# Linear Regression Pro

## 1. Summary (Introduction)

The Linear Regression Pro is a highly flexible, manually calculated version of the classic Linear Regression Channel. It is a statistically-based indicator that uses the **linear regression** (or "least squares fit") method to determine the primary trend direction.

Unlike the built-in MetaTrader 5 graphical objects, this "Pro" version offers full customization over its calculation, including the **source price** and the **channel width calculation method**. It is designed to be a robust, efficient, and precise tool for advanced technical analysis.

- The **Middle Line** is the linear regression trendline.
- The **Upper and Lower Channel Lines** are plotted based on a selected deviation method.

## 2. Mathematical Foundations and Calculation Logic

The indicator's core is the linear regression trendline, which is the straight line that best fits a series of data points over a specified period.

### Required Components

- **Period (N):** The lookback period for the regression calculation.
- **Source Price (P):** The price series used for the calculation (e.g., Close, Typical, etc.).
- **Channel Mode:** The method for calculating the channel width.
- **Deviations (M):** The multiplier for the standard deviation (used in Standard Deviation mode).

### Calculation Steps (Algorithm)

For the last `N` bars at any given point in time:

1. **Calculate the Linear Regression Line:** Using the method of least squares, find the straight line that best fits the `N` source price points.
   $\text{Regression Line}_t = a + b \times t$

2. **Calculate the Channel Width (Deviation Offset):** Based on the selected `Channel Mode`:

   - **Standard Deviation Mode:** Compute the standard deviation of the `N` price points from the calculated regression line.
     $\sigma = \sqrt{\frac{\sum_{k=1}^{N} (P_k - \text{Regression Line}_k)^2}{N}}$
     $\text{Offset} = M \times \sigma$
   - **Maximum Deviation Mode:** Find the largest absolute distance between any of the `N` price points and the regression line.
     $\text{Offset} = \text{Max}(\text{Abs}(P_k - \text{Regression Line}_k))$

3. **Calculate the Upper and Lower Channel Lines:** Add and subtract the calculated `Offset` from the regression line.
   $\text{Upper Channel}_t = \text{Regression Line}_t + \text{Offset}$
   $\text{Lower Channel}_t = \text{Regression Line}_t - \text{Offset}$

## 3. MQL5 Implementation Details

Our MQL5 implementation is a completely self-contained, robust, and accurate indicator.

- **Stability via Full Recalculation:** We employ a "brute-force" full recalculation model.
- **Fully Manual Calculations:** All calculations, including the regression line and both deviation methods, are performed **manually**. This provides maximum flexibility and ensures the indicator is independent of the limitations and potential inconsistencies of built-in graphical objects.
- **Efficient "On New Bar" Updates:** The indicator is designed to be extremely light on terminal resources. The entire complex calculation is only performed **once per bar** when a new candle forms, preventing unnecessary CPU load on every tick.
- **Clean, Non-Continuous Display:** The indicator only displays the **single, most current** regression channel calculated on the last `N` bars. This is achieved by dynamically setting the `PLOT_DRAW_BEGIN` property on each update, providing a clean, uncluttered view of the present market structure.

### Indicator Family

- **`LinearRegression_Pro.mq5`:** The standard version, which can be configured to use any `ENUM_APPLIED_PRICE`.
- **`LinearRegression_Pro_Sample.mq5`:** An educational variant that uses the **Sample Standard Deviation** (`n-1` in the denominator) instead of the Population Standard Deviation (`n`). This version produces slightly wider channels, similar to the built-in `OBJ_STDDEVCHANNEL` object in MetaTrader.
- **`LinearRegression_Pro_HeikinAshi.mq5`:** The "pure" Heikin Ashi version, where the regression is calculated on the smoothed Heikin Ashi price data.

## 4. Parameters

- **Regression Period (`InpRegressionPeriod`):** The number of bars to include in the calculation.
- **Applied Price (`InpAppliedPrice`):** The source price for the calculation (e.g., `PRICE_CLOSE`, `PRICE_TYPICAL`).
- **Channel Mode (`InpChannelMode`):** Allows the user to select the method for calculating the channel width:
  - `DEVIATION_STANDARD`: Based on the standard deviation (most common).
  - `DEVIATION_MAXIMUM`: Based on the maximum price deviation.
- **Deviations (`InpDeviations`):** The multiplier for the standard deviation, used only when `DEVIATION_STANDARD` mode is selected.

## 5. Usage and Interpretation

- **Trend Identification:** The slope of the middle line indicates the direction of the trend.
- **Dynamic Support and Resistance:** The channel lines act as dynamic support and resistance levels.
- **Overbought/Oversold:** A move to the upper channel line can be considered overbought relative to the current trend, while a move to the lower line can be considered oversold.
- **Caution:** The indicator is "repainting" by nature. It is best used for confirming the current market structure rather than for generating precise entry signals from past data.