Volume Weighted Kurtosis

This indicator analyzes market sentiment by decomposing total
Excess Kurtosis ("Fat Tails") into distinct, interpretable components:
"Between-Bar" (Trend Path) and "Within-Bar" (Microstructure).

Key Features:

1. **Moment-Based Kurtosis Decomposition:** The indicator
separates kurtosis based on the 'Estimate Bar Statistics' option.
It leverages the additive property of the Fourth Central Moment
(Cumulants) to ensure mathematical rigor:
- **Standard Mode (`Estimate Bar Statistics` = OFF):** Calculates
simple kurtosis of the selected `Source`.
- **Decomposition Mode (`Estimate Bar Statistics` = ON):** The
indicator uses a statistical model ('Estimator') to
calculate *within-bar* kurtosis.
This separates the tail risk into:
- **Between-Bar Kurtosis (Gold):** Peakedness of the price
path itself. High values indicate the trend moves in jumps
or gaps rather than a smooth progression.
- **Within-Bar Kurtosis (Blue):** Fat tails within the
microstructure. High values imply significant outliers
inside the bars (e.g., extreme wicks).
- **Interaction Variance (Dark Grey):** The comovement of variance
and mean deviations (volatility clustering relative to trend).
- **Interaction Skewness (Darker Grey):** The comovement of skewness
and mean deviations (asymmetry relative to trend).

2. **Visual Decomposition Logic:** Total Excess Kurtosis is the
primary metric displayed. Since Kurtosis coefficients are not
linearly additive, this indicator calculates the *exact* Total
Kurtosis and partitions the area/ratios based on the additive
Fourth Moment Decomposition (`M4Tot = M4Btw + M4Wtn + M4Int`). This
ensures the displayed total kurtosis remains mathematically accurate.

3. **Dual Display Modes:** The indicator offers two modes to
visualize this decomposition:
- **Absolute Mode:** Displays the *Total Kurtosis* as the main
line, with the background filled by the stacked components.
Shows the *magnitude* of the tail risk.
- **Relative Mode:** Displays the **Contribution Ratios**
of each component (-1.0 to 1.0). This isolates the
*structure/quality* of the risk (e.g., "Is the risk
driven by the trend jumps or by the candle instability?").

4. **Calculation Options:**
- **Normalization:** An optional 'Normalize' setting
transforms inputs into logarithmic space, analyzing the
kurtosis of *returns* rather than absolute prices.
(Essential for correct statistical properties).
- **Volume Weighting:** An option (`Volume weighted`) applies
volume weighting to all moment calculations, emphasizing
high-participation moves.

5. **Kurtosis Cycle Analysis:**
- **Pivot Detection:** Includes a built-in pivot detector
that identifies significant turning points (peaks/valleys) in
the *Total Kurtosis* line. (Note: This is only visible
in 'Absolute Mode').
- **Flexible Pivot Algorithms:** Supports various underlying
mathematical models for pivot detection provided by the
core library.

6. **Note on Confirmation (Lag):** Pivot signals are confirmed
using a lookback method. A pivot is only plotted *after*
the `Pivot Right Bars` input has passed, which introduces
an inherent lag.

7. **Multi-Timeframe (MTF) Capability:**
- **MTF Kurtosis Lines:** The kurtosis lines can be
calculated on a higher timeframe, with standard options
to handle gaps (`Fill Gaps`) and prevent repainting
(`Wait for...`).
- **Limitation:** The Pivot detection (`Calculate Pivots`) is
**disabled** if a Higher Timeframe (HTF) is selected.

8. **Integrated Alerts:** Includes comprehensive alerts for:
- Kurtosis magnitude (High Positive / High Negative).
- Character changes (Inter-Bar vs. Intra-Bar dominance).
- Total Kurtosis pivot (High/Low) detection.

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**DISCLAIMER**

1. **For Informational/Educational Use Only:** This indicator is
provided for informational and educational purposes only. It does
not constitute financial, investment, or trading advice, nor is
it a recommendation to buy or sell any asset.

2. **Use at Your Own Risk:** All trading decisions you make based on
the information or signals generated by this indicator are made
solely at your own risk.

3. **No Guarantee of Performance:** Past performance is not an
indicator of future results. The author makes no guarantee
regarding the accuracy of the signals or future profitability.

4. **No Liability:** The author shall not be held liable for any
financial losses or damages incurred directly or indirectly from
the use of this indicator.

5. **Signals Are Not Recommendations:** The alerts and visual signals
(e.g., crossovers) generated by this tool are not direct
recommendations to buy or sell. They are technical observations
for your own analysis and consideration.