# XCF - Cross Correlation Function

Calculates the cross-correlation function between two time series.

## Syntax

XCF(Y, X, K, Method, Return_type)

Y
is the first univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
X
is the second univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
K
is the lag order (e.g., 0 = no lag, 1 = 1st lag, etc.) to use with second-time series input (X). If missing, the default lag order of zero (i.e., no-lag) is assumed.
Method
is a switch to select the calculation method (1 = Pearson (default), 2 = Spearman, 3 = Kendall).
Order Description
1 Pearson (default).
2 Spearman.
3 Kendall.
Return_type
is a switch to select the return output (1 = correlation value (default), 2 = Std Error).
Method Description
1 Correlation Value.
2 Standard Error.

## Remarks

1. The time series is homogeneous or equally spaced.
2. The two time series must be identical in size.
3. The Pearson correlation, $r_{xy}$, is defined as follows: $$r_{xy}= \frac{\sum_{i=1}^N(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^N(x_i-\bar{x})^2\times\sum_{i=1}^N(y_i-\bar{y})^2}}$$ Where:
• $\bar{x}$ is the sample average of time series X.
• $\bar{y}$ is the sample average of time series Y.
• $x_i \in X$ is a value from the first input time series data.
• $y_i \in Y$ is a value from the second input time series data.
• $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.
4. The Spearman rank correlation, $h$, is defined as follows: $$r =1-\frac{6\sum ( x_i-y_i )^2}{N\times(N^2-1}$$ Where:
• $x_i \in X$ (e.g., $x_i$ is in the first input time series data).
• $y_i \in Y$ (e.g., $y_i$ is in the second input time series data).
• $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.
5. The Kendall tau ($\tau$) rank correlation is defined as follows: $$\tau =\frac{N_C-N_D}{\frac{1}{2}N(N-1)}$$ Where:
• $N_C$ is the number of concordant pairs of observations, $(x_i, y_i)$ and $(x_j, y_j)$, defined such that the ranks of the pairs of elements are in agreement. That is, if $x_i \gt x_j$ then $y_i \gt y_j$, and if $x_i \lt x_j$ then $y_i \lt y_j$.
• $N_D$ is the number of discordant pairs of observations, $(x_i, y_i$) and $(x_j, y_j)$, defined such that the ranks of the pairs of elements are not in agreement.
• $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.