NDK_HURST_EXPONENT

Calculates the Hurst exponent (a measure of persistence or long memory) for time series.

Returns

status code of the operation

Return values

NDK_SUCCESS	Operation successful
NDK_FAILED	Operation unsuccessful. See SFMacros.h for more details.

Parameters

[in]	pData	is the input data sample (a one dimensional array).
[in]	nSize	is the number of observations in pData.
[in]	alpha	is the statistical significance level (1%, 5%, 10%). If missing, a default of 5% is assumed.
[in]	retType	is a number that determines the type of return value: 1 = Empirical Hurst exponent (R/S method) 2 = Anis-Lloyd/Peters corrected Hurst exponent 3 = Theoretical Hurst exponent 4 = Upper limit of the confidence interval 5 = Lower limit of the confidence interval
[out]	retVal	is the calculated value of this function.

Remarks

1. The input data series may include missing values (NaN), but they will not be included in the calculations.

2. The Hurst exponent, \(h\), is defined in terms of the rescaled range as follows:

\[E \left [ \frac{R(n)}{S(n)} \right ]=C_n^H \] \[n \to \infty \]

3. Where:

• \(\left [ \frac{R(n)}{S(n)} \right ]\) is the Rescaled Range.
• \(E \left [x \right ]\) is the expected value.
• \(n\) is the time of the last observation (e.g. it corresponds to \(X_n\) in the input time series data.)
• \(h\) is a constant. of

4. The Hurst exponent is a measure autocorrelation (persistence and long memory): -A value of \(0<H<0.5\) indicates a time series with negative autocorrelation (e.g. a decrease between values will probably be followed by another decrease). -A value of \(0.5<H<1\) indicates a time series with positive autocorrelation (e.g. an increase between values will probably be followed by another increase). -A value of \(H=0.5\) indicates a "true random walk," where it is equally likely that a decrease or an increase will follow from any particular value (e.g. the time series has no memory of previous values).

5. The Hurst exponent's namesake, Harold Edwin Hurst (1880-1978), was a British hydrologist who researched reservoir capacity along the Nile river.

6. The rescaled range is calculated for a time series, \(X=X_1,X_2,\dots, X_n\), as follows:

1. Calculate the mean:
   \(m=\dfrac{1}{n} \sum_{i=1}^{n} X_i\)
2. Create a mean adjusted series:
   \(Y_t=X_{t}-m \) for \( t=1,2, \dots ,n\)
3. Calculate the cumulative deviate series Z:
   \(Z_t= \sum_{i=1}^{t} Y_{i} \) for \( t=1,2, \dots ,n\)
4. Create a range series R:
   \(R_t = max\left (Z_1, Z_2, \dots, Z_t \right )- min\left (Z_1, Z_2, \dots, Z_t \right ) \) for \( t=1,2, \dots, n\)
5. Create a standard deviation series R:
   \(S_{t}= \sqrt{\dfrac{1}{t} \sum_{i=1}^{t}\left ( X_{i} - u \right )^{2}}\) for \( t=1,2, \dots ,n\)  Where:
   \(h\) is the mean for the time series values \(X_1,X_2, \dots, X_t\)

8. Calculate the rescaled range series (R/S):
\(\left ( R/S \right )_{t} = \frac{R_{t}}{S_{t}} \) for \( t=1,2, \dots, n\)

Exceptions

Exception Type	Condition
None	N/A

Requirements

Namespace	NumXLAPI
Class	SFSDK
Scope	Public
Lifetime	Static
Package	NumXLAPI.DLL

Examples