative of two ARMA processes of the differenced time series.

\[(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}})(1-L)^d (1-L^s)^D x_t = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t\]

\[y_t = (1-L)^d (1-L^s)^D \]

Where:

• \(x_t\) is the original non-stationary output at time \(t\).
• \(y_y\) is the differenced (stationary) output at time \(t\).
• \(d\) is the non-seasonal integration order of the time series.
• \(p\) is the order of the non-seasonal AR component.
• \(P\) is the order of the seasonal AR component.
• \(q\) is the order of the non-seasonal MA component.
• \(Q\) is the order of the seasonal MA component.
• \(s\) is the seasonal length.
• \(D\) is the seasonal integration order of the time series.
• \(a_t\) is the innovation, shock, or error term at time \(t\).
• \(\{a_t\}\) time series observations are independent and identically distributed (i.e., \(i.i.d\)) and follow a Gaussian distribution (i.e., \(\Phi(0,\sigma^2)\)).

Assuming \(y_t\) follows a stationary process with a long run mean of \(\mu\), then taking the expectation from both sides, we can express \(\phi_o\) as follows:

\[\phi_o = (1-\phi_1-\phi_2-\cdots-\phi_p)(1-\Phi_1-\Phi_2-\cdots-\Phi_P)\]

Thus, the SARIMA(p, d, q)(P, D, Q)s process can now be expressed as:

\[(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}}) (y_t -\mu) = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t\]

\[z_t=y_t-\mu \]

\[(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}}) z_t = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t\]

In sum, \(z_t\) is the differenced signal after we subtract its long-run average.

Remarks

1. The variance of the shocks is constant or time-invariant.
2. The order of the seasonal or non-seasonal AR (or MA) component is solely determined by the order of the last lagged variable with a non-zero coefficient. In principle, you can have fewer parameters than the order of the component. Consider the following SARIMA(0,1,1) (0,1,1)12 process:

   \[(1-L)(1-L^{12})x_t-\mu = (1+\theta L)(1+\Theta L^{12})a_t \]

Note

This is the AIRLINE model, a special case of the SARIMA model.

Related Links

• Wikipedia - Autoregressive moving average model

References

• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics by D. S. G. Pollock; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
• James Douglas Hamilton; Time Series Analysis by James Douglas Hamilton, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series by Ruey S. Tsay, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
• Box, Jenkins and Reinsel; Time Series Analysis: Forecasting and Control by George E. P. Box & Gwilym M. Jenkins & Gregory C. Reinsel ; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
• Walter Enders; Applied Econometric Time Series by Walter Enders; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.

Function Documentation

NDK_SARIMA_FITTED()

int __stdcall NDK_SARIMA_FITTED	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ,
		FIT_RETVAL_FUNC	retType )

Returns the in-sample model fitted values of the conditional mean, volatility or residuals.

Note

1. The time series is homogeneous or equally spaced.

2. The time series may include missing values (e.g. NaN) at either end.

3. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

4. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

5. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

6. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

7. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_GOF(), NDK_SARIMA_RESID(), NDK_SARIMA_PARAM(), NDK_SARIMA_FORE(), NDK_SARIMA_VALIDATE()

Parameters

	pData	[inout] is the univariate time series data (a one dimensional array).
[in]	nSize	is the number of observations in X.
[in]	mean	is the model mean (i.e. mu).
[in]	sigma	is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
[in]	phis	are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
[in]	thetas	are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
[in]	sPhis	are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
[in]	sThetas	are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.
[in]	retType	is a switch to select a output type ( see FIT_RETVAL_FUNC).

NDK_SARIMA_FORE()

int __stdcall NDK_SARIMA_FORE	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ,
		size_t	nStep,
		FORECAST_RETVAL_FUNC	retType,
		double	alpha,
		double *	retVal )

Calculates the out-of-sample conditional forecast (i.e. mean, error, and confidence interval).

Note

1. The time series is homogeneous or equally spaced.

2. The time series may include missing values (e.g. NaN) at either end.

3. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

4. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

5. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

6. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

7. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_GOF(), NDK_SARIMA_RESID(), NDK_SARIMA_PARAM(), NDK_SARIMA_FITTED(), NDK_SARIMA_VALIDATE()

Parameters

[in]	pData	is the univariate time series data (a one dimensional array).
[in]	nSize	is the number of observations in X.
[in]	mean	is the model mean (i.e. mu).
[in]	sigma	is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
[in]	phis	are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
[in]	thetas	are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
[in]	sPhis	are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
[in]	sThetas	are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.
[in]	nStep	is the forecast time/horizon (expressed in terms of steps beyond end of the time series).
[in]	retType	is a switch to select the type of value returned (see FORECAST_RETVAL_FUNC).
[in]	alpha	is the statistical significance level. If missing, a default of 5% is assumed.
[out]	retVal	is the calculated forecast value.

NDK_SARIMA_GOF()

int __stdcall NDK_SARIMA_GOF	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ,
		GOODNESS_OF_FIT_FUNC	retType,
		double *	retVal )

Computes the log-likelihood ((LLF), Akaike Information Criterion (AIC) or other goodness of fit function of the SARIMA model.

Note

1. The time series is homogeneous or equally spaced.

2. The time series may include missing values (e.g. NaN) at either end.

3. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

4. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.

5. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

6. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

7. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

8. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_RESID(), NDK_SARIMA_PARAM(), NDK_SARIMA_FORE(), NDK_SARIMA_FITTED(), NDK_SARIMA_VALIDATE()

Parameters

[in]	pData	is the univariate time series data (a one dimensional array).
[in]	nSize	is the number of observations in X.
[in]	mean	is the model mean (i.e. mu).
[in]	sigma	is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
[in]	phis	are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
[in]	thetas	are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
[in]	sPhis	are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
[in]	sThetas	are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.
[in]	retType	is a switch to select a fitness measure ( see GOODNESS_OF_FIT_FUNC).
[out]	retVal	is the calculated goodness of fit value.

NDK_SARIMA_PARAM()

int __stdcall NDK_SARIMA_PARAM	(	double *	pData,
		size_t	nSize,
		double *	mean,
		double *	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ,
		MODEL_RETVAL_FUNC	retType,
		size_t	maxIter )

Returns the quick guess, optimal (calibrated) or std. errors of the values of model's parameters.

Note

1. The time series is homogeneous or equally spaced.

2. The time series may include missing values (e.g. NaN) at either end.

3. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

4. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

5. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

6. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

7. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_GOF(), NDK_SARIMA_RESID(), NDK_SARIMA_FORE(), NDK_SARIMA_FITTED(), NDK_SARIMA_VALIDATE()

Parameters

[in]	pData	is the univariate time series data (a one dimensional array).
[in]	nSize	is the number of observations in X.
	mean	[inout] is the mean of the ARMA process.
	sigma	[inout] is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
	phis	[inout] are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
	thetas	[inout] are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
	sPhis	[inout] are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
	sThetas	[inout] are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.
[in]	retType	is a switch to select the type of value returned: 1= Quick Guess, 2=Calibrated, 3= Std. Errors ( see MODEL_RETVAL_FUNC).
[in]	maxIter	is the maximum number of iterations used to calibrate the model. If missing or less than 100, the default maximum of 100 is assumed.

NDK_SARIMA_SIM()

int __stdcall NDK_SARIMA_SIM	(	double	mean,
		double	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ,
		double *	pData,
		size_t	nSize,
		size_t	nSeed,
		double *	retVal,
		size_t	nStep )

Returns the initial (non-optimal), optimal or standard errors of the model's parameters.

Note

1. The time series is homogeneous or equally spaced.

2. SARIMA_SIM returns an array of one simulation path starting from the end of the input data.

3. The time series may include missing values (e.g. NaN) at either end.

4. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

5. The input data argument (i.e. latest observations) is optional. If omitted, an array of zeroes is assumed.

6. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

7. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

8. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

9. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. Plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_GOF(), NDK_SARIMA_RESID(), NDK_SARIMA_FORE(), NDK_SARIMA_FITTED(), NDK_SARIMA_VALIDATE()

Parameters

[in]	mean	is the model mean (i.e. mu).
[in]	sigma	is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
[in]	phis	are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
[in]	thetas	are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
[in]	sPhis	are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
[in]	sThetas	are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.
[in]	pData	is the univariate time series data (a one dimensional array).
[in]	nSize	is the number of observations in X.
[in]	nSeed	is an unsigned integer for setting up the random number generators.
[out]	retVal	is the simulated value.
[in]	nStep	is the simulation time/horizon (expressed in terms of steps beyond end of the time series).

NDK_SARIMA_VALIDATE()

int __stdcall NDK_SARIMA_VALIDATE	(	double	mean,
		double	sigma,
		WORD	nIntegral,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	nSIntegral,
		WORD	nSPeriod,
		double *	sPhis,
		size_t	sP,
		double *	sThetas,
		size_t	sQ )

Examines the model's parameters for stability constraints (e.g. stationarity, invertibility, causality, etc.).

Note

1. The time series is homogeneous or equally spaced.

2. The time series may include missing values (e.g. NaN) at either end.

3. The residuals/innovations standard deviation (i.e. \(\sigma\)) should be greater than zero.

4. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.

5. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.

6. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.

7. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).

Returns

status code of the operation

Return values

NDK_SUCCESS	operation successful
NDK_FAILED	operation is unsuccessful ( )

See also

NDK_SARIMA_GOF(), NDK_SARIMA_RESID(), NDK_SARIMA_PARAM(), NDK_SARIMA_FORE(), NDK_SARIMA_FITTED()

Parameters

[in]	mean	is the model mean (i.e. mu).
[in]	sigma	is the standard deviation of the model's residuals/innovations.
[in]	nIntegral	is the non-seasonal difference order.
[in]	phis	are the coefficients's values of the non-seasonal AR component.
[in]	p	is the order of the non-seasonal AR component.
[in]	thetas	are the coefficients's values of the non-seasonal MA component.
[in]	q	is the order of the non-seasonal MA component.
[in]	nSIntegral	is the seasonal difference.
[in]	nSPeriod	is the number of observations per one period (e.g. 12=Annual, 4=Quarter).
[in]	sPhis	are the coefficients's values of the seasonal AR component.
[in]	sP	is the order of the seasonal AR component.
[in]	sThetas	are the coefficients's values of the seasonal MA component.
[in]	sQ	is the order of the seasonal MA component.