WAVG: Weighted Average

Weighted average assigns importance by position, giving recent or central observations a louder voice in the mean.

Property	Value
Category	Statistic
Inputs	Source (close)
Parameters	`period`
Outputs	Single series (Wavg)
Output range	`0` to `1`
Warmup	`period` bars
PineScript	wavg.pine

The Weighted Average computes a rolling linearly-weighted mean where the most recent observation receives weight N and the oldest receives weight...
Similar: WMA, EMA | Trading note: Weighted average with custom weights; flexible aggregation for composite indicators.
Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Weighted Average computes a rolling linearly-weighted mean where the most recent observation receives weight N and the oldest receives weight 1, making it mathematically identical to the Weighted Moving Average (WMA) but categorized as a statistical measure. The implementation uses a circular buffer with an O(1) incremental update scheme: rather than recomputing the full weighted sum each bar, it maintains running sums and adjusts them through add/subtract operations as values enter and exit the window. This makes WAVG one of the most efficient weighted estimators available, with constant per-bar cost regardless of the lookback period.

Historical Context

The linearly-weighted average is one of the oldest weighted estimators, predating formal statistical theory. The concept of assigning decreasing importance to older observations appears in early actuarial work (17th-18th centuries) and was formalized in weather forecasting by the mid-19th century. In technical analysis, the Weighted Moving Average became popular through the work of Martin Pring and other chartists who sought a middle ground between the SMA (equal weights, excessive lag) and the EMA (exponential weights, infinite memory).

The linear weighting scheme assigns weight w_i = i + 1 to the $i$-th sample from oldest (i = 0) to newest (i = N-1). This produces a centroid (center of mass) that is biased toward recent data: the effective lag is N/3 bars compared to (N-1)/2 for the SMA. The triangular weight distribution means the most recent value contributes 2/(N+1) times the total weight, versus 1/N for the SMA.

The O(1) update trick used in this implementation is well known in DSP: the weighted sum W = \sum i \cdot x_i can be maintained incrementally by tracking the unweighted sum S = \sum x_i and noting that when all indices shift by 1, W_{\text{new}} = W_{\text{old}} - S_{\text{old}} + N \cdot x_{\text{new}}.

Architecture and Physics

The implementation uses a circular buffer of size period with three state variables:

weightedSum: The current linearly-weighted sum \sum_{i=1}^{n} i \cdot x_{(i)} where (i) is position from oldest.
runningSum: The unweighted sum \sum x_i of all values in the buffer.
count: The current fill level (increases during warmup, equals period at steady state).

Per-bar update (O(1) operations):

Remove departing value: If the buffer position being overwritten contains a valid value, subtract it from runningSum.
Shift weights down: Subtract runningSum from weightedSum. This decrements every existing value's weight by 1 (equivalent to aging all observations).
Add new value: Add srcVal to runningSum and add count * srcVal to weightedSum (new value gets the highest weight).
Store and advance: Write to the circular buffer and advance the head pointer.

Normalization: The denominator is n(n+1)/2 where n is the current count. This handles the warmup period naturally: when only k < N values have been received, the result uses $k$-based weights.

Mathematical Foundation

The linearly-weighted average with window size n:

\text{WAVG} = \frac{\sum_{i=0}^{n-1} (i + 1) \cdot x_{n-1-i}}{\sum_{i=0}^{n-1} (i + 1)} = \frac{\sum_{i=1}^{n} i \cdot x_i}{\frac{n(n+1)}{2}}

where x_n is the most recent value (weight n) and x_1 is the oldest (weight 1).

Effective lag (centroid offset from current bar):

\text{lag} = \frac{\sum_{i=0}^{n-1} i \cdot (n - i)}{\sum_{i=0}^{n-1}(n-i)} = \frac{n-1}{3}

O(1) incremental update on arrival of new value x_{\text{new}} and departure of x_{\text{old}}:

S_{\text{new}} = S_{\text{old}} - x_{\text{old}} + x_{\text{new}} W_{\text{new}} = W_{\text{old}} - S_{\text{old}} + n \cdot x_{\text{new}} \text{WAVG} = \frac{W_{\text{new}}}{n(n+1)/2}

Weight distribution: Weight of position i from newest is \frac{n - i}{n(n+1)/2}. Most recent: \frac{2}{n+1}. Oldest: \frac{2}{n(n+1)}.

Parameter constraints: period > 0.

WAVG(source, period):
    // State variables (persistent)
    var buffer[period], head = 0, weightedSum = 0, runningSum = 0, count = 0

    srcVal = nz(source)
    oldest = buffer[head]

    if oldest is valid:
        runningSum -= oldest
    else:
        count += 1

    weightedSum -= runningSum        // shift all weights down by 1
    runningSum  += srcVal
    weightedSum += count * srcVal    // new value gets highest weight

    buffer[head] = srcVal
    head = (head + 1) % period

    denom = count * (count + 1) / 2
    return denom > 0 ? weightedSum / denom : srcVal

Performance Profile

Operation Count (Streaming Mode)

Weighted Average (WAVG) applies linearly increasing weights [1, 2, 3, ..., N] to the sliding window, using a precomputed weight sum denominator.

Operation	Count	Cost (cycles)	Subtotal
Ring buffer add/evict	1	3 cy	~3 cy
Weighted sum via FMA	N	1 cy	~N cy
Divide by weight sum	1	4 cy	~4 cy
NaN guard + state update	1	2 cy	~2 cy
Total (N=14)	O(N)	—	~23 cy

O(N) per update; weight sum denominator N(N+1)/2 precomputed in constructor. Hot path is a FMA loop over the window — amenable to vectorization.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Weight vector generation	Yes	Static precomputed array, reused
Weighted dot product	Yes	Vector FMA across window
Sliding window eviction	Partial	Ring buffer update is scalar

Batch span path benefits from Vector dot product for the weight application. AVX2 processes 4 doubles per cycle, giving ~3.5× speedup for N≥16.

Resources

Pring, M.J. "Technical Analysis Explained." 5th edition, McGraw-Hill, 2014.
Murphy, J.J. "Technical Analysis of the Financial Markets." New York Institute of Finance, 1999.
Oppenheim, A.V. & Schafer, R.W. "Discrete-Time Signal Processing." 3rd edition, Pearson, 2010.
Haykin, S. "Adaptive Filter Theory." 5th edition, Pearson, 2013.

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