QuanTAlib/lib/trends/pwma/Pwma.md

PWMA: Parabolic Weighted Moving Average

What It Does

The Parabolic Weighted Moving Average (PWMA) applies a squared weighting scheme to historical prices, assigning significantly higher importance to the most recent data points than a standard Weighted Moving Average (WMA). While WMA uses linear weights (1, 2, 3, \dots, n), PWMA uses parabolic weights (1^2, 2^2, 3^2, \dots, n^2). This results in an indicator that tracks price action with exceptional responsiveness, making it ideal for fast-moving markets and momentum calculations.

Historical Context

The concept of parabolic weighting is often associated with advanced signal processing techniques in finance, notably appearing as a core component in Jurik Research's "Velocity" indicator (Velocity = PWMA - WMA). By shifting the center of gravity even closer to the current price than a linear WMA, it minimizes lag to near-zero levels for recent price changes.

How It Works

The Core Idea

Imagine a 5-day window.

• SMA: Weights are 1, 1, 1, 1, 1.
• WMA: Weights are 1, 2, 3, 4, 5.
• PWMA: Weights are 1, 4, 9, 16, 25.

In the PWMA, the most recent price (weight 25) is 25 times more important than the oldest price (weight 1), whereas in the WMA it is only 5 times more important. This aggressive weighting allows the PWMA to turn almost instantly when the trend changes.

Mathematical Foundation

PWMA = \frac{\sum_{i=1}^{n} i^2 \cdot P_i}{\sum_{i=1}^{n} i^2}

Where:

• n = period length
• P_i = price at position i (oldest to newest)
• Denominator = \frac{n(n+1)(2n+1)}{6} (sum of squares)

Implementation Details: O(1) Streaming

Calculating the sum of i^2 \cdot P_i for every bar would be computationally expensive (O(n)). We achieve O(1) complexity using a triple running sum technique:

1. S1 (Simple Sum): \sum P_i
2. S2 (Linear Weighted Sum): \sum i \cdot P_i
3. S3 (Parabolic Weighted Sum): \sum i^2 \cdot P_i

When the window slides:

S1_{new} = S1_{old} - P_{oldest} + P_{new} S2_{new} = S2_{old} - S1_{old} + n \cdot P_{new} S3_{new} = S3_{old} - 2 \cdot S2_{old} + S1_{old} + n^2 \cdot P_{new}

This allows the indicator to update in constant time, regardless of the period length.

Configuration

Parameter	Default	Purpose	Adjustment Guidelines
Period	14	Lookback window	Shorter (5-10) for momentum; Longer (20+) for trend smoothing.

Performance Profile

Operation	Complexity	Description
Streaming update	O(1)	Constant time triple-sum update
Bar correction	O(1)	Efficient state rollback
Batch processing	O(n)	Fast sequential processing
Memory footprint	O(period)	Uses a RingBuffer to store the lookback window

Interpretation

Trading Signals

Momentum

• Rapid Turns: PWMA is excellent for identifying the exact moment a trend loses momentum, often turning before the price itself peaks or troughs.

Velocity

• PWMA - WMA: Subtracting a WMA from a PWMA of the same period creates a powerful momentum oscillator (Velocity) that is smoother than ROC but with less lag.

When It Works Best

• Fast Trends: Markets that move parabolically or have sharp V-bottoms/tops.

When It Struggles

• Noise: The extreme sensitivity to recent data means PWMA can be noisy in choppy markets. It is often best used as part of a composite indicator rather than a standalone filter.

Architecture Notes

This implementation makes specific trade-offs:

Choice: Triple Running Sums

• Implementation: Maintains S1, S2, and S3.
• Rationale: Enables O(1) updates. A naive implementation would be O(n), which is unacceptable for large periods or high-frequency trading.

Choice: Periodic Resync

• Implementation: Recalculates sums from scratch every 1,000 ticks.
• Rationale: Floating-point errors accumulate rapidly in the S3 term (which involves n^2). Periodic resync ensures long-term stability.

References

• Colby, Robert W. "The Encyclopedia of Technical Market Indicators." McGraw-Hill, 2002.
• Jurik Research. "Velocity."

C# Usage

Streaming Updates (Single Instance)

using QuanTAlib;

var pwma = new Pwma(period: 14);

// Process each new bar
TValue result = pwma.Update(new TValue(timestamp, closePrice));
Console.WriteLine($"PWMA: {result.Value:F2}");

// Check if buffer is full
if (pwma.IsHot)
{
    // Indicator is fully initialized
}

Batch Processing (Historical Data)

// TSeries API (object-oriented)
TSeries prices = ...;
TSeries pwmaValues = Pwma.Batch(prices, period: 14);

// High-performance Span API (zero allocation)
double[] prices = new double[10000];
double[] output = new double[10000];
Pwma.Calculate(prices.AsSpan(), output.AsSpan(), period: 14);

Bar Correction (isNew Parameter)

var pwma = new Pwma(14);

// New bar arrives
pwma.Update(new TValue(time, 100.5), isNew: true);

// Intra-bar price updates (real-time tick data)
pwma.Update(new TValue(time, 101.0), isNew: false); // Updates current bar
pwma.Update(new TValue(time, 100.8), isNew: false); // Updates current bar

// Next bar
pwma.Update(new TValue(time + 60, 101.2), isNew: true); // Advances state

Event-Driven Architecture

var source = new TSeries();
var pwma = new Pwma(source, period: 14);

// Subscribe to PWMA output
pwma.Pub += (value) => {
    Console.WriteLine($"New PWMA value: {value.Value}");
};

// Feeding source automatically triggers the chain
source.Add(new TValue(DateTime.Now, 105.2));