Sigmoid Alpha Bands | NAL1. Overview
Sigmoid Alpha Bands | NAL is an adaptive trend and volatility framework built around a sigmoid-weighted EMA baseline and dynamically adjusted volatility bands.
Instead of smoothing price with a fixed alpha, the indicator modifies its responsiveness using a selected market feed. Momentum, volatility, volume, or price disparity can control how quickly the baseline adapts to changing conditions.
The surrounding bands can also respond asymmetrically to bullish and bearish return shocks. This allows the upper and lower boundaries to develop independently rather than remaining equally spaced around the baseline.
2. Calculation
The indicator begins by selecting the market variable used to control the baseline’s adaptive smoothing weight.
Momentum measures changes in RSI, volatility measures changes in ATR, volume measures changes in smoothed volume, and disparity measures changes in price relative to its EMA.
sigmoidFeed = switch sigFeed
"Momentum" => ta.change(ta.rsi(src, modLen), changeL)
"Volatility" => ta.change(ta.atr(modLen), changeL)
"Volume" => ta.change(ta.ema(volume, modLen), changeL)
"Disparity" => ta.change(src / ta.ema(src, modLen), changeL)
The selected feed is passed through a sigmoid function, converting it into a bounded adaptive weight.
That weight modifies the standard EMA alpha. When the sigmoid weight increases, the baseline can respond more quickly. When it decreases, the baseline becomes more stable.
sigmoidWeight = sigmoid_function(sigmoidFeed)
baseAlpha = 2.0 / (emaLen + 1.0)
adaptiveAlpha = f_clamp(baseAlpha * (0.5 + sigmoidWeight), 0.01, 1.0)
The final adaptive baseline is calculated recursively using the changing alpha.
sigmoid_ema = f_sigmoid_ema(src, sigmoidFeed, sigLen)
The indicator then calculates its base volatility using one of four methods: standard deviation, ATR, mean absolute deviation, or median absolute deviation.
volatilityRaw = switch volFeed
"SD" => ta.stdev(src, volLen)
"ATR" => ta.atr(volLen)
"MeanAD" => ta.dev(src, volLen)
"MedianAD" => f_median_ad(src, volLen)
This raw volatility value is also processed through a sigmoid-adaptive smoothing layer. The result becomes the symmetric volatility foundation used by the bands.
volatilityFeed = ta.change(volatilityRaw / nz(ta.ema(volatilityRaw, volLen), volatilityRaw), changeL)
volatility = f_sigmoid_ema(volatilityRaw, volatilityFeed, volLen)
When asymmetric bands are enabled, positive and negative log-return shocks are separated into bullish and bearish variance components.
bullShock = math.pow(math.max(ret, 0.0), 2.0)
bearShock = math.pow(math.max(-ret, 0.0), 2.0)
totalShock = bullShock + bearShock
Each shock component is adaptively smoothed and compared with total variance. This produces separate upper and lower volatility multipliers.
The multipliers are constrained around their longer-term average so the bands can adapt without becoming unstable.
upperVol = math.max(nz(symmetricVol, syminfo.mintick) * upperMultAdj, syminfo.mintick)
lowerVol = math.max(nz(symmetricVol, syminfo.mintick) * lowerMultAdj, syminfo.mintick)
The final bands are positioned around the sigmoid-adaptive baseline.
upperBand = sigmoid_ema + finalUpper * volMul
lowerBand = sigmoid_ema - finalLower * volMul
A bullish state is established when price closes above the upper band. A bearish state is established when price closes below the lower band. While price remains between the boundaries, the existing state is preserved.
if close > upperBand
NAL := 1
if close < lowerBand
NAL := -1
3. Key Features
Sigmoid-weighted adaptive EMA baseline.
Selectable momentum, volatility, volume, or disparity adaptation feed.
Multiple volatility calculation methods.
Optional asymmetric bullish and bearish volatility bands.
Independent modeling of positive and negative return shocks.
Controlled asymmetry through long-term multiplier normalization.
State-based candle coloring, layered volatility hulls, glow effects, and directional fills.
4. Use
Sigmoid Alpha Bands is designed to identify when price expands beyond an adaptively smoothed volatility structure.
A close above the upper boundary reflects bullish expansion beyond the current baseline and volatility regime. A close below the lower boundary reflects bearish expansion beneath that structure.
The asymmetric mode allows the indicator to recognize that bullish and bearish volatility do not always develop with equal intensity. As market pressure changes, each side of the channel can adjust independently while remaining anchored to the same adaptive baseline.
This indicator is designed as a specialized component within a complete strategy architecture. Its role is to isolate the interaction between adaptive trend, changing volatility, and directional return pressure. Its full value emerges through the way this information is integrated into a broader decision framework.
Indicator
Sigmoid RSI | NAL1. Overview
Sigmoid RSI | NAL is an adaptive momentum-regime indicator built from a transformed RSI engine, sigmoid weighting, and two independently adaptive smoothing layers.
Instead of using raw RSI values directly, the indicator recenters RSI around its midpoint and passes the result through a sigmoid transformation. This reshapes the oscillator into a smoother 0–100 momentum curve while preserving directional pressure around the central regime.
The final signal is refined through volatility-sensitive smoothing and an optional momentum-driven signal line, creating a more adaptive structure than a standard RSI with fixed smoothing.
2. Calculation
The indicator starts by calculating RSI from the selected source and centering it around 50. This converts the standard RSI range into a positive and negative momentum input around the neutral midpoint.
rsiRaw = ta.rsi(src, rsiLen)
rsiNorm = rsiRaw - 50.0
The centered RSI is then passed through a sigmoid function. The sigmoid transformation compresses extreme values and smoothly maps the momentum input back into a 0–100 range.
sigmoidRSIRaw = 100.0 * sigmoid_function(rsiNorm)
The indicator then builds two separate adaptive feeds.
The volatility feed measures changes in ATR, while the momentum feed measures changes in a separate RSI calculation.
momentumFeed = ta.change(ta.rsi(src, modLen), changeL)
volatilityFeed = ta.change(ta.atr(modLen), changeL)
These feeds are used inside a sigmoid-weighted EMA. The sigmoid output adjusts the smoothing alpha dynamically, allowing the moving average to respond differently as the input environment changes.
sigmoidWeight = sigmoid_function(sigmoidFeed)
adaptiveAlpha = f_clamp(baseAlpha * (0.5 + sigmoidWeight), 0.01, 1.0)
The main Sigmoid RSI is smoothed using the volatility feed. This makes its response sensitive to changes in the volatility environment.
sigmoidRSI = f_sigmoid_ema(sigmoidRSIRaw, volatilityFeed, rsiSmoothLen)
A second adaptive smoothing layer creates the signal line using the momentum feed. This gives the indicator an internal directional reference that reacts to changes in momentum rather than using a fixed moving average alone.
sigmoidRSISignal = f_sigmoid_ema(sigmoidRSI, momentumFeed, rsiSignalLen)
The bullish state requires the Sigmoid RSI to rise above the upper threshold. When confluence is enabled, it must also remain above its adaptive signal line.
The bearish state requires the Sigmoid RSI to fall below the lower threshold and, when enabled, remain below the signal line.
NAL := sigmoidRSI > longThres and (sigmoidRSI > sigmoidRSISignal or not boolCon) ? 1 : sigmoidRSI < shortThres and (sigmoidRSI < sigmoidRSISignal or not boolCon) ? -1 : NAL
When neither condition is active, the previous directional state is preserved.
3. Key Features
Sigmoid-transformed RSI momentum engine.
Adaptive volatility-sensitive RSI smoothing.
Momentum-driven adaptive signal line.
Dynamic smoothing alpha generated through sigmoid weighting.
Upper and lower threshold regime detection.
Optional signal-line confluence.
Glow plot, threshold fills, candle coloring, bar coloring, and transition labels.
4. Use
Sigmoid RSI is designed to identify when momentum transitions into a more established bullish or bearish regime.
A move above the upper threshold reflects bullish momentum expansion, while a move below the lower threshold reflects bearish momentum deterioration. The adaptive signal line adds another layer of internal structure by measuring whether the transformed RSI remains aligned with its momentum-driven smoothing path.
The indicator’s behavior is shaped by both volatility and momentum changes, allowing its response to shift with the underlying environment rather than remaining mechanically fixed.
Sigmoid RSI is designed as a specialized momentum component within a complete strategy architecture. Its real value lies in how its adaptive regime state is integrated into a broader decision process, where several layers of market behavior are evaluated together.
Indicator
Sigmoid Transition Trailing Stop [LuxAlgo]The Sigmoid Transition Trailing Stop indicator provides a dynamic risk management tool that utilizes a mathematical sigmoid function to smoothly adjust stop levels when price trends become overextended. Unlike traditional trailing stops that move linearly or remain static until a new high/low is formed, this script proactively "catches up" to price during periods of high momentum using a non-linear transition.
🔶 USAGE
The indicator functions as a trend-following stop-loss or take-profit level. When the price crosses the trailing stop, the trend direction flips, and a new stop is calculated based on the ATR-derived distance.
The unique behavior of this tool occurs during "Adjustment" phases. When the distance between the price and the trailing stop becomes too wide (exceeding the initial ATR multiplier), the script triggers a sigmoid transition. During this phase:
The trailing stop line becomes solid (0% transparency) to indicate an active adjustment.
The stop level moves towards the price following an S-curve, providing a smooth acceleration and deceleration of the stop level.
The adjustment concludes once the sigmoid duration is reached or if the stop gets too close to the price (defined by the Min Distance setting).
🔹 Visual Cues
Green Line/Fill: Indicates a bullish trend where the stop is below the price.
Red Line/Fill: Indicates a bearish trend where the stop is above the price.
Solid Line: The sigmoid adjustment is currently active, moving the stop closer to price.
Transparent Line: The stop is in its standard state, maintaining its level or trailing highs/lows.
🔶 DETAILS
The core of the script is the sigmoid function, which creates a natural-looking transition between two price levels. Mathematically, it maps a linear time input (bars passed) to an S-shaped curve between 0 and 1.
The logic ensures that the "trailing" property is never violated; even during a sigmoid transition, the stop level can only move in the direction of the trend (up for bullish, down for bearish). If the price moves against the trend during an adjustment, the stop will simply hold its level until the price moves favorably again or the adjustment period ends.
🔶 SETTINGS
🔹 General Settings
ATR Length: The lookback period used to calculate the Average True Range, which determines the volatility-based distance of the stop.
ATR Multiplier: The base distance (in ATR units) used when the trend first flips.
Sigmoid Length (Bars): The number of bars over which the sigmoid transition occurs once triggered.
Sigmoid Amplitude (ATR Units): The maximum distance the stop is allowed to travel toward the price during an adjustment phase.
Min Distance (ATR Units): A safety buffer that prevents the sigmoid adjustment from moving the stop too close to the current price, preventing premature exits.
Indicator
Sigmoid Risk AllocatorThe Sigmoid Risk Allocator is a dynamic position sizing indicator that tells you how much of your capital to allocate based on current market conditions. Unlike simple "risk-on/risk-off" signals, this indicator gives you smooth, gradual transitions based on a sigmoid function.
Why a Sigmoid Curve?
Most position sizing approaches use fixed thresholds: "If drawdown > 20%, buy. Otherwise, don't." This creates all-or-nothing decisions.
Using the sigmoid (S-curve) makes this decision different. It creates a smooth transition where:
Small drawdowns → Stay near your baseline allocation
Moderate drawdowns → Gradually increase exposure
Large drawdowns → Approach maximum allocation
The sigmoid curve naturally "saturates" at the extremes, preventing you from going all-in too early or panicking out too fast. This is very useful to meek traders psychology and risk management in check.
What's a Sigmoid Function?
The sigmoid function is a mathematical S-curve defined as:
σ(x) = 1 / (1 + e^(-x))
This formula takes any input value and smoothly maps it to a number between 0 and 1. The curve has three key properties that make it ideal for position sizing in investing:
Smooth transitions: No sudden jumps. Allocation changes gradually.
Saturation at extremes: The curve flattens near 0 and 1, preventing overreaction and overexposure.
Sensitive in the middle: Most of the action happens around the midpoint.
To convert this into an allocation percentage, the indicator uses:
Allocation = α_min + (α_max - α_min) × σ(k × (Risk - Midpoint))
Where:
- `α_min` = Your minimum allocation (default 50%)
- `α_max` = Your maximum allocation (default 100%)
- `Risk` = Current risk metric (drawdown %, volatility, or Kelly %)
- `Midpoint` = The risk level where allocation sits halfway between min and max (default 15%)
- `k` = Steepness—how quickly allocation changes around the midpoint
Example : With defaults, if drawdown hits 15% (the midpoint), your allocation will be 75% (halfway between 50% and 100%). As the drawdown increases beyond 15%, the allocation curves toward 100%. As it decreases toward 0%, allocation curves toward 50%.
Cool, isn't it?
Asymmetric Response: Fast In, Slow Out
The indicator uses different steepness values for scaling in vs. scaling out. This is great to increase trend following. This is something I'm proud of too in this indicator.
k_increase = 30 (steep curve): When drawdowns appear, allocation ramps up quickly to catch the opportunity
k_decrease = 5 (slower curve): When conditions normalize, allocation decreases slowly to avoid selling the rebound
This asymmetry reflects how markets behave—drawdowns often overshoot fundamentals (rewarding quick entries), while recoveries tend to be more orderly (rewarding patience on exits).
Three Risk Metrics
You can choose what drives your allocation:
Drawdown (Default)
Volatility - Scales your position inversely to current market volatility.
Kelly Criterion - Automatically calculates optimal position size. The indicator applies a conservative "half Kelly" by default.
Use Cases
Position sizing for swing trading or trend following
Risk management overlay for any existing strategy
Drawdown-based DCA (dollar cost averaging) decisions
Volatility-adjusted exposure management
Feel free to provide feedback and share your thoughts!
- Henrique Centieiro Indicator
Sigmoid Allocation Indicator & DashboardTL;DR This sigmoid-based allocation indicator tells you percentage of your portfolio to invest based on how much the market has dropped.
Market at all-time high? → Stay defensive, invest less (e.g., 30%)
Market crashed hard? → Get aggressive, invest more (e.g., 100%)
The "sigmoid" part just means the transition between these two extremes follows a smooth S-shaped curve.
Description
This indicator is a sigmoid-based allocation system that dynamically adjusts a portfolio exposure based on market drawdown.
It compares multiple steepness curves (K values) to find your optimal risk profile for leveraged ETF strategies, but it can also be used to scale in-out from stocks, crypto and to understand whether to use leverage or not.
The Sigmoid Allocation Dashboard helps you to dynamically adjust a portfolio allocation based on how much a market has dropped from its all-time high.
I've implemented it using a sigmoid (S-curve) function, that dynamically calculates the optimal allocation percentages. Depending on the market conditions, the S curves transition between defensive and aggressive allocations.
The Math Behind It (if you are a geek like me)
This indicator uses the sigmoid function to create smooth S-curve transitions:
α(D) = α_min + (α_max - α_min) × σ(k × (D - D_mid))
Where:
σ(x) = 1 / (1 + e^(-x)) ← Standard sigmoid function
You can also check it here:
// Sigmoid function: σ(x) = 1 / (1 + e^(-x))
sigmoid(float x) =>
1.0 / (1.0 + math.exp(-x))
// Alpha calculation: α(D) = α_min + (α_max - α_min) × σ(k × (D - D_mid))
calcAlpha(float drawdown, float k, float a_min, float a_max, float d_midpoint) =>
sig_input = k * (drawdown - d_midpoint) / 100.0
a_min + (a_max - a_min) * sigmoid(sig_input)
User parameters (you can tweak this):
Allocation Min (%): Your baseline allocation when markets are at ATH (default: 30%)
Allocation Max (%): Your maximum allocation during deep drawdowns (default: 100%)
D_mid (%): The drawdown level where you want to be at the midpoint (default: 25%)
Why do I like sigmoid and not a linear line?
Unlike linear models, the sigmoid creates "floors" and "ceilings" for your allocation. It transitions smoothly, no sudden jumps, and you never exceed your defined min/max bounds.
Understand the K Values (Steepness)
The K parameter controls how quickly your allocation shifts from defensive to aggressive.
Lower K (for example K=5) will give you a gradual transition, but at 0% drawdown you are already at a 46% allocation.
A higher like (like K=40) will give you a sharp transition, but at 0% drawdown you are close to the minimum allocation. On the other hand, a higher K will give close to 100% allocation when the markets are at new lows.
The example below illustrates this well, then the S&P 500 reached new lows in October 2022:
Different K values will affect the sigmoid curves (and you allocations differently). The chart below illustrates well how K affects the sigmoid curves:
Read the Dashboard
The main dashboard shows:
Current drawdown from ATH
Allocation % for each K value
Suggested action (Defensive → MAX LONG)
Use the Reference Chart
The static reference panel shows what your allocation would be at various drawdown levels (0%, 10%, 20%, 30%, 40%, 50%), helping you plan ahead.
Identify Zones
The color-coded chart background shows:
- 🟢 Green Zone: Aggressive positioning - "Buy the Dip"
- 🟡 Yellow Zone: Transition zone - Scaling in/out
- 🔴 Red Zone: Defensive positioning - Protect ya gains
Use Cases
Use case 1: Leveraged ETF Portfolio Management (this is my main use case)
When holding leveraged ETFs like TQQQ or UPRO, volatility makes it important to:
- Reduce exposure near all-time highs (when crashes hurt most)
- Increase exposure during drawdowns (when recovery potential is highest)
Example Strategy:
- At ATH: Hold 30% TQQQ, 70% cash/bonds or other uncorrelated assets
- At 25% drawdown: Hold 65% TQQQ, 35% cash/bonds
- At 40%+ drawdown: Hold 100% TQQQ
Use case 2: Diversified Leveraged Portfolio
Compare different K values for different assets:
- Use K = 10 for broad market (QQQ/SPY exposure via TQQQ/UPRO)
- Use K = 25 for sector bets (TECL, SOXL, TMF) that you want to scale into faster
Use case 3: Systematic Rebalancing Signals
Use the alerts to trigger rebalancing:
- Alert when K3 allocation crosses above 90% (time to add)
- Alert when drawdown exceeds your D_mid threshold
- Alert when market returns to within 5% of ATH
Tips for Best Results
It works best in longer time frames
Adjust the ATR lookback window
Match your risk tolerance level
I use this for index investing and stocks and haven't tried with crypto
Thanks for using the indicator and let me know if you have any feedback :)
- Henrique Centieiro
Indicator
[Pandora] Error Function Treasure Trove - ERF/ERFI/Sigmoids+PRAISE:
At this time, I have to graciously thank the wonderful minds behind the new "Pine Profiler Mode" (PPM). Directly prior to this release, it allowed me to ascertain script performance even more. While I usually write mostly in highly optimized Pine code, PPM visually identified a few bottlenecks that would otherwise be hard to identify. Anyone who contributed to PPMs creation and testing before release... BRAVO!!! I commend all of those who assisted in it's state-of-the-art engineering and inception, well done!
BACKSTORY:
This script is specifically being released in defense of another member, an exceptionally unique PhD. It was brought to my attention that a script-mod-event occurred, regarding the publishing of a measly antiquated error function (ERF) calculation within his script. This sadly resulted in the now former member jumping ship after receiving unmannerly responses amidst his curious inquiries as to why his erf() was modded. To forbid rusty and rudimentary formulations because a mod-on-duty is temporally offended by a non-nefarious release of code, is in MY opinion an injustice to principles of perpetuating open-source code intended to benefit thousands to millions of community members. While Pine is the heart and soul of TV, the mathematical concepts contributed from the minds of members is the inspirational fuel of curiosity that powers it's pertinent reason to exist and evolve.
It is an indisputable fact that most members are not greatly skilled Pine Poets. Many members may be incapable of innovating robust function code in Pine, even if they have one or more PhDs. We ALL come from various disciplines of mathematical comprehension and education. Some mathematicians are not greatly skilled at coding, while some coders are not exceptional at math. So... what am I to do to attempt to resolve this circumstantial challenge??? Those who know me best are aware that I will always side with "the right side of history" in order to accomplish my primary self-defined missions I choose to accept. Serving as an algorithmic advocate, I felt compelled to intercede by compiling numerous error functions into elegant code of very high caliber that any and every TV member may choose to employ, so this ERROR never happens again.
After weeks of contemplation into algorithms I knew little about, I prioritized myself to resolve an unanticipated matter by creating advanced formulas of exquisitely crafted error functions refined to the best of my current abilities. My aversion for unresolved problems motivated me to eviscerate error function insufficiencies with many more rigid formulations beyond what is thought to exist. ERF needed a proper algorithmic exorcism anyways. In my furiosity, I contemplated an array of madMAXimum diplomatic demolition methods, choosing the chain saw massacre technique to slaughter dysfunctionalities I encountered on a battered ERF roadway. This resulted in prolific solutions that should assuredly endure the test of time. Poetically, as you will come to see, I am ripping the lid off of Pandora's box of error functions in this case to correct wrongs into a splendid bundle of rights for members.
INTENTION:
Error function (ERF) enthusiasts... PREPARE FOR GLORY!! The specific purpose of this script is to deprecate classic error functions with the creation of a fierce and formidable army of superior formulations, each having varying attributes of computational complexity with differing absolute error ranges in their results for multiple compute scenarios. This is NOT an indicator... It is intended to allow members to embark on endeavors to advance the profound knowledge base of this growing worldwide community of 60+ million inquisitive minds. For those of you who believe computational mathematics and statistics is near completion at its finest; I am here to inform you, this is ridiculous to ponder. We are no where near statistical excellence that can and will exist eventually. At this time, metaphorically speaking, we are merely scratching microns off of the surface of the skin of a statistical apple Isaac Newton once pondered.
THIS RELEASE:
Following weeks of pondering methodical experiments beyond the ordinary, I am liberating these wild notions of my error function explorations to the entire globe as copyleft code, not just Pine. This Pandora's basket of ERFs is being openly disclosed for the sake of the sanctity of mathematics, empirical science (not the garbage we are told by CONTROLocrats to blindly trust), revolutionary cutting edge engineering, cosmology, physics, information technology, artificial intelligence, and EVERY other mathematical branch of human knowledge being discovered over centuries. I do believe James Glaisher would favor my aims concerning ERF aspirations embracing the "Power of Pine".
The included functions are intended for TV members to use in any way they see fit. This is a gift to ALL members to foster future innovative excellence on this platform. Any attempt to moderate this code without notification of "self-evident clear and just cause" will be considered an irrevocable egregious action. The original foundational PURPOSE of establishing script moderation (I clearly remember) was primarily to maintain active vigilance over a growing community against intentional nefarious actions and/or behaviors in blatant disrespect to other author's works AND also thwart rampant copypasting bandit operations, all while accommodating balanced principles of fairness for an educational community cause via open source publishing that should support future algorithmic inventions well beyond my lifespan.
APPLICATIONS:
The related error functions are used in probability theory, statistics, and numerous and engineering scientific disciplines. Its key characteristics and applications are innumerable in computational realms. Its versatility and significance make it a fundamental tool in arenas of quantitative analysis and scientific research...
Probability Theory - Is widely used in probability theory to calculate probabilities and quantiles of the normal distribution.
Statistics - It's related to the Gaussian integral and plays a crucial role in statistics, especially in hypothesis testing and confidence interval calculations.
Physics - In physics, it arises in the study of diffusion equations, quantum mechanics, and heat conduction problems.
Engineering - Applications exist in engineering disciplines such as signal processing, control theory, and telecommunications.
Error Analysis - It's employed in error analysis and uncertainty quantification.
Numeric Approximations - Due to its lack of a closed-form expression, numerical methods are often employed to approximate erf/erfi().
AI, LLMs, & MACHINE LEARNING:
The error function (ERF) is indispensable to various AI applications, particularly due to its relation to Gaussian distributions and error analysis. It is used in Gaussian processes for regression and classification, probabilistic inference for Bayesian networks, soft margin computation in SVMs, neural networks involving Gaussian activation functions or noise, and clustering algorithms like Gaussian Mixture Models. Improved ERF approximations can enhance precision in these applications, reduce computational complexity, handle outliers and noise better, and improve optimization and convergence, possibly leading to more accurate, efficient, and robust AI systems.
BONUS ALGORITHMS:
While ERFs are versatile, its opposite also exists in the form of inverse error functions (ERFIs). I have also included a modified form of the inverse fisher transform along side MY sigmoid (sigmyod). I am uncertain what sigmyod() may be used for, but it's a culmination of my examinations deep into "sigmoid domains", something I am fascinated by. Whatever implications it may possess, I am unveiling it along with it's cousin functions. For curious minds, this quality of composition seen here is ideally what underlies what I would term "Pandora functionality" that empowers my Pandora indication. I go through hordes of formulations, testing, and inspection to find what appears to be the most beneficial logical/mathematical equation to apply...
SCRIPT OPERATION:
To showcase the characteristics and performance of my ERF/ERFI formulations, I devised a multi-modal script. By using bar_index , I generated a broad sequence of numeric values to input into the first ERF/ERFI parameter. These sequences allow you to inspect the contours of the error function's outputs for both ERF and ERFI. When combined with compute-intensive precision functions (CIPFs), the polynomial function output values can be subtracted from my CIPFs to obtain results of absolute error, displaying the accuracy of the many polynomial estimation functions I tuned in testing for Pine's float environment.
A host of numeric input settings are wildly adjustable to inspect values/curvatures across the range of numeric input sequences. Very large numbers, such as Divisor:100,000,100/Offset:200,000,000 for ERF modes or... Divisor:100,000,100/Offset:100,000,000 for ERFI modes, will display miniscule output values calculated from input values in close proximity to 0.0 for the various estimates, similar to a microscope. ERFI approximations very near in proximity to +/-1.0 will always yield large deviations of absolute error. Dragging/zooming your chart or using the Offset input will aid with visually clipping off those ERFI extremes where float precision functions cannot suffice.
NOTICE:
perf() and perfi() are intended for precision computation (as good as it basically gets) in a float environment. However, they are CPU intensive (especially perfi). I wouldn't recommend these being used in ANY Pine script unless it's an "absolute necessity" to do so to accomplish your goal. I only built them to obtain "absolute error curvatures" of the error functions for the polynomial approximations. These are visible in the accuracy modes in the indicator Settings.
Indicator
Dynamic Gradient Filter
Sigmoid Functions:
History and Mathematical Basis:
Sigmoid functions have a rich history in mathematics and are widely used in various fields, including statistics, machine learning, and signal processing.
The term "sigmoid" originates from the Greek words "sigma" (meaning "S-shaped") and "eidos" (meaning "form" or "type").
The sigmoid curve is characterized by its smooth S-shaped appearance, which allows it to map any real-valued input to a bounded output range, typically between 0 and 1.
The most common form of the sigmoid function is the logistic function:
Logistic Function (σ):
Defined as σ(x) = 1 / (1 + e^(-x)), where:
'x' is the input value,
'e' is Euler's number (approximately 2.71828).
This function was first introduced by Belgian mathematician Pierre François Verhulst in the 1830s to model population growth with limiting factors.
It gained popularity in the early 20th century when statisticians like Ronald Fisher began using it in regression analysis.
Specific Sigmoid Functions Used in the Indicator:
sig(val):
The 'sig' function in this indicator is a modified version of the logistic function, clamping a value between 0 and 1 on the sigmoid curve.
siga(val):
The 'siga' function adjusts values between -1 and 1 on the sigmoid curve, offering a centered variation of the sigmoid effect.
sigmoid(val):
The 'sigmoid' function provides a standard implementation of the logistic function, calculating the sigmoid value of the input data.
Adaptive Smoothing Factor:
The ' adaptiveSmoothingFactor(gradient, k)' function computes a dynamic smoothing factor for the filter based on the gradient of the price data and the user-defined sensitivity parameter 'k' .
Gradient:
The gradient represents the rate of change in price, calculated as the absolute difference between the current and previous close prices.
Sensitivity (k):
The 'k' parameter adjusts how quickly the filter reacts to changes in the gradient. Higher values of 'k' lead to a more responsive filter, while lower values result in smoother outputs.
Usage in the Indicator:
The "close" value refers to the closing price of each period in the chart's time frame
The indicator calculates the gradient by measuring the absolute difference between the current "close" price and the previous "close" price.
This gradient represents the strength or magnitude of the price movement within the chosen time frame.
The "close" value plays a pivotal role in determining the dynamic behavior of the "Dynamic Gradient Filter," as it directly influences the smoothing factor.
What Makes This Special:
The "Dynamic Gradient Filter" indicator stands out due to its adaptive nature and responsiveness to changing market conditions.
Dynamic Smoothing Factor:
The indicator's dynamic smoothing factor adjusts in real-time based on the rate of change in price (gradient) and the user-defined sensitivity '(k)' parameter.
This adaptability allows the filter to respond promptly to both minor fluctuations and significant price movements.
Smoothed Price Action:
The final output of the filter is a smoothed representation of the price action, aiding traders in identifying trends and potential reversals.
Customizable Sensitivity:
Traders can adjust the 'Sensitivity' parameter '(k)' to suit their preferred trading style, making the indicator versatile for various strategies.
Visual Clarity:
The plotted "Dynamic Gradient Filter" line on the chart provides a clear visual guide, enhancing the understanding of market dynamics.
Usage:
Traders and analysts can utilize the "Dynamic Gradient Filter" to:
Identify trends and reversals in price movements.
Filter out noise and highlight significant price changes.
Fine-tune trading strategies by adjusting the sensitivity parameter.
Enhance visual analysis with a dynamically adjusting filter line on the chart.
Literature:
en.wikipedia.org
medium.com
en.wikipedia.org
Indicator
Indicator
Indicator
Inverse Fisher Z-Score Introduction
The inverse fisher transform or hyperbolic tangent function is a type os sigmoid function (sometime called squashing function) , those types of functions can rescale a result in a certain range and are widely used in artificial intelligence. More in depth the fisher transform can make the correlation coefficient of a time series normally distributed, in practice if you apply the fisher transform to the correlation coefficient between a time series and a linear function you will end up with an estimate of the z-score of the time series. The inverse transform however can do the contrary, it can take the z-score and transform it into a rough estimate of the correlation coefficient, if your z-score is not smooth then you will have a non-smooth estimate of the correlation coefficient, that's quite nice no ?
The Indicator
The inverse fisher transform of the z-score will produce results in a range of 1/-1, here however i will rescale in a range of 100/0 because its a standard range for oscillators in technical analysis. Values over 80 indicate an overbought market, under 20 an oversold market. The smooth option in the indicator settings will make the indicator use a linearly weighted moving average as input thus resulting in a smoother result.
The indicator with smooth option.
Conclusion
I presented a new oscillator indicator who use the inverse fisher transform of a z-score. Using the fisher transform and its inverse can give a new shape to your indicator, make sure to control the scale of your indicator before applying the fisher transform, the inverse transform should be applied to values in range of 1/-1 but you can use higher limits (2/-2,3/-3...) , however remember that higher limits will approximate an heavy side step function (square shape) . I hope you will find an use to this indicator.
Thanks for reading ! Indicator
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