Indicator
Tail-Risk & Fragility OscillatorTail-Risk & Fragility Oscillator
A risk-shape meter — not a buy/sell oscillator. It measures the asymmetry (skewness) and tail-fatness (excess kurtosis) of recent returns to answer one question your momentum and trend tools cannot: is fragile, fat-tailed risk building right now, and on which side? It reads this in plain language, optionally from higher-resolution "realized" data, and forward-calibrates whether those fragile states have actually preceded moves on the symbol you're viewing.
Why these parts are combined (not a mashup for show). Skewness says which tail is heavier (where a violent move is more likely to come from). Excess kurtosis says how heavy the tails are (how likely an outsized move is at all). Neither alone is "fragility" — fragility is one tail heavy AND tails fattening together — so both are combined into one reading. Realized estimation (optional) measures these moments from intrabar returns rather than one value per bar, far less noisy. And a Cornish-Fisher value-at-risk turns skew + kurtosis into a single "expected bad-case move" you can size against. Together they form one coherent tail-risk tool.
How it works. Returns feed central moments over a window: skew = m3/σ³, excess kurtosis = m4/m2² − 3. With "Realized" on, the moments come from a lower-timeframe return stream (confirmed only). The signed wave is standardized skewness; a percentile of kurtosis flags fat tails; a Cornish-Fisher quantile combines both, scaled to the chart timeframe over the calibration horizon, into a value-at-risk. A "fragile-down" state (negative-skew extreme + fat tails) is then scored for a subsequent down move and "fragile-up" for an up move, using triple-barrier outcomes, split into in-sample and recent out-of-sample, with a confidence interval and a multiple-testing check.
How to use. Read the Verdict row — "Downside-fragile" / "Upside-fragile" means a heavy tail is building on that side; "Tails normal" means no fragility. The Conviction row says whether that state has actually preceded the expected move on this symbol. Use it to size down, widen stops, or expect violence on the fragile side — layered on your own directional tool, never as a standalone entry. The "Bad-case move" row is the modelled adverse move over the next N bars at your chosen confidence.
What's original. Realized higher-moment estimation, a horizon-scaled Cornish-Fisher VaR readout, the forward triple-barrier calibration with an out-of-sample split, and a conviction read that openly admits when there's no proven edge — on a risk axis (fragility) that most indicators ignore entirely.
Inputs. Price source (change it for any market), reading mode (Simple/Pro), moment and realized-estimation controls, risk thresholds and VaR confidence, full calibration settings, and an auto-adapting dashboard legible on dark or light charts. Defaults are tuned for NSE:NIFTY1! intraday.
Honesty & limitations. Moments from short windows are noisy and the VaR is a model estimate, not a guarantee. Edge figures are computed on this chart's own history with overlapping windows and no costs — context, not a backtest; past behaviour doesn't predict the future.
Disclaimer: for research and education only. Not financial advice. Trading carries risk of loss; manage your own size. Indicator
Volume Weighted Intra Bar LR KurtosisThis indicator analyzes market character by decomposing total
Excess Kurtosis ("Fat Tails") of a SINGLE BAR into four distinct,
interpretable components based on a Linear Regression model.
Key Features:
1. **Intra-Bar LR Kurtosis Decomposition:** For each bar on the chart,
the indicator analyzes the underlying price action on a smaller
timeframe ('Intra-Bar Timeframe'). It fits a Linear Regression
line through the intra-bar data to decompose the 4th Moment:
- **Trend Kurtosis (Gold):** Peakedness of the regression line
itself. High values indicate the price path within the bar
moves in sudden jumps, steps, or gaps (discontinuous path).
- **Residual Kurtosis (Red):** Excess Kurtosis of the noise
around the regression line. Captures "Hidden Tail Risk" or
extreme outliers within the bar relative to the trend.
- **Within-Bar Kurtosis (Blue):** Fat tails derived from the
microstructure of individual intra-bar candles.
- **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 statistical moments are additive,
this indicator calculates the *exact* Total Kurtosis and partitions
the columns based on the Law of Total Moments.
3. **Dual Display Modes:** The indicator offers two modes to
visualize this decomposition:
- **Absolute Mode:** Plots the *total* kurtosis as a
stacked column chart. Stacking logic groups components to
ensure visual clarity of the magnitude.
- **Relative Mode:** Plots the direct *contribution ratio*
(proportion) of each component relative to the total sum,
ideal for identifying the dominant driver (Trend vs. Noise).
4. **Calculation Options:**
- **Normalization:** An optional 'Normalize' setting
transforms inputs into logarithmic space, analyzing the
kurtosis of *returns* rather than absolute prices.
- **Volume Weighting:** An option (`Volume weighted`) applies
volume weighting to all regression and 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 Analysis Lines:** The entire intra-bar analysis can be
run on a higher timeframe (using the `Timeframe` input),
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 (Trend Jumps vs. Noise Outliers).
- Total Kurtosis pivot (High/Low) detection.
**Caution: Real-Time Data Behavior (Intra-Bar Repainting)**
This indicator uses high-resolution intra-bar data. As a result, the
values on the **current, unclosed bar** (the real-time bar) will
update dynamically as new intra-bar data arrives. This behavior is
normal and necessary for this type of analysis. Signals should only
be considered final **after the main chart bar has closed.**
---
**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. Indicator
Volume Weighted Intra Bar KurtosisThis indicator analyzes market sentiment by providing a detailed
view of Excess Kurtosis ("Fat Tails"). It uses data from a lower,
intra-bar timeframe to separate the total kurtosis of a single bar
into distinct, interpretable components.
Key Features:
1. **Intra-Bar Kurtosis Decomposition:** For each bar on the chart,
the indicator analyzes the underlying price action on a smaller
timeframe ('Intra-Bar Timeframe'). Unlike Variance, the Fourth
Central Moment (Kurtosis) decomposes into three parts:
- **Between-Bar Kurtosis (Gold):** Peakedness of the price
path *between* the intra-bar candles. High values indicate
that the macro movement happened in jumps or gaps rather
than a smooth progression.
- **Within-Bar Kurtosis (Blue):** Fat tails derived from the
microstructure (extreme wicks) *inside* the intra-bar candles.
- **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 columns based on the additive
Fourth Moment Decomposition (`M4Tot = M4Btw + M4Wtn + M4Int`).
3. **Dual Display Modes:** The indicator offers two modes to
visualize this information:
- **Absolute Mode:** Plots the *total* kurtosis as a
stacked column chart, showing the *absolute magnitude* of
tail risk and the contribution of each component.
- **Relative Mode:** Plots the components as a 100% stacked
column chart (scaled from 0 to 1), focusing purely on the
*energy ratio* of the components.
4. **Calculation Options:**
- **Normalization:** An optional 'Normalize' setting
calculates an **Exponential Regression Curve** (log-space),
making the analysis suitable for comparing assets with
different scales (e.g., BTC vs EURUSD).
- **Volume Weighting:** An option (`Volume weighted`) applies
volume weighting to all mean and moment calculations.
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 Analysis Lines:** The entire intra-bar analysis can be
run on a higher timeframe (using the `Timeframe` input),
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 alerts for:
- Kurtosis magnitude (High Positive / High Negative).
- Kurtosis character changes/emerging/fading.
- Total Kurtosis pivot (High/Low) detection.
**Caution: Real-Time Data Behavior (Intra-Bar Repainting)**
This indicator uses high-resolution intra-bar data. As a result, the
values on the **current, unclosed bar** (the real-time bar) will
update dynamically as new intra-bar data arrives. This behavior is
normal and necessary for this type of analysis. Signals should only
be considered final **after the main chart bar has closed.**
---
**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. Indicator
Volume Weighted LR KurtosisThis indicator analyzes market character by decomposing total
Excess Kurtosis ("Fat Tails") into four distinct, interpretable
components based on a Linear Regression model.
Key Features:
1. **Four-Component Kurtosis Decomposition:** The indicator
separates market tail risk based on the 'Estimate Bar Statistics' option.
It leverages the Law of Total Moments to provide an additive
breakdown of the 4th Statistical Moment:
- **Trend Kurtosis (Gold):** Peakedness of the regression line
itself. High values indicate the trend moves in sudden jumps,
steps, or gaps (discontinuous path).
- **Residual Kurtosis (Red):** Excess Kurtosis of the noise
around the regression line. This captures the "Hidden Tail Risk"
(extreme outliers relative to the trend).
- **Within-Bar Kurtosis (Blue):** Fat tails derived from the
microstructure of individual bars (requires 'Estimate Bar Statistics').
- **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 statistical moments are additive,
this indicator calculates the *exact* Total Kurtosis and partitions
the area to visualize the contribution (weight) of each
structural source to the overall tail risk.
3. **Dual Display Modes:** The indicator offers two modes to
visualize this decomposition:
- **Absolute Mode:** Displays the *total* kurtosis as a
stacked area chart, allowing to see the magnitude of tail risk.
Stacking logic groups components to ensure visual clarity.
- **Relative Mode:** Displays the direct *contribution ratio*
(proportion) of each component relative to the total sum,
ideal for identifying the dominant driver of the risk.
4. **Calculation Options:**
- **Normalization:** An optional 'Normalize' setting
transforms inputs into logarithmic space, analyzing the
kurtosis of *returns* rather than absolute prices.
- **Volume Weighting:** An option (`Volume weighted`) applies
volume weighting to all regression and 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. This helps identify extremes in
market fragility or structural changes.
- **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).
- Kurtosis character changes/emerging/fading.
- Total Kurtosis pivot (High/Low) detection.
---
**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. Indicator
Volume Weighted KurtosisThis 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.
---
**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. Indicator
N-Degree Moment-Based Adaptive Detection🙏🏻 N-Degree Moment-Based Adaptive Detection (NDMBAD) method is a generalization of MBAD since the horizontal line fit passing through the data's mean can be simply treated as zero-degree polynomial regression. We can extend the MBAD logic to higher-degree polynomial regression.
I don't think I need to talk a lot about the thing there; the logic is really the same as in MBAD, just hit the link above and read if you want. The only difference is now we can gather cumulants not only from the horizontal mean fit (degree = 0) but also from higher-order polynomial regression fit, including linear regression (degree = 1).
Why?
Simply because residuals from the 0-degree model don't contain trend information, and while in some cases that's exactly what you need, in other cases, you want to model your trend explicitly. Imagine your underlying process trends in a steady manner, and you want to control the extreme deviations from the process's core. If you're going to use 0-degree, you'll be treating this beautiful steady trend as a residual itself, which "constantly deviates from the process mean." It doesn't make much sense.
How?
First, if you set the length to 0, you will end up with the function incrementally applied to all your data starting from bar_index 0. This can be called the expanding window mode. That's the functionality I include in all my scripts lately (where it makes sense). As I said in the MBAD description, choosing length is a matter of doing business & applied use of my work, but I think I'm open to talk about it.
I don't see much sense in using degree > 1 though (still in research on it). If you have dem curves, you can use Fourier transform -> spectral filtering / harmonic regression (regression with Fourier terms). The job of a degree > 0 is to model the direction in data, and degree 1 gets it done. In mean reversion strategies, it means that you don't wanna put 0-degree polynomial regression (i.e., the mean) on non-stationary trending data in moving window mode because, this way, your residuals will be contaminated with the trend component.
By the way, you can send thanks to @aaron294c , he said like mane MBAD is dope, and it's gonna really complement his work, so I decided to drop NDMBAD now, gonna be more useful since it covers more types of data.
I wanned to call it N-Order Moment Adaptive Detection because it abbreviates to NOMAD, which sounds cool and suits me well, because when I perform as a fire dancer, nomad style is one of my outfits. Burning Man stuff vibe, you know. But the problem is degree and order really mean two different things in the polynomial context, so gotta stay right & precise—that's the priority.
∞ Indicator
Moment-Based Adaptive DetectionMBAD (Moment-Based Adaptive Detection) : a method applicable to a wide range of purposes, like outlier or novelty detection, that requires building a sensible interval/set of thresholds. Unlike other methods that are static and rely on optimizations that inevitably lead to underfitting/overfitting, it dynamically adapts to your data distribution without any optimizations, MLE, or stuff, and provides a set of data-driven adaptive thresholds, based on closed-form solution with O(n) algo complexity.
1.5 years ago, when I was still living in Versailles at my friend's house not knowing what was gonna happen in my life tomorrow, I made a damn right decision not to give up on one idea and to actually R&D it and see what’s up. It allowed me to create this one.
The Method Explained
I’ve been wandering about z-values, why exactly 6 sigmas, why 95%? Who decided that? Why would you supersede your opinion on data? Based on what? Your ego?
Then I consciously noticed a couple of things:
1) In control theory & anomaly detection, the popular threshold is 3 sigmas (yet nobody can firmly say why xD). If your data is Laplace, 3 sigmas is not enough; you’re gonna catch too many values, so it needs a higher sigma.
2) Yet strangely, the normal distribution has kurtosis of 3, and 6 for Laplace.
3) Kurtosis is a standardized moment, a moment scaled by stdev, so it means "X amount of something measured in stdevs."
4) You generate synthetic data, you check on real data (market data in my case, I am a quant after all), and you see on both that:
lower extension = mean - standard deviation * kurtosis ≈ data minimum
upper extension = mean + standard deviation * kurtosis ≈ data maximum
Why not simply use max/min?
- Lower info gain: We're not using all info available in all data points to estimate max/min; we just pick the current higher and lower values. Lol, it’s the same as dropping exponential smoothing with alpha = 0 on stationary data & calling it a day.
You can’t update the estimates of min and max when new data arrives containing info about the matter. All you can do is just extend min and max horizontally, so you're not using new info arriving inside new data.
- Mixing order and non-order statistics is a bad idea; we're losing integrity and coherence. That's why I don't like the Hurst exponent btw (and yes, I came up with better metrics of my own).
- Max & min are not even true order statistics, unlike a percentile (finding which requires sorting, which requires multiple passes over your data). To find min or max, you just need to do one traversal over your data. Then with or without any weighting, 100th percentile will equal max. So unlike a weighted percentile, you can’t do weighted max. Then while you can always check max and min of a geometric shape, now try to calculate the 56th percentile of a pentagram hehe.
TL;DR max & min are rather topological characteristics of data, just as the difference between starting and ending points. Not much to do with statistics.
Now the second part of the ballet is to work with data asymmetry:
1) Skewness is also scaled by stdev -> so it must represent a shift from the data midrange measured in stdevs -> given asymmetric data, we can include this info in our models. Unlike kurtosis, skewness has a sign, so we add it to both thresholds:
lower extension = mean - standard deviation * kurtosis + standard deviation * skewness
upper extension = mean + standard deviation * kurtosis + standard deviation * skewness
2) Now our method will work with skewed data as well, omg, ain’t it cool?
3) Hold up, but what about 5th and 6th moments (hyperskewness & hyperkurtosis)? They should represent something meaningful as well.
4) Perhaps if extensions represent current estimated extremums, what goes beyond? Limits, beyond which we expect data not to be able to pass given the current underlying process generating the data?
When you extend this logic to higher-order moments, i.e., hyperskewness & hyperkurtosis (5th and 6th moments), they measure asymmetry and shape of distribution tails, not its core as previous moments -> makes no sense to mix 4th and 3rd moments (skewness and kurtosis) with 5th & 6th, so we get:
lower limit = mean - standard deviation * hyperkurtosis + standard deviation * hyperskewness
upper limit = mean + standard deviation * hyperkurtosis + standard deviation * hyperskewness
While extensions model your data’s natural extremums based on current info residing in the data without relying on order statistics, limits model your data's maximum possible and minimum possible values based on current info residing in your data. If a new data point trespasses limits, it means that a significant change in the data-generating process has happened, for sure, not probably—a confirmed structural break.
And finally we use time and volume weighting to include order & process intensity information in our model.
I can't stress it enough: despite the popularity of these non-weighted methods applied in mainstream open-access time series modeling, it doesn’t make ANY sense to use non-weighted calculations on time series data . Time = sequence, it matters. If you reverse your time series horizontally, your means, percentiles, whatever, will stay the same. Basically, your calculations will give the same results on different data. When you do it, you disregard the order of data that does have order naturally. Does it make any sense to you? It also concerns regressions applied on time series as well, because even despite the slope being opposite on your reversed data, the centroid (through which your regression line always comes through) will be the same. It also might concern Fourier (yes, you can do weighted Fourier) and even MA and AR models—might, because I ain’t researched it extensively yet.
I still can’t believe it’s nowhere online in open access. No chance I’m the first one who got it. It’s literally in front of everyone’s eyes for centuries—why no one tells about it?
How to use
That’s easy: can be applied to any, even non-stationary and/or heteroscedastic time series to automatically detect novelties, outliers, anomalies, structural breaks, etc. In terms of quant trading, you can try using extensions for mean reversion trades and limits for emergency exits, for example. The market-making application is kinda obvious as well.
The only parameter the model has is length, and it should NOT be optimized but picked consciously based on the process/system you’re applying it to and based on the task. However, this part is not about sharing info & an open-access instrument with the world. This is about using dem instruments to do actual business, and we can’t talk about it.
∞ Indicator
GaussianDistributionLibrary "GaussianDistribution"
This library defines a custom type `distr` representing a Gaussian (or other statistical) distribution.
It provides methods to calculate key statistical moments and scores, including mean, median, mode, standard deviation, variance, skewness, kurtosis, and Z-scores.
This library is useful for analyzing probability distributions in financial data.
Disclaimer:
I am not a mathematician, but I have implemented this library to the best of my understanding and capacity. Please be indulgent as I tried to translate statistical concepts into code as accurately as possible. Feedback, suggestions, and corrections are welcome to improve the reliability and robustness of this library.
mean(source, length)
Calculate the mean (average) of the distribution
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
Returns: Mean (μ)
stdev(source, length)
Calculate the standard deviation (σ) of the distribution
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
Returns: Standard deviation (σ)
skewness(source, length, mean, stdev)
Calculate the skewness (γ₁) of the distribution
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
@return Skewness (γ₁)
skewness(source, length)
Overloaded skewness to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Skewness (γ₁)
mode(mean, stdev, skewness)
Estimate mode - Most frequent value in the distribution (approximation based on skewness)
Parameters:
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
skewness (float) : the skewness (γ₁) of the distribution
@return Mode
mode(source, length)
Overloaded mode to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Mode
median(mean, stdev, skewness)
Estimate median - Middle value of the distribution (approximation)
Parameters:
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
skewness (float) : the skewness (γ₁) of the distribution
@return Median
median(source, length)
Overloaded median to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Median
variance(stdev)
Calculate variance (σ²) - Square of the standard deviation
Parameters:
stdev (float) : the standard deviation (σ) of the distribution
@return Variance (σ²)
variance(source, length)
Overloaded variance to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Variance (σ²)
kurtosis(source, length, mean, stdev)
Calculate kurtosis (γ₂) - Degree of "tailedness" in the distribution
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
@return Kurtosis (γ₂)
kurtosis(source, length)
Overloaded kurtosis to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Kurtosis (γ₂)
normal_score(source, mean, stdev)
Calculate Z-score (standard score) assuming a normal distribution
Parameters:
source (float) : Distribution source (typically a price or indicator series)
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
@return Z-Score
normal_score(source, length)
Overloaded normal_score to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Z-Score
non_normal_score(source, mean, stdev, skewness, kurtosis)
Calculate adjusted Z-score considering skewness and kurtosis
Parameters:
source (float) : Distribution source (typically a price or indicator series)
mean (float) : the mean (average) of the distribution
stdev (float) : the standard deviation (σ) of the distribution
skewness (float) : the skewness (γ₁) of the distribution
kurtosis (float) : the "tailedness" in the distribution
@return Z-Score
non_normal_score(source, length)
Overloaded non_normal_score to calculate from source and length
Parameters:
source (float) : Distribution source (typically a price or indicator series)
length (int) : Window length for the distribution (must be >= 30 for meaningful statistics)
@return Z-Score
method init(this)
Initialize all statistical fields of the `distr` type
Namespace types: distr
Parameters:
this (distr)
method init(this, source, length)
Overloaded initializer to set source and length
Namespace types: distr
Parameters:
this (distr)
source (float)
length (int)
distr
Custom type to represent a Gaussian distribution
Fields:
source (series float) : Distribution source (typically a price or indicator series)
length (series int) : Window length for the distribution (must be >= 30 for meaningful statistics)
mode (series float) : Most frequent value in the distribution
median (series float) : Middle value separating the greater and lesser halves of the distribution
mean (series float) : μ (1st central moment) - Average of the distribution
stdev (series float) : σ or standard deviation (square root of the variance) - Measure of dispersion
variance (series float) : σ² (2nd central moment) - Squared standard deviation
skewness (series float) : γ₁ (3rd central moment) - Asymmetry of the distribution
kurtosis (series float) : γ₂ (4th central moment) - Degree of "tailedness" relative to a normal distribution
normal_score (series float) : Z-score assuming normal distribution
non_normal_score (series float) : Adjusted Z-score considering skewness and kurtosis Library
NormalDistributionFunctionsLibrary "NormalDistributionFunctions"
The NormalDistributionFunctions library encompasses a comprehensive suite of statistical tools for financial market analysis. It provides functions to calculate essential statistical measures such as mean, standard deviation, skewness, and kurtosis, alongside advanced functionalities for computing the probability density function (PDF), cumulative distribution function (CDF), Z-score, and confidence intervals. This library is designed to assist in the assessment of market volatility, distribution characteristics of asset returns, and risk management calculations, making it an invaluable resource for traders and financial analysts.
meanAndStdDev(source, length)
Calculates and returns the mean and standard deviation for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
length (int) : int: The lookback period for the calculation.
Returns: Returns an array where the first element is the mean and the second element is the standard deviation of the data series for the given period.
skewness(source, mean, stdDev, length)
Calculates and returns skewness for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
length (int) : int: The lookback period for the calculation.
Returns: Returns skewness value
kurtosis(source, mean, stdDev, length)
Calculates and returns kurtosis for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
length (int) : int: The lookback period for the calculation.
Returns: Returns kurtosis value
pdf(x, mean, stdDev)
pdf: Calculates the probability density function for a given value within a normal distribution.
Parameters:
x (float) : float: The value to evaluate the PDF at.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
Returns: Returns the probability density function value for x.
cdf(x, mean, stdDev)
cdf: Calculates the cumulative distribution function for a given value within a normal distribution.
Parameters:
x (float) : float: The value to evaluate the CDF at.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
Returns: Returns the cumulative distribution function value for x.
confidenceInterval(mean, stdDev, size, confidenceLevel)
Calculates the confidence interval for a data series mean.
Parameters:
mean (float) : float: The mean of the data series.
stdDev (float) : float: The standard deviation of the data series.
size (int) : int: The sample size.
confidenceLevel (float) : float: The confidence level (e.g., 0.95 for 95% confidence).
Returns: Returns the lower and upper bounds of the confidence interval. Library
Library
ArrayStatisticsLibrary "ArrayStatistics"
Statistic Functions using arrays.
rms(sample) Root Mean Squared
Parameters:
sample : float array, data sample points.
Returns: float
skewness_pearson1(sample) Pearson's 1st Coefficient of Skewness.
Parameters:
sample : float array, data sample.
Returns: float
skewness_pearson2(sample) Pearson's 2nd Coefficient of Skewness.
Parameters:
sample : float array, data sample.
Returns: float
pearsonr(sample_a, sample_b) Pearson correlation coefficient measures the linear relationship between two datasets.
Parameters:
sample_a : float array, sample with data.
sample_b : float array, sample with data.
Returns: float p
kurtosis(sample) Kurtosis of distribution.
Parameters:
sample : float array, data sample.
Returns: float
range_int(sample, percent) Get range around median containing specified percentage of values.
Parameters:
sample : int array, Histogram array.
percent : float, Values percentage around median.
Returns: tuple with , Returns the range which containes specifies percentage of values. Library
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