Pascal Weighted Moving Average (Pascal WMA) Professional

1. Summary (Introduction)

The Pascal Weighted Moving Average (Pascal WMA) is a unique type of weighted moving average that derives its weights from the coefficients of Pascal's triangle. This produces a set of weights that are perfectly symmetrical and follow a smooth, bell-shaped (Gaussian-like) curve.

The Pascal WMA is a symmetrical smoothing filter. Its primary purpose is not to follow trends with minimal lag, but to provide an exceptionally smooth and stable representation of the market's central tendency.

Our PascalWMA_Pro implementation is a unified, professional version that allows the calculation to be based on either standard or Heikin Ashi price data, selectable from a single input parameter.

2. Mathematical Foundations and Calculation Logic

The Pascal WMA calculates a weighted average where the weights are the binomial coefficients found in a row of Pascal's triangle.

Required Components

Period (N): The lookback period for the moving average.
Source Price: The price series used for calculation.

Calculation Steps (Algorithm)

Generate Pascal Weights: For a given period N, the weights are the coefficients of the binomial expansion of (x+y)^{N-1}. The k-th weight is calculated using the combination formula:
- Weight_k = C(N-1, k) = \frac{(N-1)!}{k! \cdot (N-1-k)!}
Calculate the Weighted Sum: For each bar t, multiply the last N prices by the corresponding Pascal coefficients.
- \text{Weighted Sum}_t = \sum_{i=0}^{N-1} (\text{Price}_{t-i} \cdot Weight_i)
Calculate the Sum of Weights: Sum all the generated Pascal weights. The sum of the n-th row is 2^n.
- \text{Sum of Weights} = 2^{N-1}
Calculate the Final WMA Value: Divide the weighted sum of prices by the sum of the weights.
- \text{Pascal WMA}_t = \frac{\text{Weighted Sum}_t}{\text{Sum of Weights}}

3. MQL5 Implementation Details

Our MQL5 implementation follows a modern, object-oriented design to ensure stability, reusability, and maintainability.

Modular Calculation Engine (PascalWMA_Calculator.mqh): The entire calculation logic is encapsulated within a reusable include file.
- CPascalWMACalculator: The base class that performs the full calculation on a given source price.
- CPascalWMACalculator_HA: A child class that inherits all the complex logic and only overrides the initial data preparation step to use smoothed Heikin Ashi prices as its input.
Optimized Incremental Calculation (O(1)): Unlike basic implementations that recalculate the entire history on every tick, this indicator employs an intelligent incremental algorithm.
- State Tracking: It utilizes prev_calculated to process only new bars.
- Persistent Buffers: Internal buffers persist their state between ticks.
Robust Weight Generation: The Pascal's triangle coefficients are calculated only once during initialization using an iterative method to prevent integer overflow for large periods.

4. Parameters

Period (InpPeriod): The lookback period for the moving average. A longer period results in a smoother line. (Default: 21).
Applied Price (InpSourcePrice): The source price for the calculation. (Standard or Heikin Ashi).

5. Usage and Interpretation

The Pascal WMA should be interpreted as a high-quality smoothing filter and a "mean" or "center of gravity" line, not as a traditional trend-following moving average.

Noise Reduction and Trend Clarity: The primary use is to filter out market noise and provide a clearer picture of the underlying price movement.
Mean Reversion Signals: The line acts as a "magnet" for the price. When the price moves significantly away from the Pascal WMA, it can be considered over-extended, increasing the probability of a reversion back towards the line.
Caution: Due to its inherent nature as a centered, smoothing filter, the Pascal WMA will always lag the price. It should not be used for fast crossover signals. Its strength lies in its exceptional smoothness and its ability to define the market's equilibrium point.

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