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Indicators/MyIndicators/Authors/Blau/Blau_Ergodic_DTI.md

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+# Ergodic Directional Trend Index (DTI)
+
+## 1. Summary (Introduction)
+
+The Ergodic Directional Trend Index (DTI) is a sophisticated trend oscillator developed by William Blau. It is designed to measure the direction and strength of a trend in a manner similar to Wilder's ADX, but it utilizes Blau's signature "Ergodic" double-smoothing methodology to produce a much smoother and less noisy output.
+
+The indicator is based on the momentum of high and low prices. By double-smoothing this directional momentum, the DTI provides a clear, responsive, and reliable oscillator that fluctuates around a zero line, helping traders to identify the prevailing trend and its strength.
+
+The **Ergodic DTI Oscillator** is a supplementary indicator that displays the difference between the main DTI line and its signal line as a histogram, providing a clearer visual of accelerating and decelerating trend momentum.
+
+## 2. Mathematical Foundations and Calculation Logic
+
+The Ergodic DTI calculation applies the double-smoothing logic of the True Strength Index (TSI) to a unique directional momentum input.
+
+### Required Components
+
+- **Momentum Period (N_mom):** The lookback period for calculating directional momentum.
+- **Slow EMA Period (N_slow):** The period for the first, longer-term EMA smoothing.
+- **Fast EMA Period (N_fast):** The period for the second, shorter-term EMA smoothing.
+- **Signal Period:** The period for the optional moving average signal line.
+- **Price Data:** The `High` and `Low` of each bar.
+
+### Calculation Steps (Algorithm)
+
+1. **Calculate Directional Momentum:** For each bar, calculate the upward and downward momentum based on the change in Highs and Lows over the `N_mom` period.
+
+   - $\text{Up Momentum}_i = \text{Max}(0, \text{High}_i - \text{High}_{i - N_{\text{mom}}})$
+   - $\text{Down Momentum}_i = \text{Max}(0, \text{Low}_{i - N_{\text{mom}}} - \text{Low}_i)$
+
+2. **Calculate Composite High/Low Momentum (HLM):** Subtract the Down Momentum from the Up Momentum.
+   $\text{HLM}_i = \text{Up Momentum}_i - \text{Down Momentum}_i$
+
+3. **First EMA Smoothing (Slow Period):** Apply an `N_slow`-period EMA to both the `HLM` and its absolute value.
+   $\text{EMA}_{\text{slow}}(\text{HLM})_i = \text{EMA}(\text{HLM}, N_{\text{slow}})_i$
+   $\text{EMA}_{\text{slow}}(\text{AbsHLM})_i = \text{EMA}(\text{Abs}(\text{HLM}), N_{\text{slow}})_i$
+
+4. **Second EMA Smoothing (Fast Period):** Apply an `N_fast`-period EMA to the results of the first smoothing step.
+   $\text{EMA}_{\text{fast}}(\text{EMA}_{\text{slow}}(\text{HLM}))_i = \text{EMA}(\text{EMA}_{\text{slow}}(\text{HLM}), N_{\text{fast}})_i$
+   $\text{EMA}_{\text{fast}}(\text{EMA}_{\text{slow}}(\text{AbsHLM}))_i = \text{EMA}(\text{EMA}_{\text{slow}}(\text{AbsHLM}), N_{\text{fast}})_i$
+
+5. **Calculate the Ergodic DTI Value:** Divide the double-smoothed HLM by the double-smoothed absolute HLM and scale the result to 100.
+   $\text{DTI}_i = 100 \times \frac{\text{EMA}_{\text{fast}}(\text{EMA}_{\text{slow}}(\text{HLM}))_i}{\text{EMA}_{\text{fast}}(\text{EMA}_{\text{slow}}(\text{AbsHLM}))_i}$
+
+6. **Calculate the Signal Line & Oscillator:** The signal line is a moving average of the DTI line, and the oscillator is the difference between the two.
+
+## 3. MQL5 Implementation Details
+
+Our MQL5 implementations are self-contained, robust, and accurate representations of the Ergodic DTI and its oscillator.
+
+- **Stability via Full Recalculation:** We employ a "brute-force" full recalculation within the `OnCalculate` function for maximum stability.
+
+- **Fully Manual EMA Calculations:** All EMA calculations are performed **manually**. Each recursive EMA calculation is carefully initialized with a **manual Simple Moving Average (SMA)** to provide a stable starting point for the calculation chain and to prevent floating-point overflows.
+
+- **Clear, Staged Calculation:** The `OnCalculate` function is structured into clear, sequential steps, each handled by a dedicated `for` loop, making the complex, multi-stage logic easy to follow.
+
+- **Flexible Signal Line:** The `Blau_Ergodic_DTI.mq5` indicator includes a user-configurable moving average signal line, with a robust `switch` block that correctly handles all MA types (SMA, EMA, SMMA, LWMA).
+
+- **Heikin Ashi Variant (`Blau_Ergodic_DTI_HeikinAshi.mq5`):**
+  - Our toolkit also includes a "pure" Heikin Ashi version. The calculation logic is identical, but it uses the smoothed Heikin Ashi `ha_high` and `ha_low` values to calculate the initial directional momentum.
+  - This results in an exceptionally smooth trend oscillator, ideal for filtering out market noise.
+
+## 4. Parameters
+
+- **Momentum Period (`InpMomentumPeriod`):** The lookback period for the initial High/Low momentum calculation. Default is `1`.
+- **Slow Period (`InpSlowPeriod`):** The period for the first, longer-term EMA smoothing. Default is `20`.
+- **Fast Period (`InpFastPeriod`):** The period for the second, shorter-term EMA smoothing. Default is `5`.
+- **Signal Line Settings:**
+  - `InpSignalPeriod`: The lookback period for the signal line. Default is `3`.
+  - `InpSignalMAType`: The type of moving average for the signal line. Default is `MODE_EMA`.
+
+## 5. Usage and Interpretation
+
+The Ergodic DTI is a powerful trend-following oscillator.
+
+- **Zero Line Crossovers:** This is a primary trend signal.
+  - **Bullish:** A crossover of the DTI line above the zero line indicates that upward directional momentum is now dominant.
+  - **Bearish:** A crossover below the zero line indicates that downward directional momentum is dominant.
+- **Signal Line Crossovers:** Provide earlier, shorter-term signals about shifts in trend momentum.
+- **Overbought/Oversold Levels:** The +25 and -25 levels can be used to identify periods of strong, potentially overextended trend momentum.
+- **Divergence:** Divergences between price and the DTI can be powerful signals of a weakening trend. For example, if the price makes a new high but the DTI fails to do so, it suggests the underlying upward momentum is fading.
+- **Oscillator (Histogram):** The `Blau_Ergodic_DTI_Oscillator` provides a clear visual of the relationship between the DTI and its signal line, highlighting the acceleration and deceleration of the trend's momentum.