NDK_ACFCI

Calculates the confidence interval limits (upper/lower) for the autocorrelation function.

Returns

status code of the operation

Return values

NDK_SUCCESS	Operation successful
NDK_FAILED	Operation unsuccessful. See Macros for full list.

Parameters

[in]	X	is the univariate time series data (a one dimensional array).
[in]	N	is the number of observations in X.
[in]	K	is the lag order (e.g. k=0 (no lag), k=1 (1st lag), etc.).
[in]	alpha	is the statistical significance level. If missing, a default of 5% is assumed.
[out]	ULCI	is the upper limit value of the confidence interval
[out]	LLCI	is the lower limit value of the confidence interval.

Remarks

1. The time series is homogeneous or equally spaced.

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

3. The lag order (k) must be less than the time series size, or else an error value (NDK_FAILED) is returned.

4. The ACFCI function calculates the confidence limits as:

• \(\hat\rho_k - Z_{\alpha/2}\times \sigma_{\rho_k} \leq \rho_k \leq \hat\rho_k+ Z_{\alpha/2}\times \sigma_{\rho_k}\)
  />, where:
  • \(rho_k\) is the population autocorrelation function.
  • \(\sigma_{\rho_k}\) is the standard error of the sample autocorrelation.
  • \(\hat{\rho_{k}}\) is the sample autocorrelation function for lag k.
  • \(Z\sim N(0,1)\)
  • \(P(\left|Z\right|\geq Z_{\alpha/2}) = \alpha\)

5. For the case in which the underlying population distribution is normal, the sample autocorrelation also has a normal distribution:

• \(\hat \rho_k \sim N(\rho_k,\sigma_{\rho_k}^2)\), where:
  • \(\hat \rho_k\) is the sample autocorrelation for lag k.
  • \(\rho_k\) is the population autocorrelation for lag k.
  • \(\sigma_{\rho_k}\) is the standard error of the sample autocorrelation for lag k.

6. Bartlett proved that the variance of the sample autocorrelation of a stationary normal stochastic process (i.e. independent, identically normal distributed errors) can be formulated as:

• \(\sigma_{\rho_k}^2 = \frac{\sum_{j=-\infty}^{\infty}\rho_j^2+\rho_{j+k}\rho_{j-k}-4\rho_j\rho_k\rho_{i-k}+2\rho_j^2\rho_k^2}{T}\)

7. Furthermore, the variance of the sample autocorrelation is reformulated:

• \(\sigma_{\rho_k}^2 = \frac{1+\sum_{j=1}^{k-1}\hat\rho_j^2}{T}\), where:
  • \(\sigma_{\rho_k}\) is the standard error of the sample autocorrelation for lag k.
  • \(T\) is the sample data size.
  • \(\hat\rho_j\) is the sample autocorrelation function for lag j.
  • \(k\) is the lag order.

Requirements

Header	SFSDK.H
Library	SFSDK.LIB
DLL	SFSDK.DLL

Example

#include "SFMacros.h"
#include "SFSDK.h"

// Input time series: 110 observation 
double data[110]={0.23, 0.24, 0.45, ..., 0.95}

int nRet = NDK_FAILED;
double alpha = 0.05f;
double UL = -2.0f;
double LL = -2.0f;

nRet = NDK_ACFCI(data, 110, 1, alpha, &UL,  &LL);
if( nRet < NDK_SUCCESS){
    // Error occured
    // Call NDK_MSG to retrieve description of the error, and write it to the log file
    ....
}