is a vector of kernel functions and is an affine transform. Specifically, can be circular translation transform, rotation transform, or scale transform, etc. The kernel cross-correlation extends cross-correlation from linear space to kernel space. Cross-correlation is equivariant to translation; kernel cross-correlation is equivariant to any affine transforms, including translation, rotation, and scale, etc. Explanation As an example, consider two real valued functions and differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much must be shifted along the x-axis to make it identical to . The formula essentially slides the function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive. With complex-valued functions and , taking the conjugate of ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral. In econometrics, lagged cross-correlation is sometimes referred to as cross-autocorrelation. Properties Cross-correlation of random vectors Definition For random vectors and , each containing random elements whose expected value and variance exist, the cross-correlation matrix of and is defined byand has dimensions . Written component-wise:The random vectors and need not have the same dimension, and either might be a scalar value. Example For example, if and are random vectors, then is a matrix whose -th entry is . Definition for complex random vectors If and are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of and is defined bywhere denotes Hermitian transposition. Cross-correlation of stochastic processes In time series analysis and statistics, the cross-correlation of a pair of random process is the correlation between values of the processes at different times, as a function of the two times. Let be a pair of random processes, and be any point in time ( may be an integer for a discrete-time process or a real number for a continuous-time process). Then is the value (or realization) produced by a given run of the process at time . Cross-correlation function Suppose that the process has means and and variances and at time , for each . Then the definition of