first-order difference

The first-order difference in a time series or sequence of numerical data is the series of differences between each item and its preceeding item. That is, if we start with the series:

x₀, x₁, x₂, x₃ ...

the first order differences are :

(x₁ &munus; ₀), (x₂ &munus; x₁), (x₃ &munus; x₂) ...

For a constant series, such as 4,4,4,4,... the first-order differences are zero. For a linear series such as 3,5,7,9,11, the first-order differences are constant. For a quadratic series such as 1,4,9,16..., the first-order differences grow linearly. They thus incrementally simplify polynomial series and were the basis for the Difference Engine. there are limits to this approach, for example for expenential series, such as 1,2,4,8,16..., the first-order differences are themselves still exponential.

Note the 'first' in the term is in contrast to second-order differences which are the differences of the first-order differences, and so on for higher-order differences.