deltasdifferences

Deltasdifferences is a concept in data analysis and mathematics that refers to a class of measures used to quantify how two sequences diverge by examining the changes between consecutive elements, or deltas, in each sequence. It focuses on the comparison of rates of change rather than on absolute values.

Definition: Let X = (x1, ..., xn) and Y = (y1, ..., yn) be two real-valued sequences of equal length. The first-difference sequences are ΔX_i = x_{i+1} − x_i and ΔY_i = y_{i+1} − y_i for i = 1, ..., n−1. The deltasdifferences at position i are D_i = ΔX_i − ΔY_i. A global deltasdifferences score can be defined by applying a norm to D, for example M1 = sum_{i=1}^{n−1} |D_i| (L1 norm) or M2 = sqrt(sum D_i^2) (L2 norm).

Interpretation: The score reflects how the rate of change between the two series differs over time, emphasizing changes in trend rather than levels. The concept can be extended to multi-dimensional data by comparing corresponding delta vectors or by using weighted differences.

Computation and variants: One may also consider second differences (differences of deltas) or alternative cross-difference forms. Normalization can be applied by scaling the deltas by their standard deviation or by the range of values to improve comparability.

Applications: Deltasdifferences is used in time-series analysis, change-point detection, financial data comparison, sensor data fusion, quality control, and anomaly detection to identify regimes where the dynamics of one series diverge from another.

Example: X = [1,3,4,8], Y = [2,4,4,7]. ΔX = [2,1,4], ΔY = [2,0,3], D = [0,1,1]. L2 norm ≈ 1.414; L1 norm = 2.