Grid by Volatility (Expo)█ Overview
The Grid by Volatility is designed to provide a dynamic grid overlay on your price chart. This grid is calculated based on the volatility and adjusts in real-time as market conditions change. The indicator uses Standard Deviation to determine volatility and is useful for traders looking to understand price volatility patterns, determine potential support and resistance levels, or validate other trading signals.
█ How It Works
The indicator initiates its computations by assessing the market volatility through an established statistical model: the Standard Deviation. Following the volatility determination, the algorithm calculates a central equilibrium line—commonly referred to as the "mid-line"—on the chart to serve as a baseline for additional computations. Subsequently, upper and lower grid lines are algorithmically generated and plotted equidistantly from the central mid-line, with the distance being dictated by the previously calculated volatility metrics.
█ How to Use
Trend Analysis: The grid can be used to analyze the underlying trend of the asset. For example, if the price is above the Average Line and moves toward the Upper Range, it indicates a strong bullish trend.
Support and Resistance: The grid lines can act as dynamic support and resistance levels. Price tends to bounce off these levels or breakthrough, providing potential trade opportunities.
Volatility Gauge: The distance between the grid lines serves as a measure of market volatility. Wider lines indicate higher volatility, while narrower lines suggest low volatility.
█ Settings
Volatility Length: Number of bars to calculate the Standard Deviation (Default: 200)
Squeeze Adjustment: Multiplier for the Standard Deviation (Default: 6)
Grid Confirmation Length: Number of bars to calculate the weighted moving average for smoothing the grid lines (Default: 2)
-----------------
Disclaimer
The information contained in my Scripts/Indicators/Ideas/Algos/Systems does not constitute financial advice or a solicitation to buy or sell any securities of any type. I will not accept liability for any loss or damage, including without limitation any loss of profit, which may arise directly or indirectly from the use of or reliance on such information.
All investments involve risk, and the past performance of a security, industry, sector, market, financial product, trading strategy, backtest, or individual's trading does not guarantee future results or returns. Investors are fully responsible for any investment decisions they make. Such decisions should be based solely on an evaluation of their financial circumstances, investment objectives, risk tolerance, and liquidity needs.
My Scripts/Indicators/Ideas/Algos/Systems are only for educational purposes!
Search in scripts for "algo"
Machine Learning: kNN (New Approach)Description:
kNN is a very robust and simple method for data classification and prediction. It is very effective if the training data is large. However, it is distinguished by difficulty at determining its main parameter, K (a number of nearest neighbors), beforehand. The computation cost is also quite high because we need to compute distance of each instance to all training samples. Nevertheless, in algorithmic trading KNN is reported to perform on a par with such techniques as SVM and Random Forest. It is also widely used in the area of data science.
The input data is just a long series of prices over time without any particular features. The value to be predicted is just the next bar's price. The way that this problem is solved for both nearest neighbor techniques and for some other types of prediction algorithms is to create training records by taking, for instance, 10 consecutive prices and using the first 9 as predictor values and the 10th as the prediction value. Doing this way, given 100 data points in your time series you could create 10 different training records. It's possible to create even more training records than 10 by creating a new record starting at every data point. For instance, you could take the first 10 data points and create a record. Then you could take the 10 consecutive data points starting at the second data point, the 10 consecutive data points starting at the third data point, etc.
By default, shown are only 10 initial data points as predictor values and the 6th as the prediction value.
Here is a step-by-step workthrough on how to compute K nearest neighbors (KNN) algorithm for quantitative data:
1. Determine parameter K = number of nearest neighbors.
2. Calculate the distance between the instance and all the training samples. As we are dealing with one-dimensional distance, we simply take absolute value from the instance to value of x (| x – v |).
3. Rank the distance and determine nearest neighbors based on the K'th minimum distance.
4. Gather the values of the nearest neighbors.
5. Use average of nearest neighbors as the prediction value of the instance.
The original logic of the algorithm was slightly modified, and as a result at approx. N=17 the resulting curve nicely approximates that of the sma(20). See the description below. Beside the sma-like MA this algorithm also gives you a hint on the direction of the next bar move.
LinearRegressionLibraryLibrary "LinearRegressionLibrary" contains functions for fitting a regression line to the time series by means of different models, as well as functions for estimating the accuracy of the fit.
Linear regression algorithms:
RepeatedMedian(y, n, lastBar) applies repeated median regression (robust linear regression algorithm) to the input time series within the selected interval.
Parameters:
y :: float series, source time series (e.g. close)
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
Output:
mSlope :: float, slope of the regression line
mInter :: float, intercept of the regression line
TheilSen(y, n, lastBar) applies the Theil-Sen estimator (robust linear regression algorithm) to the input time series within the selected interval.
Parameters:
y :: float series, source time series
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
Output:
tsSlope :: float, slope of the regression line
tsInter :: float, intercept of the regression line
OrdinaryLeastSquares(y, n, lastBar) applies the ordinary least squares regression (non-robust) to the input time series within the selected interval.
Parameters:
y :: float series, source time series
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
Output:
olsSlope :: float, slope of the regression line
olsInter :: float, intercept of the regression line
Model performance metrics:
metricRMSE(y, n, lastBar, slope, intercept) returns the Root-Mean-Square Error (RMSE) of the regression. The better the model, the lower the RMSE.
Parameters:
y :: float series, source time series (e.g. close)
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
slope :: float, slope of the evaluated linear regression line
intercept :: float, intercept of the evaluated linear regression line
Output:
rmse :: float, RMSE value
metricMAE(y, n, lastBar, slope, intercept) returns the Mean Absolute Error (MAE) of the regression. MAE is is similar to RMSE but is less sensitive to outliers. The better the model, the lower the MAE.
Parameters:
y :: float series, source time series
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
slope :: float, slope of the evaluated linear regression line
intercept :: float, intercept of the evaluated linear regression line
Output:
mae :: float, MAE value
metricR2(y, n, lastBar, slope, intercept) returns the coefficient of determination (R squared) of the regression. The better the linear regression fits the data (compared to the sample mean), the closer the value of the R squared is to 1.
Parameters:
y :: float series, source time series
n :: integer, the length of the selected time interval
lastBar :: integer, index of the last bar of the selected time interval (defines the position of the interval)
slope :: float, slope of the evaluated linear regression line
intercept :: float, intercept of the evaluated linear regression line
Output:
Rsq :: float, R-sqared score
Usage example:
//@version=5
indicator('ExampleLinReg', overlay=true)
// import the library
import tbiktag/LinearRegressionLibrary/1 as linreg
// define the studied interval: last 100 bars
int Npoints = 100
int lastBar = bar_index
int firstBar = bar_index - Npoints
// apply repeated median regression to the closing price time series within the specified interval
{square bracket}slope, intercept{square bracket} = linreg.RepeatedMedian(close, Npoints, lastBar)
// calculate the root-mean-square error of the obtained linear fit
rmse = linreg.metricRMSE(close, Npoints, lastBar, slope, intercept)
// plot the line and print the RMSE value
float y1 = intercept
float y2 = intercept + slope * (Npoints - 1)
if barstate.islast
{indent} line.new(firstBar,y1, lastBar,y2)
{indent} label.new(lastBar,y2,text='RMSE = '+str.format("{0,number,#.#}", rmse))
Pulu's 3 Moving Averages
Pulu's 3 Moving Averages
Release version 1, date 2021-09-28
This script allows you to customize three sets of moving averages, turn on/off, set color and parameters. It also tags the start date of the last set of moving average if there is. This, release version 1, supports eight moving average algorithms:
ALMA, Arnaud Legoux Moving Average
EMA, Exponential Moving Average
RMA, Adjusted exponential moving average (aka Wilder’s EMA)
SMA, Simple Moving Average
SWMA, Symmetrically-Weighted Moving Average
VWAP, Volume-Weighted Average Price
VWMA, Volume-Weighted Moving Average
WMA, Weighted Moving Average
The availability and function parameters
Func. Availability Parameters
ALMA
MA1, MA2, MA3
source
length
offset
sigma
EMA
RMA
SMA
VWMA
WMA
MA1, MA2, MA3
source
length
SWMA
VWAP
MA1
source
Parameters
Parameter Description
source the series of values to process. The default is to use the closing price to calculate the moving average.
length an integer value that defines the number of bars to calculate the moving average on. The SWMA and VWAP do not use this parameter.
ALMA offset a floating-point value that controls the tradeoff between smoothness (with a value closer to 1) and responsiveness (with a value closer to 0). This parameter is only used by ALMA.
ALMA sigma a floating-point value that specifies the ALMA’s smoothness. The larger this value, the smoother the moving average is. This parameter is only used by ALMA.
I'm not sure if it is needed, so I do not let the three Moving Averages of the script to have indivial algorithm setting. Because that will involve much complicated condition testing and use up more TradingView script lines limit. If you need to combine different algorithms in the three sets of moving averages, or have other ideas, leave a message to let me know; maybe I will try it in the next update.
我不確定是否需要，所以我沒有讓腳本的三組移動平均線有各別的算法設置。因為這將涉及更多複雜的條件測試，並使用更多 TradingView 腳本列數限制。如果您需要在三組均線中組合不同的算法，或者有其他想法，請留言告訴我；也許我會在下一次更新中嘗試。
Cycles StrategyThis is back-testable strategy is a modified version of the Stochastic strategy. It is meant to accompany the modified Stochastic indicator: "Cycles".
Modifications to the Stochastic strategy include;
1. Programmable settings for the Stochastic Periods (%K, %D and Smooth %K).
2. Programmable settings for the MACD Periods (Fast, Slow, Smoothing)
3. Programmable thresholds for %K, to qualify a potential entry strategy.
4. Programmable thresholds for %D, to qualify a potential exit strategy.
5. Buttons to choose which components to use in the trading algorithm.
6. Choose the month and year to back test.
The trading algorithm:
1. When %K exceeds the upper/lower threshold and then hooks down/up, in the direction of the Moving Average (MA). This is the minimum entry qualification.
2. When %D exceeds the lower/upper threshold and angled in the direction of the trade, is the exit qualification.
3. Additional entry filters include the direction of MACD, Signal and %D. Also, the "cliff", being a long entry is a higher high or a short entry is a lower low.
4. Strategy can only go "Long" or "Short" depending on the selected setting.
5. By matching the settings in the "Cycles" indicator, you can (almost) see what the strategy is doing.
6. Be sure to select the "Recalculate" buttons, to recalculate on every new Tick, for best results.
Please click the Like button and leave a comment if you appreciate this script. Improvements will be implemented as time goes on.
I am not a licensed trade advisor. This strategy is for entertainment only. Use at your own risk!
gfg//@version=5
indicator("Lux Gainz Style Algo", overlay=true)
// User Inputs
fastLen = input.int(12, "Fast MA Length", minval=1)
slowLen = input.int(26, "Slow MA Length", minval=1)
rsiLen = input.int(14, "RSI Length", minval=1)
rsiOverbought = input.float(70, "RSI Overbought Level")
rsiOversold = input.float(30, "RSI Oversold Level")
sensitivity = input.float(1.5, "Signal Sensitivity", minval=0.1, step=0.1)
// Moving Averages for Trend
fastMA = ta.ema(close, fastLen)
slowMA = ta.ema(close, slowLen)
// RSI for Momentum
rsi = ta.rsi(close, rsiLen)
// Trend Conditions
bullTrend = fastMA > slowMA
bearTrend = fastMA < slowMA
// Confirmation Signals
longSignal = ta.crossover(fastMA, slowMA) and rsi < rsiOversold * sensitivity
shortSignal = ta.crossunder(fastMA, slowMA) and rsi > rsiOverbought / sensitivity
// Plot Moving Averages
plot(fastMA, color=color.new(color.green, 0), title="Fast EMA")
plot(slowMA, color=color.new(color.red, 0), title="Slow EMA")
// Candle Coloring for Trend Strength
barcolor(bullTrend ? color.new(color.green, 70) : bearTrend ? color.new(color.red, 70) : color.gray)
// Plot Buy/Sell Signals
plotshape(longSignal, title="Buy Signal", location=location.belowbar, color=color.green, style=shape.labelup, size=size.small)
plotshape(shortSignal, title="Sell Signal", location=location.abovebar, color=color.red, style=shape.labeldown, size=size.small)
// Alerts
alertcondition(longSignal, title="Long Entry", message="Lux Gainz Algo: Long Entry Signal")
alertcondition(shortSignal, title="Short Entry", message="Lux Gainz Algo: Short Entry Signal")
[blackcat] L3 Improved Dual Ehlers BPF for Volatility DetectionOVERVIEW
This script implements an advanced L3 Improved Dual Ehlers Bandpass Filter (BPF) for volatility detection, combining both L1 and L2 calculation methods to create a comprehensive trading signal. The script leverages John Ehlers' sophisticated digital signal processing techniques to identify market cycles and extract meaningful trading signals from price action. By combining multiple cycle detection methods and filtering approaches, it provides traders with a powerful tool for identifying trend changes, momentum shifts, and potential reversal points across various market conditions and timeframes. The L3 approach uniquely combines the outputs of both L1 (01 range) and L2 (-11 range) methods, creating a signal that ranges from -1~2 and provides enhanced sensitivity to market dynamics.
FEATURES
🔄 Dual Calculation Methods: Choose between L1 (01 range), L2 (-11 range), or combine both for L3 signal (-1~2 range) to match your trading style
📊 Multiple Cycle Detection: Seven different dominant cycle calculation methods including HoDyDC (Hilbert Transform Dominant Cycle), PhAcDC (Phase Accumulation Dominant Cycle), DuDiDC (Duane Dominant Cycle), CycPer (Cycle Period), BPZC (Bandpass Zero Crossing), AutoPer (Autocorrelation Period), and DFTDC (Discrete Fourier Transform Dominant Cycle)
🎛️ Flexible Mixing Options: Six sophisticated mixing methods including weighted averaging, simple sum, difference extraction, dominant-only, subdominant-only, and adaptive mixing that adjusts based on signal strength
🌊 Bandpass Filtering: Precise bandwidth control for both dominant and subdominant filters, allowing fine-tuning of frequency response characteristics
📈 Advanced Divergence Detection: Robust algorithm for identifying bullish and bearish divergences with customizable lookback periods and range constraints
🎨 Comprehensive Visualization: Extensive customization options for all signals, colors, plot styles, and display elements
🔔 Comprehensive Alert System: Built-in alerts for divergence signals, zero line crosses, and various market conditions
📊 Real-time Cycle Information: Optional display of dominant and subdominant cycle periods for educational purposes
🔄 Adaptive Signal Processing: Dynamic adjustment of parameters based on market conditions and volatility
🎯 Multiple Signal Outputs: Simultaneous generation of L1, L2, and L3 signals for different trading strategies
HOW TO USE
Select Calculation Method: Choose between "l1" (01 range), "l2" (-11 range), or "both" (L3, -1~2 range) in the Calculation Method settings based on your preferred signal characteristics
Configure Cycle Detection: Select your preferred Dominant Cycle Method from the seven available options and adjust the Cycle Part parameter (0.1-0.9) to fine-tune cycle sensitivity
Set Subdominant Parameters: Configure the subdominant cycle either as a ratio of the dominant cycle or as a fixed period, depending on your analysis approach
Adjust Filter Bandwidth: Fine-tune the bandwidth settings for both dominant and subdominant filters (0.1-1.0) to control the frequency response and signal smoothing
Choose Mixing Method: Select how to combine the filters - weighted averaging for balance, sum for maximum sensitivity, difference for trend isolation, or adaptive mixing for dynamic response
Configure Smoothing: Select from SMA, EMA, or HMA smoothing methods with adjustable length (1-20 bars) to reduce noise in the final signal
Customize Visualization: Enable/disable individual plots, divergence detection, zero line, fill areas, and customize all colors to match your chart preferences
Set Divergence Parameters: Configure lookback ranges (5-60 bars) for divergence detection to match your trading timeframe and style
Monitor Signals: Watch for crosses above/below zero line and divergence patterns, paying attention to signal strength and consistency
Set Up Alerts: Configure alerts for divergence signals, zero line crosses, and other market conditions to stay informed of trading opportunities
LIMITATIONS
The script requires the dc_ta library from blackcat1402 for several advanced cycle calculation methods (HoDyDC, PhAcDC, DuDiDC, CycPer, BPZC, AutoPer, DFTDC)
L1 method operates in 01 range while L2 method uses -11 range, requiring different interpretation approaches
Combined L3 signal ranges from -1~2 when both methods are selected, creating unique signal characteristics that traders must adapt to
Divergence detection accuracy depends on proper lookback period settings and market volatility conditions
Performance may be impacted with very long lookback ranges (>60 bars) or when multiple plots are simultaneously enabled
The script is designed for non-overlay use and may not display correctly on certain chart types or with conflicting indicators
Adaptive mixing method requires careful threshold tuning to avoid excessive signal fluctuation
Cycle detection algorithms may produce unreliable results during low volatility or highly choppy market conditions
The script assumes regular price data and may not perform optimally with irregular or gapped price sequences
NOTES
The script implements advanced mathematical calculations including bandpass filters, Hilbert transforms, and various cycle detection algorithms developed by John Ehlers
For optimal results, experiment with different cycle detection methods and bandwidth settings across various market conditions and timeframes
The adaptive mixing method automatically adjusts weights based on signal strength, providing dynamic response to changing market conditions
Divergence detection works best when the "Plot Divergence" option is enabled and when combined with other technical analysis tools
Zero line crosses can indicate potential trend changes or momentum shifts, especially when confirmed by volume or other indicators
The script includes commented code for cycle information display that can be enabled if you want to monitor cycle periods in real-time
Different calculation methods may perform better in different market environments - L1 tends to be smoother while L2 is more sensitive
The subdominant cycle helps filter out noise and provides additional confirmation for signals generated by the dominant cycle
Bandwidth settings control the filter's frequency response - lower values provide more smoothing while higher values increase sensitivity
Mixing methods offer different approaches to combining signals - weighted averaging is generally most reliable for most trading applications
THANKS
Special thanks to John Ehlers for his pioneering work in cycle analysis and digital signal processing for financial markets. This script implements and significantly improves upon his bandpass filter methodology, incorporating multiple advanced techniques from his extensive body of work. Also heartfelt thanks to blackcat1402 for the dc_ta library that provides essential cycle calculation methods and for maintaining such a valuable resource for the Pine Script community. Additional appreciation to the TradingView platform for providing the tools and environment that make sophisticated technical analysis accessible to traders worldwide. This script represents a collaborative effort in advancing the field of algorithmic trading and technical analysis.
Langlands-Operadic Möbius Vortex (LOMV)Langlands-Operadic Möbius Vortex (LOMV)
Where Pure Mathematics Meets Market Reality
A Revolutionary Synthesis of Number Theory, Category Theory, and Market Dynamics
🎓 THEORETICAL FOUNDATION
The Langlands-Operadic Möbius Vortex represents a groundbreaking fusion of three profound mathematical frameworks that have never before been combined for market analysis:
The Langlands Program: Harmonic Analysis in Markets
Developed by Robert Langlands (Fields Medal recipient), the Langlands Program creates bridges between number theory, algebraic geometry, and harmonic analysis. In our indicator:
L-Function Implementation:
- Utilizes the Möbius function μ(n) for weighted price analysis
- Applies Riemann zeta function convergence principles
- Calculates quantum harmonic resonance between -2 and +2
- Measures deep mathematical patterns invisible to traditional analysis
The L-Function core calculation employs:
L_sum = Σ(return_val × μ(n) × n^(-s))
Where s is the critical strip parameter (0.5-2.5), controlling mathematical precision and signal smoothness.
Operadic Composition Theory: Multi-Strategy Democracy
Category theory and operads provide the mathematical framework for composing multiple trading strategies into a unified signal. This isn't simple averaging - it's mathematical composition using:
Strategy Composition Arity (2-5 strategies):
- Momentum analysis via RSI transformation
- Mean reversion through Bollinger Band mathematics
- Order Flow Polarity Index (revolutionary T3-smoothed volume analysis)
- Trend detection using Directional Movement
- Higher timeframe momentum confirmation
Agreement Threshold System: Democratic voting where strategies must reach consensus before signal generation. This prevents false signals during market uncertainty.
Möbius Function: Number Theory in Action
The Möbius function μ(n) forms the mathematical backbone:
- μ(n) = 1 if n is a square-free positive integer with even number of prime factors
- μ(n) = -1 if n is a square-free positive integer with odd number of prime factors
- μ(n) = 0 if n has a squared prime factor
This creates oscillating weights that reveal hidden market periodicities and harmonic structures.
🔧 COMPREHENSIVE INPUT SYSTEM
Langlands Program Parameters
Modular Level N (5-50, default 30):
Primary lookback for quantum harmonic analysis. Optimized by timeframe:
- Scalping (1-5min): 15-25
- Day Trading (15min-1H): 25-35
- Swing Trading (4H-1D): 35-50
- Asset-specific: Crypto 15-25, Stocks 30-40, Forex 35-45
L-Function Critical Strip (0.5-2.5, default 1.5):
Controls Riemann zeta convergence precision:
- Higher values: More stable, smoother signals
- Lower values: More reactive, catches quick moves
- High frequency: 0.8-1.2, Medium: 1.3-1.7, Low: 1.8-2.3
Frobenius Trace Period (5-50, default 21):
Galois representation lookback for price-volume correlation:
- Measures harmonic relationships in market flows
- Scalping: 8-15, Day Trading: 18-25, Swing: 25-40
HTF Multi-Scale Analysis:
Higher timeframe context prevents trading against major trends:
- Provides market bias and filters signals
- Improves win rates by 15-25% through trend alignment
Operadic Composition Parameters
Strategy Composition Arity (2-5, default 4):
Number of algorithms composed for final signal:
- Conservative: 4-5 strategies (higher confidence)
- Moderate: 3-4 strategies (balanced approach)
- Aggressive: 2-3 strategies (more frequent signals)
Category Agreement Threshold (2-5, default 3):
Democratic voting minimum for signal generation:
- Higher agreement: Fewer but higher quality signals
- Lower agreement: More signals, potential false positives
Swiss-Cheese Mixing (0.1-0.5, default 0.382):
Golden ratio φ⁻¹ based blending of trend factors:
- 0.382 is φ⁻¹, optimal for natural market fractals
- Higher values: Stronger trend following
- Lower values: More contrarian signals
OFPI Configuration:
- OFPI Length (5-30, default 14): Order Flow calculation period
- T3 Smoothing (3-10, default 5): Advanced exponential smoothing
- T3 Volume Factor (0.5-1.0, default 0.7): Smoothing aggressiveness control
Unified Scoring System
Component Weights (sum ≈ 1.0):
- L-Function Weight (0.1-0.5, default 0.3): Mathematical harmony emphasis
- Galois Rank Weight (0.1-0.5, default 0.2): Market structure complexity
- Operadic Weight (0.1-0.5, default 0.3): Multi-strategy consensus
- Correspondence Weight (0.1-0.5, default 0.2): Theory-practice alignment
Signal Threshold (0.5-10.0, default 5.0):
Quality filter producing:
- 8.0+: EXCEPTIONAL signals only
- 6.0-7.9: STRONG signals
- 4.0-5.9: MODERATE signals
- 2.0-3.9: WEAK signals
🎨 ADVANCED VISUAL SYSTEM
Multi-Dimensional Quantum Aura Bands
Five-layer resonance field showing market energy:
- Colors: Theme-matched gradients (Quantum purple, Holographic cyan, etc.)
- Expansion: Dynamic based on score intensity and volatility
- Function: Multi-timeframe support/resistance zones
Morphism Flow Portals
Category theory visualization showing market topology:
- Green/Cyan Portals: Bullish mathematical flow
- Red/Orange Portals: Bearish mathematical flow
- Size/Intensity: Proportional to signal strength
- Recursion Depth (1-8): Nested patterns for flow evolution
Fractal Grid System
Dynamic support/resistance with projected L-Scores:
- Multiple Timeframes: 10, 20, 30, 40, 50-period highs/lows
- Smart Spacing: Prevents level overlap using ATR-based minimum distance
- Projections: Estimated signal scores when price reaches levels
- Usage: Precise entry/exit timing with mathematical confirmation
Wick Pressure Analysis
Rejection level prediction using candle mathematics:
- Upper Wicks: Selling pressure zones (purple/red lines)
- Lower Wicks: Buying pressure zones (purple/green lines)
- Glow Intensity (1-8): Visual emphasis and line reach
- Application: Confluence with fractal grid creates high-probability zones
Regime Intensity Heatmap
Background coloring showing market energy:
- Black/Dark: Low activity, range-bound markets
- Purple Glow: Building momentum and trend development
- Bright Purple: High activity, strong directional moves
- Calculation: Combines trend, momentum, volatility, and score intensity
Six Professional Themes
- Quantum: Purple/violet for general trading and mathematical focus
- Holographic: Cyan/magenta optimized for cryptocurrency markets
- Crystalline: Blue/turquoise for conservative, stability-focused trading
- Plasma: Gold/magenta for high-energy volatility trading
- Cosmic Neon: Bright neon colors for maximum visibility and aggressive trading
📊 INSTITUTIONAL-GRADE DASHBOARD
Unified AI Score Section
- Total Score (-10 to +10): Primary decision metric
- >5: Strong bullish signals
- <-5: Strong bearish signals
- Quality ratings: EXCEPTIONAL > STRONG > MODERATE > WEAK
- Component Analysis: Individual L-Function, Galois, Operadic, and Correspondence contributions
Order Flow Analysis
Revolutionary OFPI integration:
- OFPI Value (-100% to +100%): Real buying vs selling pressure
- Visual Gauge: Horizontal bar chart showing flow intensity
- Momentum Status: SHIFTING, ACCELERATING, STRONG, MODERATE, or WEAK
- Trading Application: Flow shifts often precede major moves
Signal Performance Tracking
- Win Rate Monitoring: Real-time success percentage with emoji indicators
- Signal Count: Total signals generated for frequency analysis
- Current Position: LONG, SHORT, or NONE with P&L tracking
- Volatility Regime: HIGH, MEDIUM, or LOW classification
Market Structure Analysis
- Möbius Field Strength: Mathematical field oscillation intensity
- CHAOTIC: High complexity, use wider stops
- STRONG: Active field, normal position sizing
- MODERATE: Balanced conditions
- WEAK: Low activity, consider smaller positions
- HTF Trend: Higher timeframe bias (BULL/BEAR/NEUTRAL)
- Strategy Agreement: Multi-algorithm consensus level
Position Management
When in trades, displays:
- Entry Price: Original signal price
- Current P&L: Real-time percentage with risk level assessment
- Duration: Bars in trade for timing analysis
- Risk Level: HIGH/MEDIUM/LOW based on current exposure
🚀 SIGNAL GENERATION LOGIC
Balanced Long/Short Architecture
The indicator generates signals through multiple convergent pathways:
Long Entry Conditions:
- Score threshold breach with algorithmic agreement
- Strong bullish order flow (OFPI > 0.15) with positive composite signal
- Bullish pattern recognition with mathematical confirmation
- HTF trend alignment with momentum shifting
- Extreme bullish OFPI (>0.3) with any positive score
Short Entry Conditions:
- Score threshold breach with bearish agreement
- Strong bearish order flow (OFPI < -0.15) with negative composite signal
- Bearish pattern recognition with mathematical confirmation
- HTF trend alignment with momentum shifting
- Extreme bearish OFPI (<-0.3) with any negative score
Exit Logic:
- Score deterioration below continuation threshold
- Signal quality degradation
- Opposing order flow acceleration
- 10-bar minimum between signals prevents overtrading
⚙️ OPTIMIZATION GUIDELINES
Asset-Specific Settings
Cryptocurrency Trading:
- Modular Level: 15-25 (capture volatility)
- L-Function Precision: 0.8-1.3 (reactive to price swings)
- OFPI Length: 10-20 (fast correlation shifts)
- Cascade Levels: 5-7, Theme: Holographic
Stock Index Trading:
- Modular Level: 25-35 (balanced trending)
- L-Function Precision: 1.5-1.8 (stable patterns)
- OFPI Length: 14-20 (standard correlation)
- Cascade Levels: 4-5, Theme: Quantum
Forex Trading:
- Modular Level: 35-45 (smooth trends)
- L-Function Precision: 1.6-2.1 (high smoothing)
- OFPI Length: 18-25 (disable volume amplification)
- Cascade Levels: 3-4, Theme: Crystalline
Timeframe Optimization
Scalping (1-5 minute charts):
- Reduce all lookback parameters by 30-40%
- Increase L-Function precision for noise reduction
- Enable all visual elements for maximum information
- Use Small dashboard to save screen space
Day Trading (15 minute - 1 hour):
- Use default parameters as starting point
- Adjust based on market volatility
- Normal dashboard provides optimal information density
- Focus on OFPI momentum shifts for entries
Swing Trading (4 hour - Daily):
- Increase lookback parameters by 30-50%
- Higher L-Function precision for stability
- Large dashboard for comprehensive analysis
- Emphasize HTF trend alignment
🏆 ADVANCED TRADING STRATEGIES
The Mathematical Confluence Method
1. Wait for Fractal Grid level approach
2. Confirm with projected L-Score > threshold
3. Verify OFPI alignment with direction
4. Enter on portal signal with quality ≥ STRONG
5. Exit on score deterioration or opposing flow
The Regime Trading System
1. Monitor Aether Flow background intensity
2. Trade aggressively during bright purple periods
3. Reduce position size during dark periods
4. Use Möbius Field strength for stop placement
5. Align with HTF trend for maximum probability
The OFPI Momentum Strategy
1. Watch for momentum shifting detection
2. Confirm with accelerating flow in direction
3. Enter on immediate portal signal
4. Scale out at Fibonacci levels
5. Exit on flow deceleration or reversal
⚠️ RISK MANAGEMENT INTEGRATION
Mathematical Position Sizing
- Use Galois Rank for volatility-adjusted sizing
- Möbius Field strength determines stop width
- Fractal Dimension guides maximum exposure
- OFPI momentum affects entry timing
Signal Quality Filtering
- Trade only STRONG or EXCEPTIONAL quality signals
- Increase position size with higher agreement levels
- Reduce risk during CHAOTIC Möbius field periods
- Respect HTF trend alignment for directional bias
🔬 DEVELOPMENT JOURNEY
Creating the LOMV was an extraordinary mathematical undertaking that pushed the boundaries of what's possible in technical analysis. This indicator almost didn't happen. The theoretical complexity nearly proved insurmountable.
The Mathematical Challenge
Implementing the Langlands Program required deep research into:
- Number theory and the Möbius function
- Riemann zeta function convergence properties
- L-function analytical continuation
- Galois representations in finite fields
The mathematical literature spans decades of pure mathematics research, requiring translation from abstract theory to practical market application.
The Computational Complexity
Operadic composition theory demanded:
- Category theory implementation in Pine Script
- Multi-dimensional array management for strategy composition
- Real-time democratic voting algorithms
- Performance optimization for complex calculations
The Integration Breakthrough
Bringing together three disparate mathematical frameworks required:
- Novel approaches to signal weighting and combination
- Revolutionary Order Flow Polarity Index development
- Advanced T3 smoothing implementation
- Balanced signal generation preventing directional bias
Months of intensive research culminated in breakthrough moments when the mathematics finally aligned with market reality. The result is an indicator that reveals market structure invisible to conventional analysis while maintaining practical trading utility.
🎯 PRACTICAL IMPLEMENTATION
Getting Started
1. Apply indicator with default settings
2. Select appropriate theme for your markets
3. Observe dashboard metrics during different market conditions
4. Practice signal identification without trading
5. Gradually adjust parameters based on observations
Signal Confirmation Process
- Never trade on score alone - verify quality rating
- Confirm OFPI alignment with intended direction
- Check fractal grid level proximity for timing
- Ensure Möbius field strength supports position size
- Validate against HTF trend for bias confirmation
Performance Monitoring
- Track win rate in dashboard for strategy assessment
- Monitor component contributions for optimization
- Adjust threshold based on desired signal frequency
- Document performance across different market regimes
🌟 UNIQUE INNOVATIONS
1. First Integration of Langlands Program mathematics with practical trading
2. Revolutionary OFPI with T3 smoothing and momentum detection
3. Operadic Composition using category theory for signal democracy
4. Dynamic Fractal Grid with projected L-Score calculations
5. Multi-Dimensional Visualization through morphism flow portals
6. Regime-Adaptive Background showing market energy intensity
7. Balanced Signal Generation preventing directional bias
8. Professional Dashboard with institutional-grade metrics
📚 EDUCATIONAL VALUE
The LOMV serves as both a practical trading tool and an educational gateway to advanced mathematics. Traders gain exposure to:
- Pure mathematics applications in markets
- Category theory and operadic composition
- Number theory through Möbius function implementation
- Harmonic analysis via L-function calculations
- Advanced signal processing through T3 smoothing
⚖️ RESPONSIBLE USAGE
This indicator represents advanced mathematical research applied to market analysis. While the underlying mathematics are rigorously implemented, markets remain inherently unpredictable.
Key Principles:
- Use as part of comprehensive trading strategy
- Implement proper risk management at all times
- Backtest thoroughly before live implementation
- Understand that past performance does not guarantee future results
- Never risk more than you can afford to lose
The mathematics reveal deep market structure, but successful trading requires discipline, patience, and sound risk management beyond any indicator.
🔮 CONCLUSION
The Langlands-Operadic Möbius Vortex represents a quantum leap forward in technical analysis, bringing PhD-level pure mathematics to practical trading while maintaining visual elegance and usability.
From the harmonic analysis of the Langlands Program to the democratic composition of operadic theory, from the number-theoretic precision of the Möbius function to the revolutionary Order Flow Polarity Index, every component works in mathematical harmony to reveal the hidden order within market chaos.
This is more than an indicator - it's a mathematical lens that transforms how you see and understand market structure.
Trade with mathematical precision. Trade with the LOMV.
*"Mathematics is the language with which God has written the universe." - Galileo Galilei*
*In markets, as in nature, profound mathematical beauty underlies apparent chaos. The LOMV reveals this hidden order.*
— Dskyz, Trade with insight. Trade with anticipation.
MLExtensions_CoreLibrary "MLExtensions_Core"
A set of extension methods for a novel implementation of a Approximate Nearest Neighbors (ANN) algorithm in Lorentzian space, focused on computation.
normalizeDeriv(src, quadraticMeanLength)
Returns the smoothed hyperbolic tangent of the input series.
Parameters:
src (float) : The input series (i.e., the first-order derivative for price).
quadraticMeanLength (int) : The length of the quadratic mean (RMS).
Returns: nDeriv The normalized derivative of the input series.
normalize(src, min, max)
Rescales a source value with an unbounded range to a target range.
Parameters:
src (float) : The input series
min (float) : The minimum value of the unbounded range
max (float) : The maximum value of the unbounded range
Returns: The normalized series
rescale(src, oldMin, oldMax, newMin, newMax)
Rescales a source value with a bounded range to anther bounded range
Parameters:
src (float) : The input series
oldMin (float) : The minimum value of the range to rescale from
oldMax (float) : The maximum value of the range to rescale from
newMin (float) : The minimum value of the range to rescale to
newMax (float) : The maximum value of the range to rescale to
Returns: The rescaled series
getColorShades(color)
Creates an array of colors with varying shades of the input color
Parameters:
color (color) : The color to create shades of
Returns: An array of colors with varying shades of the input color
getPredictionColor(prediction, neighborsCount, shadesArr)
Determines the color shade based on prediction percentile
Parameters:
prediction (float) : Value of the prediction
neighborsCount (int) : The number of neighbors used in a nearest neighbors classification
shadesArr (array) : An array of colors with varying shades of the input color
Returns: shade Color shade based on prediction percentile
color_green(prediction)
Assigns varying shades of the color green based on the KNN classification
Parameters:
prediction (float) : Value (int|float) of the prediction
Returns: color
color_red(prediction)
Assigns varying shades of the color red based on the KNN classification
Parameters:
prediction (float) : Value of the prediction
Returns: color
tanh(src)
Returns the the hyperbolic tangent of the input series. The sigmoid-like hyperbolic tangent function is used to compress the input to a value between -1 and 1.
Parameters:
src (float) : The input series (i.e., the normalized derivative).
Returns: tanh The hyperbolic tangent of the input series.
dualPoleFilter(src, lookback)
Returns the smoothed hyperbolic tangent of the input series.
Parameters:
src (float) : The input series (i.e., the hyperbolic tangent).
lookback (int) : The lookback window for the smoothing.
Returns: filter The smoothed hyperbolic tangent of the input series.
tanhTransform(src, smoothingFrequency, quadraticMeanLength)
Returns the tanh transform of the input series.
Parameters:
src (float) : The input series (i.e., the result of the tanh calculation).
smoothingFrequency (int)
quadraticMeanLength (int)
Returns: signal The smoothed hyperbolic tangent transform of the input series.
n_rsi(src, n1, n2)
Returns the normalized RSI ideal for use in ML algorithms.
Parameters:
src (float) : The input series (i.e., the result of the RSI calculation).
n1 (simple int) : The length of the RSI.
n2 (simple int) : The smoothing length of the RSI.
Returns: signal The normalized RSI.
n_cci(src, n1, n2)
Returns the normalized CCI ideal for use in ML algorithms.
Parameters:
src (float) : The input series (i.e., the result of the CCI calculation).
n1 (simple int) : The length of the CCI.
n2 (simple int) : The smoothing length of the CCI.
Returns: signal The normalized CCI.
n_wt(src, n1, n2)
Returns the normalized WaveTrend Classic series ideal for use in ML algorithms.
Parameters:
src (float) : The input series (i.e., the result of the WaveTrend Classic calculation).
n1 (simple int)
n2 (simple int)
Returns: signal The normalized WaveTrend Classic series.
n_adx(highSrc, lowSrc, closeSrc, n1)
Returns the normalized ADX ideal for use in ML algorithms.
Parameters:
highSrc (float) : The input series for the high price.
lowSrc (float) : The input series for the low price.
closeSrc (float) : The input series for the close price.
n1 (simple int) : The length of the ADX.
regime_filter(src, threshold, useRegimeFilter)
Parameters:
src (float)
threshold (float)
useRegimeFilter (bool)
filter_adx(src, length, adxThreshold, useAdxFilter)
filter_adx
Parameters:
src (float) : The source series.
length (simple int) : The length of the ADX.
adxThreshold (int) : The ADX threshold.
useAdxFilter (bool) : Whether to use the ADX filter.
Returns: The ADX.
filter_volatility(minLength, maxLength, sensitivityMultiplier, useVolatilityFilter)
filter_volatility
Parameters:
minLength (simple int) : The minimum length of the ATR.
maxLength (simple int) : The maximum length of the ATR.
sensitivityMultiplier (float) : Multiplier for the historical ATR to control sensitivity.
useVolatilityFilter (bool) : Whether to use the volatility filter.
Returns: Boolean indicating whether or not to let the signal pass through the filter.
Volume Predictor [PhenLabs]📊 Volume Predictor
Version: PineScript™ v6
📌 Description
The Volume Predictor is an advanced technical indicator that leverages machine learning and statistical modeling techniques to forecast future trading volume. This innovative tool analyzes historical volume patterns to predict volume levels for upcoming bars, providing traders with valuable insights into potential market activity. By combining multiple prediction algorithms with pattern recognition techniques, the indicator delivers forward-looking volume projections that can enhance trading strategies and market analysis.
🚀 Points of Innovation:
Machine learning pattern recognition using Lorentzian distance metrics
Multi-algorithm prediction framework with algorithm selection
Ensemble learning approach combining multiple prediction methods
Real-time accuracy metrics with visual performance dashboard
Dynamic volume normalization for consistent scale representation
Forward-looking visualization with configurable prediction horizon
🔧 Core Components
Pattern Recognition Engine : Identifies similar historical volume patterns using Lorentzian distance metrics
Multi-Algorithm Framework : Offers five distinct prediction methods with configurable parameters
Volume Normalization : Converts raw volume to percentage scale for consistent analysis
Accuracy Tracking : Continuously evaluates prediction performance against actual outcomes
Advanced Visualization : Displays actual vs. predicted volume with configurable future bar projections
Interactive Dashboard : Shows real-time performance metrics and prediction accuracy
🔥 Key Features
The indicator provides comprehensive volume analysis through:
Multiple Prediction Methods : Choose from Lorentzian, KNN Pattern, Ensemble, EMA, or Linear Regression algorithms
Pattern Matching : Identifies similar historical volume patterns to project future volume
Adaptive Predictions : Generates volume forecasts for multiple bars into the future
Performance Tracking : Calculates and displays real-time prediction accuracy metrics
Normalized Scale : Presents volume as a percentage of historical maximums for consistent analysis
Customizable Visualization : Configure how predictions and actual volumes are displayed
Interactive Dashboard : View algorithm performance metrics in a customizable information panel
🎨 Visualization
Actual Volume Columns : Color-coded green/red bars showing current normalized volume
Prediction Columns : Semi-transparent blue columns representing predicted volume levels
Future Bar Projections : Forward-looking volume predictions with configurable transparency
Prediction Dots : Optional white dots highlighting future prediction points
Reference Lines : Visual guides showing the normalized volume scale
Performance Dashboard : Customizable panel displaying prediction method and accuracy metrics
📖 Usage Guidelines
History Lookback Period
Default: 20
Range: 5-100
This setting determines how many historical bars are analyzed for pattern matching. A longer period provides more historical data for pattern recognition but may reduce responsiveness to recent changes. A shorter period emphasizes recent market behavior but might miss longer-term patterns.
🧠 Prediction Method
Algorithm
Default: Lorentzian
Options: Lorentzian, KNN Pattern, Ensemble, EMA, Linear Regression
Selects the algorithm used for volume prediction:
Lorentzian: Uses Lorentzian distance metrics for pattern recognition, offering excellent noise resistance
KNN Pattern: Traditional K-Nearest Neighbors approach for historical pattern matching
Ensemble: Combines multiple methods with weighted averaging for robust predictions
EMA: Simple exponential moving average projection for trend-following predictions
Linear Regression: Projects future values based on linear trend analysis
Pattern Length
Default: 5
Range: 3-10
Defines the number of bars in each pattern for machine learning methods. Shorter patterns increase sensitivity to recent changes, while longer patterns may identify more complex structures but require more historical data.
Neighbors Count
Default: 3
Range: 1-5
Sets the K value (number of nearest neighbors) used in KNN and Lorentzian methods. Higher values produce smoother predictions by averaging more historical patterns, while lower values may capture more specific patterns but could be more susceptible to noise.
Prediction Horizon
Default: 5
Range: 1-10
Determines how many future bars to predict. Longer horizons provide more forward-looking information but typically decrease accuracy as the prediction window extends.
📊 Display Settings
Display Mode
Default: Overlay
Options: Overlay, Prediction Only
Controls how volume information is displayed:
Overlay: Shows both actual volume and predictions on the same chart
Prediction Only: Displays only the predictions without actual volume
Show Prediction Dots
Default: false
When enabled, adds white dots to future predictions for improved visibility and clarity.
Future Bar Transparency (%)
Default: 70
Range: 0-90
Controls the transparency of future prediction bars. Higher values make future bars more transparent, while lower values make them more visible.
📱 Dashboard Settings
Show Dashboard
Default: true
Toggles display of the prediction accuracy dashboard. When enabled, shows real-time accuracy metrics.
Dashboard Location
Default: Bottom Right
Options: Top Left, Top Right, Bottom Left, Bottom Right
Determines where the dashboard appears on the chart.
Dashboard Text Size
Default: Normal
Options: Small, Normal, Large
Controls the size of text in the dashboard for various display sizes.
Dashboard Style
Default: Solid
Options: Solid, Transparent
Sets the visual style of the dashboard background.
Understanding Accuracy Metrics
The dashboard provides key performance metrics to evaluate prediction quality:
Average Error
Shows the average difference between predicted and actual values
Positive values indicate the prediction tends to be higher than actual volume
Negative values indicate the prediction tends to be lower than actual volume
Values closer to zero indicate better prediction accuracy
Accuracy Percentage
A measure of how close predictions are to actual outcomes
Higher percentages (>70%) indicate excellent prediction quality
Moderate percentages (50-70%) indicate acceptable predictions
Lower percentages (<50%) suggest weaker prediction reliability
The accuracy metrics are color-coded for quick assessment:
Green: Strong prediction performance
Orange: Moderate prediction performance
Red: Weaker prediction performance
✅ Best Use Cases
Anticipate upcoming volume spikes or drops
Identify potential volume divergences from price action
Plan entries and exits around expected volume changes
Filter trading signals based on predicted volume support
Optimize position sizing by forecasting market participation
Prepare for potential volatility changes signaled by volume predictions
Enhance technical pattern analysis with volume projection context
⚠️ Limitations
Volume predictions become less accurate over longer time horizons
Performance varies based on market conditions and asset characteristics
Works best on liquid assets with consistent volume patterns
Requires sufficient historical data for pattern recognition
Sudden market events can disrupt prediction accuracy
Volume spikes may be muted in predictions due to normalization
💡 What Makes This Unique
Machine Learning Approach : Applies Lorentzian distance metrics for robust pattern matching
Algorithm Selection : Offers multiple prediction methods to suit different market conditions
Real-time Accuracy Tracking : Provides continuous feedback on prediction performance
Forward Projection : Visualizes multiple future bars with configurable display options
Normalized Scale : Presents volume as a percentage of maximum volume for consistent analysis
Interactive Dashboard : Displays key metrics with customizable appearance and placement
🔬 How It Works
The Volume Predictor processes market data through five main steps:
1. Volume Normalization:
Converts raw volume to percentage of maximum volume in lookback period
Creates consistent scale representation across different timeframes and assets
Stores historical normalized volumes for pattern analysis
2. Pattern Detection:
Identifies similar volume patterns in historical data
Uses Lorentzian distance metrics for robust similarity measurement
Determines strength of pattern match for prediction weighting
3. Algorithm Processing:
Applies selected prediction algorithm to historical patterns
For KNN/Lorentzian: Finds K nearest neighbors and calculates weighted prediction
For Ensemble: Combines multiple methods with optimized weighting
For EMA/Linear Regression: Projects trends based on statistical models
4. Accuracy Calculation:
Compares previous predictions to actual outcomes
Calculates average error and prediction accuracy
Updates performance metrics in real-time
5. Visualization:
Displays normalized actual volume with color-coding
Shows current and future volume predictions
Presents performance metrics through interactive dashboard
💡 Note:
The Volume Predictor performs optimally on liquid assets with established volume patterns. It’s most effective when used in conjunction with price action analysis and other technical indicators. The multi-algorithm approach allows adaptation to different market conditions by switching prediction methods. Pay special attention to the accuracy metrics when evaluating prediction reliability, as sudden market changes can temporarily reduce prediction quality. The normalized percentage scale makes the indicator consistent across different assets and timeframes, providing a standardized approach to volume analysis.
Accurate Bollinger Bands mcbw_ [True Volatility Distribution]The Bollinger Bands have become a very important technical tool for discretionary and algorithmic traders alike over the last decades. It was designed to give traders an edge on the markets by setting probabilistic values to different levels of volatility. However, some of the assumptions that go into its calculations make it unusable for traders who want to get a correct understanding of the volatility that the bands are trying to be used for. Let's go through what the Bollinger Bands are said to show, how their calculations work, the problems in the calculations, and how the current indicator I am presenting today fixes these.
--> If you just want to know how the settings work then skip straight to the end or click on the little (i) symbol next to the values in the indicator settings window when its on your chart <--
--------------------------- What Are Bollinger Bands ---------------------------
The Bollinger Bands were formed in the 1980's, a time when many retail traders interacted with their symbols via physically printed charts and computer memory for personal computer memory was measured in Kb (about a factor of 1 million smaller than today). Bollinger Bands are designed to help a trader or algorithm see the likelihood of price expanding outside of its typical range, the further the lines are from the current price implies the less often they will get hit. With a hands on understanding many strategies use these levels for designated levels of breakout trades or to assist in defining price ranges.
--------------------------- How Bollinger Bands Work ---------------------------
The calculations that go into Bollinger Bands are rather simple. There is a moving average that centers the indicator and an equidistant top band and bottom band are drawn at a fixed width away. The moving average is just a typical moving average (or common variant) that tracks the price action, while the distance to the top and bottom bands is a direct function of recent price volatility. The way that the distance to the bands is calculated is inspired by formulas from statistics. The standard deviation is taken from the candles that go into the moving average and then this is multiplied by a user defined value to set the bands position, I will call this value 'the multiple'. When discussing Bollinger Bands, that trading community at large normally discusses 'the multiple' as a multiplier of the standard deviation as it applies to a normal distribution (gaußian probability). On a normal distribution the number of standard deviations away (which trades directly use as 'the multiple') you are directly corresponds to how likely/unlikely something is to happen:
1 standard deviation equals 68.3%, meaning that the price should stay inside the 1 standard deviation 68.3% of the time and be outside of it 31.7% of the time;
2 standard deviation equals 95.5%, meaning that the price should stay inside the 2 standard deviation 95.5% of the time and be outside of it 4.5% of the time;
3 standard deviation equals 99.7%, meaning that the price should stay inside the 3 standard deviation 99.7% of the time and be outside of it 0.3% of the time.
Therefore when traders set 'the multiple' to 2, they interpret this as meaning that price will not reach there 95.5% of the time.
---------------- The Problem With The Math of Bollinger Bands ----------------
In and of themselves the Bollinger Bands are a great tool, but they have become misconstrued with some incorrect sense of statistical meaning, when they should really just be taken at face value without any further interpretation or implication.
In order to explain this it is going to get a bit technical so I will give a little math background and try to simplify things. First let's review some statistics topics (distributions, percentiles, standard deviations) and then with that understanding explore the incorrect logic of how Bollinger Bands have been interpreted/employed.
---------------- Quick Stats Review ----------------
.
(If you are comfortable with statistics feel free to skip ahead to the next section)
.
-------- I: Probability distributions --------
When you have a lot of data it is helpful to see how many times different results appear in your dataset. To visualize this people use "histograms", which just shows how many times each element appears in the dataset by stacking each of the same elements on top of each other to form a graph. You may be familiar with the bell curve (also called the "normal distribution", which we will be calling it by). The normal distribution histogram looks like a big hump around zero and then drops off super quickly the further you get from it. This shape (the bell curve) is very nice because it has a lot of very nifty mathematical properties and seems to show up in nature all the time. Since it pops up in so many places, society has developed many different shortcuts related to it that speed up all kinds of calculations, including the shortcut that 1 standard deviation = 68.3%, 2 standard deviations = 95.5%, and 3 standard deviations = 99.7% (these only apply to the normal distribution). Despite how handy the normal distribution is and all the shortcuts we have for it are, and how much it shows up in the natural world, there is nothing that forces your specific dataset to look like it. In fact, your data can actually have any possible shape. As we will explore later, economic and financial datasets *rarely* follow the normal distribution.
-------- II: Percentiles --------
After you have made the histogram of your dataset you have built the "probability distribution" of your own dataset that is specific to all the data you have collected. There is a whole complicated framework for how to accurately calculate percentiles but we will dramatically simplify it for our use. The 'percentile' in our case is just the number of data points we are away from the "middle" of the data set (normally just 0). Lets say I took the difference of the daily close of a symbol for the last two weeks, green candles would be positive and red would be negative. In this example my dataset of day by day closing price difference is:
week 1:
week 2:
sorting all of these value into a single dataset I have:
I can separate the positive and negative returns and explore their distributions separately:
negative return distribution =
positive return distribution =
Taking the 25th% percentile of these would just be taking the value that is 25% towards the end of the end of these returns. Or akin the 100%th percentile would just be taking the vale that is 100% at the end of those:
negative return distribution (50%) = -5
positive return distribution (50%) = +4
negative return distribution (100%) = -10
positive return distribution (100%) = +20
Or instead of separating the positive and negative returns we can also look at all of the differences in the daily close as just pure price movement and not account for the direction, in this case we would pool all of the data together by ignoring the negative signs of the negative reruns
combined return distribution =
In this case the 50%th and 100%th percentile of the combined return distribution would be:
combined return distribution (50%) = 4
combined return distribution (100%) = 10
Sometimes taking the positive and negative distributions separately is better than pooling them into a combined distribution for some purposes. Other times the combined distribution is better.
Most financial data has very different distributions for negative returns and positive returns. This is encapsulated in sayings like "Price takes the stairs up and the elevator down".
-------- III: Standard Deviation --------
The formula for the standard deviation (refereed to here by its shorthand 'STDEV') can be intimidating, but going through each of its elements will illuminate what it does. The formula for STDEV is equal to:
square root ( (sum ) / N )
Going back the the dataset that you might have, the variables in the formula above are:
'mean' is the average of your entire dataset
'x' is just representative of a single point in your dataset (one point at a time)
'N' is the total number of things in your dataset.
Going back to the STDEV formula above we can see how each part of it works. Starting with the '(x - mean)' part. What this does is it takes every single point of the dataset and measure how far away it is from the mean of the entire dataset. Taking this value to the power of two: '(x - mean) ^ 2', means that points that are very far away from the dataset mean get 'penalized' twice as much. Points that are very close to the dataset mean are not impacted as much. In practice, this would mean that if your dataset had a bunch of values that were in a wide range but always stayed in that range, this value ('(x - mean) ^ 2') would end up being small. On the other hand, if your dataset was full of the exact same number, but had a couple outliers very far away, this would have a much larger value since the square par of '(x - mean) ^ 2' make them grow massive. Now including the sum part of 'sum ', this just adds up all the of the squared distanced from the dataset mean. Then this is divided by the number of values in the dataset ('N'), and then the square root of that value is taken.
There is nothing inherently special or definitive about the STDEV formula, it is just a tool with extremely widespread use and adoption. As we saw here, all the STDEV formula is really doing is measuring the intensity of the outliers.
--------------------------- Flaws of Bollinger Bands ---------------------------
The largest problem with Bollinger Bands is the assumption that price has a normal distribution. This is assumption is massively incorrect for many reasons that I will try to encapsulate into two points:
Price return do not follow a normal distribution, every single symbol on every single timeframe has is own unique distribution that is specific to only itself. Therefore all the tools, shortcuts, and ideas that we use for normal distributions do not apply to price returns, and since they do not apply here they should not be used. A more general approach is needed that allows each specific symbol on every specific timeframe to be treated uniquely.
The distributions of price returns on the positive and negative side are almost never the same. A more general approach is needed that allows positive and negative returns to be calculated separately.
In addition to the issues of the normal distribution assumption, the standard deviation formula (as shown above in the quick stats review) is essentially just a tame measurement of outliers (a more aggressive form of outlier measurement might be taking the differences to the power of 3 rather than 2). Despite this being a bit of a philosophical question, does the measurement of outlier intensity as defined by the STDEV formula really measure what we want to know as traders when we're experiencing volatility? Or would adjustments to that formula better reflect what we *experience* as volatility when we are actively trading? This is an open ended question that I will leave here, but I wanted to pose this question because it is a key part of what how the Bollinger Bands work that we all assume as a given.
Circling back on the normal distribution assumption, the standard deviation formula used in the calculation of the bands only encompasses the deviation of the candles that go into the moving average and have no knowledge of the historical price action. Therefore the level of the bands may not really reflect how the price action behaves over a longer period of time.
------------ Delivering Factually Accurate Data That Traders Need------------
In light of the problems identified above, this indicator fixes all of these issue and delivers statistically correct information that discretionary and algorithmic traders can use, with truly accurate probabilities. It takes the price action of the last 2,000 candles and builds a huge dataset of distributions that you can directly select your percentiles from. It also allows you to have the positive and negative distributions calculated separately, or if you would like, you can pool all of them together in a combined distribution. In addition to this, there is a wide selection of moving averages directly available in the indicator to choose from.
Hedge funds, quant shops, algo prop firms, and advanced mechanical groups all employ the true return distributions in their work. Now you have access to the same type of data with this indicator, wherein it's doing all the lifting for you.
------------------------------ Indicator Settings ------------------------------
.
---- Moving average ----
Select the type of moving average you would like and its length
---- Bands ----
The percentiles that you enter here will be pulled directly from the return distribution of the last 2,000 candles. With the typical Bollinger Bands, traders would select 2 standard deviations and incorrectly think that the levels it highlights are the 95.5% levels. Now, if you want the true 95.5% level, you can just enter 95.5 into the percentile value here. Each of the three available bands takes the true percentile you enter here.
---- Separate Positive & Negative Distributions----
If this box is checked the positive and negative distributions are treated indecently, completely separate from each other. You will see that the width of the top and bottom bands will be different for each of the percentiles you enter.
If this box is unchecked then all the negative and positive distributions are pooled together. You will notice that the width of the top and bottom bands will be the exact same.
---- Distribution Size ----
This is the number of candles that the price return is calculated over. EG: to collect the price return over the last 33 candles, the difference of price from now to 33 candles ago is calculated for the last 2,000 candles, to build a return distribution of 2000 points of price differences over 33 candles.
NEGATIVE NUMBERS(<0) == exact number of candles to include;
EG: setting this value to -20 will always collect volatility distributions of 20 candles
POSITIVE NUMBERS(>0) == number of candles to include as a multiple of the Moving Average Length value set above;
EG: if the Moving Average Length value is set to 22, setting this value to 2 will use the last 22*2 = 44 candles for the collection of volatility distributions
MORE candles being include will generally make the bands WIDER and their size will change SLOWER over time.
I wish you focus, dedication, and earnest success on your journey.
Happy trading :)
Intellect_city - Halvings Bitcoin CycleWhat is halving?
The halving timer shows when the next Bitcoin halving will occur, as well as the dates of past halvings. This event occurs every 210,000 blocks, which is approximately every 4 years. Halving reduces the emission reward by half. The original Bitcoin reward was 50 BTC per block found.
Why is halving necessary?
Halving allows you to maintain an algorithmically specified emission level. Anyone can verify that no more than 21 million bitcoins can be issued using this algorithm. Moreover, everyone can see how much was issued earlier, at what speed the emission is happening now, and how many bitcoins remain to be mined in the future. Even a sharp increase or decrease in mining capacity will not significantly affect this process. In this case, during the next difficulty recalculation, which occurs every 2014 blocks, the mining difficulty will be recalculated so that blocks are still found approximately once every ten minutes.
How does halving work in Bitcoin blocks?
The miner who collects the block adds a so-called coinbase transaction. This transaction has no entry, only exit with the receipt of emission coins to your address. If the miner's block wins, then the entire network will consider these coins to have been obtained through legitimate means. The maximum reward size is determined by the algorithm; the miner can specify the maximum reward size for the current period or less. If he puts the reward higher than possible, the network will reject such a block and the miner will not receive anything. After each halving, miners have to halve the reward they assign to themselves, otherwise their blocks will be rejected and will not make it to the main branch of the blockchain.
The impact of halving on the price of Bitcoin
It is believed that with constant demand, a halving of supply should double the value of the asset. In practice, the market knows when the halving will occur and prepares for this event in advance. Typically, the Bitcoin rate begins to rise about six months before the halving, and during the halving itself it does not change much. On average for past periods, the upper peak of the rate can be observed more than a year after the halving. It is almost impossible to predict future periods because, in addition to the reduction in emissions, many other factors influence the exchange rate. For example, major hacks or bankruptcies of crypto companies, the situation on the stock market, manipulation of “whales,” or changes in legislative regulation.
---------------------------------------------
Table - Past and future Bitcoin halvings:
---------------------------------------------
Date: Number of blocks: Award:
0 - 03-01-2009 - 0 block - 50 BTC
1 - 28-11-2012 - 210000 block - 25 BTC
2 - 09-07-2016 - 420000 block - 12.5 BTC
3 - 11-05-2020 - 630000 block - 6.25 BTC
4 - 20-04-2024 - 840000 block - 3.125 BTC
5 - 24-03-2028 - 1050000 block - 1.5625 BTC
6 - 26-02-2032 - 1260000 block - 0.78125 BTC
7 - 30-01-2036 - 1470000 block - 0.390625 BTC
8 - 03-01-2040 - 1680000 block - 0.1953125 BTC
9 - 07-12-2043 - 1890000 block - 0.09765625 BTC
10 - 10-11-2047 - 2100000 block - 0.04882813 BTC
11 - 14-10-2051 - 2310000 block - 0.02441406 BTC
12 - 17-09-2055 - 2520000 block - 0.01220703 BTC
13 - 21-08-2059 - 2730000 block - 0.00610352 BTC
14 - 25-07-2063 - 2940000 block - 0.00305176 BTC
15 - 28-06-2067 - 3150000 block - 0.00152588 BTC
16 - 01-06-2071 - 3360000 block - 0.00076294 BTC
17 - 05-05-2075 - 3570000 block - 0.00038147 BTC
18 - 08-04-2079 - 3780000 block - 0.00019073 BTC
19 - 12-03-2083 - 3990000 block - 0.00009537 BTC
20 - 13-02-2087 - 4200000 block - 0.00004768 BTC
21 - 17-01-2091 - 4410000 block - 0.00002384 BTC
22 - 21-12-2094 - 4620000 block - 0.00001192 BTC
23 - 24-11-2098 - 4830000 block - 0.00000596 BTC
24 - 29-10-2102 - 5040000 block - 0.00000298 BTC
25 - 02-10-2106 - 5250000 block - 0.00000149 BTC
26 - 05-09-2110 - 5460000 block - 0.00000075 BTC
27 - 09-08-2114 - 5670000 block - 0.00000037 BTC
28 - 13-07-2118 - 5880000 block - 0.00000019 BTC
29 - 16-06-2122 - 6090000 block - 0.00000009 BTC
30 - 20-05-2126 - 6300000 block - 0.00000005 BTC
31 - 23-04-2130 - 6510000 block - 0.00000002 BTC
32 - 27-03-2134 - 6720000 block - 0.00000001 BTC
Adaptive Fisherized Z-scoreHello Fellas,
It's time for a new adaptive fisherized indicator of me, where I apply adaptive length and more on a classic indicator.
Today, I chose the Z-score, also called standard score, as indicator of interest.
Special Features
Advanced Smoothing: JMA, T3, Hann Window and Super Smoother
Adaptive Length Algorithms: In-Phase Quadrature, Homodyne Discriminator, Median and Hilbert Transform
Inverse Fisher Transform (IFT)
Signals: Enter Long, Enter Short, Exit Long and Exit Short
Bar Coloring: Presents the trade state as bar colors
Band Levels: Changes the band levels
Decision Making
When you create such a mod you need to think about which concepts are the best to conclude. I decided to take Inverse Fisher Transform instead of normalization to make a version which fits to a fixed scale to avoid the usual distortion created by normalization.
Moreover, I chose JMA, T3, Hann Window and Super Smoother, because JMA and T3 are the bleeding-edge MA's at the moment with the best balance of lag and responsiveness. Additionally, I chose Hann Window and Super Smoother because of their extraordinary smoothing capabilities and because Ehlers favours them.
Furthermore, I decided to choose the half length of the dominant cycle instead of the full dominant cycle to make the indicator more responsive which is very important for a signal emitter like Z-score. Signal emitters always need to be faster or have the same speed as the filters they are combined with.
Usage
The Z-score is a low timeframe scalper which works best during choppy/ranging phases. The direction you should trade is determined by the last trend change. E.g. when the last trend change was from bearish market to bullish market and you are now in a choppy/ranging phase confirmed by e.g. Chop Zone or KAMA slope you want to do long trades.
Interpretation
The Z-score indicator is a momentum indicator which shows the number of standard deviations by which the value of a raw score (price/source) is above or below the mean value of what is being observed or measured. Easily explained, it is almost the same as Bollinger Bands with another visual representation form.
Signals
B -> Buy -> Z-score crosses above lower band
S -> Short -> Z-score crosses below upper band
BE -> Buy Exit -> Z-score crosses above 0
SE -> Sell Exit -> Z-score crosses below 0
If you were reading till here, thank you already. Now, follows a bunch of knowledge for people who don't know the concepts I talk about.
T3
The T3 moving average, short for "Tim Tillson's Triple Exponential Moving Average," is a technical indicator used in financial markets and technical analysis to smooth out price data over a specific period. It was developed by Tim Tillson, a software project manager at Hewlett-Packard, with expertise in Mathematics and Computer Science.
The T3 moving average is an enhancement of the traditional Exponential Moving Average (EMA) and aims to overcome some of its limitations. The primary goal of the T3 moving average is to provide a smoother representation of price trends while minimizing lag compared to other moving averages like Simple Moving Average (SMA), Weighted Moving Average (WMA), or EMA.
To compute the T3 moving average, it involves a triple smoothing process using exponential moving averages. Here's how it works:
Calculate the first exponential moving average (EMA1) of the price data over a specific period 'n.'
Calculate the second exponential moving average (EMA2) of EMA1 using the same period 'n.'
Calculate the third exponential moving average (EMA3) of EMA2 using the same period 'n.'
The formula for the T3 moving average is as follows:
T3 = 3 * (EMA1) - 3 * (EMA2) + (EMA3)
By applying this triple smoothing process, the T3 moving average is intended to offer reduced noise and improved responsiveness to price trends. It achieves this by incorporating multiple time frames of the exponential moving averages, resulting in a more accurate representation of the underlying price action.
JMA
The Jurik Moving Average (JMA) is a technical indicator used in trading to predict price direction. Developed by Mark Jurik, it’s a type of weighted moving average that gives more weight to recent market data rather than past historical data.
JMA is known for its superior noise elimination. It’s a causal, nonlinear, and adaptive filter, meaning it responds to changes in price action without introducing unnecessary lag. This makes JMA a world-class moving average that tracks and smooths price charts or any market-related time series with surprising agility.
In comparison to other moving averages, such as the Exponential Moving Average (EMA), JMA is known to track fast price movement more accurately. This allows traders to apply their strategies to a more accurate picture of price action.
Inverse Fisher Transform
The Inverse Fisher Transform is a transform used in DSP to alter the Probability Distribution Function (PDF) of a signal or in our case of indicators.
The result of using the Inverse Fisher Transform is that the output has a very high probability of being either +1 or –1. This bipolar probability distribution makes the Inverse Fisher Transform ideal for generating an indicator that provides clear buy and sell signals.
Hann Window
The Hann function (aka Hann Window) is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing.
Super Smoother
The Super Smoother uses a special mathematical process for the smoothing of data points.
The Super Smoother is a technical analysis indicator designed to be smoother and with less lag than a traditional moving average.
Adaptive Length
Length based on the dominant cycle length measured by a "dominant cycle measurement" algorithm.
Happy Trading!
Best regards,
simwai
---
Credits to
@cheatcountry
@everget
@loxx
@DasanC
@blackcat1402
Support & Resistance AI (K means/median) [ThinkLogicAI]█ OVERVIEW
K-means is a clustering algorithm commonly used in machine learning to group data points into distinct clusters based on their similarities. While K-means is not typically used directly for identifying support and resistance levels in financial markets, it can serve as a tool in a broader analysis approach.
Support and resistance levels are price levels in financial markets where the price tends to react or reverse. Support is a level where the price tends to stop falling and might start to rise, while resistance is a level where the price tends to stop rising and might start to fall. Traders and analysts often look for these levels as they can provide insights into potential price movements and trading opportunities.
█ BACKGROUND
The K-means algorithm has been around since the late 1950s, making it more than six decades old. The algorithm was introduced by Stuart Lloyd in his 1957 research paper "Least squares quantization in PCM" for telecommunications applications. However, it wasn't widely known or recognized until James MacQueen's 1967 paper "Some Methods for Classification and Analysis of Multivariate Observations," where he formalized the algorithm and referred to it as the "K-means" clustering method.
So, while K-means has been around for a considerable amount of time, it continues to be a widely used and influential algorithm in the fields of machine learning, data analysis, and pattern recognition due to its simplicity and effectiveness in clustering tasks.
█ COMPARE AND CONTRAST SUPPORT AND RESISTANCE METHODS
1) K-means Approach:
Cluster Formation: After applying the K-means algorithm to historical price change data and visualizing the resulting clusters, traders can identify distinct regions on the price chart where clusters are formed. Each cluster represents a group of similar price change patterns.
Cluster Analysis: Analyze the clusters to identify areas where clusters tend to form. These areas might correspond to regions of price behavior that repeat over time and could be indicative of support and resistance levels.
Potential Support and Resistance Levels: Based on the identified areas of cluster formation, traders can consider these regions as potential support and resistance levels. A cluster forming at a specific price level could suggest that this level has been historically significant, causing similar price behavior in the past.
Cluster Standard Deviation: In addition to looking at the means (centroids) of the clusters, traders can also calculate the standard deviation of price changes within each cluster. Standard deviation is a measure of the dispersion or volatility of data points around the mean. A higher standard deviation indicates greater price volatility within a cluster.
Low Standard Deviation: If a cluster has a low standard deviation, it suggests that prices within that cluster are relatively stable and less likely to exhibit sudden and large price movements. Traders might consider placing tighter stop-loss orders for trades within these clusters.
High Standard Deviation: Conversely, if a cluster has a high standard deviation, it indicates greater price volatility within that cluster. Traders might opt for wider stop-loss orders to allow for potential price fluctuations without getting stopped out prematurely.
Cluster Density: Each data point is assigned to a cluster so a cluster that is more dense will act more like gravity and
2) Traditional Approach:
Trendlines: Draw trendlines connecting significant highs or lows on a price chart to identify potential support and resistance levels.
Chart Patterns: Identify chart patterns like double tops, double bottoms, head and shoulders, and triangles that often indicate potential reversal points.
Moving Averages: Use moving averages to identify levels where the price might find support or resistance based on the average price over a specific period.
Psychological Levels: Identify round numbers or levels that traders often pay attention to, which can act as support and resistance.
Previous Highs and Lows: Identify significant previous price highs and lows that might act as support or resistance.
The key difference lies in the approach and the foundation of these methods. Traditional methods are based on well-established principles of technical analysis and market psychology, while the K-means approach involves clustering price behavior without necessarily incorporating market sentiment or specific price patterns.
It's important to note that while the K-means approach might provide an interesting way to analyze price data, it should be used cautiously and in conjunction with other traditional methods. Financial markets are influenced by a wide range of factors beyond just price behavior, and the effectiveness of any method for identifying support and resistance levels should be thoroughly tested and validated. Additionally, developments in trading strategies and analysis techniques could have occurred since my last update.
█ K MEANS ALGORITHM
The algorithm for K means is as follows:
Initialize cluster centers
assign data to clusters based on minimum distance
calculate cluster center by taking the average or median of the clusters
repeat steps 1-3 until cluster centers stop moving
█ LIMITATIONS OF K MEANS
There are 3 main limitations of this algorithm:
Sensitive to Initializations: K-means is sensitive to the initial placement of centroids. Different initializations can lead to different cluster assignments and final results.
Assumption of Equal Sizes and Variances: K-means assumes that clusters have roughly equal sizes and spherical shapes. This may not hold true for all types of data. It can struggle with identifying clusters with uneven densities, sizes, or shapes.
Impact of Outliers: K-means is sensitive to outliers, as a single outlier can significantly affect the position of cluster centroids. Outliers can lead to the creation of spurious clusters or distortion of the true cluster structure.
█ LIMITATIONS IN APPLICATION OF K MEANS IN TRADING
Trading data often exhibits characteristics that can pose challenges when applying indicators and analysis techniques. Here's how the limitations of outliers, varying scales, and unequal variance can impact the use of indicators in trading:
Outliers are data points that significantly deviate from the rest of the dataset. In trading, outliers can represent extreme price movements caused by rare events, news, or market anomalies. Outliers can have a significant impact on trading indicators and analyses:
Indicator Distortion: Outliers can skew the calculations of indicators, leading to misleading signals. For instance, a single extreme price spike could cause indicators like moving averages or RSI (Relative Strength Index) to give false signals.
Risk Management: Outliers can lead to overly aggressive trading decisions if not properly accounted for. Ignoring outliers might result in unexpected losses or missed opportunities to adjust trading strategies.
Different Scales: Trading data often includes multiple indicators with varying units and scales. For example, prices are typically in dollars, volume in units traded, and oscillators have their own scale. Mixing indicators with different scales can complicate analysis:
Normalization: Indicators on different scales need to be normalized or standardized to ensure they contribute equally to the analysis. Failure to do so can lead to one indicator dominating the analysis due to its larger magnitude.
Comparability: Without normalization, it's challenging to directly compare the significance of indicators. Some indicators might have a larger numerical range and could overshadow others.
Unequal Variance: Unequal variance in trading data refers to the fact that some indicators might exhibit higher volatility than others. This can impact the interpretation of signals and the performance of trading strategies:
Volatility Adjustment: When combining indicators with varying volatility, it's essential to adjust for their relative volatilities. Failure to do so might lead to overemphasizing or underestimating the importance of certain indicators in the trading strategy.
Risk Assessment: Unequal variance can impact risk assessment. Indicators with higher volatility might lead to riskier trading decisions if not properly taken into account.
█ APPLICATION OF THIS INDICATOR
This indicator can be used in 2 ways:
1) Make a directional trade:
If a trader thinks price will go higher or lower and price is within a cluster zone, The trader can take a position and place a stop on the 1 sd band around the cluster. As one can see below, the trader can go long the green arrow and place a stop on the one standard deviation mark for that cluster below it at the red arrow. using this we can calculate a risk to reward ratio.
Calculating risk to reward: targeting a risk reward ratio of 2:1, the trader could clearly make that given that the next resistance area above that in the orange cluster exceeds this risk reward ratio.
2) Take a reversal Trade:
We can use cluster centers (support and resistance levels) to go in the opposite direction that price is currently moving in hopes of price forming a pivot and reversing off this level.
Similar to the directional trade, we can use the standard deviation of the cluster to place a stop just in case we are wrong.
In this example below we can see that shorting on the red arrow and placing a stop at the one standard deviation above this cluster would give us a profitable trade with minimal risk.
Using the cluster density table in the upper right informs the trader just how dense the cluster is. Higher density clusters will give a higher likelihood of a pivot forming at these levels and price being rejected and switching direction with a larger move.
█ FEATURES & SETTINGS
General Settings:
Number of clusters: The user can select from 3 to five clusters. A good rule of thumb is that if you are trading intraday, less is more (Think 3 rather than 5). For daily 4 to 5 clusters is good.
Cluster Method: To get around the outlier limitation of k means clustering, The median was added. This gives the user the ability to choose either k means or k median clustering. K means is the preferred method if the user things there are no large outliers, and if there appears to be large outliers or it is assumed there are then K medians is preferred.
Bars back To train on: This will be the amount of bars to include in the clustering. This number is important so that the user includes bars that are recent but not so far back that they are out of the scope of where price can be. For example the last 2 years we have been in a range on the sp500 so 505 days in this setting would be more relevant than say looking back 5 years ago because price would have to move far to get there.
Show SD Bands: Select this to show the 1 standard deviation bands around the support and resistance level or unselect this to just show the support and resistance level by itself.
Features:
Besides the support and resistance levels and standard deviation bands, this indicator gives a table in the upper right hand corner to show the density of each cluster (support and resistance level) and is color coded to the cluster line on the chart. Higher density clusters mean price has been there previously more than lower density clusters and could mean a higher likelihood of a reversal when price reaches these areas.
█ WORKS CITED
Victor Sim, "Using K-means Clustering to Create Support and Resistance", 2020, towardsdatascience.com
Chris Piech, "K means", stanford.edu
█ ACKNOLWEDGMENTS
@jdehorty- Thanks for the publish template. It made organizing my thoughts and work alot easier.
Recursive Micro Zigzag🎲 Overview
Zigzag is basic building block for any pattern recognition algorithm. This indicator is a research-oriented tool that combines the concepts of Micro Zigzag and Recursive Zigzag to facilitate a comprehensive analysis of price patterns. This indicator focuses on deriving zigzag on multiple levels in more efficient and enhanced manner in order to support enhanced pattern recognition.
The Recursive Micro Zigzag Indicator utilises the Micro Zigzag as the foundation and applies the Recursive Zigzag technique to derive higher-level zigzags. By integrating these techniques, this indicator enables researchers to analyse price patterns at multiple levels and gain a deeper understanding of market behaviour.
🎲 Concept:
Micro Zigzag Base : The indicator utilises the Micro Zigzag concept to capture detailed price movements within each candle. It allows for the visualisation of the sequential price action within the candle, aiding in pattern recognition at a micro level.
Basic implementation of micro zigzag can be found in this link - Micro-Zigzag
Recursive Zigzag Expansion : Building upon the Micro Zigzag base, the indicator applies the Recursive Zigzag concept to derive higher-level zigzags. Through recursive analysis of the Micro Zigzag's pivots, the indicator uncovers intricate patterns and trends that may not be evident in single-level zigzags.
Earlier implementations of recursive zigzag can be found here:
Recursive Zigzag
Recursive Zigzag - Trendoscope
And the libraries
rZigzag
ZigzagMethods
The major differences in this implementation are
Micro Zigzag Base - Earlier implementation made use of standard zigzag as base whereas this implementation uses Micro Zigzag as base
Not cap on Pivot depth - Earlier implementation was limited by the depth of level 0 zigzag. In this implementation, we are trying to build the recursive algorithm progressively so that there is no cap on the depth of level 0 zigzag. But, if we go for higher levels, there is chance of program timing out due to pine limitations.
These algorithms are useful in automatically spotting patterns on the chart including Harmonic Patterns, Chart Patterns, Elliot Waves and many more.
Channel Based Zigzag [HeWhoMustNotBeNamed]🎲 Concept
Zigzag is built based on the price and number of offset bars. But, in this experiment, we build zigzag based on different bands such as Bollinger Band, Keltner Channel and Donchian Channel. The process is simple:
🎯 Derive bands based on input parameters
🎯 High of a bar is considered as pivot high only if the high price is above or equal to upper band.
🎯 Similarly low of a bar is considered as pivot low only if low price is below or equal to lower band.
🎯 Adding the pivot high/low follows same logic as that of regular zigzag where pivot high is always followed by pivot low and vice versa.
🎯 If the new pivot added is of same direction as that of last pivot, then both pivots are compared with each other and only the extreme one is kept. (Highest in case of pivot high and lowest in case of pivot low)
🎯 If a bar has both pivot high and pivot low - pivot with same direction as previous pivot is added to the list first before adding the pivot with opposite direction.
🎲 Use Cases
Can be used for pattern recognition algorithms instead of standard zigzag. This will help derive patterns which are relative to bands and channels.
Example: John Bollinger explains how to manually scan double tap using Bollinger Bands in this video: www.youtube.com This modified zigzag base can be used to achieve the same using algorithmic means.
🎲 Settings
Few simple configurations which will let you select the band properties. Notice that there is no zigzag length here. All the calculations depend on the bands.
With bands display, indicator looks something like this
Note that pivots do not always represent highest/lowest prices. They represent highest/lowest price relative to bands.
As mentioned many times, application of zigzag is not for buying at lower price and selling at higher price. It is mainly used for pattern recognition either manually or via algorithms. Lets build new Harmonic, Chart patterns, Trend Lines using the new zigzag?
cbndLibrary "cbnd"
Description:
A standalone Cumulative Bivariate Normal Distribution (CBND) functions that do not require any external libraries.
This includes 3 different CBND calculations: Drezner(1978), Drezner and Wesolowsky (1990), and Genz (2004)
Comments:
The standardized cumulative normal distribution function returns the probability that one random
variable is less than a and that a second random variable is less than b when the correlation
between the two variables is p. Since no closed-form solution exists for the bivariate cumulative
normal distribution, we present three approximations. The first one is the well-known
Drezner (1978) algorithm. The second one is the more efficient Drezner and Wesolowsky (1990)
algorithm. The third is the Genz (2004) algorithm, which is the most accurate one and therefore
our recommended algorithm. West (2005b) and Agca and Chance (2003) discuss the speed and
accuracy of bivariate normal distribution approximations for use in option pricing in
ore detail.
Reference:
The Complete Guide to Option Pricing Formulas, 2nd ed. (Espen Gaarder Haug)
CBND1(A, b, rho)
Returns the Cumulative Bivariate Normal Distribution (CBND) using Drezner 1978 Algorithm
Parameters:
A : float,
b : float,
rho : float,
Returns: float.
CBND2(A, b, rho)
Returns the Cumulative Bivariate Normal Distribution (CBND) using Drezner and Wesolowsky (1990) function
Parameters:
A : float,
b : float,
rho : float,
Returns: float.
CBND3(x, y, rho)
Returns the Cumulative Bivariate Normal Distribution (CBND) using Genz (2004) algorithm (this is the preferred method)
Parameters:
x : float,
y : float,
rho : float,
Returns: float.
STD-Stepped Fast Cosine Transform Moving Average [Loxx]STD-Stepped Fast Cosine Transform Moving Average is an experimental moving average that uses Fast Cosine Transform to calculate a moving average. This indicator has standard deviation stepping in order to smooth the trend by weeding out low volatility movements.
What is the Discrete Cosine Transform?
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT, which is often called simply "the DCT". This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply "the inverse DCT" or "the IDCT". Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to MD signals. There are several algorithms to compute MD DCT. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT, : ix, xiii, 1, 141–304 used in several ISO/IEC and ITU-T international standards.
Notable settings
windowper = period for calculation, restricted to powers of 2: "16", "32", "64", "128", "256", "512", "1024", "2048", this reason for this is FFT is an algorithm that computes DFT (Discrete Fourier Transform) in a fast way, generally in 𝑂(𝑁⋅log2(𝑁)) instead of 𝑂(𝑁2). To achieve this the input matrix has to be a power of 2 but many FFT algorithm can handle any size of input since the matrix can be zero-padded. For our purposes here, we stick to powers of 2 to keep this fast and neat. read more about this here: Cooley–Tukey FFT algorithm
smthper = smoothing count, this smoothing happens after the first FCT regular pass. this zeros out frequencies from the previously calculated values above SS count. the lower this number, the smoother the output, it works opposite from other smoothing periods
Included
Alerts
Signals
Loxx's Expanded Source Types
Additional reading
A Fast Computational Algorithm for the Discrete Cosine Transform by Chen et al.
Practical Fast 1-D DCT Algorithms With 11 Multiplications by Loeffler et al.
Cooley–Tukey FFT algorithm
Weighted Burg AR Spectral Estimate Extrapolation of Price [Loxx]Weighted Burg AR Spectral Estimate Extrapolation of Price is an indicator that uses an autoregressive spectral estimation called the Weighted Burg Algorithm. This method is commonly used in speech modeling and speech prediction engines. This method also includes Levinson–Durbin algorithm. As was already discussed previously in the following indicator:
Levinson-Durbin Autocorrelation Extrapolation of Price
What is Levinson recursion or Levinson–Durbin recursion?
In many applications, the duration of an uninterrupted measurement of a time series is limited. However, it is often possible to obtain several separate segments of data. The estimation of an autoregressive model from this type of data is discussed. A straightforward approach is to take the average of models estimated from each segment separately. In this way, the variance of the estimated parameters is reduced. However, averaging does not reduce the bias in the estimate. With the Burg algorithm for segments, both the variance and the bias in the estimated parameters are reduced by fitting a single model to all segments simultaneously. As a result, the model estimated with the Burg algorithm for segments is more accurate than models obtained with averaging. The new weighted Burg algorithm for segments allows combining segments of different amplitudes.
The Burg algorithm estimates the AR parameters by determining reflection coefficients that minimize the sum of for-ward and backward residuals. The extension of the algorithm to segments is that the reflection coefficients are estimated by minimizing the sum of forward and backward residuals of all segments taken together. This means a single model is fitted to all segments in one time. This concept is also used for prediction error methods in system identification, where the input to the system is known, like in ARX modeling
Data inputs
Source Settings: -Loxx's Expanded Source Types. You typically use "open" since open has already closed on the current active bar
LastBar - bar where to start the prediction
PastBars - how many bars back to model
LPOrder - order of linear prediction model; 0 to 1
FutBars - how many bars you want to forward predict
BurgWin - weighing function index, rectangular, hamming, or parabolic
Things to know
Normally, a simple moving average is calculated on source data. I've expanded this to 38 different averaging methods using Loxx's Moving Avreages.
This indicator repaints
Included
Bar color muting
Further reading
Performance of the weighted burg methods of ar spectral estimation for pitch-synchronous analysis of voiced speech
The Burg algorithm for segments
Techniques for the Enhancement of Linear Predictive Speech Coding in Adverse Conditions
Related Indicators
Synthetic Price Action GeneratorNOTICE:
First thing you need to know, it "DOES NOT" reflect the price of the ticker you will load it on. THIS IS NOT AN INDICATOR FOR TRADING! It's a developer tool solely generating random values that look exactly like the fractals we observe every single day. This script's generated candles are as fake as the never ending garbage news cycles we are often force fed and expected to believe by using carefully scripted narratives peddled as hypnotic truth to psychologically and emotionally influence you to the point of control by coercion and subjugation. I wanted to make the script's synthetic nature very clear using that analogy, it's dynamically artificial. Do not accidentally become disillusioned by this scripts values, make trading decisions from it, and lastly don't become victim to predatory media magic ministry parrots with pretty, handsome smiles, compelling you to board their ferris wheel of fear. Now, on to the good stuff...
BACKSTORY:
Occasionally I find myself in situations where I have to build analyzers in Pine to actually build novel quantitative analytic indicators and tools worthy of future use. These analyzers certainly don't exist on this platform, but usually are required to engineer and tweak algorithms of the highest quality with the finest computational caliber. I have numerous other synthesizers to publish besides this one.
For many reasons, I needed a synthetic environment to utilize the analyzers I built in Pine, to even pursue building some exotic indicators and algorithms. Pine doesn't allow sourcing of tuples. Not to mention, I required numerous Pine advancements to make long held dreams into tangible realities. Many Pine upgrades have arrived and MANY, MANY more are in need of implementation for all. Now that I have this, intending to use it in the future often when in need, you can now use it too. I do anticipate some skilled Pine poets will employ this intended handy utility to design and/or improved indicators for trading.
ORIGIN:
This was inspired by the brilliance from the world renowned ALGOmist John F. Ehlers, but it's taken on a completely alien form from its original DNA. Browsing on the internet for something else, I came across an article with a small code snippet, and I remembered an old wish of mine. I have long known that by flipping back and forth on specific tickers and timeframes in my Watchlist is not the most efficient way to evaluate indicators in multiple theatres of price action. I realized, I always wanted to possess and use this sort of tool, so... I put it into Pine form, but now have decided to inject it with Pine Script steroids. The outcome is highly mutable candle formations in a reusable mutagenic package, observable above and masquerading as genuine looking price candles.
OVERVIEW:
I guess you could call it a price action synthesizer, but I entitled it "Synthetic Price Action Generator" for those who may be searching for such a thing. You may find this more useful on the All or 5Y charts initially to witness indication from beginning (barstate.isfirst === barindex==0) to end (last_bar_index), but you may also use keyboard shortcuts + + to view the earliest plottable bars on any timeframe. I often use that keyboard shortcut to qualify an indicator through the entirety of it's runtime.
A lot can go wrong unexpectedly with indicator initialization, and you will never know it if you don't inspect it. Many recursively endowed Infinite Impulse Response (IIR) Filters can initialize with unintended results that minutely ring in slightly erroneous fashion for the entire runtime, beginning to end, causing deviations from "what should of been..." values with false signals. Looking closely at spg(), you will recognize that 3 EMAs are employed to manage and maintain randomness of CLOSE, HIGH, and LOW. In fact, any indicator's barindex==0 initialization can be inspected with the keyboard shortcuts above. If you see anything obviously strange in an authors indicator, please contact the developer if possible and respectfully notify them.
PURPOSE:
The primary intended application of this script, is to offer developers from advanced to even novice skill levels assistance with building next generation indicators. Mostly, it's purpose is for testing and troubleshooting indicators AND evaluating how they perform in a "manageable" randomized environment. Some times indicators flake out on rare but problematic price fluctuations, and this may help you with finding your issues/errata sooner than later. While the candles upon initial loading look pristine, by tweaking it to the minval/maxval parameters limits OR beyond with a few code modifications, you can generate unusual volatility, for instance... huge wicks. Limits of minval= and maxval= of are by default set to a comfort zone of operation. Massive wicks or candle bodies will undoubtedly affect your indication and often render them useless on tickers that exhibit that behavior, like WGMCF intraday currently.
Copy/paste boundaries are provided for relevant insertion into another script. Paste placement should happen at the very top of a script. Note that by overwriting the close, open, high, etc... values, your compiler will give you generous warnings of "variable shadowing" in abundance, but this is an expected part of applying it to your novel script, no worries. plotcandle() can be copied over too and enabled/disabled in Settings->Style. Always remember to fully remove this scripts' code and those assignments properly before actual trading use of your script occurs, AND specifically when publishing. The entirety of this provided code should never, never exist in a published indicator.
OTHER INTENTIONS:
Even though these are 100% synthetic generated price points, you will notice ALL of the fractal pseudo-patterns that commonly exist in the markets, are naturally occurring with this generator too. You can also swiftly immerse yourself in pattern recognition exercises with increased efficiency in real time by clicking any SPAG Setting in focus and then using the up/down arrow keys. I hope I explained potential uses adequately...
On a personal note, the existence of fractal symmetry often makes me wonder, do we truly live in a totality chaotic universe or is it ordered mathematically for some outcomes to a certain extent. I think both. My observations, it's a pre-deterministic reality completely influenced by infinitesimal amounts of sentient free will with unimaginable existing and emerging quantities. Some how an unknown mysterious mechanism governing the totality of universal physics and mathematics counts this 100.0% flawlessly and perpetually. Anyways, you can't change the past that long existed before your birth or even yesterday, but you can choose to dream, create, and forge the future into your desires and hopes. As always, shite always happens when your not looking for it. What you choose to do after stepping in it unintentionally... is totally up to you. :) Maybe this tool and tips provided will aid you in not stepping in an algo cachucha up to your ankles somehow.
SCRIPTING LESSONS PORTRAYED IN THIS SCRIPT:
Pine etiquette and code cleanliness
Overwrite capabilities of built-in Pine variables for testing indicators
Various techniques to organize Settings panel while providing ease of adjustment utility
Use of tooltip= to provide users adequate valuable information. Most people want to trade with indicators, not blindly make adjustments to them without any knowledge of their intended operation/effects
When available time provides itself, I will consider your inquiries, thoughts, and concepts presented below in the comments section, should you have any questions or comments regarding this indicator. When my indicators achieve more prevalent use by TV members , I may implement more ideas when they present themselves as worthy additions. Have a profitable future everyone!
the "fasle" hull moving averageThere is a little different between my "fasle hull moving average" the "correct one".
the correct algorithm:
hma = wma((2*wma(close,n/2) - wma(close,n),sqrt(n))
the "fasle" algorithm:
=wma((2*wma(close,n/4) - wma(close,n),sqrt(n))
Amazing! Why the "fasle" describe the trend so accurate!?
Harmonic Patterns + Fib [CRT Trader]Overview
The Harmonic Patterns Fibonacci indicator is an advanced technical analysis tool designed to automatically detect and visualize Fibonacci-based harmonic patterns on financial charts. This indicator helps traders identify high-probability reversal zones and potential entry/exit points based on precise mathematical relationships.
Supported Patterns
5-Point Patterns (X-A-B-C-D Structure)
Gartley Pattern: The most common harmonic pattern with reliable reversal signals
AB/XA = 0.618, BC/AB = 0.618, CD/BC = 1.272, AD/XA = 0.786
Butterfly Pattern: Strong reversal pattern indicating potential trend changes
AB/XA = 0.786, BC/AB = 0.618, CD/BC = 1.618, AD/XA = 1.270
Bat Pattern: Medium-term reversal pattern with high accuracy
AB/XA = 0.382, BC/AB = 0.886, CD/BC = 1.618, AD/XA = 0.886
Crab Pattern: Aggressive reversal pattern with extended D point
AB/XA = 0.618, BC/AB = 0.886, CD/BC = 2.240, AD/XA = 1.618
Shark Pattern: Trend continuation or reversal pattern
AB/XA = 0.618, BC/AB = 1.130, CD/BC = 1.618, AD/XA = 0.886
4-Point Pattern (A-B-C-D Structure)
ABCD Pattern: Basic harmonic structure forming the foundation of all patterns
BC/AB = 0.382-0.886, CD/BC = 1.130-2.618
Key Features
Fibonacci Validation
Each pattern is validated against precise Fibonacci ratios with customizable tolerance
Mathematical accuracy ensures reliable pattern recognition
Eliminates false signals through strict ratio requirements
Performance Optimization
Pivot Detection: Automatically identifies significant highs and lows
Scan Frequency Control: Adjustable scanning intervals to optimize performance
Early Exit Algorithms: Efficient computation to reduce processing load
Pattern Limit: Control maximum number of patterns displayed
Visual Elements
Pattern Lines: Clear visualization of pattern structure with colored lines
Fill Areas: Highlighted zones between pattern legs
Point Labels: X, A, B, C, D markers for easy identification
Fibonacci Levels: Optional Fibonacci retracement/extension levels
Bullish/Bearish Colors: Green for bullish, red for bearish patterns
Customizable Settings
Pattern Selection: Enable/disable specific pattern types
Tolerance Adjustment: Fine-tune pattern recognition sensitivity (5-30%)
Color Customization: Personalize visual appearance
Information Table: Optional statistics display
Trading Applications
Entry Signals
Reversal Zones: Identify high-probability reversal areas at pattern completion
Confluence Trading: Combine with other technical indicators for confirmation
Risk Management: Use pattern structure to define stop-loss levels
Market Analysis
Support/Resistance: Pattern points often act as future S/R levels
Price Targets: Fibonacci extensions provide potential profit targets
Market Structure: Understand underlying market geometry and rhythm
Strategy Integration
Swing Trading: Ideal for medium-term position entries
Position Trading: Long-term trend reversal identification
Day Trading: Intraday reversal patterns on lower timeframes
How to Use
Add to Chart: Apply the indicator to any timeframe and instrument
Configure Settings: Adjust tolerance, colors, and pattern types as needed
Wait for Completion: Patterns are valid only when D point is formed
Confirm with Volume: Look for volume confirmation at pattern completion
Set Stop Loss: Place stops beyond X point for 5-point patterns, or A point for ABCD
Target Levels: Use Fibonacci extensions for profit targets
Important Notes
Pattern Completion: Wait for full pattern formation before taking action
Market Context: Consider overall market trend and conditions
Risk Management: Always use appropriate position sizing and stops
Backtesting: Test the indicator on historical data before live trading
Multiple Timeframes: Analyze patterns across different timeframes for confirmation
Technical Requirements
Lookback Period: Adjustable pivot detection sensitivity
Depth Setting: Controls how far back the algorithm searches for patterns
Memory Efficient: Optimized for real-time performance without lag
This indicator is suitable for all experience levels, from beginners learning harmonic patterns to advanced traders seeking automated pattern recognition. The combination of mathematical precision and visual clarity makes it an essential tool for harmonic trading strategies.
Auto-Fit Growth Trendline# **Theoretical Algorithmic Principles of the Auto-Fit Growth Trendline (AFGT)**
## **🎯 What Does This Algorithm Do?**
The Auto-Fit Growth Trendline is an advanced technical analysis system that **automates the identification of long-term growth trends** and **projects future price levels** based on historical cyclical patterns.
### **Primary Functionality:**
- **Automatically detects** the most significant lows in regular periods (monthly, quarterly, semi-annually, annually)
- **Constructs a dynamic trendline** that connects these historical lows
- **Projects the trend into the future** with high mathematical precision
- **Generates Fibonacci bands** that act as dynamic support and resistance levels
- **Automatically adapts** to different timeframes and market conditions
### **Strategic Purpose:**
The algorithm is designed to identify **fundamental value zones** where price has historically found support, enabling traders to:
- Identify optimal entry points for long positions
- Establish realistic price targets based on mathematical projections
- Recognize dynamic support and resistance levels
- Anticipate long-term price movements
---
## **🧮 Core Mathematical Foundations**
### **Adaptive Temporal Segmentation Theory**
The algorithm is based on **dynamic temporal partition theory**, where time is divided into mathematically coherent uniform intervals. It uses modular transformations to create bijective mappings between continuous timestamps and discrete periods, ensuring each temporal point belongs uniquely to a specific period.
**What does this achieve?** It allows the algorithm to automatically identify natural market cycles (annual, quarterly, etc.) without manual intervention, adapting to the inherent periodicity of each asset.
The temporal mapping function implements a **discrete affine transformation** that normalizes different frequencies (monthly, quarterly, semi-annual, annual) to a space of unique identifiers, enabling consistent cross-temporal comparative analysis.
---
## **📊 Local Extrema Detection Theory**
### **Multi-Point Retrospective Validation Principle**
Local minima detection is founded on **relative extrema theory with sliding window**. Instead of using a simple minimum finder, it implements a cross-validation system that examines the persistence of the extremum across multiple historical periods.
**What problem does this solve?** It eliminates false minima caused by temporal volatility, identifying only those points that represent true historical support levels with statistical significance.
This approach is based on the **statistical confirmation principle**, where a minimum is only considered valid if it maintains its extremum condition during a defined observation period, significantly reducing false positives caused by transitory volatility.
---
## **🔬 Robust Interpolation Theory with Outlier Control**
### **Contextual Adaptive Interpolation Model**
The mathematical core uses **piecewise linear interpolation with adaptive outlier correction**. The key innovation lies in implementing a **contextual anomaly detector** that identifies not only absolute extreme values, but relative deviations to the local context.
**Why is this important?** Financial markets contain extreme events (crashes, bubbles) that can distort projections. This system identifies and appropriately weights them without completely eliminating them, preserving directional information while attenuating distortions.
### **Implicit Bayesian Smoothing Algorithm**
When an outlier is detected (deviation >300% of local average), the system applies a **simplified Kalman filter** that combines the current observation with a local trend estimation, using a weight factor that preserves directional information while attenuating extreme fluctuations.
---
## **📈 Stabilized Extrapolation Theory**
### **Exponential Growth Model with Dampening**
Extrapolation is based on a **modified exponential growth model with progressive dampening**. It uses multiple historical points to calculate local growth ratios, implements statistical filtering to eliminate outliers, and applies a dampening factor that increases with extrapolation distance.
**What advantage does this offer?** Long-term projections in finance tend to be exponentially unrealistic. This system maintains short-to-medium term accuracy while converging toward realistic long-term projections, avoiding the typical "exponential explosions" of other methods.
### **Asymptotic Convergence Principle**
For long-term projections, the algorithm implements **controlled asymptotic convergence**, where growth ratios gradually converge toward pre-established limits, avoiding unrealistic exponential projections while preserving short-to-medium term accuracy.
---
## **🌟 Dynamic Fibonacci Projection Theory**
### **Continuous Proportional Scaling Model**
Fibonacci bands are constructed through **uniform proportional scaling** of the base curve, where each level represents a linear transformation of the main curve by a constant factor derived from the Fibonacci sequence.
**What is its practical utility?** It provides dynamic resistance and support levels that move with the trend, offering price targets and profit-taking points that automatically adapt to market evolution.
### **Topological Preservation Principle**
The system maintains the **topological properties** of the base curve in all Fibonacci projections, ensuring that spatial and temporal relationships are consistently preserved across all resistance/support levels.
---
## **⚡ Adaptive Computational Optimization**
### **Multi-Scale Resolution Theory**
It implements **automatic multi-resolution analysis** where data granularity is dynamically adjusted according to the analysis timeframe. It uses the **adaptive Nyquist principle** to optimize the signal-to-noise ratio according to the temporal observation scale.
**Why is this necessary?** Different timeframes require different levels of detail. A 1-minute chart needs more granularity than a monthly one. This system automatically optimizes resolution for each case.
### **Adaptive Density Algorithm**
Calculation point density is optimized through **adaptive sampling theory**, where calculation frequency is adjusted according to local trend curvature and analysis timeframe, balancing visual precision with computational efficiency.
---
## **🛡️ Robustness and Fault Tolerance**
### **Graceful Degradation Theory**
The system implements **multi-level graceful degradation**, where under error conditions or insufficient data, the algorithm progressively falls back to simpler but reliable methods, maintaining basic functionality under any condition.
**What does this guarantee?** That the indicator functions consistently even with incomplete data, new symbols with limited history, or extreme market conditions.
### **State Consistency Principle**
It uses **mathematical invariants** to guarantee that the algorithm's internal state remains consistent between executions, implementing consistency checks that validate data structure integrity in each iteration.
---
## **🔍 Key Theoretical Innovations**
### **A. Contextual vs. Absolute Outlier Detection**
It revolutionizes traditional outlier detection by considering not only the absolute magnitude of deviations, but their relative significance within the local context of the time series.
**Practical impact:** It distinguishes between legitimate market movements and technical anomalies, preserving important events like breakouts while filtering noise.
### **B. Extrapolation with Weighted Historical Memory**
It implements a memory system that weights different historical periods according to their relevance for current prediction, creating projections more adaptable to market regime changes.
**Competitive advantage:** It automatically adapts to fundamental changes in asset dynamics without requiring manual recalibration.
### **C. Automatic Multi-Timeframe Adaptation**
It develops an automatic temporal resolution selection system that optimizes signal extraction according to the intrinsic characteristics of the analysis timeframe.
**Result:** A single indicator that functions optimally from 1-minute to monthly charts without manual adjustments.
### **D. Intelligent Asymptotic Convergence**
It introduces the concept of controlled asymptotic convergence in financial extrapolations, where long-term projections converge toward realistic limits based on historical fundamentals.
**Added value:** Mathematically sound long-term projections that avoid the unrealistic extremes typical of other extrapolation methods.
---
## **📊 Complexity and Scalability Theory**
### **Optimized Linear Complexity Model**
The algorithm maintains **linear computational complexity** O(n) in the number of historical data points, guaranteeing scalability for extensive time series analysis without performance degradation.
### **Temporal Locality Principle**
It implements **temporal locality**, where the most expensive operations are concentrated in the most relevant temporal regions (recent periods and near projections), optimizing computational resource usage.
---
## **🎯 Convergence and Stability**
### **Probabilistic Convergence Theory**
The system guarantees **probabilistic convergence** toward the real underlying trend, where projection accuracy increases with the amount of available historical data, following **law of large numbers** principles.
**Practical implication:** The more history an asset has, the more accurate the algorithm's projections will be.
### **Guaranteed Numerical Stability**
It implements **intrinsic numerical stability** through the use of robust floating-point arithmetic and validations that prevent overflow, underflow, and numerical error propagation.
**Result:** Reliable operation even with extreme-priced assets (from satoshis to thousand-dollar stocks).
---
## **💼 Comprehensive Practical Application**
**The algorithm functions as a "financial GPS"** that:
1. **Identifies where we've been** (significant historical lows)
2. **Determines where we are** (current position relative to the trend)
3. **Projects where we're going** (future trend with specific price levels)
4. **Provides alternative routes** (Fibonacci bands as alternative targets)
This theoretical framework represents an innovative synthesis of time series analysis, approximation theory, and computational optimization, specifically designed for long-term financial trend analysis with robust and mathematically grounded projections.
