1989issue C101-8

Constructing lag-matched triangular moving averages

A moving average reduces the influence of any one observation by combining a set of periods, which lowers high-frequency jitter while adding lag and still transmitting residual noise. Equal-lag construction retunes each smoother to a five-period delay, writes a center-peaked triangular weight schedule, and can place that numeric average inside a relative-strength index.

A moving average combines a set of periods so no single observation dominates, which lowers high-frequency jitter while adding lag and still transmitting residual noise.
Noise reaches the output through a jump-on effect when a new value first enters the window and through a drop-off effect when an aged value leaves it.
Equal-lag comparisons targeted the five-period delay of an 11-period simple moving average, then measured average noise throughput of 12.856 percent for that simple average, 7.7 percent for an exponential moving average, and 10 percent for a linear weighted moving average.
A triangular moving average raises weights to the middle of the window and lowers them to the end, and that same numeric average can replace the exponential average inside a relative-strength index.

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A moving average trades jitter for lag

A moving average reduces the influence of any one observation by combining a set of periods, which lowers high-frequency jitter while adding lag and still transmitting some residual noise. Lag is the delay between an event in the raw series and the period in which that event peaks in the smoothed output. Noise is high-frequency variation that appears and disappears inside a single sampling interval and is not a move sustained across several periods.

How noise still enters the window

Noise reaches a moving-average output through a jump-on effect when a new value first enters the window and through a drop-off effect when an aged value leaves it. The jump-on effect is the output jolt that occurs when a new observation first receives weight inside the averaging window. The drop-off effect is the opposite output jolt that occurs when an observation ages out of the averaging window.

Lock lag before comparing smoothers

Equal-lag comparisons were constructed by targeting a five-period delay, the delay of an 11-period simple moving average, and retuning each smoother's window until that lag was matched. Editorial: lock that shared delay before ranking filter shapes, otherwise unequal lag is mixed into the comparison.

Write the triangular weight schedule

A triangular moving average is a numeric filter. It assigns an explicit weight to each observation in the window, sums the weighted values, and divides by the sum of the weights. Filter shape is the sequence of weights applied across the lookback: a simple moving average is rectangular, while a triangular moving average peaks near the center. Construction raises weights linearly to the middle of the window, lowers them linearly to the end, sums each observation times its weight, and divides by the sum of the weights.

Measure how a spike leaks through

Lag and noise throughput were read from an artificial series containing one 100-unit spike. Average noise used the square root of the summed squares of the largest positive and negative one-period output peaks, and worst-case noise used the sum of those peaks. Under that equal-lag protocol, measured average noise throughput was 12.856 percent for the 11-period simple moving average, 7.7 percent for an exponential moving average, and 10 percent for a linear weighted moving average. Editorial: read those figures as spike leakage at matched lag, not as a performance claim.

Residual one-day movement versus multi-day response

A 14-day triangular moving average and a 14-day simple moving average were applied to period-to-period Treasury-bond closing changes to illustrate residual one-day movement versus multi-day response. When the averaging window is shortened to seven periods or less, a triangular filter can show more intermediate-frequency peaks than a simple moving average even while its highest-frequency response stays lower. Editorial: that short-window contrast is a filter-shape observation, not a rule for choosing a live lookback.

Reuse the triangular average inside a relative-strength index

The exponential average inside a relative-strength-index calculation can be replaced by a triangular numeric average, leaving the rest of the oscillator construction intact. A relative-strength index is a bounded momentum construction, and swapping that internal smoother does not change the surrounding ratio logic. Editorial: reuse the same triangular schedule so the comparison is between filters rather than chart folklore.

Educational research material, not investment advice. Historical source context does not establish present-day performance.

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19901-1 pp.Next on Triangular moving averageLag-aligned MACD from triangular moving averagesA triangular moving average is a symmetric smoother whose weight peaks at the center of a fixed lookback window and declines toward both ends, so lag sits near the midpoint and equals half the window length minus one.

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