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)
	Computes the goodness of fit measure (e.g., log-likelihood function (LLF), AIC, etc.) of the estimated ARIMA model.

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.

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

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

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

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.).

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

The time series is homogeneous or equally spaced.
The time series may include missing values (e.g. NaN) at either end.
For the input argument - ( \( [\phi] \)) (parameters of the non-seasonal AR component):
- The input argument is optional and can be omitted, so no non-seasonal AR component is included.
- The order of the parameters starts with the lowest lag.
- One or more parameters may have missing values (i.e., NaN).
- The order of the last lagged variable (i.e., \( \phi_p \)) with non-zero coefficient solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
For the input argument - ( \( [\theta] \)) (parameters of the non-seasonal MA component)::
- The input argument is optional and can be omitted, so no non-seasonal MA component is included.
- The order of the parameters starts with the lowest lag.
- One or more values in the input argument can be missing (i.e., NaN).
- The order of the last lagged variable (i.e., \( \theta_q \)) with non-zero coefficient solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
For the input argument - ( \( [\Phi] \)) (parameters of the seasonal AR component):
- The input argument is optional and can be omitted, so no seasonal AR component is included.
- The order of the parameters starts with the lowest lag.
- One or more parameters may have missing values (i.e., NaN).
- The order of the last lagged variable (i.e., \( \Phi_P \)) with non-zero coefficient solely determines the order of the seasonal AR component model in the array with a numeric value (vs. missing or error).
For the input argument - ( \( [\Theta] \)) (parameters of the seasonal MA component)::
- The input argument is optional and can be omitted, so no seasonal MA component is included.
- The order of the parameters starts with the lowest lag.
- One or more values in the input argument can be missing (i.e., NaN).
- The order of the last lagged variable (i.e., \( \Theta_Q \)) with non-zero coefficient solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
The non-seasonal integration order - \( d \)- is optional and can be omitted, in which case d is assumed to be zero.
The seasonal integration order - \( D \)- is optional and can be omitted, in which case sD is assumed to be zero.
The season length - \( s \) - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).
The long-run mean (i.e., \( \mu \)) can take any value or be omitted, in which a zero value is assumed.
The residuals/innovations standard deviation ( \( \sigma \)) must be greater than zero.
The variance of the shocks is constant or time-invariant.

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

Parameters

[in,out] pData (Required) a univariate time series data.

[in] nSize (Required) the number of observations in pData.

[in] mean (Optional) the ARMA model mean, i.e., \(\mu \).

[in] sigma (Required) the standard deviation of the model's residuals/innovations (i.e., \(\sigma \))..

[in] nIntegral (Required) the non-seasonal difference order (i.e., \( d \)).

[in] phis (Optional) the parameters of the AR(p) component model (starting with the lowest lag), i.e., \( [\phi] \).

[in] p (Optional) the number of elements in phis (order of AR component, i.e., \( p \)).

[in] thetas (Optional) the parameters of the MA(q) component model (starting with the lowest lag), i.e., \( [\theta] \).

[in] q (Optional) the number of elements in thetas (order of MA component, i.e., \( q \)).

[in] nSIntegral (Optional) the seasonal difference (i.e., \( D \)).

[in] nSPeriod (Optional) the number of observations per one period (e.g. 12=Annual, 4=Quarter), i.e., \( s \).

[in] sPhis (Optional) the parameters of of the seasonal AR component , i.e., \( [\Phi] \).

[in] sP (Optional) the order of the seasonal AR component, i.e., \( P \).

[in] sThetas (Optional) the parameters of the seasonal MA component, i.e., \( [\Theta] \).

[in] sQ (Optional) the order of the seasonal MA component, i.e., \( Q \).

[in]

retType

(Required) a switch to select the desired output type (see FIT_RETVAL_FUNC).

Value	Output Type
1	Fitted conditional mean
2	Fitted conditional volatility or standard deviation
3	Raw residuals (actual - fitted mean)
4	Standardized residuals - (actual - fitted mean)/fitted volatility.

Returns: status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

The underlying model is described here.
The time series is homogeneous or equally spaced.
The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
the function returns an array of one simulation path starting from the end of the input data.
The function removes any observations with missing values from the dataset before calculating.
If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll SFLUC.dll SFLOG.dll

Since: v1.63

See also

NDK_SARIMA_GOF
NDK_SARIMA_PARAM
NDK_SARIMA_SIM
NDK_SARIMA_FORE
NDK_SARIMA_VALIDATE

◆ 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 forecast statistics.

Parameters

[in] pData (Required) is the univariate time series data.

[in] nSize (Required) iis the number of observations in pData.