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Kurtosis 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. ∞
Pine Script® indicator
by gorx1
2
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. ∞
Pine Script® indicator
by gorx1
3
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
Pine Script® library
by Horrorin
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.
Pine Script® library
by tkarolak
4
MomentsLibrary "Moments" Based on Moments (Mean,Variance,Skewness,Kurtosis) . Rewritten for Pinescript v5. logReturns(src) Calculates log returns of a series (e.g log percentage change)   Parameters:      src : Source to use for the returns calculation (e.g. close).   Returns: Log percentage returns of a series mean(src, length) Calculates the mean of a series using ta.sma   Parameters:      src : Source to use for the mean calculation (e.g. close).      length : Length to use mean calculation (e.g. 14).   Returns: The sma of the source over the length provided. variance(src, length) Calculates the variance of a series   Parameters:      src : Source to use for the variance calculation (e.g. close).      length : Length to use for the variance calculation (e.g. 14).   Returns: The variance of the source over the length provided. standardDeviation(src, length) Calculates the standard deviation of a series   Parameters:      src : Source to use for the standard deviation calculation (e.g. close).      length : Length to use for the standard deviation calculation (e.g. 14).   Returns: The standard deviation of the source over the length provided. skewness(src, length) Calculates the skewness of a series   Parameters:      src : Source to use for the skewness calculation (e.g. close).      length : Length to use for the skewness calculation (e.g. 14).   Returns: The skewness of the source over the length provided. kurtosis(src, length) Calculates the kurtosis of a series   Parameters:      src : Source to use for the kurtosis calculation (e.g. close).      length : Length to use for the kurtosis calculation (e.g. 14).   Returns: The kurtosis of the source over the length provided. skewnessStandardError(sampleSize) Estimates the standard error of skewness based on sample size   Parameters:      sampleSize : The number of samples used for calculating standard error.   Returns: The standard error estimate for skewness based on the sample size provided. kurtosisStandardError(sampleSize) Estimates the standard error of kurtosis based on sample size   Parameters:      sampleSize : The number of samples used for calculating standard error.   Returns: The standard error estimate for kurtosis based on the sample size provided. skewnessCriticalValue(sampleSize) Estimates the critical value of skewness based on sample size   Parameters:      sampleSize : The number of samples used for calculating critical value.   Returns: The critical value estimate for skewness based on the sample size provided. kurtosisCriticalValue(sampleSize) Estimates the critical value of kurtosis based on sample size   Parameters:      sampleSize : The number of samples used for calculating critical value.   Returns: The critical value estimate for kurtosis based on the sample size provided.
Pine Script® library
by nolantait
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.
Pine Script® library
by RicardoSantos
1
Combo Backtest 123 Reversal & FSK (Fast and Slow Kurtosis) This is combo strategies for get a cumulative signal. First strategy This System was created from the Book "How I Tripled My Money In The Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies. The strategy buys at market, if close price is higher than the previous close during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50. The strategy sells at market, if close price is lower than the previous close price during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50. Second strategy This indicator plots the Fast & Slow Kurtosis. The Kurtosis is a market sentiment indicator. The Kurtosis is constructed from three different parts. The Kurtosis, the Fast Kurtosis(FK), and the Fast/Slow Kurtosis(FSK). WARNING: - For purpose educate only - This script to change bars colors.
Pine Script® strategy
by HPotter
1
Combo Strategy 123 Reversal & FSK (Fast and Slow Kurtosis) This is combo strategies for get a cumulative signal. First strategy This System was created from the Book "How I Tripled My Money In The Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies. The strategy buys at market, if close price is higher than the previous close during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50. The strategy sells at market, if close price is lower than the previous close price during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50. Second strategy This indicator plots the Fast & Slow Kurtosis. The Kurtosis is a market sentiment indicator. The Kurtosis is constructed from three different parts. The Kurtosis, the Fast Kurtosis(FK), and the Fast/Slow Kurtosis(FSK). WARNING: - For purpose educate only - This script to change bars colors.
Pine Script® indicator
by HPotter
1