Logarithmic Regression AlternativeLogarithmic regression is typically used to model situations where growth or decay accelerates rapidly at first and then slows over time. Bitcoin is a good example.
𝑦 = 𝑎 + 𝑏 * ln(𝑥)
With this logarithmic regression (log reg) formula 𝑦 (price) is calculated with constants 𝑎 and 𝑏, where 𝑥 is the bar_index .
Instead of using the sum of log x/y values, together with the dot product of log x/y and the sum of the square of log x-values, to calculate a and b, I wanted to see if it was possible to calculate a and b differently.
In this script, the log reg is calculated with several different assumed a & b values, after which the log reg level is compared to each Swing. The log reg, where all swings on average are closest to the level, produces the final 𝑎 & 𝑏 values used to display the levels.
🔶 USAGE
The script shows the calculated logarithmic regression value from historical swings, provided there are enough swings, the price pattern fits the log reg model, and previous swings are close to the calculated Top/Bottom levels.
When the price approaches one of the calculated Top or Bottom levels, these levels could act as potential cycle Top or Bottom.
Since the logarithmic regression depends on swing values, each new value will change the calculation. A well-fitted model could not fit anymore in the future.
Swings are based on Weekly bars. A Top Swing, for example, with Swing setting 30, is the highest value in 60 weeks. Thirty bars at the left and right of the Swing will be lower than the Top Swing. This means that a confirmation is triggered 30 weeks after the Swing. The period will be automatically multiplied by 7 on the daily chart, where 30 becomes 210 bars.
Please note that the goal of this script is not to show swings rapidly; it is meant to show the potential next cycle's Top/Bottom levels.
🔹 Multiple Levels
The script includes the option to display 3 Top/Bottom levels, which uses different values for the swing calculations.
Top: 'high', 'maximum open/close' or 'close'
Bottom: 'low', 'minimum open/close' or 'close'
These levels can be adjusted up/down with a percentage.
Lastly, an "Average" is included for each set, which will only be visible when "AVG" is enabled, together with both Top and Bottom levels.
🔹 Notes
Users have to check the validity of swings; the above example only uses 1 Top Swing for its calculations, making the Top level unreliable.
Here, 1 of the Bottom Swings is pretty far from the bottom level, changing the swing settings can give a more reliable bottom level where all swings are close to that level.
Note the display was set at "Logarithmic", it can just as well be shown as "Regular"
In the example below, the price evolution does not fit the logarithmic regression model, where growth should accelerate rapidly at first and then slows over time.
Please note that this script can only be used on a daily timeframe or higher; using it at a lower timeframe will show a warning. Also, it doesn't work with bar-replay.
🔶 DETAILS
The code gathers data from historical swings. At the last bar, all swings are calculated with different a and b values. The a and b values which results in the smallest difference between all swings and Top/Bottom levels become the final a and b values.
The ranges of a and b are between -20.000 to +20.000, which means a and b will have the values -20.000, -19.999, -19.998, -19.997, -19.996, ... -> +20.000.
As you can imagine, the number of calculations is enormous. Therefore, the calculation is split into parts, first very roughly and then very fine.
The first calculations are done between -20 and +20 (-20, -19, -18, ...), resulting in, for example, 4.
The next set of calculations is performed only around the previous result, in this case between 3 (4-1) and 5 (4+1), resulting in, for example, 3.9. The next set goes even more in detail, for example, between 3.8 (3.9-0.1) and 4.0 (3.9 + 0.1), and so on.
1) -20 -> +20 , then loop with step 1 (result (example): 4 )
2) 4 - 1 -> 4 +1 , then loop with step 0.1 (result (example): 3.9 )
3) 3.9 - 0.1 -> 3.9 +0.1 , then loop with step 0.01 (result (example): 3.93 )
4) 3.93 - 0.01 -> 3.93 +0.01, then loop with step 0.001 (result (example): 3.928)
This ensures complicated calculations with less effort.
These calculations are done at the last bar, where the levels are displayed, which means you can see different results when a new swing is found.
Also, note that this indicator has been developed for a daily (or higher) timeframe chart.
🔶 SETTINGS
Three sets
High/Low
• color setting
• Swing Length settings for 'High' & 'Low'
• % adjustment for 'High' & 'Low'
• AVG: shows average (when both 'High' and 'Low' are enabled)
Max/Min (maximum open/close, minimum open/close)
• color setting
• Swing Length settings for 'Max' & 'Min'
• % adjustment for 'Max' & 'Min'
• AVG: shows average (when both 'Max' and 'Min' are enabled)
Close H/Close L (close Top/Bottom level)
• color setting
• Swing Length settings for 'Close H' & 'Close L'
• % adjustment for 'Close H' & 'Close L'
• AVG: shows average (when both 'Close H' and 'Close L' are enabled)
Show Dashboard, including Top/Bottom levels of the desired source and calculated a and b values.
Show Swings + Dot size
在腳本中搜尋"Cycle"
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
90cycle @joshuuu90 minute cycle is a concept about certain time windows of the day.
This indicator has two different options. One uses the 90 minute cycle times mentioned by traderdaye, the other uses the cls operational times split up into 90 minutes session.
e.g. we can often see a fake move happening in the 90 minute window between 2.30am and 4am ny time.
The indicator draws vertical lines at the start/end of each session and the user is able to only display certain sessions (asia, london, new york am and pm)
For the traderdayes option, the indicator also counts the windows from 1 to 4 and calls them q1,q2,q3,q4 (q-quarter)
⚠️ Open Source ⚠️
Coders and TV users are authorized to copy this code base, but a paid distribution is prohibited. A mention to the original author is expected, and appreciated.
⚠️ Terms and Conditions ⚠️
This financial tool is for educational purposes only and not financial advice. Users assume responsibility for decisions made based on the tool's information. Past performance doesn't guarantee future results. By using this tool, users agree to these terms.
inverse_fisher_transform_adaptive_stochastic█ Description
The indicator is the implementation of inverse fisher transform an indicator transform of the adaptive stochastic (dominant cycle), as in the Cycle Analytics for Trader pg. 198 (John F. Ehlers). Indicator transformation in brief means reshaping the indicator to be more interpretable. The inverse fisher transform is achieved by compressing values near the extremes many extraneous and irrelevant wiggles are removed from the indicator, as cited.
█ Inverse Fisher Transform
input = 2*(adaptive_stoc - .5)
output = e(2*k*input) -1 / e(2*k*input) +1
█ Feature:
iFish i.e. output value
trigger i.e. previous 1 bar of iFish * 0.90
if iFish crosses above the trigger, consider a buy indicated with the green line
while, iFish crosses below the trigger, consider a sell indicate by the red line
in addition iFish needs to be greater than the previous iFish
timing marketIntraday time cycle . it is valid for nifty and banknifty .just add this on daily basis . ignore previous day data
BTC Pi MultipleThe Pi Multiple is a function of 350 and 111-day moving average. When both intersect and the 111-day MA crosses above, it has historically coincided with a cycle top with a 3-day margin.
With the Pi Multiple, this intersection is visible when the line crosses zero upwards.
The indicator is called the Pi Multiple because 350/111 is close to Pi. It is based on the Pi Cycle Top Indicator developed by Philip Swift and has been modified for better readability by David Bertho.
Cycle Dynamic Composite AverageThis MA uses the formula of simple cycle indicator to find 2 cycles periods length's .
The CDCA is the result of 8 different ma to control and filter the price. The regression line is the signal , don t need to look candles, but just the cross between MA and reg lin.
Election Year GainsShows the yearly gains of the chart in U.S. Election years.
Use the options to turn on other years in the cycle.
For use with the 12M chart.
Will show non-sensical data with other intervals.
TASC 2025.06 Cybernetic Oscillator█ OVERVIEW
This script implements the Cybernetic Oscillator introduced by John F. Ehlers in his article "The Cybernetic Oscillator For More Flexibility, Making A Better Oscillator" from the June 2025 edition of the TASC Traders' Tips . It cascades two-pole highpass and lowpass filters, then scales the result by its root mean square (RMS) to create a flexible normalized oscillator that responds to a customizable frequency range for different trading styles.
█ CONCEPTS
Oscillators are indicators widely used by technical traders. These indicators swing above and below a center value, emphasizing cyclic movements within a frequency range. In his article, Ehlers explains that all oscillators share a common characteristic: their calculations involve computing differences . The reliance on differences is what causes these indicators to oscillate about a central point.
The difference between two data points in a series acts as a highpass filter — it allows high frequencies (short wavelengths) to pass through while significantly attenuating low frequencies (long wavelengths). Ehlers demonstrates that a simple difference calculation attenuates lower-frequency cycles at a rate of 6 dB per octave. However, the difference also significantly amplifies cycles near the shortest observable wavelength, making the result appear noisier than the original series. To mitigate the effects of noise in a differenced series, oscillators typically smooth the series with a lowpass filter, such as a moving average.
Ehlers highlights an underlying issue with smoothing differenced data to create oscillators. He postulates that market data statistically follows a pink spectrum , where the amplitudes of cyclic components in the data are approximately directly proportional to the underlying periods. Specifically, he suggests that cyclic amplitude increases by 6 dB per octave of wavelength.
Because some conventional oscillators, such as RSI, use differencing calculations that attenuate cycles by only 6 dB per octave, and market cycles increase in amplitude by 6 dB per octave, such calculations do not have a tangible net effect on larger wavelengths in the analyzed data. The influence of larger wavelengths can be especially problematic when using these oscillators for mean reversion or swing signals. For instance, an expected reversion to the mean might be erroneous because oscillator's mean might significantly deviate from its center over time.
To address the issues with conventional oscillator responses, Ehlers created a new indicator dubbed the Cybernetic Oscillator. It uses a simple combination of highpass and lowpass filters to emphasize a specific range of frequencies in the market data, then normalizes the result based on RMS. The process is as follows:
Apply a two-pole highpass filter to the data. This filter's critical period defines the longest wavelength in the oscillator's passband.
Apply a two-pole SuperSmoother (lowpass filter) to the highpass-filtered data. This filter's critical period defines the shortest wavelength in the passband.
Scale the resulting waveform by its RMS. If the filtered waveform follows a normal distribution, the scaled result represents amplitude in standard deviations.
The oscillator's two-pole filters attenuate cycles outside the desired frequency range by 12 dB per octave. This rate outweighs the apparent rate of amplitude increase for successively longer market cycles (6 dB per octave). Therefore, the Cybernetic Oscillator provides a more robust isolation of cyclic content than conventional oscillators. Best of all, traders can set the periods of the highpass and lowpass filters separately, enabling fine-tuning of the frequency range for different trading styles.
█ USAGE
The "Highpass period" input in the "Settings/Inputs" tab specifies the longest wavelength in the oscillator's passband, and the "Lowpass period" input defines the shortest wavelength. The oscillator becomes more responsive to rapid movements with a smaller lowpass period. Conversely, it becomes more sensitive to trends with a larger highpass period. Ehlers recommends setting the smallest period to a value above 8 to avoid aliasing. The highpass period must not be smaller than the lowpass period. Otherwise, it causes a runtime error.
The "RMS length" input determines the number of bars in the RMS calculation that the indicator uses to normalize the filtered result.
This indicator also features two distinct display styles, which users can toggle with the "Display style" input. With the "Trend" style enabled, the indicator plots the oscillator with one of two colors based on whether its value is above or below zero. With the "Threshold" style enabled, it plots the oscillator as a gray line and highlights overbought and oversold areas based on the user-specified threshold.
Below, we show two instances of the script with different settings on an equities chart. The first uses the "Threshold" style with default settings to pass cycles between 20 and 30 bars for mean reversion signals. The second uses a larger highpass period of 250 bars and the "Trend" style to visualize trends based on cycles spanning less than one year:
z-score-calkusi-v1.143z-scores incorporate the moment of N look-back bars to allow future price projection.
z-score = (X - mean)/std.deviation ; X = close
z-scores update with each new close print and with each new bar. Each new bar augments the mean and std.deviation for the N bars considered. The old Nth bar falls away from consideration with each new historical bar.
The indicator allows two other options for X: RSI or Moving Average.
NOTE: While trading use the "price" option only.
The other two options are provided for visualisation of RSI and Moving Average as z-score curves.
Use z-scores to identify tops and bottoms in the future as well as intermediate intersections through which a z-score will pass through with each new close and each new bar.
Draw lines from peaks and troughs in the past through intermediate peaks and troughs to identify projected intersections in the future. The most likely intersections are those that are formed from a line that comes from a peak in the past and another line that comes from a trough in the past. Try getting at least two lines from historical peaks and two lines from historical troughs to pass through a future intersection.
Compute the target intersection price in the future by clicking on the z-score indicator header to see a drag-able horizontal line to drag over the intersection. The target price is the last value displayed in the indicator's status bar after the closing price.
When the indicator header is clicked, a white horizontal drag-able line will appear to allow dragging the line over an intersection that has been drawn on the indicator for a future z-score projection and the associated future closing price.
With each new bar that appears, it is necessary to repeat the procedure of clicking the z-score indicator header to be able to drag the drag-able horizontal line to see the new target price for the selected intersection. The projected price will be different from the current close price providing a price arbitrage in time.
New intermediate peaks and troughs that appear require new lines be drawn from the past through the new intermediate peak to find a new intersection in the future and a new projected price. Since z-score curves are sort of cyclical in nature, it is possible to see where one has to locate a future intersection by drawing lines from past peaks and troughs.
Do not get fixated on any one projected price as the market decides which projected price will be realised. All prospective targets should be manually updated with each new bar.
When the z-score plot moves outside a channel comprised of lines that are drawn from the past, be ready to adjust to new market conditions.
z-score plots that move above the zero line indicate price action that is either rising or ranging. Similarly, z-score plots that move below the zero line indicate price action that is either falling or ranging. Be ready to adjust to new market conditions when z-scores move back and forth across the zero line.
A bar with highest absolute z-score for a cycle screams "reversal approaching" and is followed by a bar with a lower absolute z-score where close price tops and bottoms are realised. This can occur either on the next bar or a few bars later.
The indicator also displays the required N for a Normal(0,1) distribution that can be set for finer granularity for the z-score curve.This works with the Confidence Interval (CI) z-score setting. The default z-score is 1.96 for 95% CI.
Common Confidence Interval z-scores to find N for Normal(0,1) with a Margin of Error (MOE) of 1:
70% 1.036
75% 1.150
80% 1.282
85% 1.440
90% 1.645
95% 1.960
98% 2.326
99% 2.576
99.5% 2.807
99.9% 3.291
99.99% 3.891
99.999% 4.417
9-Jun-2025
Added a feature to display price projection labels at z-score levels 3, 2, 1, 0, -1, -2, 3.
This provides a range for prices available at the current time to help decide whether it is worth entering a trade. If the range of prices from say z=|2| to z=|1| is too narrow, then a trade at the current time may not be worth the risk.
Added plot for z-score moving average.
28-Jun-2025
Added Settings option for # of Std.Deviation level Price Labels to display. The default is 3. Min is 2. Max is 6.
This feature allows likelihood assessment for Fibonacci price projections from higher time frames at lower time frames. A Fibonacci price projection that falls outside |3.x| Std.Deviations is not likely.
Added Settings option for Chart Bar Count and Target Label Offset to allow placement of price labels for the standard z-score levels to the right of the window so that these are still visible in the window.
Target Label Offset allows adjustment of placement of Target Price Label in cases when the Target Price Label is either obscured by the price labels for the standard z-score levels or is too far right to be visible in the window.
9-Jul-2025
z-score 1.142 updates:
Displays in the status line before the close price the range for the selected Std. Deviation levels specified in Settings and |z-zMa|.
When |z-zMa| > |avg(z-zMa)| and zMa rising, |z-zMa| and zMa displays in aqua.
When |z-zMa| > |avg(z-zMa)| and zMa falling, |z-zMa| and zMa displays in red.
When |z-zMa| <= |avg(z-zMa)|, z and zMa display in gray.
z usually crosses over zMa when zMa is gray but not always. So if cross-over occurs when zMa is not gray, it implies a strong move in progress.
Practice makes perfect.
Use this indicator at your own risk
Omori Law Recovery PhasesWhat is the Omori Law?
Originally a seismological model, the Omori Law describes how earthquake aftershocks decay over time. It follows a power law relationship: the frequency of aftershocks decreases roughly proportionally to 1/(t+c)^p, where:
t = time since the main shock
c = time offset constant
p = power law exponent (typically around 1.0)
Application to the markets
Financial markets experience "aftershocks" similar to earthquakes:
Market Crashes as Main Shocks: Major market declines (crashes) represent the initial shock event.
Volatility Decay: After a crash, market volatility typically declines following a power law pattern rather than a linear or exponential one.
Behavioral Components: The decay pattern reflects collective market psychology - initial panic gives way to uncertainty, then stabilization, and finally normalization.
The Four Recovery Phases
The Omori decay pattern in markets can be divided into distinct phases:
Acute Phase: Immediately after the crash, characterized by extreme volatility, panic selling, and sharp reversals. Trading is hazardous.
Reaction Phase: Volatility begins decreasing, but markets test previous levels. False rallies and retests of lows are common.
Repair Phase: Structure returns to the market. Volatility approaches normal levels, and traditional technical analysis becomes more reliable.
Recovery Phase: The final stage where market behavior normalizes completely. The impact of the original shock has fully decayed.
Why It Matters for Traders
Understanding where the market stands in this recovery cycle provides valuable context:
Risk Management: Adjust position sizing based on the current phase
Strategy Selection: Different strategies work in different phases
Psychological Preparation: Know what to expect based on the phase
Time Horizon Guidance: Each phase suggests appropriate time frames for trading
QG-Particle OscillatorThis is an advanced oscillator based on auxiliary particle filter. It separates signal from noise and uses smoothing algorithm similar to JMA.
The main oscillator line is a smoothed and detrended version of the price series similar to detrended oscillator line. The purple/aqua lines are a prediction based on an additional adaptive smoothing technique and current volatility.
The prediction is smoothed twice and is supposed to represent the true signal without any noise, thus the prediction should always be less than the raw detrend line. However, certain volatile conditions will cause the prediction to cross above/below the detrend line. When this happens the likelihood of a reversal or pullback is extremely high.
There are 3 dots on the zero line- Red, Green and Yellow. The yellow dots warn of an eminent pullback 2 bars before it actually occurs. This is a non-repainting indicator.
One can also use this indicator to trade CCI signals, similar to zero line rejection in existing trend.
The indicator has 2 settings- Period and Phase. The phase represents cycle phase and Period represents oscillator period.
Credits: This indicator has been originally published for Ninjatrader and this is conversion into pinescript.
Lyapunov Market Instability (LMI)Lyapunov Market Instability (LMI)
What is Lyapunov Market Instability?
Lyapunov Market Instability (LMI) is a revolutionary indicator that brings chaos theory from theoretical physics into practical trading. By calculating Lyapunov exponents—a measure of how rapidly nearby trajectories diverge in phase space—LMI quantifies market sensitivity to initial conditions. This isn't another oscillator or trend indicator; it's a mathematical lens that reveals whether markets are in chaotic (trending) or stable (ranging) regimes.
Inspired by the meditative color field paintings of Mark Rothko, this indicator transforms complex chaos mathematics into an intuitive visual experience. The elegant simplicity of the visualization belies the sophisticated theory underneath—just as Rothko's seemingly simple color blocks contain profound depth.
Theoretical Foundation (Chaos Theory & Lyapunov Exponents)
In dynamical systems, the Lyapunov exponent (λ) measures the rate of separation of infinitesimally close trajectories:
λ > 0: System is chaotic—small changes lead to dramatically different outcomes (butterfly effect)
λ < 0: System is stable—trajectories converge, perturbations die out
λ ≈ 0: Edge of chaos—transition between regimes
Phase Space Reconstruction
Using Takens' embedding theorem , we reconstruct market dynamics in higher dimensions:
Time-delay embedding: Create vectors from price at different lags
Nearest neighbor search: Find historically similar market states
Trajectory evolution: Track how these similar states diverged over time
Divergence rate: Calculate average exponential separation
Market Application
Chaotic markets (λ > threshold): Strong trends emerge, momentum dominates, use breakout strategies
Stable markets (λ < threshold): Mean reversion dominates, fade extremes, range-bound strategies work
Transition zones: Market regime about to change, reduce position size, wait for confirmation
How LMI Works
1. Phase Space Construction
Each point in time is embedded as a vector using historical prices at specific delays (τ). This reveals the market's hidden attractor structure.
2. Lyapunov Calculation
For each current state, we:
- Find similar historical states within epsilon (ε) distance
- Track how these initially similar states evolved
- Measure exponential divergence rate
- Average across multiple trajectories for robustness
3. Signal Generation
Chaos signals: When λ crosses above threshold, market enters trending regime
Stability signals: When λ crosses below threshold, market enters ranging regime
Divergence detection: Price/Lyapunov divergences signal potential reversals
4. Rothko Visualization
Color fields: Background zones represent market states with Rothko-inspired palettes
Glowing line: Lyapunov exponent with intensity reflecting market state
Minimalist design: Focus on essential information without clutter
Inputs:
📐 Lyapunov Parameters
Embedding Dimension (default: 3)
Dimensions for phase space reconstruction
2-3: Simple dynamics (crypto/forex) - captures basic momentum patterns
4-5: Complex dynamics (stocks/indices) - captures intricate market structures
Higher dimensions need exponentially more data but reveal deeper patterns
Time Delay τ (default: 1)
Lag between phase space coordinates
1: High-frequency (1m-15m charts) - captures rapid market shifts
2-3: Medium frequency (1H-4H) - balances noise and signal
4-5: Low frequency (Daily+) - focuses on major regime changes
Match to your timeframe's natural cycle
Initial Separation ε (default: 0.001)
Neighborhood size for finding similar states
0.0001-0.0005: Highly liquid markets (major forex pairs)
0.0005-0.002: Normal markets (large-cap stocks)
0.002-0.01: Volatile markets (crypto, small-caps)
Smaller = more sensitive to chaos onset
Evolution Steps (default: 10)
How far to track trajectory divergence
5-10: Fast signals for scalping - quick regime detection
10-20: Balanced for day trading - reliable signals
20-30: Slow signals for swing trading - major regime shifts only
Nearest Neighbors (default: 5)
Phase space points for averaging
3-4: Noisy/fast markets - adapts quickly
5-6: Balanced (recommended) - smooth yet responsive
7-10: Smooth/slow markets - very stable signals
📊 Signal Parameters
Chaos Threshold (default: 0.05)
Lyapunov value above which market is chaotic
0.01-0.03: Sensitive - more chaos signals, earlier detection
0.05: Balanced - optimal for most markets
0.1-0.2: Conservative - only strong trends trigger
Stability Threshold (default: -0.05)
Lyapunov value below which market is stable
-0.01 to -0.03: Sensitive - quick stability detection
-0.05: Balanced - reliable ranging signals
-0.1 to -0.2: Conservative - only deep stability
Signal Smoothing (default: 3)
EMA period for noise reduction
1-2: Raw signals for experienced traders
3-5: Balanced - recommended for most
6-10: Very smooth for position traders
🎨 Rothko Visualization
Rothko Classic: Deep reds for chaos, midnight blues for stability
Orange/Red: Warm sunset tones throughout
Blue/Black: Cool, meditative ocean depths
Purple/Grey: Subtle, sophisticated palette
Visual Options:
Market Zones : Background fields showing regime areas
Transitions: Arrows marking regime changes
Divergences: Labels for price/Lyapunov divergences
Dashboard: Real-time state and trading signals
Guide: Educational panel explaining the theory
Visual Logic & Interpretation
Main Elements
Lyapunov Line: The heart of the indicator
Above chaos threshold: Market is trending, follow momentum
Below stability threshold: Market is ranging, fade extremes
Between thresholds: Transition zone, reduce risk
Background Zones: Rothko-inspired color fields
Red zone: Chaotic regime (trending)
Gray zone: Transition (uncertain)
Blue zone: Stable regime (ranging)
Transition Markers:
Up triangle: Entering chaos - start trend following
Down triangle: Entering stability - start mean reversion
Divergence Signals:
Bullish: Price makes low but Lyapunov rising (stability breaking down)
Bearish: Price makes high but Lyapunov falling (chaos dissipating)
Dashboard Information
Market State: Current regime (Chaotic/Stable/Transitioning)
Trading Bias: Specific strategy recommendation
Lyapunov λ: Raw value for precision
Signal Strength: Confidence in current regime
Last Change: Bars since last regime shift
Action: Clear trading directive
Trading Strategies
In Chaotic Regime (λ > threshold)
Follow trends aggressively: Breakouts have high success rate
Use momentum strategies: Moving average crossovers work well
Wider stops: Expect larger swings
Pyramid into winners: Trends tend to persist
In Stable Regime (λ < threshold)
Fade extremes: Mean reversion dominates
Use oscillators: RSI, Stochastic work well
Tighter stops: Smaller expected moves
Scale out at targets: Trends don't persist
In Transition Zone
Reduce position size: Uncertainty is high
Wait for confirmation: Let regime establish
Use options: Volatility strategies may work
Monitor closely: Quick changes possible
Advanced Techniques
- Multi-Timeframe Analysis
- Higher timeframe LMI for regime context
- Lower timeframe for entry timing
- Alignment = highest probability trades
- Divergence Trading
- Most powerful at regime boundaries
- Combine with support/resistance
- Use for early reversal detection
- Volatility Correlation
- Chaos often precedes volatility expansion
- Stability often precedes volatility contraction
- Use for options strategies
Originality & Innovation
LMI represents a genuine breakthrough in applying chaos theory to markets:
True Lyapunov Calculation: Not a simplified proxy but actual phase space reconstruction and divergence measurement
Rothko Aesthetic: Transforms complex math into meditative visual experience
Regime Detection: Identifies market state changes before price makes them obvious
Practical Application: Clear, actionable signals from theoretical physics
This is not a combination of existing indicators or a visual makeover of standard tools. It's a fundamental rethinking of how we measure and visualize market dynamics.
Best Practices
Start with defaults: Parameters are optimized for broad market conditions
Match to your timeframe: Adjust tau and evolution steps
Confirm with price action: LMI shows regime, not direction
Use appropriate strategies: Chaos = trend, Stability = reversion
Respect transitions: Reduce risk during regime changes
Alerts Available
Chaos Entry: Market entering chaotic regime - prepare for trends
Stability Entry: Market entering stable regime - prepare for ranges
Bullish Divergence: Potential bottom forming
Bearish Divergence: Potential top forming
Chart Information
Script Name: Lyapunov Market Instability (LMI) Recommended Use: All markets, all timeframes Best Performance: Liquid markets with clear regimes
Academic References
Takens, F. (1981). "Detecting strange attractors in turbulence"
Wolf, A. et al. (1985). "Determining Lyapunov exponents from a time series"
Rosenstein, M. et al. (1993). "A practical method for calculating largest Lyapunov exponents"
Note: After completing this indicator, I discovered @loxx's 2022 "Lyapunov Hodrick-Prescott Oscillator w/ DSL". While both explore Lyapunov exponents, they represent independent implementations with different methodologies and applications. This indicator uses phase space reconstruction for regime detection, while his combines Lyapunov concepts with HP filtering.
Disclaimer
This indicator is for research and educational purposes only. It does not constitute financial advice or provide direct buy/sell signals. Chaos theory reveals market character, not future prices. Always use proper risk management and combine with your own analysis. Past performance does not guarantee future results.
See markets through the lens of chaos. Trade the regime, not the noise.
Bringing theoretical physics to practical trading through the meditative aesthetics of Mark Rothko
Trade with insight. Trade with anticipation.
— Dskyz , for DAFE Trading Systems
Bitcoin as % Global M2 signalThis script provides signal system:
Buy signal: each time the YoY of the Global M2 rises more than 2.5% while the distance between the bitcoin price as a percentage of the Global M2 is below its yearly SMA.
Sell signal: the distance between the bitcoin price as a percentage of the Global M2 and its yearly SMA is > 0.7
This is a very simple system, but it seems to work pretty well to ride the bitcoin price cycle wave.
The parameters are hard coded but they can be easily changed to test different levels for both the buy and sell signals.
ONE RING 8 MA Bands with RaysCycle analysis tool ...
MAs: Eight moving averages (MA1–MA8) with customizable lengths, types (RMA, WMA, EMA, SMA), and offsets
Bands: Upper/lower bands for each MA, calculated based on final_pctX (Percentage mode) or final_ptsX (Points mode), scaled by multiplier
Rays: Forward-projected lines for bands, with customizable start points, styles (Solid, Dashed, Dotted), and lengths (up to 500 bars)
Band Choices
Manual: Uses individual inputs for band offsets
Uniform: Sets all offsets to base_pct (e.g., 0.1%) or base_pts (e.g., 0.1 points)
Linear: Scales linearly (e.g., base_pct * 1, base_pct * 2, base_pct * 3 ..., base_pct * 8)
Exponential: Scales exponentially (e.g., base_pct * 1, base_pct * 2, base_pct * 4, base_pct * 8 ..., base_pct * 128)
ATR-Based: Offsets are derived from the Average True Range (ATR), scaled by a linear factor. Dynamic bands that adapt to market conditions, useful for breakout or mean-reversion strategies. (final_pct1 = base_pct * atr, final_pct2 = base_pct * atr * 2, ..., final_pct8 = base_pct * atr * 8)
Geometric: Offsets follow a geometric progression (e.g., base_pct * r^0, base_pct * r^1, base_pct * r^2, ..., where r is a ratio like 1.5) This is less aggressive than Exponential (which uses powers of 2) and provides a smoother progression.
Example: If base_pct = 0.1, r = 1.5, then final_pct1 = 0.1%, final_pct2 = 0.15%, final_pct3 = 0.225%, ..., final_pct8 ≈ 1.71%
Harmonic: Offsets are based on harmonic flavored ratios. final_pctX = base_pct * X / (9 - X), final_ptsX = base_pts * X / (9 - X) for X = 1 to 8 This creates a harmonic-like progression where offsets increase non-linearly, ensuring MA8 bands are wider than MA1 bands, and avoids duplicating the Linear choice above.
Ex. offsets for base_pct = 0.1: MA1: ±0.0125% (0.1 * 1/8), MA2: ±0.0286% (0.1 * 2/7), MA3: ±0.05% (0.1 * 3/6), MA4: ±0.08% (0.1 * 4/5), MA5: ±0.125% (0.1 * 5/4), MA6: ±0.2% (0.1 * 6/3), MA7: ±0.35% (0.1 * 7/2), MA8: ±0.8% (0.1 * 8/1)
Square Root: Offsets grow with the square root of the band index (e.g., base_pct * sqrt(1), base_pct * sqrt(2), ..., base_pct * sqrt(8)). This creates a gradual widening, less aggressive than Linear or Exponential. Set final_pct1 = base_pct * sqrt(1), final_pct2 = base_pct * sqrt(2), ..., final_pct8 = base_pct * sqrt(8).
Example: If base_pct = 0.1, then final_pct1 = 0.1%, final_pct2 ≈ 0.141%, final_pct3 ≈ 0.173%, ..., final_pct8 ≈ 0.283%.
Fibonacci: Uses Fibonacci ratios (e.g., base_pct * 1, base_pct * 1.618, base_pct * 2.618
Percentage vs. Points Toggle:
In Percentage mode, bands are calculated as ma * (1 ± (final_pct / 100) * multiplier)
In Points mode, bands are calculated as ma ± final_pts * multiplier, where final_pts is in price units.
Threshold Setting for Slope:
Threshold setting for determining when the slope would be significant enough to call it a change in direction. Can check efficiency by setting MA1 to color on slope temporarily
Arrow table: Shows slope direction of 8 MAs using an Up or Down triangle, or shows Flat condition if no triangle.
Price Position Percentile (PPP)
Price Position Percentile (PPP)
A statistical analysis tool that dynamically measures where current price stands within its historical distribution. Unlike traditional oscillators, PPP adapts to market conditions by calculating percentile ranks, creating a self-adjusting framework for identifying extremes.
How It Works
This indicator analyzes the last 200 price bars (customizable) and calculates the percentile rank of the current price within this distribution. For example, if the current price is at the 80th percentile, it means the price is higher than 80% of all prices in the lookback period.
The indicator creates five dynamic zones based on percentile thresholds:
Extremely Low Zone (<5%) : Prices in the lowest 5% of the distribution, indicating potential oversold conditions.
Low Zone (5-25%) : Accumulation zone where prices are historically low but not extreme.
Neutral Zone (25-75%) : Fair value zone representing the middle 50% of the price distribution.
High Zone (75-95%) : Distribution zone where prices are historically high but not extreme.
Extremely High Zone (>95%) : Prices in the highest 5% of the distribution, suggesting potential bubble conditions.
Mathematical Foundation
Unlike fixed-threshold indicators, PPP uses a non-parametric approach:
// Core percentile calculation
percentile = (count_of_prices_below_current / total_prices) * 100
// Threshold calculation using built-in function
p_extremely_low = ta.percentile_linear_interpolation(source, lookback, 5)
p_low = ta.percentile_linear_interpolation(source, lookback, 25)
p_neutral_high = ta.percentile_linear_interpolation(source, lookback, 75)
p_extremely_high = ta.percentile_linear_interpolation(source, lookback, 95)
Key Features
Dynamic Adaptation : All zones adjust automatically as price distribution changes
Statistical Robustness : Works on any timeframe and any market, including highly volatile cryptocurrencies
Visual Clarity : Color-coded zones provide immediate visual context
Non-parametric Analysis : Makes no assumptions about price distribution shape
Historical Context : Shows how zones evolved over time, revealing market regime changes
Practical Applications
PPP provides objective statistical context for price action, helping traders make more informed decisions based on historical price distribution rather than arbitrary levels.
Value Investment : Identify statistically significant low prices for potential entry points
Risk Management : Recognize when prices reach historical extremes for profit taking
Cycle Analysis : Observe how percentile zones expand and contract during different market phases
Market Regime Detection : Identify transitions between accumulation, markup, distribution, and markdown phases
Usage Guidelines
This indicator is particularly effective when:
- Used across multiple timeframes for confirmation
- Combined with volume analysis for validation of extremes
- Applied in conjunction with trend identification tools
- Monitored for divergences between price action and percentile ranking
Timed Reversion Markers (Custom Session Alerts)This script plots vertical histogram markers at specific intraday time points defined by the user. It is designed for traders who follow time-based reversion or breakout setups tied to predictable market behavior at key clock times, such as institutional opening moves, midday reversals, or end-of-day volatility.
Unlike traditional price-action indicators, this tool focuses purely on time-based triggers, a technique often used in time cycle analysis, market internals, and volume-timing strategies.
The indicator includes eight fully customizable time inputs, allowing users to mark any intraday minute with precision using a decimal hour format (for example, 9.55 for 9:55 AM). Each input is automatically converted into hour and minute format, and a visual histogram marker is plotted once per day at that exact time.
Example use cases:
Mark institutional session opens (e.g., 9:30, 10:00, 15:30)
Time-based mean reversion or volatility windows
Backtest recurring time-based reactions
Highlight algorithmic spike zones
The vertical plots serve as non-intrusive, high-contrast visual markers for scalping setups, session analysis, and decision-making checkpoints. All markers are displayed at the top of the chart without interfering with price candles.
TASC 2025.02 Autocorrelation Indicator█ OVERVIEW
This script implements the Autocorrelation Indicator introduced by John Ehlers in the "Drunkard's Walk: Theory And Measurement By Autocorrelation" article from the February 2025 edition of TASC's Traders' Tips . The indicator calculates the autocorrelation of a price series across several lags to construct a periodogram , which traders can use to identify market cycles, trends, and potential reversal patterns.
█ CONCEPTS
Drunkard's walk
A drunkard's walk , formally known as a random walk , is a type of stochastic process that models the evolution of a system or variable through successive random steps.
In his article, John Ehlers relates this model to market data. He discusses two first- and second-order partial differential equations, modified for discrete (non-continuous) data, that can represent solutions to the discrete random walk problem: the diffusion equation and the wave equation. According to Ehlers, market data takes on a mixture of two "modes" described by these equations. He theorizes that when "diffusion mode" is dominant, trading success is almost a matter of luck, and when "wave mode" is dominant, indicators may have improved performance.
Pink spectrum
John Ehlers explains that many recent academic studies affirm that market data has a pink spectrum , meaning the power spectral density of the data is proportional to the wavelengths it contains, like pink noise . A random walk with a pink spectrum suggests that the states of the random variable are correlated and not independent. In other words, the random variable exhibits long-range dependence with respect to previous states.
Autocorrelation function (ACF)
Autocorrelation measures the correlation of a time series with a delayed copy, or lag , of itself. The autocorrelation function (ACF) is a method that evaluates autocorrelation across a range of lags , which can help to identify patterns, trends, and cycles in stochastic market data. Analysts often use ACF to detect and characterize long-range dependence in a time series.
The Autocorrelation Indicator evaluates the ACF of market prices over a fixed range of lags, expressing the results as a color-coded heatmap representing a dynamic periodogram. Ehlers suggests the information from the periodogram can help traders identify different market behaviors, including:
Cycles : Distinguishable as repeated patterns in the periodogram.
Reversals : Indicated by sharp vertical changes in the periodogram when the indicator uses a short data length .
Trends : Indicated by increasing correlation across lags, starting with the shortest, over time.
█ USAGE
This script calculates the Autocorrelation Indicator on an input "Source" series, smoothed by Ehlers' UltimateSmoother filter, and plots several color-coded lines to represent the periodogram's information. Each line corresponds to an analyzed lag, with the shortest lag's line at the bottom of the pane. Green hues in the line indicate a positive correlation for the lag, red hues indicate a negative correlation (anticorrelation), and orange or yellow hues mean the correlation is near zero.
Because Pine has a limit on the number of plots for a single indicator, this script divides the periodogram display into three distinct ranges that cover different lags. To see the full periodogram, add three instances of this script to the chart and set the "Lag range" input for each to a different value, as demonstrated in the chart above.
With a modest autocorrelation length, such as 20 on a "1D" chart, traders can identify seasonal patterns in the price series, which can help to pinpoint cycles and moderate trends. For instance, on the daily ES1! chart above, the indicator shows repetitive, similar patterns through fall 2023 and winter 2023-2024. The green "triangular" shape rising from the zero lag baseline over different time ranges corresponds to seasonal trends in the data.
To identify turning points in the price series, Ehlers recommends using a short autocorrelation length, such as 2. With this length, users can observe sharp, sudden shifts along the vertical axis, which suggest potential turning points from upward to downward or vice versa.
MCDX_SignalThe MCDX indicator (Market Cycle Dynamic Index) is a technical indicator developed by Trung Pham. It is a tool used for analyzing the stock market, often utilized to identify big money flow (Big Money) and evaluate the strength of individual stocks or the overall market.
MCDX is known for its distinctive histogram chart with red and green bars. The red bars typically represent the inflow of big money, while the green bars indicate small money flow or outflows.
Pivot Highs/Lows with Bar CountsWhat does the indicator do?
This indicator adds labels to a chart at swing (a.k.a., "pivot") highs and lows. Each label may contain a date, the closing price at the swing, the number of bars since the last swing in the same direction, and the number of bars from the last swing in the opposite direction. A table is also added to the chart that shows the average, min, and max number of bars between swings.
OK, but how do I use it?
Many markets -- especially sideways-moving ones -- commonly cycle between swing highs and lows at regular time intervals. By measuring the number of bars between highs and lows -- both same-sided swings (i.e., H-H and L-L) and opposite-sided swings (i.e., H-L and L-H) -- you can then project the averages of those bar counts from the last high or low swing to make predictions about where the next swing high or low should occur. Note that this indicator does not make the projection for you. You have to determine which swing you want to project from and then use the bar counts from the indicator to draw a line, place a label, etc.
Example: Chart of BTC/USD
The indicator shows pivot highs and lows with bar counts, and it displays a table of stats on those pivots.
If you focus on the center section of the chart, you can see that prices were moving in a sideways channel with very regular highs and lows. This indicator counts the bars between these pivots, and you could have used those counts to predict when the next high or low may have occurred.
The bar counts do not work as well on the more recent section of the chart because there are no regularly time swings.
RSI w/Hann WindowingThis RSI by John Ehlers of "Yet Another" Improved RSI. Taking advantage of the Hann windowing. As seen on PRC and published by John Ehlers, it has a zero mean and appears smoother than the classic RSI. In his own words " I prefer oscillator-type indicators to have a zero mean. We can achieve this simply by multiplying the classic RSI by 2 so it swings from 0 to 2, and then subtract 1 from the product so the indicator swings from -1 to +1." Ehlers goes on to say " Bear in mind 14 may not be the best length to analysis. So, the best length to use for the RSIH indicator is on the order of the dominant cycle period of the data."
This indicator works well with both bullish and bearish divergences. It also works well with oversold and overbought indications. Shown by the Red zone on top (Overbought) and the green zone on the bottom(oversold). Each which have an adjustable buffer zone. You may need to adjust the length of the RSIH to suit your asset. There are also multiply signal line's to choose from. Also take note of when the RSIH crosses up or down on the signal line.
None of this is financial advice.
MVRVZ - MVRVZ Top and Bottom Indicator for BTC [Logue]Market Value-Realized Value Z-score (MVRVZ) - The MVRV-Z score measures the value of the bitcoin network by comparing the market cap to the realized value and dividing by the standard deviation of the market cap (market cap – realized cap) / std(market cap)). When the market value is significantly higher than the realized value, the bitcoin network is "overvalued". Very high values have signaled cycle tops in the past and low values have signaled bottoms. For tops, the default trigger value is above 6.85. For bottoms, the indicator is triggered when the MVRVZ is below -0.25 (default).
NUPL - Net Unrealized Profit-Loss BTC Tops/Bottoms [Logue]Net Unrealized Profit Loss (NUPL) - The NUPL measures the profit state of the bitcoin network to determine if past transfers of BTC are currently in an unrealized profit or loss state.
Values above zero indicate that the network is in overall profit, while values below zero indicate the network is in overall loss. Highly positive NUPL values indicate overvaluation of the BTC network and relatively negative NUPL values indicate an undervaluation of the BTC network.
For tops: The default setting for tops is based on decreasing "strength" of BTC tops. A decreasing linear function (trigger = slope * time + intercept) was fit to past cycle tops for this indicator and is used as the default to signal macro tops. The user can change the slope and intercept of the line by changing the slope and/or intercept factor. The user also has the option to indicate tops based on a horizontal line via a settings selection. This horizontal line default value is 73. This indicator is triggered for a top when the NUPL is above the trigger value.
For bottoms: Bottoms are displayed based on a horizontal line with a default setting of -13. The indicator is triggered for a bottom when the NUPL is below the bottom trigger value.
