Lorentzian Length Adaptive Moving Average [LLAMA] Adaptation of "Machine Learning: Lorentzian Classification" by
Gradient color by base on work by
LLAMA: A regime-aware adaptive moving average that bends with the market.
Start with a problem traders know:
Traditional moving averages are either too slow (EMA200) or too fast (EMA9)
Adaptive MAs exist, but they often hug price too tightly or smooth too much, failing to balance bias and tactics
LLAMA uses a Lorentzian distance function to adapt its length dynamically. Instead of a fixed smoothing window, it stretches or contracts depending on market conditions. This distortion reduces lag while still providing a clear bias line.
The indicator looks back at recent bars and measures how similar they are using a Lorentzian distance (a log‑scaled absolute difference). It keeps track of the “nearest neighbors” — bars that most resemble the current regime. Each neighbor carries a label (long, short, neutral) based on simple price comparisons. By averaging these labels, LLAMA predicts whether the market is leaning bullish or bearish. That prediction is then mapped into a dynamic length between and .
Bullish bias -> length stretches toward max (smoother, more stable).
Bearish bias -> length contracts toward min (snappier, more reactive).
During breakouts, LLAMA tightens and comes into contact with bars, giving actionable signals. During chop, it stretches to avoid false triggers. It covers both ends of the spectrum (bias and tactics) in one line, something static MA's can't do.
Think of LLAMA as a lens that bends with the market:
Wide lens (max length) for big picture bias.
Narrow lens (min length) for tactical precision.
The "Lorentzian Loop" is the math that decides when to widen or narrow. Indicator
DynamicMALibrary "DynamicMA"
Dynamic Moving Averages Library
Introduction
The Dynamic Moving Averages Library is a specialized collection of custom built functions designed to calculate moving averages dynamically, beginning from the first available bar. Unlike standard moving averages, which rely on fixed length lookbacks, this library ensures that indicators remain fully functional from the very first data point, making it an essential tool for analysing assets with short time series or limited historical data.
This approach allows traders and developers to build robust indicators that do not require a preset amount of historical data before generating meaningful outputs. It is particularly advantageous for:
Newly listed assets with minimal price history.
High-timeframe trading, where large lookback periods can lead to delayed or missing data.
By eliminating the constraints of fixed lookback periods, this library enables the seamless construction of trend indicators, smoothing functions, and hybrid models that adapt instantly to market conditions.
Comprehensive Set of Custom Moving Averages
The library includes a wide range of custom dynamic moving averages, each designed for specific analytical use cases:
SMA (Simple Moving Average) – The fundamental moving average, dynamically computed.
EMA (Exponential Moving Average) – Adaptive smoothing for better trend tracking.
DEMA (Double Exponential Moving Average) – Faster trend detection with reduced lag.
TEMA (Triple Exponential Moving Average) – Even more responsive than DEMA.
WMA (Weighted Moving Average) – Emphasizes recent price action while reducing noise.
VWMA (Volume Weighted Moving Average) – Accounts for volume to give more weight to high-volume periods.
HMA (Hull Moving Average) – A superior smoothing method with low lag.
SMMA (Smoothed Moving Average) – A hybrid approach between SMA and EMA.
LSMA (Least Squares Moving Average) – Uses linear regression for trend detection.
RMA (Relative Moving Average) – Used in RSI-based calculations for smooth momentum readings.
ALMA (Arnaud Legoux Moving Average) – A Gaussian-weighted MA for superior signal clarity.
Hyperbolic MA (HyperMA) – A mathematically optimized averaging method with dynamic weighting.
Each function dynamically adjusts its calculation length to match the available bar count, ensuring instant functionality on all assets.
Fully Optimized for Pine Script v6
This library is built on Pine Script v6, ensuring compatibility with modern PulseWire indicators and scripts. It includes exportable functions for seamless integration into custom indicators, making it easy to develop trend-following models, volatility filters, and adaptive risk-management systems.
Why Use Dynamic Moving Averages?
Traditional moving averages suffer from a common limitation: they require a fixed historical window to generate meaningful values. This poses several problems:
New Assets Have No Historical Data - If an asset has only been trading for a short period, traditional moving averages may not be able to generate valid signals.
High Timeframes Require Massive Lookbacks - On 1W or 1M charts, a 200-period SMA would require 200 weeks or months of data, making it unusable on newer assets.
Delayed Signal Initialization - Standard indicators often take dozens of bars to stabilize, reducing effectiveness when trading new trends.
The Dynamic Moving Averages Library eliminates these issues by ensuring that every function:
Starts calculation from bar one, using available data instead of waiting for a lookback period.
Adapts dynamically across timeframes, making it equally effective on low or high timeframes.
Allows smoother, more responsive trend tracking, particularly useful for volatile or low-liquidity assets.
This flexibility makes it indispensable for custom script developers, quantitative analysts, and discretionary traders looking to build more adaptive and resilient indicators.
Final Summary
The Dynamic Moving Averages Library is a versatile and powerful set of functions designed to overcome the limitations of fixed-lookback indicators. By dynamically adjusting the calculation length from the first bar, this library ensures that moving averages remain fully functional across all timeframes and asset types, making it an essential tool for traders and developers alike.
With built-in adaptability, low-lag smoothing, and support for multiple moving average types, this library unlocks new possibilities for quantitative trading and strategy development - especially for assets with short price histories or those traded on higher timeframes.
For traders looking to enhance signal reliability, minimize lag, and build adaptable trading systems, the Dynamic Moving Averages Library provides an efficient and flexible solution.
SMA(sourceData, maxLength)
Dynamic SMA
Parameters:
sourceData (float)
maxLength (int)
EMA(src, length)
Dynamic EMA
Parameters:
src (float)
length (int)
DEMA(src, length)
Dynamic DEMA
Parameters:
src (float)
length (int)
TEMA(src, length)
Dynamic TEMA
Parameters:
src (float)
length (int)
WMA(src, length)
Dynamic WMA
Parameters:
src (float)
length (int)
HMA(src, length)
Dynamic HMA
Parameters:
src (float)
length (int)
VWMA(src, volsrc, length)
Dynamic VWMA
Parameters:
src (float)
volsrc (float)
length (int)
SMMA(src, length)
Dynamic SMMA
Parameters:
src (float)
length (int)
LSMA(src, length, offset)
Dynamic LSMA
Parameters:
src (float)
length (int)
offset (int)
RMA(src, length)
Dynamic RMA
Parameters:
src (float)
length (int)
ALMA(src, length, offset_sigma, sigma)
Dynamic ALMA
Parameters:
src (float)
length (int)
offset_sigma (float)
sigma (float)
HyperMA(src, length)
Dynamic HyperbolicMA
Parameters:
src (float)
length (int)
Library
DynamicFunctionsLibrary "DynamicFunctions"
Custom Dynamic functions that allow an adaptive calculation beginning from the first bar
RoC(src, period)
Dynamic RoC
Parameters:
src (float) : and period
Custom function to calculate the actual period considering non-na source values
period (int)
dynamicMedian(src, length)
Dynamic Median
Parameters:
src (float) : and length
length (int)
kernelRegression(src, bandwidth, kernel_type)
Dynamic Kernel Regression Calculation Uses either of the following inputs for kernel_type: Epanechnikov Logistic Wave
Parameters:
src (float)
bandwidth (int)
kernel_type (string)
waveCalculation(source, bandwidth, width)
Use together with kernelRegression function to get chart applicable band
Parameters:
source (float)
bandwidth (int)
width (float)
Rsi(src, length)
Dynamic RSI function
Parameters:
src (float)
length (int)
dynamicStdev(src, period)
Dynamic SD function
Parameters:
src (float)
period (int)
stdv_bands(src, length, mult)
Dynamic SD Bands
Parameters:
src (float)
length (int)
mult (float)
Returns: Basis, Positive SD, Negative SD
Adx(dilen, adxlen)
Dynamic ADX
Parameters:
dilen (int)
adxlen (int)
Returns: adx
Atr(length)
Dynamic ATR
Parameters:
length (int)
Returns: ATR
Macd(source, fastLength, slowLength, signalSmoothing)
Dynamic MACD
Parameters:
source (float)
fastLength (int)
slowLength (int)
signalSmoothing (int)
Returns: macdLine, signalLine, histogram Library
Optimal Length BackTester [YinYangAlgorithms]This Indicator allows for a ‘Optimal Length’ to be inputted within the Settings as a Source. Unlike most Indicators and/or Strategies that rely on either Static Lengths or Internal calculations for the length, this Indicator relies on the Length being derived from an external Indicator in the form of a Source Input.
This may not sound like much, but this application may allows limitless implementations of such an idea. By allowing the input of a Length within a Source Setting you may have an ‘Optimal Length’ that adjusts automatically without the need for manual intervention. This may allow for Traditional and Non-Traditional Indicators and/or Strategies to allow modifications within their settings as well to accommodate the idea of this ‘Optimal Length’ model to create an Indicator and/or Strategy that adjusts its length based on the top performing Length within the current Market Conditions.
This specific Indicator aims to allow backtesting with an ‘Optimal Length’ inputted as a ‘Source’ within the Settings.
This ‘Optimal Length’ may be used to display and potentially optimize multiple different Traditional Indicators within this BackTester. The following Traditional Indicators are included and available to be backtested with an ‘Optimal Length’ inputted as a Source in the Settings:
Moving Average; expressed as either a: Simple Moving Average, Exponential Moving Average or Volume Weighted Moving Average
Bollinger Bands; expressed based on the Moving Average Type
Donchian Channels; expressed based on the Moving Average Type
Envelopes; expressed based on the Moving Average Type
Envelopes Adjusted; expressed based on the Moving Average Type
All of these Traditional Indicators likewise may be displayed with multiple ‘Optimal Lengths’. They have the ability for multiple different ‘Optimal Lengths’ to be inputted and displayed, such as:
Fast Optimal Length
Slow Optimal Length
Neutral Optimal Length
By allowing for the input of multiple different ‘Optimal Lengths’ we may express the ‘Optimal Movement’ of such an expressed Indicator based on different Time Frames and potentially also movement based on Fast, Slow and Neutral (Inclusive) Lengths.
This in general is a simple Indicator that simply allows for the input of multiple different varieties of ‘Optimal Lengths’ to be displayed in different ways using Tradition Indicators. However, the idea and model of accepting a Length as a Source is unique and may be adopted in many different forms and endless ideas.
Tutorial:
You may add an ‘Optimal Length’ within the Settings as a ‘Source’ as followed in the example above. This Indicator allows for the input of a:
Neutral ‘Optimal Length’
Fast ‘Optimal Length’
Slow ‘Optimal Length’
It is important to account for all three as they generally encompass different min/max length values and therefore result in varying ‘Optimal Length’s’.
For instance, say you’re calculating the ‘Optimal Length’ and you use:
Min: 1
Max: 400
This would therefore be scanning for 400 (inclusive) lengths.
As a general way of calculating you may assume the following for which lengths are being used within an ‘Optimal Length’ calculation:
Fast: 1 - 199
Slow: 200 - 400
Neutral: 1 - 400
This allows for the calculation of a Fast and Slow length within the predetermined lengths allotted. However, it likewise allows for a Neutral length which is inclusive to all lengths alloted and may be deemed the ‘Most Accurate’ for these reasons. However, just because the Neutral is inclusive to all lengths, doesn’t mean the Fast and Slow lengths are irrelevant. The Fast and Slow length inputs may be useful for seeing how specifically zoned lengths may fair, and likewise when they cross over and/or under the Neutral ‘Optimal Length’.
This Indicator features the ability to display multiple different types of Traditional Indicators within the ‘Display Type’.
We will go over all of the different ‘Display Types’ with examples on how using a Fast, Slow and Neutral length would impact it:
Simple Moving Average:
In this example above have the Fast, Slow and Neutral Optimal Length formatted as a Slow Moving Average. The first example is on the 15 minute Time Frame and the second is on the 1 Day Time Frame, demonstrating how the length changes based on the Time Frame and the effects it may have.
Here we can see that by inputting ‘Optimal Lengths’ as a Simple Moving Average we may see moving averages that change over time with their ‘Optimal Lengths’. These lengths may help identify Support and/or Resistance locations. By using an 'Optimal Length' rather than a static length, we may create a Moving Average which may be more accurate as it attempts to be adaptive to current Market Conditions.
Bollinger Bands:
Bollinger Bands are a way to see a Simple Moving Average (SMA) that then uses Standard Deviation to identify how much deviation has occurred. This Deviation is then Added and Subtracted from the SMA to create the Bollinger Bands which help Identify possible movement zones that are ‘within range’. This may mean that the price may face Support / Resistance when it reaches the Outer / Inner bounds of the Bollinger Bands. Likewise, it may mean the Price is ‘Overbought’ when outside and above or ‘Underbought’ when outside and below the Bollinger Bands.
By applying All 3 different types of Optimal Lengths towards a Traditional Bollinger Band calculation we may hope to see different ranges of Bollinger Bands and how different lookback lengths may imply possible movement ranges on both a Short Term, Long Term and Neutral perspective. By seeing these possible ranges you may have the ability to identify more levels of Support and Resistance over different lengths and Trading Styles.
Donchian Channels:
Above you’ll see two examples of Machine Learning: Optimal Length applied to Donchian Channels. These are displayed with both the 15 Minute Time Frame and the 1 Day Time Frame.
Donchian Channels are a way of seeing potential Support and Resistance within a given lookback length. They are a way of withholding the High’s and Low’s of a specific lookback length and looking for deviation within this length. By applying a Fast, Slow and Neutral Machine Learning: Optimal Length to these Donchian Channels way may hope to achieve a viable range of High’s and Low’s that one may use to Identify Support and Resistance locations for different ranges of Optimal Lengths and likewise potentially different Trading Strategies.
Envelopes / Envelopes Adjusted:
Envelopes are an interesting one in the sense that they both may be perceived as useful; however we deem that with the use of an ‘Optimal Length’ that the ‘Envelopes Adjusted’ may work best. We will start with examples of the Traditional Envelope then showcase the Adjusted version.
Envelopes:
As you may see, a Traditional form of Envelopes even produced with a Machine Learning: Optimal Length may not produce optimal results. Unfortunately this may occur with some Traditional Indicators and they may need some adjustments as you’ll notice with the ‘Envelopes Adjusted’ version. However, even without the adjustments, these Envelopes may be useful for seeing ‘Overbought’ and ‘Oversold’ locations within a Machine Learning: Optimal Length standpoint.
Envelopes Adjusted:
By adding an adjustment to these Envelopes, we may hope to better reflect our Optimal Length within it. This is caused by adding a ratio reflection towards the current length of the Optimal Length and the max Length used. This allows for the Fast and Neutral (and potentially Slow if Neutral is greater) to achieve a potentially more accurate result.
Envelopes, much like Bollinger Bands are a way of seeing potential movement zones along with potential Support and Resistance. However, unlike Bollinger Bands which are based on Standard Deviation, Envelopes are based on percentages +/- from the Simple Moving Average.
We will conclude our Tutorial here. Hopefully this has given you some insight into how useful adding a ‘Optimal Length’ within an external (secondary) Indicator as a Source within the Settings may be. Likewise, how useful it may be for automation sake in the sense that when the ‘Optimal Length’ changes, it doesn’t rely on an alert where you need to manually update it yourself; instead it will update Automatically and you may reap the benefits of such with little manual input needed (aside from the initial setup).
If you have any questions, comments, ideas or concerns please don't hesitate to contact us.
HAPPY TRADING! Indicator
Indicator
Indicator
True Adaptive-Lookback Phase Change Index [Loxx]Previously I posted a Phase Change Index using Ehlers Autocorrelation Periodogram Algorithm to tease out the adaptive periods. You can find the previous version here: . This new version is also adaptive but uses a different method to derive the adaptive length inputs. This adaptive method derives period inputs by counting pivots from past candles. This version also relies on Jurik Smoothing to generate the final signal. I named this one "true" because I should have specified in the previous PCI's title that it's powered by Ehlers Autocorrelation Periodogram. Additionally, you'll notice the ALB algorithm has changed from other indicators, This is restrict the range of possible ALB period outputs to a specific range so the indicator is usable.
And remember, this is an inverse indicator. This means that small values on the oscillator indicate bullish sentiment and higher values on the oscillator indicate bearish sentiment.
What is the Phase Change Index?
Based on the M.H. Pee's TASC article "Phase Change Index".
Prices at any time can be up, down, or unchanged. A period where market prices remain relatively unchanged is referred to as a consolidation. A period that witnesses relatively higher prices is referred to as an uptrend, while a period of relatively lower prices is called a downtrend.
The Phase Change Index ( PCI ) is an indicator designed specifically to detect changes in market phases.
This indicator is made as he describes it with one deviation: if we follow his formula to the letter then the "trend" is inverted to the actual market trend. Because of that an option to display inverted (and more logical) values is added.
What is the Jurik Moving Average?
Have you noticed how moving averages add some lag (delay) to your signals? ... especially when price gaps up or down in a big move, and you are waiting for your moving average to catch up? Wait no more! JMA eliminates this problem forever and gives you the best of both worlds: low lag and smooth lines.
Ideally, you would like a filtered signal to be both smooth and lag-free. Lag causes delays in your trades, and increasing lag in your indicators typically result in lower profits. In other words, late comers get what's left on the table after the feast has already begun.
That's why investors, banks and institutions worldwide ask for the Jurik Research Moving Average ( JMA ). You may apply it just as you would any other popular moving average. However, JMA's improved timing and smoothness will astound you.
What is adaptive Jurik volatility
One of the lesser known qualities of Juirk smoothing is that the Jurik smoothing process is adaptive. "Jurik Volty" (a sort of market volatility ) is what makes Jurik smoothing adaptive. The Jurik Volty calculation can be used as both a standalone indicator and to smooth other indicators that you wish to make adaptive.
Included:
Bar coloring
2 signal variations w/ alerts
Indicator
Indicator
+ Ultimate MAWhat is the "Ultimate MA" exactly, you ask? Simple. It actually takes as its influence the Rex Dog Moving Average (which I have included as an MA in some of my other indicators), an invention by xkavalis that is simply an average of different length moving averages.
It's available for free on his account, so take a look at it.
I've recently become drawn to using fibonacci sequence numbers as lookbacks for moving averages, and they work really well (I'm honestly beginning to think the number doesn't matter).
You can see where this is going. The Ultimate MA is an average of several (eight) moving averages of varying lengths (5 - 144) all of fibonacci numbers. Sounds pretty basic, right? That's not actually the case, however.
If you were to take all these numbers, add them up, then average them by eight you'd get ~46. Now, stick a 46 period moving average on the chart and compare it to this one and see what you get. They track price very differently. Still, this all sort of sounds like I'm copying the RDMA, which isn't a sin in itself but is hardly grounds for releasing a new MA into the wild.
The actual initial problem I wanted to tackle was how to take in to account for the entire range of price action in a candle in a moving average. ohlc4 sort of does this, but it's still just one line that is an average of all these prices, and I thought there might be a better way not claiming that what I came upon is, but I like it).
My solution was to plot two moving averages: one an average of price highs, and the other an average of lows, thus creating a high/low price channel. Perhaps this is not a new thing at all. I don't know. This is just an idea I had that I figured I could implement easily enough.
Originally I had just applied this to a 21 period EMA, but then the idea sort of expanded into what you see here. I kept thinking "is 21 the best?" What about faster or slower? Then I thought about the RDMA and decided on this implimentation.
Further, I take the high and low moving averages and divide them by two in order to get a basis. You can turn all this stuff on or off, though I do like the default settings.
After that I wanted to add bands to it to measure volatility. There is an RDMA version that utilizes ATR bands, but I could never find myself happy with these.
I just wanted something... else. I also, actually made my own version of xkavalis' RDMA bands with some of the extra stuff I included here, but obviously didn't feel comfortable releasing it as an indicator as I hadn't changed it enough significantly in my mind to fairly do so. I eventually settled on Bollinger Bands as an appropriate solution to apply to the situation. I really like them. It took some fiddling because I had to create a standard deviation for both the high and low MAs instead of just one, and then figure out the best combination of moving averages and standard deviations to add and subtract to get the bands right.
Then I decided I wanted to add a few different moving averages to choose from instead of just an EMA even though I think it's the "best." I didn't want to make things too complicated, so I just went with the standards--EMA, SMA, WMA, HMA-- + 1, the ALMA (which gives some adjustability with its offset and sigma).
Also, you can run more than one moving average at a time (try running an HMA with a slower one).
Oh yeah, the bands? You can set them, in a dropdown box, to be based on which ever moving average you want.
Furthermore, this is a multi-timeframe indicator, so if you want to run it on a higher time frame than the one you are trading on, it's great for that.
ALSO, I actually have the basis color setup as multi-timeframe. What this means is that if you are looking at an hourly chart, you can set the color to a 4h (or higher) chart if you want, and if the current candle is above or below the previous close of the basis on that higher timeframe you will know simply by looking at the color of it ((while still being on the hourly chart). It's just a different way of utilizing higher timeframe information, but without the indicator itself plotted as higher timeframe.
I'm nearly finished. Almost last thing is a 233 period moving average. It's plotted as an average of the SMA, EMA, and Kijun-sen.
Lastly, there are alerts for price crossing the inner border of the bands, or the 233 MA.
Below is a zoomed in look at a chart.
Much credit and gratitude to xkavalis for coming up with the idea of an average of moving averages. Indicator
