A statistical technique comparing two datasets to find matching patterns, used here on sky maps.Cross-correlation is a statistical method that measures the similarity between two datasets as one is shifted relative to the other. It is used to find patterns that appear in both datasets, even if they are offset in time or space. In the context of sky maps, cross-correlation helps astronomers compare maps of different cosmic signals—like temperature fluctuations and galaxy distributions—to see if they share common features, which can reveal underlying physical connections.
The concept of cross-correlation has roots in early 20th-century statistics and signal processing, with formal development by statisticians like Norbert Wiener in the 1940s, though similar ideas appeared earlier in time-series analysis.
Given two sequences or maps, cross-correlation computes a value for each possible shift: it multiplies the values at corresponding positions, sums the products, and repeats for every shift. A high value at a particular shift indicates that the two datasets align well when offset by that amount. In practice, for large datasets like sky maps, this is often done efficiently using Fourier transforms, where cross-correlation becomes a simple multiplication in frequency space.
Cross-correlation is widely used in astronomy to compare cosmic microwave background maps with galaxy surveys, in engineering for radar and sonar signal detection, and in finance to analyze relationships between economic time series.
Cross-correlation is the mathematical basis behind how your smartphone's GPS receiver locks onto satellite signals by matching the incoming signal with a known code.
| c | variable | cross-correlation value — cross correlation |
| v | variable | Doppler shift velocity — velocity metre per second |
| λ | variable | wavelength — wavelength metre |
| S1 | variable | first spectrum — spectrum |
| S2 | variable | second spectrum — spectrum |
| S̄1 | variable | mean of first spectrum over wavelength — mean spectrum |
| S̄2 | variable | mean of second spectrum over wavelength — mean spectrum |
| σ | variable | standard deviation of spectrum — standard deviation |
| ∑ | operator | sum over wavelength bins — summation |
