Zero Lag Trend Signals (MTF) [AlgoAlpha]Zero Lag Trend Signals 🚀📈
Ready to take your trend-following strategy to the next level? Say hello to Zero Lag Trend Signals , a precision-engineered Pine Script™ indicator designed to eliminate lag and provide rapid trend insights across multiple timeframes. 💡 This tool blends zero-lag EMA (ZLEMA) logic with volatility bands, trend-shift markers, and dynamic alerts. The result? Timely signals with minimal noise for clearer decision-making, whether you're trading intraday or on longer horizons. 🔄
🟢 Zero-Lag Trend Detection : Uses a zero-lag EMA (ZLEMA) to smooth price data while minimizing delay.
⚡ Multi-Timeframe Signals : Displays trends across up to 5 timeframes (from 5 minutes to daily) on a sleek table.
📊 Volatility-Based Bands : Adaptive upper and lower bands, helping you identify trend reversals with reduced false signals.
🔔 Custom Alerts : Get notified of key trend changes instantly with built-in alert conditions.
🎨 Color-Coded Visualization : Bullish and bearish signals pop with clear color coding, ensuring easy chart reading.
⚙️ Fully Configurable : Modify EMA length, band multiplier, colors, and timeframe settings to suit your strategy.
How to Use 📚
⭐ Add the Indicator : Add the indicator to favorites by pressing the star icon. Set your preferred EMA length and band multiplier. Choose your desired timeframes for multi-frame trend monitoring.
💻 Watch the Table & Chart : The top-right table dynamically updates with bullish or bearish signals across multiple timeframes. Colored arrows on the chart indicate potential entry points when the price crosses the ZLEMA with confirmation from volatility bands.
🔔 Enable Alerts : Configure alerts for real-time notifications when trends shift—no need to monitor charts constantly.
How It Works 🧠
The script calculates the zero-lag EMA (ZLEMA) by compensating for data lag, giving traders more responsive moving averages. It checks for volatility shifts using the Average True Range (ATR), multiplied to create upper and lower deviation bands. If the price crosses above or below these bands, it marks the start of new trends. Additionally, the indicator aggregates trend data from up to five configurable timeframes and displays them in a neat summary table. This helps you confirm trends across different intervals—ideal for multi-timeframe analysis. The visual signals include upward and downward arrows on the chart, denoting potential entries or exits when trends align across timeframes. Traders can use these cues to make well-timed trades and avoid lag-related pitfalls.

# Zerolag

Nasan Moving AverageNasan Moving Average belong to the group of moving average which provides a high degree of smoothness with very low lag.
The calculation process involves several steps to analyze the typical price of a financial asset over specific periods. It starts by computing a simple moving average and standard deviation of the typical price. Then, it standardizes (differencing TP - Average Typical price over previous n periods) the price and applies an inverse hyperbolic sine transformation to the standardized value. The transformed values are summed cumulatively, and various weighted moving averages are calculated to adjust and smooth the data. The final output is a smoothed signal with reduced lag.
Input Parameters:
len: Differencing length (default 21, Use a minimum of 5 and for lower time frames less than 15 min use values between 300 -3000)
len1: Correction Factor Length 1 (default 21, this determines the length of the MA you want , eg. 10 MA, 50 MA, 100 MA, )
len2: Correction Factor Length 2 (default 9, this works best if it is ~ </=1/2 of len1 )
len3: Smoothing Length (default 5, I would not change this and only use if I want to introduce lag where you want to use it for cross over strategies).
Differencing and Standardization:
The code calculates the standardized price a by differencing the typical price and normalizing it using the mean and standard deviation. This step standardizes the price changes.
Transformation:
The transformation using logarithms and square roots (b) aim to stabilize the variance and make the distribution more normal-like, improving the robustness of the cumulative sum c.
Cumulative Sum:
The cumulative sum c of the transformed series helps in integrating the series over time, capturing the overall trend and movement.
Correction Factors:
Correction factors c1 and c4 adjust the cumulative sum based on weighted averages, to correct any biases or to align it with the typical price.
Smoothing:
The final result c6 is smoothed using a weighted moving average, reducing noise and making it easier to interpret trends.

Nonlinear Regression, Zero-lag Moving Average [Loxx]Nonlinear Regression and Zero-lag Moving Average
Technical indicators are widely used in financial markets to analyze price data and make informed trading decisions. This indicator presents an implementation of two popular indicators: Nonlinear Regression and Zero-lag Moving Average (ZLMA). Let's explore the functioning of these indicators and discuss their significance in technical analysis.
Nonlinear Regression
The Nonlinear Regression indicator aims to fit a nonlinear curve to a given set of data points. It calculates the best-fit curve by minimizing the sum of squared errors between the actual data points and the predicted values on the curve. The curve is determined by solving a system of equations derived from the data points.
We define a function "nonLinearRegression" that takes two parameters: "src" (the input data series) and "per" (the period over which the regression is calculated). It calculates the coefficients of the nonlinear curve using the least squares method and returns the predicted value for the current period. The nonlinear regression curve provides insights into the overall trend and potential reversals in the price data.
Zero-lag Moving Average (ZLMA)
Moving averages are widely used to smoothen price data and identify trend directions. However, traditional moving averages introduce a lag due to the inclusion of past data. The Zero-lag Moving Average (ZLMA) overcomes this lag by dynamically adjusting the weights of past values, resulting in a more responsive moving average.
We create a function named "zlma" that calculates the ZLMA. It takes two parameters: "src" (the input data series) and "per" (the period over which the ZLMA is calculated). The ZLMA is computed by first calculating a weighted moving average (LWMA) using a linearly decreasing weight scheme. The LWMA is then used to calculate the ZLMA by applying the same weight scheme again. The ZLMA provides a smoother representation of the price data while reducing lag.
Combining Nonlinear Regression and ZLMA
The ZLMA is applied to the input data series using the function "zlma(src, zlmaper)". The ZLMA values are then passed as input to the "nonLinearRegression" function, along with the specified period for nonlinear regression. The output of the nonlinear regression is stored in the variable "out".
To enhance the visual representation of the indicator, colors are assigned based on the relationship between the nonlinear regression value and a signal value (sig) calculated from the previous period's nonlinear regression value. If the current "out" value is greater than the previous "sig" value, the color is set to green; otherwise, it is set to red.
The indicator also includes optional features such as coloring the bars based on the indicator's values and displaying signals for potential long and short positions. The signals are generated based on the crossover and crossunder of the "out" and "sig" values.
Wrapping Up
This indicator combines two important concepts: Nonlinear Regression and Zero-lag Moving Average indicators, which are valuable tools for technical analysis in financial markets. These indicators help traders identify trends, potential reversals, and generate trading signals. By combining the nonlinear regression curve with the zero-lag moving average, this indicator provides a comprehensive view of the price dynamics. Traders can customize the indicator's settings and use it in conjunction with other analysis techniques to make well-informed trading decisions.

RedK DIY ZLMA: Customizable Zero-Lag MA (Educational / Utility)This script is more of an educational / utility piece rather than a fully-fledged indicator - It provides an easy way to customize and produce a zero-lag Moving average that can then be used in various scenarios
What is DIY_ZLMA?
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The DIY ZLMA is for fans and enthusiasts of researching Moving Averages (like me) - the script enables the user to play around with one of the common approaches used to reduce lag in moving averages - which was explained in this old post below
Suggested uses of the DIY_ZLMA
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* The Zero-lag approach here applies 3 moving average passes to a source data series - I'll refer to these 3 passes as Base MA Pass , De-lagging Pass, and Smoothing Pass - these "passes" can be customized from the indicator settings in terms of MA Length and type. The first pass allows the choice of a "source", and the second pass allows additional fine tuning by playing around with the magnification factor. The 3rd pass (smoothing) is optional and can be skipped altogether when needed. (as noted in the script, HMA and TEMA, which are very common low-lag MA's use slightly different approach in the calculation than the one used here .. so we can't get an equivalent of either of these MA's with the customization of DIY_ZLMA parameters)
* After the user experiments with the various settings for the 3 passes, and finds a "preferred combination", the script not only plots the resulting My_ZLMA - it also produces the "1-line Pine script formula" that the user can then use in any other script, maybe to smoothen some data series, or to combine with other types of moving averages to create multi-MA cross-over trading signals... and so on.
* The DIY_ZLMA can also be added to another indicator as a signal line using the Indicator-on-Indicator feature of TradingView (review this post for step-by-step -->
)
* the script also showcases couple of recent (and very neat) Pine features: the use of User-defined Types (UDT) and User-defined Methods - which are awesome and a lot of fun to work with :)
Since this is more of a utility piece, I added as many comments as possible to the script to explain the way it works - so it's more valuable if someone finds it by searching the "Add Indicator" feature in TradingView charts
Please feel free to play around with this new toy :) and share comments and feedback below if you find this useful. I truly hope you do.

Zero-lag, 3-Pole Super Smoother [Loxx]Zero-lag, 3-Pole Super Smoother is an Ehlers 3-pole smoother with lag reduction
What is 3-pole Super Smoother?
A SuperSmoother filter is used anytime a moving average of any type would otherwise be used, with the result that the SuperSmoother filter output would have substantially less lag for an equivalent amount of smoothing produced by the moving average. For example, a five-bar SMA has a cutoff period of approximately 10 bars and has two bars of lag. A SuperSmoother filter with a cutoff period of 10 bars has a lag a half bar larger than the two-pole modified Butterworth filter. Therefore, such a SuperSmoother filter has a maximum lag of approximately 1.5 bars and even less lag into the attenuation band of the filter. The differential in lag between moving average and SuperSmoother filter outputs becomes even larger when the cutoff periods are larger.
Included:
-Color bars
-Loxx's Expanded Source Types

Zero Lag Detrended Price Oscillator (ZL DPO)This indicator attempts to create a zero lag Detrended price oscillator using 2 different scripts. I actually really like the results so far. I hope you all find it useful too.
Green>Red = long
Red>Green = short
The lines on the example chart are some of the signals that the indicator gave on default settings.
The greens are wins, reds are outright losses, and blues are "scratch" trades(signal for other side before hitting stop loss).
All of the signals tested were using NNFX money management to see if they are wins or losses(1.5x atr for SL and 1 Atr for TP).

Hybrid, Zero lag, Adaptive cycle MACD [Loxx]TASC's March 2008 edition Traders' Tips includes an article by John Ehlers titled "Measuring Cycle Periods," and describes the use of bandpass filters to estimate the length, in bars, of the currently dominant price cycle.
What are Dominant Cycles and Why should we use them?
Even the most casual chart reader will be able to spot times when the market is cycling and other times when longer-term trends are in play. Cycling markets are ideal for swing trading however attempting to “trade the swing” in a trending market can be a recipe for disaster. Similarly, applying trend trading techniques during a cycling market can equally wreak havoc in your account. Cycle or trend modes can readily be identified in hindsight. But it would be useful to have an objective scientific approach to guide you as to the current market mode.
There are a number of tools already available to differentiate between cycle and trend modes. For example, measuring the trend slope over the cycle period to the amplitude of the cyclic swing is one possibility.
We begin by thinking of cycle mode in terms of frequency or its inverse, periodicity. Since the markets are fractal; daily, weekly, and intraday charts are pretty much indistinguishable when time scales are removed. Thus it is useful to think of the cycle period in terms of its bar count. For example, a 20 bar cycle using daily data corresponds to a cycle period of approximately one month.
When viewed as a waveform, slow-varying price trends constitute the waveform's low frequency components and day-to-day fluctuations (noise) constitute the high frequency components. The objective in cycle mode is to filter out the unwanted components--both low frequency trends and the high frequency noise--and retain only the range of frequencies over the desired swing period. A filter for doing this is called a bandpass filter and the range of frequencies passed is the filter's bandwidth .
Indicator Features
-Zero lag or Regular MACD/signal calculation
- Fixed or Band-pass Dominant Cycle for MACD and Signal MA period inputs
-10 different moving average options for both MACD and Signal MA calculations
-Separate Band-pass Dominant Cycle calculations for both MACD and Signal MA calculations
- Slow-to-Fast Band-pass Dominant Cycle input to tweak the ratio of MACD MA input periods as they relate to each other

Supertrend Ext1st it exactly looks like the original Supertrend indicator. But if you see the options, you can see it is totally different:
It uses my other indicator inside: Zero Lag Keltner Channels, so you can use smoothed ATR bands for calculation instead of the raw ATR. By default it's length is 1 so it works like the original Supertrend.
You can choose different sources and moving averages for Keltner Channel calculation
It can filter outliers in ATR calculation
The indicator code is in my TAExt library, so it can easily be used in custom strategies.

Ehlers Average Error Filter [CC]The Average Error Filter was created by John Ehlers and this is a variation of a Zero Lag Exponential Moving Average that uses a Super Smoother to filter out the noise and then uses a second Super Smoother of the difference between the current price and the filtered data. This works well as a trendline and does give out a few false signals like all indicators inevitably do but most signals do a good job of keeping up with the trend and providing clear entries and exits when the trend changes. I have included strong buy and sell signals in addition to normal ones so like always darker colors are strong signals and lighter colors are normal ones. Buy when the line turns green and sell when it turns red.
Let me know if there are any other scripts you would like to see me publish!

RedK Dual-Color Zero-Lag Moving Average (RedK ZLMA v1)Here's my Dual-color Zero-Lag Moving Average indicator - with alerts - as a separate study
This is published in response to couple of requests i received. Please refer to previous posts on TA Basics on creating zero-lag MAs for more background.
This version adds couple of extras
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- the ability to choose the price being used (close vs hl2, hlc3..etc)
- ability to change resolution
the example chart below on the right uses a weekly resolution for the ZLMA on top of a daily chart - the left chart shows the same view but with the ZLMA resolution set to "same as chart"
- Alerts-enabled for when trend change is detected
- optional smoothing - keep the smoothing low (3 or 4) else it introduces lagging again
i hope some will find this useful
- the code is light and can easily be integrated into other indicators of your own - or you can add this ZLMA to your charts in combination with other MA's you use to easily identify trends and swings.
let me know your feedback.
Good luck!

AM_ZLWMA Crossover_xDear All,
Please find updated version of Zero lag WMA crossover. In addition to this added 20/50 EMA to get better results.
Settings:
ZLWMA = 8 identified by Blue line
Fast WMA = 21 identified by Red line
EMA Period 20/50
How to Trade:
Buy: Wait for positive crossover (Blue cross above Red) to happen trade can be taken there but to get better results post crossover once price move above EMA 20 then buy can be initiated.
Sell: Wait for negative crossover (Blue cross below Red) to happen trade can be taken there but to get better results post crossover once price move below EMA 20 then sell can be initiated.
Reason to wait for confirmation of price action above / Below moving average is most of the time it happens that even after crossover happens sometimes it was just a pull back and from there it could be a possibility for reversal. Once price action confirms moving above / below EMA20 then trade will be initiated.
Please comment and ask for any doubt

[blackcat] L2 Ehlers Zero-lag EMACircumstance Remarks: Because of my carelessness, the script of the same name that I posted before was banned and hidden because the description contained content that violated the TradingView House Rule. After communicating with the MOD, I corrected the description and obtained permission to publish it again. I hereby declare. Sorry for the inconvenience!
Level: 2
Background
John F. Ehlers introuced Zero-lag EMA Indicator in Nov, 2010.
Function
In “Zero Lag (Well, Almost)” article, authors John Ehlers and Ric Way presented their zero-lag exponential moving average indicator and strategy. They have adapted their zero-lag EMA by extending the functionality in an additional chart indicator named “Zero-Lag EMA”. Labels were added so that the user can be alerted when a crossing of the averages occurs.
The authors created an error-correcting filter for an exponential moving average ( EMA ) that seeks to minimize the lag effect of increasing periods. Increasing the gain parameter from zero changes the filter from an EMA with lag to effectively zero lag (albeit with zero smoothing also). The crossover of these lines can be used to form a trading strategy, with the addition of some threshold value for the difference between the Price and error-correcting line.
Key Signal
ZLEMA ---> Zero-lag EMA fast line
Trigger ---> Zero-lag EMA slow line
Pros and Cons
100% John F. Ehlers definition translation, even variable names are the same. This help readers who would like to use pine to read his book.
Remarks
The 76th script for Blackcat1402 John F. Ehlers Week publication.
Readme
In real life, I am a prolific inventor. I have successfully applied for more than 60 international and regional patents in the past 12 years. But in the past two years or so, I have tried to transfer my creativity to the development of trading strategies. Tradingview is the ideal platform for me. I am selecting and contributing some of the hundreds of scripts to publish in Tradingview community. Welcome everyone to interact with me to discuss these interesting pine scripts.
The scripts posted are categorized into 5 levels according to my efforts or manhours put into these works.
Level 1 : interesting script snippets or distinctive improvement from classic indicators or strategy. Level 1 scripts can usually appear in more complex indicators as a function module or element.
Level 2 : composite indicator/strategy. By selecting or combining several independent or dependent functions or sub indicators in proper way, the composite script exhibits a resonance phenomenon which can filter out noise or fake trading signal to enhance trading confidence level.
Level 3 : comprehensive indicator/strategy. They are simple trading systems based on my strategies. They are commonly containing several or all of entry signal, close signal, stop loss, take profit, re-entry, risk management, and position sizing techniques. Even some interesting fundamental and mass psychological aspects are incorporated.
Level 4 : script snippets or functions that do not disclose source code. Interesting element that can reveal market laws and work as raw material for indicators and strategies. If you find Level 1~2 scripts are helpful, Level 4 is a private version that took me far more efforts to develop.
Level 5 : indicator/strategy that do not disclose source code. private version of Level 3 script with my accumulated script processing skills or a large number of custom functions. I had a private function library built in past two years. Level 5 scripts use many of them to achieve private trading strategy.

[blackcat] L2 Ehlers Zero-lag SmootherLevel: 2
Background
John F. Ehlers introuced Zero-Lag Data Smoothers in Jul, 2002.
Function
John Ehlers introduced "Zero-Lag Data Smoothers", the infinite impulse response (IIR) filter and finite impulse response (FIR) filter.
In his article this issue on zero-lag smoothing, John Ehlers notes that his favorite filter is the symmetrically weighted six-bar finite impulse response (FIR) filter. This is also known as a triangular moving average, and can be conveniently implemented as a double-smoothed simple moving average. Per Ehlers, since this filter has six elements, its lag is 2.5 bars. Via further processing, this lag can be reduced to zero, but this produces too much overshoot. As a compromise, Ehlers suggests reducing the lag to one bar. To enable a user to adjust the lag easily, I provide the pine v4 code for an Adjustable Lag Filter indicator below. The first input, Price, should typically be set to OHLC, hl2, hl3, ohlc4 etc. The second input, LagReduction, should be set to a value in the zero-to-2.5 range. Setting it to zero will result in no adjustment, and the output will match that of the raw triangular average. Setting it to 2.5 will reduce the lag to zero. Setting it to 1.5 will reduce the lag to one bar.
Key Signal
Filter--> Zero-Lag Data Smoother fast line
Trigger--> Zero-Lag Data Smoother slow line
Pros and Cons
100% John F. Ehlers definition translation, even variable names are the same. This help readers who would like to use pine to read his book.
Remarks
The 67th script for Blackcat1402 John F. Ehlers Week publication.
Readme
In real life, I am a prolific inventor. I have successfully applied for more than 60 international and regional patents in the past 12 years. But in the past two years or so, I have tried to transfer my creativity to the development of trading strategies. Tradingview is the ideal platform for me. I am selecting and contributing some of the hundreds of scripts to publish in Tradingview community. Welcome everyone to interact with me to discuss these interesting pine scripts.
The scripts posted are categorized into 5 levels according to my efforts or manhours put into these works.
Level 1 : interesting script snippets or distinctive improvement from classic indicators or strategy. Level 1 scripts can usually appear in more complex indicators as a function module or element.
Level 2 : composite indicator/strategy. By selecting or combining several independent or dependent functions or sub indicators in proper way, the composite script exhibits a resonance phenomenon which can filter out noise or fake trading signal to enhance trading confidence level.
Level 3 : comprehensive indicator/strategy. They are simple trading systems based on my strategies. They are commonly containing several or all of entry signal, close signal, stop loss, take profit, re-entry, risk management, and position sizing techniques. Even some interesting fundamental and mass psychological aspects are incorporated.
Level 4 : script snippets or functions that do not disclose source code. Interesting element that can reveal market laws and work as raw material for indicators and strategies. If you find Level 1~2 scripts are helpful, Level 4 is a private version that took me far more efforts to develop.
Level 5 : indicator/strategy that do not disclose source code. private version of Level 3 script with my accumulated script processing skills or a large number of custom functions. I had a private function library built in past two years. Level 5 scripts use many of them to achieve private trading strategy.

[blackcat] L3 Ehlers ZeroLag Intraday Trading SystemLevel: 3
Background
John F. Ehlers introuced ZeroLag Intraday Trading System in his "Rocket Science for Traders" chapter 16.
Function
blackcat L3 EhlersZeroLag Intraday Trading System is used to find proper long and short entries. Dr. Ehlers developed a completely automatic ZeroLag Intraday Trading System. The concepts of the Instantaneous Trendline and the ZeroLag EMA are very powerful. To demonstrate just how profound these concepts are, Dr. Ehlers designed an intraday trading system. An intraday trade is defined as any active trade that is traded and then closed at the end of the day.
Key Signal
Smooth --> 4 bar WMA w/ 1 bar lag
Detrender --> The amplitude response of a minimum-length HT can be improved by adjusting the filter coefficients by
trial and error. HT does not allow DC component at zero frequency for transformation. So, Detrender is used to remove DC component/ trend component.
Q1 --> Quadrature phase signal
I1 --> In-phase signal
Period --> Dominant Cycle in bars
SmoothPeriod --> Period with complex averaging
DCPeriod ---> Dominant Cycle Period
Trendline ---> IT fast line
ZeroLag ---> Zero Lag Filter
long ---> long entry signal
short ---> short entry signal
Pros and Cons
100% John F. Ehlers definition translation of original work, even variable names are the same. This help readers who would like to use pine to read his book. If you had read his works, then you will be quite familiar with my code style.
NOTE: This version of Trading System has better preformance than "Automatic SineTrend Trading System".
Remarks
The 12th script for Blackcat1402 John F. Ehlers Week publication.
Readme
In real life, I am a prolific inventor. I have successfully applied for more than 60 international and regional patents in the past 12 years. But in the past two years or so, I have tried to transfer my creativity to the development of trading strategies. Tradingview is the ideal platform for me. I am selecting and contributing some of the hundreds of scripts to publish in Tradingview community. Welcome everyone to interact with me to discuss these interesting pine scripts.
The scripts posted are categorized into 5 levels according to my efforts or manhours put into these works.
Level 1 : interesting script snippets or distinctive improvement from classic indicators or strategy. Level 1 scripts can usually appear in more complex indicators as a function module or element.
Level 2 : composite indicator/strategy. By selecting or combining several independent or dependent functions or sub indicators in proper way, the composite script exhibits a resonance phenomenon which can filter out noise or fake trading signal to enhance trading confidence level.
Level 3 : comprehensive indicator/strategy. They are simple trading systems based on my strategies. They are commonly containing several or all of entry signal, close signal, stop loss, take profit, re-entry, risk management, and position sizing techniques. Even some interesting fundamental and mass psychological aspects are incorporated.
Level 4 : script snippets or functions that do not disclose source code. Interesting element that can reveal market laws and work as raw material for indicators and strategies. If you find Level 1~2 scripts are helpful, Level 4 is a private version that took me far more efforts to develop.
Level 5 : indicator/strategy that do not disclose source code. private version of Level 3 script with my accumulated script processing skills or a large number of custom functions. I had a private function library built in past two years. Level 5 scripts use many of them to achieve private trading strategy.

Zalligator - Zero-Lag Alligator IndicatorI applied the zero-lag moving average theory to the Alligator Indicator. It seems like some different rules would would be required versus the traditional Alligator. Let me know what you think!

Vervoort SmoothingThis script has both the Triple Exponential Moving Average (TEMA) and zero lag sma written as functions. Both from Capturing Profit with Technical Analysis (24-25) by Sylvain Vervoort.
Best regards,
capam

Forward Backward EMA [Repaint]Perform forward-backward filtering using exponential averaging, thus providing a zero-phase exponential moving average. The output repaint and cannot be used as input for other indicators.
Settings
Length : moving average period
Src : data input of the moving average
Plot Color : the color of the displayed plot
Line Width : width of the plotted line
Usages
The main usage of moving averages is to provide an estimate of the underlying trend in the price by removing higher term variations from it. Non-causal (repainting) indicators are limited to offline applications, as such, they are most useful for summary analyses, note that it is still possible to infer from the output of repainting indicators, however since past outputs are subject to changes, it is extremely difficult to track the effectiveness of such indicators, and in online applications they only track the price, making them equally useful for predictive applications than following the direction of an individual candle.
Non-causal filters can be useful in order to have a better view of symbols with a relatively uninformative evolution.
Details
Causal filters have lag, this is the cost of using past observations as inputs, the more past observations you use, the more lag you will obtain (assuming these past observations have non-zero weights). There are various solutions to reduce the lag of a moving average, the most simple one relying on giving higher weights to more recent observations, another one relies on introducing gain in the filter passband, that is amplifying certain variations in the input signal while attenuating/removing higher term ones, finally, we can use adaptive moving averages to avoid excessive lag.
All these previous solutions can be used causally, but they are far from being perfect, as the lag reduction is often done at the cost of smoothness, if we were to keep the original smoothness of the filter while having no lag we would need to use non-causal solutions. The most common solution is to directly use future values as inputs, such moving averages are called "two-sided" moving averages since they use past values as input (left side) as well as future input values (right side), this is equivalent to shifting the results of a moving average backward.
The advantages of two-sided moving averages is that they conserve the original amplitude response of the moving average, however, it won't be possible to compute the most recent values of the moving average (since we won't have access to future values at a certain point), an alternative method heavily used in digital signal processing is forward-backward filtering.
The method consists of applying a filter forward in time, then we apply it once again backward. In order for you to have an easier understanding of this process think about applying a moving average normally starting at time t = 0 , then apply that moving average once again using the previous results as input but start from t = N-1 , that is from the most recent point, and proceed backward, plotting the result from left to right until you get back to t = 0 .
From this, it follows that forward-backward filtering applies a filter twice, the resulting filter is thus a two-passes filter, this results in an even smoother output (more precisely the filter amplitude response is squared).
Forward-backward filtering can be done in Pinescript by using the function "line.new" inside a loop, an exponential moving average is applied forward first, then once again backward inside the loop, "line.new" is used to plot the results backward.
Notes
It is important to note that forward-backward filtering is a repainting process, all the results of the indicator you see on the chart are subject to change over time. Since the method make use of line.new you will have around only 54 visible observations, with the impossibility of using them as input for other indicators. If you see indicators in the future with the same characteristics be aware that they will repaint.
Never purchase/rent filters that appear as having no lag, they are either repainting or the results are coming from a lucky shot or from an overfitted model, it is impossible to make both zero-lag and causal moving averages with pinescript, if you have doubt don't test your luck, better safe than sorry.

Trendflex - Another new Ehlers indicatorSource: Stocks and Commodities V38
Hooray! Another new John Ehlers indicator!
John claims this indicator is lag-less and uses the SPY on the Daily as an example.
This indicator is a slight modification of Reflex, which I have posted here
I think it's better for Stocks and ETFs than Reflex since it factors in long trends. It tends to keep you in winning trades for a long time.
I believe this indicator can be used for entries or exits, potentially both.
Entry
1. Entering Long positions at the pivot low points (Stocks and ETFs)
2. Entering Long when the Reflex crosses above the zero lines (Stocks, ETFs, Commodities )
Exit
1. Exiting Long positions at a new pivot high point (Stocks and ETFs)
2. Exiting Long when the Reflex crosses below the zero lines (Stocks, ETFs, Commodities )
In this example, I place a Long order on the SPY every time the Reflex crosses above the zero level and exit when it crosses below or pops my stop loss, set at 1.5 * Daily ATR.
2/3 Wins
+16.05%
Let me know in the comment section if you're able to use this in a strategy.

Reflex - A new Ehlers indicatorSource: Stocks and Commodities V38
Hooray! A new John Ehlers indicator!
John claims this indicator is lag-less and uses the SPY on the Daily as an example.
He states that drawing a line from peak to peak (or trough to trough) will correspond perfectly with the Asset.
I have to say I agree! There is typically one bar of lag or no lag at all!
I believe this indicator can be used for either entries or exits, but not both.
Entry
1. Entering Long positions at the pivot low points (Stocks and ETFs)
2. Entering Long when the Reflex crosses above the zero lines (Stocks, ETFs, Commodities)
Exit
1. Exiting Long positions at a new pivot high point (Stocks and ETFs)
2. Exiting Long when the Reflex crosses below the zero lines (Stocks, ETFs, Commodities)
In this example, I place a Long order on the SPY every time the Reflex crosses above the zero level and exit when it crosses below or pops my stop loss, set at 1.5 * Daily ATR.
4/6 Wins
+10.76%
For me, that's good enough to create a strategy and backtest on several Indices and ETFs, which is what I have a hunch this will work on.
I think there is a lot of promise from a single Indicator!
Let me know in the comment section if you're able to use this in a strategy.

ZLMA - Low-Lag Moving Average Based On An Alternative SMA DesignThere can be many ways to make a simple moving average, you can either sum the current and the n-1 previous data points and divide the result by n , or you can do it more efficiently by first taking the cumulative sum of your data points, and subtracting the current cumulative sum result with the cumulative sum results n bars ago, then divide the result by n . This can be described by the following formulas:
a(t) = a(t-1) + price(t)
b(t) = (a(t) - a(t-n))/n
This method is the one used in order to allow the user to use a series as SMA period, more info here:
Today we use this design in order to provide a pretty efficient low-lag moving average where the amount of lag of the moving average can be increased/decreased by the user.
THE INDICATOR
length control the period of the moving average, with larger value of length returning larger filtering amount. The lag setting in the other hand control the amount of lag of the moving average, with larger value of lag returning a moving average with less lag. The lag setting can't be lower than 1 or greater than 2, but values lower than 1 and greater than 0 would just return a moving average with larger filtering amount while values greater than 2 would create crazy wild overshoots.
In blue lag = 1.8, in red lag = 1.4, when lag = 1 the moving average is equal to a simple moving average of period length. Remember that larger values of lag will return greater over/undershoots.
Approximate amplitude response of the moving average, like all low-lag moving averages you can see frequencies amplified (the ones on the left greater than 1) .
SUMMARY
We proposed a low-lag moving average based on the cumulative/change SMA design where the lag of the moving average can be controlled by the user. There are tons of low-lag moving averages already, and they don't necessarily provide different results from each others, however this one is still relatively interesting as you can switch from a simple MA from a low-lagging one, other indicators are ready using this design and will be posted soon.

Shapeshifting Moving Average - Switching From Low-Lag To SmoothThe term "shapeshifting" is more appropriate when used with something with a shape that isn't supposed to change, this is not the case of a moving average whose shape can be altered by the length setting or even by an external factor in the case of adaptive moving averages, but i'll stick with it since it describe the purpose of the proposed moving average pretty well.
In the case of moving averages based on convolution, their properties are fully described by the moving average kernel ( set of weights ), smooth moving averages tend to have a symmetrical bell shaped kernel, while low lag moving averages have negative weights. One of the few moving averages that would let the user alter the shape of its kernel is the Arnaud Legoux moving average, which convolve the input signal with a parametric gaussian function in which the center and width can be changed by the user, however this moving average is not a low-lagging one, as the weights don't include negative values.
Other moving averages where the user can change the kernel from user settings where already presented, i posted a lot of them, but they only focused on letting the user decrease or increase the lag of the moving average, and didn't included specific parameters controlling its smoothness. This is why the shapeshifting moving average is proposed, this parametric moving average will let the user switch from a smooth moving average to a low-lagging one while controlling the amount of lag of the moving average.
Settings/Kernel Interaction
Note that it could be possible to design a specific kernel function in order to provide a more efficient approach to today goal, but the original indicator was a simple low-lag moving average based on a modification of the second derivative of the arc tangent function and because i judged the indicator a bit boring i decided to include this parametric particularity.
As said the moving average "kernel", who refer to the set of weights used by the moving average, is based on a modification of the second derivative of the arc tangent function, the arc tangent function has a "S" shaped curve, "S" shaped functions are called sigmoid functions, the first derivative of a sigmoid function is bell shaped, which is extremely nice in order to design smooth moving averages, the second derivative of a sigmoid function produce a "sinusoid" like shape ( i don't have english words to describe such shape, let me know if you have an idea ) and is great to design bandpass filters.
We modify this 2nd derivative in order to have a decreasing function with negative values near the end, and we end up with:
The function is parametric, and the user can change it ( thus changing the properties of the moving average ) by using the settings, for example an higher power value would reduce the lag of the moving average while increasing overshoots. When power < 3 the moving average can act as a slow moving average in a moving average crossover system, as weights would not include negative values.
Here power = 0 and length = 50. The shapeshifting moving average can approximate a simple moving average with very low power values, as this would make the kernel approximate a rectangular function, however this is only a curiosity and not something you should do.
As A Smooth Moving Average
“So smooth, and so tranquil. It doesn't get any quieter than this”
A smooth moving average kernel should be : symmetrical, not to width and not to sharp, bell shaped curve are often appropriates, the proposed moving average kernel can be symmetrical and can return extremely smooth results. I will use the Blackman filter as comparison.
The smooth version of the moving average can be used when the "smooth" setting is selected. Here power can only be an even number, if power is odd, power will be equal to the nearest lowest even number. When power = 0, the kernel is simply a parabola:
More smoothness can be achieved by using power = 2
In red the shapeshifting moving average, in green a Blackman filter of both length = 100. Higher values of power will create lower negative values near the border of the kernel shape, this often allow to retain information about the peaks and valleys in the input signal. Power = 6 approximate the Blackman filter pretty well.
Conclusion
A moving average using a modification of the 2nd derivative of the arc tangent function as kernel has been presented, the kernel is parametric and allow the user to switch from a low-lag moving average where the lag can be increased/decreased to a really smooth moving average.
As you can see once you get familiar with a function shape, you can know what would be the characteristics of a moving average using it as kernel, this is where you start getting intimate with moving averages.
On a side note, have you noticed that the views counter in posted ideas/indicators has been removed ? This is truly a marvelous idea don't you think ?
Thanks for reading !

Right Sided Ricker Moving Average And The Gaussian DerivativesIn general gaussian related indicators are built by using the gaussian function in one way or another, for example a gaussian filter is built by using a truncated gaussian function as filter kernel (kernel refer to the set weights) and has many great properties, note that i say truncated because the gaussian function is not supposed to be finite. In general the gaussian function is represented by a symmetrical bell shaped curve, however the gaussian function is parametric, and the user might adjust the position of the peak as well as the width of the curve, an indicator using this parametric approach is the Arnaud Legoux moving average (ALMA) who posses a length parameter controlling the filter length, a peak parameter controlling the position of the peak of the gaussian function as well as a width parameter, those parameters can increase/decrease the lag and smoothness of the moving average output.
However what about the derivatives of the gaussian function ? We don't talk much about them and thats a pity because they are extremely interesting and have many great properties as well, therefore in this post i'll present a low lag moving average based on the modification of the 2nd order derivative of the gaussian function, i believe this post will be extremely informative and i hope you will enjoy reading it, if you are not a math person you can skip the introduction on gaussian derivatives and their properties used as filter kernel.
Gaussian Derivatives And The Ricker Wavelet
The notion of derivative is continuous, so we will stick with the term discrete derivative instead, which just refer to the rate of change in the function, we have a change function in pinescript, and we will be using it to show an approximation of the gaussian function derivatives.
Earlier i used the term 2nd order derivative, here the derivative order refer to the order of differentiation, that is the number of time we apply the change function. For example the 0 (zeroth) order derivative mean no differentiation, the 1st order derivative mean we use differentiation 1 time, that is change(f) , 2nd order mean we use differentiation 2 times, that is change(change(f)) , derivates based on multiple differentiation are called "higher derivative". It will be easier to show a graphic :
Here we can see a normal gaussian function in blue, its scaled 1st order derivative in orange, and its scaled 2nd derivative in green, note that i use scaled because i used multiplication in order for you to see each curve, else it would have been less easy to observe them. The number of time a gaussian function derivative cross 0 is based on the order of differentiation, that is 2nd order = the function crossing 0 two times.
Now we can explain what is the Ricker wavelet, the Ricker wavelet is just the normalized 2nd order derivative of a gaussian function with inverted sign, and unlike the gaussian function the only thing you can change is the width parameter. The formula of the Ricker wavelet is show'n here en.wikipedia.org , where sigma is the width parameter.
The Ricker wavelet has this look :
Because she is shaped like a sombrero the Ricker wavelet is also called "mexican hat wavelet", now what would happen if we used a Ricker wavelet as filter kernel ? The response is that we would end-up with a bandpass filter, in fact the derivatives of the gaussian function would all give the kernel of a bandpass filter, with higher order derivatives making the frequency response of the filter approximate a symmetrical gaussian function, if i recall a filter using the first order derivative of a gaussian function would give a frequency response that is left skewed, this skewness is removed when using higher order derivatives.
The Indicator
I didn't wanted to make a bandpass filter, as lately i'am more interested in low-lag filters, so how can we use the Ricker wavelet to make a low-lag low-pass filter ? The response is by taking the right side of the Ricker wavelet, and since values of the wavelets are negatives near the border we know that the filter passband is non-monotonic, that is we know that the filter will have low-lag as frequencies in the passband will be amplified.
So taking the right side of the Ricker wavelet only mean that t has to be greater than 0 and linearly increasing, thats easy, however the width parameter can be tricky to use, this was already the case with ALMA, so how can we work with it ? First it can be seen that values of width needs to be adjusted based on the filter length.
In red width = 14, in green width = 5. We can see that an higher values of width would give really low weights, when the number of negative weights is too important the filter can have a negative group delay thus becoming predictive, this simply mean that the overshoots/undershoots will be crazy wild and that a great fit will be impossible.
Here two moving averages using the previous described kernels, they don't fit the price well at all ! In order to fix this we can simply define width as a function of the filter length, therefore the parameter "Percentage Width" was introduced, and simply set the width of the Ricker wavelet as p percent of the filter length. Lower values of percent width reduce the lag of the moving average, but lets see precisely how this parameter influence the filter output :
Here the filter length is equal to 100, and the percent width is equal to 60, the fit is quite great, lower values of percent width will increase overshoots, in fact the filter become predictive once the percent width is equal or lower to 50.
Here the percent width is equal to 50. Higher values of percent width reduce the overshoots, and a value of 100 return a filter with no overshoots that is suited to act as a lagging moving average.
Above percent width is set to 100. In order to make use of the predictive side of the filter, it would be great to introduce a forecast option, however this require to find the best forecast horizon period based on length and width, this is no easy task.
Finally lets estimate a least squares moving average with the proposed moving average, you know me...a percent width set to 63 will return a relatively good estimate of the LSMA.
LSMA in green and the proposed moving in red with percent width = 63 and both length = 100.
Conclusion
A new low-lag moving average using a right sided Ricker wavelet as filter kernel has been introduced, we have also seen some properties of gaussian derivatives. You can see that lately i published more moving averages where the user can adjust certain properties of the filter kernel such as curve width for example, if you like those moving averages you can check the Parametric Corrective Linear Moving Averages indicator published last month :
I don't exclude working with pure forms of gaussian derivatives in the future, as i didn't published much oscillators lately.
Thx for reading !

Parametric Corrective Linear Moving AveragesImpulse responses can fully describe their associated systems, for example a linearly weighted moving average (WMA) has a linearly decaying impulse response, therefore we can deduce that lag is reduced since recent values are the ones with the most weights, the Blackman moving average (or Blackman filter) has a bell shaped impulse response, that is mid term values are the ones with the most weights, we can deduce that such moving average is pretty smooth, the least squares moving average has negative weights, we can therefore deduce that it aim to heavily reduce lag, and so on. We could even estimate the lag of a moving average by looking at its impulse response (calculating the lag of a moving average is the aim of my next article with Pinescripters) .
Today a new moving average is presented, such moving average use a parametric rectified linear unit function as weighting function, we will see that such moving average can be used as a low lag moving average as well as a signal moving average, thus creating a moving average crossover system. Finally we will estimate the LSMA using the proposed moving average.
Correctivity And The Parametric Rectified Linear Unit Function
Lot of terms are used, each representing one thing, lets start with the easiest one,"corrective". In some of my posts i may have used the term "underweighting", which refer to the process of attributing negative weights to the input of a moving average, a corrective moving average is simply a moving average underweighting oldest values of the input, simply put most of the low lag moving averages you'll find are corrective. This term was used by Aistis Raudys in its paper "Optimal Negative Weight Moving Average for Stock Price Series Smoothing" and i felt like it was a more elegant term to use instead of "low-lag".
Now we will describe the parametric rectified linear unit function (PReLU), this function is the one used as weighting function and is not that complex. This function has two inputs, alpha , and x , in short if x is greater than 0, x remain unchanged, however if x is lower than 0, then the function output is alpha × x , if alpha is equal to 1 then the function is equivalent to an identity function, if alpha is equal to 0 then the function is equivalent to a rectified unit function.
PReLU is mostly used in neural network design as an activation function, i wont explain to you how neural networks works but remember that neural networks aim to mimic the neural networks in the brain, and the activation function mimic the process of neuron firing. Its a super interesting topic because activation functions regroup many functions that can be used for technical indicators, one example being the inverse fisher RSI who make use of the hyperbolic tangent function.
Finally the term parametric used here refer to the ability of the user to change the aspect of the weighting function thanks to certain settings, thinking about it, it isn't a common things for moving averages indicators to let the user modify the characteristics of the weighting function, an exception being the Arnaud Legoux moving average (ALMA) which weighting function is a gaussian function, the user can control the peak and width of the function.
The Indicator
The indicator has two moving averages displayed on the chart, a trigger moving average (in blue) and a signal moving average (in red), their crosses can generate signals. The length parameter control the filter length, with higher values of length filtering longer term price fluctuations.
The percentage of negative weights parameter aim to determine the percentage of negative weights in the weighting function, note that the signal moving average won't use the same amount and will use instead : 100 - Percentage , this allow to reverse the weighting function thus creating a more lagging output for signal. Note that this parameter is caped at 50, this is because values higher than 50 would make the trigger moving average become the signal moving average, in short it inverse the role of the moving averages, that is a percentage of 25 would be the same than 75.
In red the moving average using 25% of negative weights, in blue the same moving average using 14% percent of negative weights. In theory, more negative weights = less lag = more overshoots.
Here the trigger MA in blue has 0% of negative weights, the trigger MA in green has however 35% of negative weights, the difference in lag can be clearly seen. In the case where there is 0% of negative weights the trigger become a simple WMA while the signal one become a moving average with linearly increasing weights.
The corrective factor is the same as alpha in PReLU, and determine the steepness of the negative weights line, this parameter is constrained in a range of (0,1), lower values will create a less steep negative weights line, this parameter is extremely useful when we want to reduce overshoots, an example :
here the corrective factor is equal to 1 (so the weighting function is an identity function) and we use 45% of negative weights, this create lot of overshoots, however a corrective factor of 0.5 reduce them drastically :
Center Of Linearity
The impulse response of the signal moving average is inverse to the impulse response of the trigger moving average, if we where to show them together we would see that they would crosses at a point, denoted center of linearity, therefore the crosses of each moving averages correspond to the cross of the center of linearity oscillator and 0 of same period.
This is also true with the center of gravity oscillator, linear covariance oscillator and linear correlation oscillator. Of course the center of linearity oscillator is way more efficient than the proposed indicator, and if a moving average crossover system is required, then the wma/sma pair is equivalent and way more efficient, who would know that i would propose something with more efficient alternatives ? xD
Estimating A Least Squares Moving Average
I guess...yeah...but its not my fault you know !!! Its a linear weighting function ! What can i do about it ?
The least squares moving average is corrective, its weighting function is linearly decreasing and posses negative weights with an amount of negative weights inferior to 50%, now we only need to find the exact percentage amount of negative weights. How to do it ? Well its not complicated if we recall the estimation with the WMA/SMA combination.
So, an LSMA of period p is equal to : 3WMA(p) - 2SMA(p) , each coefficient of the combination can give us this percentage, that is 2/3*100 = 33.333 , so there are 33.33% percent of negative weights in the weighting function of the least squares moving average.
In blue the trigger moving average with percentage of negative values et to 33.33, and in green the lsma of both period 50.
Conclusion
Altho inefficient, the proposed moving averages remain extremely interesting. They make use of the PReLU function as weighting function and allow the user to have a more accurate control over the characteristics of the moving averages output such as lag and overshoot amount, such parameters could even be made adaptive.
We have also seen how to estimate the least squares moving average, we have seen that the lsma posses 33.333...% of negative weights in its weighting function, another useful information.
The lsma is always behind me, not letting me focus on cryptobot super profit indicators using massive amount of labels, its like each time i make an indicator, the lsma come back, like a jealous creature, she want the center of attention, but you know well that the proposed indicator is inefficient ! Inefficient elegance (effect of the meds) .
Thanks for reading !