sameign

Sameign is a term used in time-series analysis and signal processing to describe the extent to which two real-valued sequences share the same directional movement over time. In practice, it captures how often the signs of the two sequences agree, ignoring their magnitudes.

Formally, for sequences x_t and y_t defined over a common index set T, the sign function sgn(z) returns -1 for negative values, 0 for zero, and 1 for positive values. The sameign measure S is defined (for the nonzero observations) as S = (1/N) sum_{t in T} I[sgn(x_t) = sgn(y_t) and sgn(x_t) != 0], where N is the number of indices with both sgn(x_t) and sgn(y_t) nonzero, and I is the indicator function. Some treatments include zeros by counting them as matches or excluding them; such conventions affect the value of S.

Variants include a windowed sameign, computed over a sliding window to reflect local alignment, and a magnitude-weighted sameign, where each match is weighted by a function of the magnitudes, such as min(|x_t|, |y_t|) or |x_t y_t|.

Interpretation: S ranges from 0 to 1. Values near 1 indicate strong directional coherence between the two sequences; values near 0 indicate little or opposite movement. The measure is complementary to correlation coefficients and can be useful when magnitude information is unreliable or when directionality is of primary interest.

Applications: in finance to compare asset moves, in neuroscience to compare the direction of neural signals, in engineering for cross-signal alignment, and in anomaly detection where the sign of changes matters more than their size.

See also: sign correlation, cross-correlation, directional similarity.