Step 03 · Stochastic Series
Autocorrelation and the Shape of Decay
Slide a record against a shifted copy of itself and measure how well the two agree. Repeat for every shift and you obtain the autocorrelation, the clearest single portrait of a record's memory.
Comparing a record with itself
Take the record and a copy displaced by one step. Pair each value with the value that preceded it, and measure how closely the pairs move together. The measure is a correlation, so it lies between minus one and one. Repeat with a displacement of two steps, then three, and so on, and record each result against its displacement. For a stationary record this list of numbers, called the autocorrelation function, depends only on the displacement and not on where the window was taken.
Reading the decay
Three shapes cover most cases. A curve that drops immediately into the bronze band belongs to a record with no memory; the shift has already destroyed any resemblance. A curve that falls smoothly toward zero, crossing the band after some shift, belongs to a record whose memory fades gradually, with a characteristic time. A curve that falls and then rises again, oscillating, belongs to a record with a rhythm, such as a seasonal swing, where resemblance returns at a fixed interval.
The rate of decay as a half-life
The speed of the fall is informative in itself. A curve that halves at each step describes a record whose memory is very short. A curve that stays high for many steps describes one whose memory is long. In an exponential decay the rate can be summarised by a single number, the half-life, the displacement at which the resemblance drops to half its starting value. Quoting that one number is a compact and honest way to describe how quickly the past stops mattering.
The band is a guide, not a wall
The bronze band marks the range of displacements where a memoryless record would wander by chance alone. Values inside the band are indistinguishable from chance; values outside it are evidence of genuine memory. A single excursion just outside the band is weak evidence, while a broad run of them is strong, and the compendium treats the distinction with care.
Why this is the pivot
Autocorrelation is the pivot of the compendium. Once it is measured, a reader can say whether a record has memory, how long that memory lasts, and whether the memory carries a rhythm. The next step in the compendium translates this same information into the language of frequency, giving the memory of a record a second portrait that will prove even more revealing.
Reading a stem plot of the function
The autocorrelation function is usually drawn as stems rising from a horizontal axis, one stem for each displacement. The height of a stem is the resemblance at that displacement, and the pattern of heights is the whole story. A reader should scan the first few stems first, because that is where memory is strongest, and then follow the descent until the stems fall inside the guide band. Where they fall inside, resemblance has ceased to be distinguishable from chance.
The stems also reveal whether the descent is smooth or interrupted. A smooth descent belongs to a record whose memory fades evenly; an interrupted one, with a stem leaping up after several quiet stems, belongs to a record with a returning rhythm. Distinguishing the two by eye takes a moment, and it saves a great deal of arithmetic that would only recover the same distinction later.