Integer construction of moving averages and exponential smoothing

Integer arithmetic and moving-average construction

Integer arithmetic can be much faster than floating-point arithmetic on many computer systems, but overflow and results smaller than one make moving-average construction harder.

A simple moving average equally weights a fixed lookback of ordered observations and is usually accumulated first, then divided once at the end. A linear moving average is a recency-weighted average whose weights rise in a straight line, so the sum of the weights grows quickly with lookback and can overflow an integer accumulator. Exponential smoothing is a recursive forecast that blends the latest observation with the previous smoothed value, conventionally through a factor of two divided by lookback plus one.

Overflow in long-lookback linear averages

Integer overflow is a running product or sum that exceeds the largest representable integer, including the 32-bit signed ceiling of 2147483647. Overflow is most likely in long-lookback linear moving averages because the summed weights can reach the tens of thousands.

A 200-period linear moving average of a price series stored as integers between 200000 and 300000, with total weight 20100, can produce an intermediate weighted sum above 4 billion, which exceeds that 32-bit signed maximum.

Construction can limit overflow by dividing a running weighted sum before it exceeds the integer range, by trapping overflow and switching to floating-point arithmetic, or by scaling stored prices so the product-sum cannot overflow.

When multiply and divide change places

In integer arithmetic the order of multiply and divide changes the result. With factors 2, 1, and 2, multiplying first then dividing yields 1, while dividing first yields 0.

That early divide is truncation: loss of a fractional result when integer division is performed before later multiplications, collapsing a nonzero quantity to zero. Accurate integer construction postpones division until the last step. Deferred division rearranges a moving-average or smoothing formula so every multiplication finishes before a single terminal division, which simple and linear moving averages typically already do.

An integer update for exponential smoothing

The usual exponential-smoothing factor of 2 divided by lookback plus one is a fraction between 0 and 1 and cannot be stored in an integer variable.

An equivalent integer exponential update adds the previous smoothed value to twice the current residual and divides last by lookback plus one, a denominator that can be precomputed. That keeps the smoother inside the same deferred-division discipline used for the averages.