Trend Forecasting - The Quant Science🌏 Trend Forecasting | ENG 🌏
This plug-in acts as a statistical filter, adding new information to your chart that will allow you to quickly verify the direction of a trend and the probability with which the price will be above or below the average in the future, helping you to uncover probable market inefficiencies.
🧠 Model calculation
The model calculates the arithmetic mean in relation to positive and negative events within the available sample for the selected time series. Where a positive event is defined as a closing price greater than the average, and a negative event as a closing price less than the average. Once all events have been calculated, the probabilities are extrapolated by relating each event.
Example
Positive event A: 70
Negative event B: 30
Total events: 100
Probabilities A: (100 / 70) x 100 = 70%
Probabilities B: (100 / 30) x 100 = 30%
Event A has a 70% probability of occurring compared to Event B which has a 30% probability.
🔍 Information Filter
The data on the graph show the future probabilities of prices being above average (default in green) and the probabilities of prices being below average (default in red).
The information that can be quickly retrieved from this indicator is:
1. Trend: Above-average prices together with a constant of data in green greater than 50% + 1 indicate that the observed historical series shows a bullish trend. The probability is correlated proportionally to the value of the data; the higher and increasing the expected value, the greater the observed bullish trend. On the other hand, a below-average price together with a red-coloured data constant show quantitative data regarding the presence of a bearish trend.
2. Future Probability: By analysing the data, it is possible to find the probability with which the price will be above or below the average in the future. In green are classified the probabilities that the price will be higher than the average, in red are classified the probabilities that the price will be lower than the average.
🔫 Operational Filter .
The indicator can be used operationally in the search for investment or trading opportunities given its ability to identify an inefficiency within the observed data sample.
⬆ Bullish forecast
For bullish trades, the inefficiency will appear as a historical series with a bullish trend, with high probability of a bullish trend in the future that is currently below the average.
⬇ Bearish forecast
For short trades, the inefficiency will appear as a historical series with a bearish trend, with a high probability of a bearish trend in the future that is currently above the average.
📚 Settings
Input: via the Input user interface, it is possible to adjust the periods (1 to 500) with which the average is to be calculated. By default the periods are set to 200, which means that the average is calculated by taking the last 200 periods.
Style: via the Style user interface it is possible to adjust the colour and switch a specific output on or off.
🇮🇹Previsione Della Tendenza Futura | ITA 🇮🇹
Questo plug-in funge da filtro statistico, aggiungendo nuove informazioni al tuo grafico che ti permetteranno di verificare rapidamente tendenza di un trend, probabilità con la quale il prezzo si troverà sopra o sotto la media in futuro aiutandoti a scovare probabili inefficienze di mercato.
🧠 Calcolo del modello
Il modello calcola la media aritmetica in relazione con gli eventi positivi e negativi all'intero del campione disponibile per la serie storica selezionata. Dove per evento positivo si intende un prezzo alla chiusura maggiore della media, mentre per evento negativo si intende un prezzo alla chiusura minore della media. Calcolata la totalità degli eventi le probabilità vengono estrapolate rapportando ciascun evento.
Esempio
Evento positivo A: 70
Evento negativo B: 30
Totale eventi : 100
Formula A: (100 / 70) x 100 = 70%
Formula B: (100 / 30) x 100 = 30%
Evento A ha una probabilità del 70% di realizzarsi rispetto all' Evento B che ha una probabilità pari al 30%.
🔍 Filtro informativo
I dati sul grafico mostrano le probabilità future che i prezzi siano sopra la media (di default in verde) e le probabilità che i prezzi siano sotto la media (di default in rosso).
Le informazioni che si possono rapidamente reperire da questo indicatore sono:
1. Trend: I prezzi sopra la media insieme ad una costante di dati in verde maggiori al 50% + 1 indicano che la serie storica osservata presenta un trend rialzista. La probabilità è correlata proporzionalmente al valore del dato; tanto più sarà alto e crescente il valore atteso e maggiore sarà la tendenza rialzista osservata. Viceversa, un prezzo sotto la media insieme ad una costante di dati classificati in colore rosso mostrano dati quantitativi riguardo la presenza di una tendenza ribassista.
2. Probabilità future: analizzando i dati è possibile reperire la probabilità con cui il prezzo si troverà sopra o sotto la media in futuro. In verde vengono classificate le probabilità che il prezzo sarà maggiore alla media, in rosso vengono classificate le probabilità che il prezzo sarà minore della media.
🔫 Filtro operativo
L' indicatore può essere utilizzato a livello operativo nella ricerca di opportunità di investimento o di trading vista la capacità di identificare un inefficienza all'interno del campione di dati osservato.
⬆ Previsione rialzista
Per operatività di tipo rialzista l'inefficienza apparirà come una serie storica a tendenza rialzista, con alte probabilità di tendenza rialzista in futuro che attualmente si trova al di sotto della media.
⬇ Previsione ribassista
Per operatività di tipo short l'inefficienza apparirà come una serie storica a tendenza ribassista, con alte probabilità di tendenza ribassista in futuro che si trova attualmente sopra la media.
📚 Impostazioni
Input: tramite l'interfaccia utente Input è possibile regolare i periodi (da 1 a 500) con cui calcolare la media. Di default i periodi sono impostati sul valore di 200, questo significa che la media viene calcolata prendendo gli ultimi 200 periodi.
Style: tramite l'interfaccia utente Style è possibile regolare il colore e attivare o disattivare un specifico output.

# Statisticalprobability

Moving Average Cross Probability [AlgoAlpha]Moving Average Cross Probability 📈✨
The Moving Average Cross Probability by AlgoAlpha calculates the probability of a cross-over or cross-under between the fast and slow values of a user defined Moving Average type before it happens, allowing users to benefit by front running the market.
✨ Key Features:
📊 Probability Histogram: Displays the Probability of MA cross in the form of a histogram.
🔄 Data Table: Displays forecast information for quick analysis.
🎨 Customizable MAs: Choose from various moving averages and customize their length.
🚀 How to Use:
🛠 Add Indicator: Add the indicator to favorites, and customize the settings to suite your trading style.
📊 Analyze Market: Watch the indicator to look for trend shifts early or for trend continuations.
🔔 Set Alerts: Get notified of bullish/bearish points.
✨ How It Works:
The Moving Average Cross Probability Indicator by AlgoAlpha determines the probability by looking at a probable range of values that the price can take in the next bar and finds out what percentage of those possibilities result in the user defined moving average crossing each other. This is done by first using the HMA to predict what the next price value will be, a standard deviation based range is then calculated. The range is divided by the user defined resolution and is split into multiple levels, each of these levels represent a possible value for price in the next bar. These possible predicted values are used to calculate the possible MA values for both the fast and slow MAs that may occur in the next bar and are then compared to see how many of those possible MA results end up crossing each other.
Stay ahead of the market with the Moving Average Cross Probability Indicator AlgoAlpha! 📈💡

Vwap Z-Score with Signals [UAlgo]The "VWAP Z-Score with Signals " is a technical analysis tool designed to help traders identify potential buy and sell signals based on the Volume Weighted Average Price (VWAP) and its Z-Score. This indicator calculates the VWAP Z-Score to show how far the current price deviates from the VWAP in terms of standard deviations. It highlights overbought and oversold conditions with visual signals, aiding in the identification of potential market reversals. The tool is customizable, allowing users to adjust parameters for their specific trading needs.
🔶 Features
VWAP Z-Score Calculation: Measures the deviation of the current price from the VWAP using standard deviations.
Customizable Parameters: Allows users to set the length of the VWAP Z-Score calculation and define thresholds for overbought and oversold levels.
Reversal Signals: Provides visual signals when the Z-Score crosses the specified thresholds, indicating potential buy or sell opportunities.
🔶 Usage
Extreme Z-Score values (both positive and negative) highlight significant deviations from the VWAP, useful for identifying potential reversal points.
The indicator provides visual signals when the Z-Score crosses predefined thresholds:
A buy signal (🔼) appears when the Z-Score crosses above the lower threshold, suggesting the price may be oversold and a potential upward reversal.
A sell signal (🔽) appears when the Z-Score crosses below the upper threshold, suggesting the price may be overbought and a potential downward reversal.
These signals can help you identify potential entry and exit points in your trading strategy.
🔶 Disclaimer
The "VWAP Z-Score with Signals " indicator is designed for educational purposes and to assist traders in their technical analysis. It does not guarantee profitable trades and should not be considered as financial advice.
Users should conduct their own research and use this indicator in conjunction with other tools and strategies.
Trading involves significant risk, and it is possible to lose more than your initial investment.

BTC Valuation
The BTC Valuation indicator
is a powerful tool designed to assist traders and analysts in evaluating the current state of Bitcoin's market valuation. By leveraging key moving averages and a logarithmic trendline, this indicator offers valuable insights into potential buying or selling opportunities based on historical price value.
Key Features:
200MA/P (200-day Moving Average to Price Ratio):
Provides a perspective on Bitcoin's long-term trend by comparing the current price to its 200-day Simple Moving Average (SMA).
A positive value suggests potential undervaluation, while a negative value may indicate overvaluation.
50MA/P (50-day Moving Average to Price Ratio):
Focuses on short-term trends, offering insights into the relationship between Bitcoin's current price and its 50-day SMA.
Helps traders identify potential bullish or bearish trends in the near term.
LTL/P (Logarithmic TrendLine to Price Ratio):
Incorporates a logarithmic trendline, considering Bitcoin's historical age in days.
Assists in evaluating whether the current price aligns with the long-term logarithmic trend, signaling potential overvaluation or undervaluation.
How to Use:
Z Score Indicator Integration:
The BTC Valuation indicator leverages the Z Score Indicator to score the ratios in a statistical way.
Statistical scoring provides a standardized measure of how far each ratio deviates from the mean, aiding in a more nuanced and objective evaluation.
Z Score Indicator
This BTC Valuation indicator provides a comprehensive view of Bitcoin's valuation dynamics, allowing traders to make informed decisions.
While indicators like BTC Valuation provide valuable insights, it's crucial to remember that no indicator guarantees market predictions.
Traders should use indicators as part of a comprehensive strategy and consider multiple factors before making trading decisions.
Historical performance is not indicative of future results. Exercise caution and continually refine your approach based on market dynamics.

Commitments of Traders Report [Advanced]This indicator displays the Commitment of Traders (COT) report data in a clear, table format similar to an Excel spreadsheet, with additional functionalities to analyze open interest and position changes. The COT report, published weekly by the Commodity Futures Trading Commission (CFTC), provides valuable insights into market sentiment by revealing the positioning of various trader categories.
Display:
Release Date: When the data was released.
Open Interest: Shows the total number of open contracts for the underlying instrument held by selected trader category.
Net Contracts: Shows the difference between long and short positions for selected trader category.
Long/Short OI: Displays the long and short positions held by selected trader category.
Change in Long/Short OI: Displays the change in long and short positions since the previous reporting period. This can highlight buying or selling pressure.
Long & Short Percentage: Displays the percentage of total long and short positions held by each category.
Trader Categories (Configurable)
Commercials: Hedgers who use futures contracts to manage risk associated with their underlying business (e.g., producers, consumers).
Non-Commercials (Large Speculators): Speculative traders with large positions who aim to profit from price movements (e.g., hedge funds, investment banks).
Non-Reportable (Small Speculators/Retail Traders): Smaller traders with positions below the CFTC reporting thresholds.
CFTC Code: If the indicator fails to retrieve data, you can manually enter the CFTC code for the specific instrument. The code for instrument can be found on CFTC's website.
Using the Indicator Effectively
Market Sentiment Gauge: Analyze the positioning of each trader category to gauge overall market sentiment.
High net longs by commercials might indicate a bullish outlook, while high net shorts could suggest bearish sentiment.
Changes in open interest and long/short positions can provide additional insights into buying and selling pressure.
Trend Confirmation: Don't rely solely on COT data for trade signals. Use it alongside price action and other technical indicators for confirmation.
Identify Potential Turning Points: Extreme readings in COT data, combined with significant changes in open interest or positioning, might precede trend reversals, but exercise caution and combine with other analysis tools.
Disclaimer
Remember, the COT report is just one piece of the puzzle. It should not be used for making isolated trading decisions. Consider incorporating it into a comprehensive trading strategy that factors in other technical and fundamental analysis.
Credit
A big shoutout to Nick from Transparent FX ! His expertise and thoughtful analysis have been a major inspiration in developing this COT Report indicator. To know more about this indicator and how to use it, be sure to check out his work.

Likelihood of Winning - Probability Density FunctionIn developing the "Likelihood of Winning - Probability Density Function (PDF)" indicator, my aim was to offer traders a statistical tool to quantify the probability of reaching target prices. This indicator, grounded in risk assessment principles, enables users to analyze potential outcomes based on the normal distribution, providing insights into market dynamics.
The tool's flexibility allows for customization of the data series, lookback periods, and target settings for both long and short scenarios. It features a color-coded visualization to easily distinguish between probabilities of hitting specified targets, enhancing decision-making in trading strategies.
I'm excited to share this indicator with the trading community, hoping it will enhance data-driven decision-making and offer a deeper understanding of market risks and opportunities. My goal is to continuously improve this tool based on user feedback and market evolution, contributing to more informed trading practices.
This indicator leverages the "NormalDistributionFunctions" library, enabling easy integration into other indicators or strategies. Users can readily embed advanced statistical analysis into their trading tools, fostering innovation within the Pine Script community.

Mean Reversion Watchlist [Z score]Hi Traders !
What is the Z score:
The Z score measures a values variability factor from the mean, this value is denoted by z and is interpreted as the number of standard deviations from the mean.
The Z score is often applied to the normal distribution to “standardize” the values; this makes comparison of normally distributed random variables with different units possible.
This popular reversal based indicator makes an assumption that the sample distribution (in this case the sample of price values) is normal, this allows for the interpretation that values with an extremely high or low percentile or “Z” value will likely be reversal zones.
This is because in the population data (the true distribution) which is known, anomaly values are very rare, therefore if price were to take a z score factor of 3 this would mean that price lies 3 standard deviations from the mean in the positive direction and is in the ≈99% percentile of all values. We would take this as a sign of a negative reversal as it is very unlikely to observe a consecutive equal to or more extreme than this percentile or Z value.
The z score normalization equation is given by
In Pine Script the Z score can be computed very easily using the below code.
// Z score custom function
Zscore(source, lookback) =>
sma = ta.sma(source, lookback)
stdev = ta.stdev(source, lookback, true)
zscore = (source - sma) / stdev
zscore
The Indicator:
This indicator plots the Z score for up to 20 different assets ( Note the maximum is 40 however the utility of 40 plots in one indicator is not much, there is a diminishing marginal return of the number of plots ).
Z score threshold levels can also be specified, the interpretation is the same as stated above.
The timeframe can also be fixed, by toggling the “Time frame lock” user input under the “TIME FRAME LOCK” user input group ( Note this indicator does not repain t).

VWMA/SMA Delta Volatility (Statistical Anomaly Detector)The "VWMA/SMA Delta Volatility (Statistical Anomaly Detector)" indicator is a tool designed to detect and visualize volatility in a financial market's price data. The indicator calculates the difference (delta) between two moving averages (VWMA/SMA) and uses statistical analysis to identify anomalies or extreme price movements. Here's a breakdown of its components:
Hypothesis:
The hypothesis behind this indicator is that extreme price movements or anomalies in the market can be detected by analyzing the difference between two moving averages and comparing it to a statistically derived normal distribution. When the MA delta (the difference between two MAs: VWMA/SMA) exceeds a certain threshold based on standard deviation and the Z-score coefficient, it may indicate increased market volatility or potential trading opportunities.
Calculation of MA Delta:
The indicator calculates the MA delta by subtracting a simple moving average (SMA) from a volume-weighted moving average (VWMA) of a selected price source. This calculation represents the difference in the market's short-term and long-term trends.
Statistical Analysis:
To detect anomalies, the indicator performs statistical analysis on the MA delta. It calculates a moving average (MA) of the MA delta and its standard deviation over a specified sample size. This MA acts as a baseline, and the standard deviation is used to measure how much the MA delta deviates from the mean.
Delta Normalization:
The MA delta, lower filter, and upper filter are normalized using a function that scales them to a specific range, typically from -100 to 100. Normalization helps in comparing these values on a consistent scale and enhances their visual representation.
Visual Representation:
The indicator visualizes the results through histograms and channels:
The histogram bars represent the normalized MA delta. Red bars indicate negative and below-lower-filter values, green bars indicate positive and above-upper-filter values, and silver bars indicate values within the normal range.
It also displays a Z-score channel, which represents the upper and lower filters after normalization. This channel helps traders identify price levels that are statistically significant and potentially indicative of market volatility.
In summary, the "MA Delta Volatility (Statistical Anomaly Detector)" indicator aims to help traders identify abnormal price movements in the market by analyzing the difference between two moving averages and applying statistical measures. It can be a valuable tool for traders looking to spot potential opportunities during periods of increased volatility or to identify potential market anomalies.

Normal Distribution CurveThis Normal Distribution Curve is designed to overlay a simple normal distribution curve on top of any TradingView indicator. This curve represents a probability distribution for a given dataset and can be used to gain insights into the likelihood of various data levels occurring within a specified range, providing traders and investors with a clear visualization of the distribution of values within a specific dataset. With the only inputs being the variable source and plot colour, I think this is by far the simplest and most intuitive iteration of any statistical analysis based indicator I've seen here!
Traders can quickly assess how data clusters around the mean in a bell curve and easily see the percentile frequency of the data; or perhaps with both and upper and lower peaks identify likely periods of upcoming volatility or mean reversion. Facilitating the identification of outliers was my main purpose when creating this tool, I believed fixed values for upper/lower bounds within most indicators are too static and do not dynamically fit the vastly different movements of all assets and timeframes - and being able to easily understand the spread of information simplifies the process of identifying key regions to take action.
The curve's tails, representing the extreme percentiles, can help identify outliers and potential areas of price reversal or trend acceleration. For example using the RSI which typically has static levels of 70 and 30, which will be breached considerably more on a less liquid or more volatile asset and therefore reduce the actionable effectiveness of the indicator, likewise for an asset with little to no directional volatility failing to ever reach this overbought/oversold areas. It makes considerably more sense to look for the top/bottom 5% or 10% levels of outlying data which are automatically calculated with this indicator, and may be a noticeable distance from the 70 and 30 values, as regions to be observing for your investing.
This normal distribution curve employs percentile linear interpolation to calculate the distribution. This interpolation technique considers the nearest data points and calculates the price values between them. This process ensures a smooth curve that accurately represents the probability distribution, even for percentiles not directly present in the original dataset; and applicable to any asset regardless of timeframe. The lookback period is set to a value of 5000 which should ensure ample data is taken into calculation and consideration without surpassing any TradingView constraints and limitations, for datasets smaller than this the indicator will adjust the length to just include all data. The labels providing the percentile and average levels can also be removed in the style tab if preferred.
Additionally, as an unplanned benefit is its applicability to the underlying price data as well as any derived indicators. Turning it into something comparable to a volume profile indicator but based on the time an assets price was within a specific range as opposed to the volume. This can therefore be used as a tool for identifying potential support and resistance zones, as well as areas that mark market inefficiencies as price rapidly accelerated through. This may then give a cleaner outlook as it eliminates the potential drawbacks of volume based profiles that maybe don't collate all exchange data or are misrepresented due to large unforeseen increases/decreases underlying capital inflows/outflows.
Thanks to @ALifeToMake, @Bjorgum, vgladkov on stackoverflow (and possibly some chatGPT!) for all the assistance in bringing this indicator to life. I really hope every user can find some use from this and help bring a unique and data driven perspective to their decision making. And make sure to please share any original implementaions of this tool too! If you've managed to apply this to the average price change once you've entered your position to better manage your trade management, or maybe overlaying on an implied volatility indicator to identify potential options arbitrage opportunities; let me know! And of course if anyone has any issues, questions, queries or requests please feel free to reach out! Thanks and enjoy.

High of Day Low of Day hourly timings: Statistics. Time of day %High of Day (HoD) & Low of Day (LoD) hourly timings: Statistics. Time of day % likelihood for high and low.
//Purpose:
To collect stats on the hourly occurrences of HoD and LoD in an asset, to see which times of day price is more likely to form its highest and lowest prices.
//How it works:
Each day, HoD and LoD are calculated and placed in hourly 'buckets' from 0-23. Frequencies and Percentages are then calculated and printed/tabulated based on the full asset history available.
//User Inputs:
-Timezone (default is New York); important to make sure this matches your chart's timezone
-Day start time: (default is Tradingview's standard). Toggle Custom input box to input your own custom day start time.
-Show/hide day-start vertical lines; show/hide previous day's 'HoD hour' label (default toggled on). To be used as visual aid for setting up & verifying timezone settings are correct and table is populating correctly).
-Use historical start date (default toggled off): Use this along with bar-replay to backtest specific periods in price (i.e. consolidated vs trending, dull vs volatile).
-Standard formatting options (text color/size, table position, etc).
-Option to show ONLY on hourly chart (default toggled off): since this indicator is of most use by far on the hourly chart (most history, max precision).
// Notes & Tips:
-Make sure Timezone settings match (input setting & chart timezone).
-Play around with custom input day start time. Choose a 'dead' time (overnight) so as to ensure stats are their most meaningful (if you set a day start time when price is likely to be volatile or trending, you may get a biased / misleadingly high readout for the start-of-day/ end-of-day hour, due to price's tendency for continuation through that time.
-If you find a time of day with significantly higher % and it falls either side of your day start time. Try adjusting day start time to 'isolate' this reading and thereby filter out potential 'continuation bias' from the stats.
-Custom input start hour may not match to your chart at first, but this is not a concern: simply increment/decrement your input until you get the desired start time line on the chart; assuming your timezone settings for chart and indicator are matching, all will then work properly as designed.
-Use the the lines and labels along with bar-replay to verify HoD/LoD hours are printing correctly and table is populating correctly.
-Hour 'buckets' represent the start of said hour. i.e. hour 14 would be populated if HoD or LoD formed between 14:00 and 15:00.
-Combined % is simply the average of HoD % and LoD %. So it is the % likelihood of 'extreme of day' occurring in that hour.
-Best results from using this on Hourly charts (sub-hourly => less history; above hourly => less precision).
-Note that lower tier Tradingview subscriptions will get less data history. Premium acounts get 20k bars history => circa 900 days history on hourly chart for ES1!
-Works nicely on Btc/Usd too: any 24hr assets this will give meaningful data (whereas some commodities, such as Lean Hogs which only trade 5hrs in a day, will yield less meaningful data).
Example usage on S&P (ES1! 1hr chart): manual day start time of 11pm; New York timezone; Visual aid lines and labels toggled on. HoD LoD hour timings with 920 days history:

Price Legs: Average Heights; 'Smart ATR'Price Legs: Average Heights; 'Smart ATR'. Consol Range Gauge
~~ Indicator to show small and large price legs (based on short and long input pivot lengths), and calculating the average heights of these price legs; counting legs from user-input start time ~~
//Premise: Wanted to use this as something like a 'Smart ATR': where the average/typical range of a distinct & dynamic price leg could be calculated based on a user-input time interval (as opposed to standard ATR, which is simply the average range over a consistent repeating period, with no regard to market structure). My instinct is that this would be most useful for consolidated periods & range trading: giving the trader an idea of what the typical size of a price leg might be in the current market state (hence in the title, Consol Range gauge)
//Features & User inputs:
-Start time: confirm input when loading indicator by clicking on the chart. Then drag the vertical line to change start time easily.
-Large Legs (toggle on/off) and user-input pivot lookback/lookforward length (larger => larger legs)
-Small Legs (toggle on/off) and user-input pivot lookback/lookforward length (smaller => smaller legs)
-Display Stats table: toggle on/off: simple view- shows the averages of large (up & down), small (up & down), and combined (for each).
-Extended stats table: toggle on/off option to show the averages of the last 3 legs of each category (up/down/large/small/combined)
-Toggle on/off Time & Price chart text labels of price legs (time in mins/hours/days; price in $ or pips; auto assigned based on asset)
-Table position: user choice.
//Notes & tips:
-Using custom start time along with replay mode, you can select any arbitrary chunk of price for the purpose of backtesting.
-Play around with the pivot lookback lengths to find price legs most suitable to the current market regime (consolidating/trending; high volatility/ low volatility)
-Single bar price legs will never be counted: they must be at least 2 bars from H>>L or L>>H.
//Credits: Thanks to @crypto_juju for the idea of applying statistics to this simple price leg indicator.
Simple View: showing only the full averages (counting from Start time):
View showing ONLY the large legs, with Time & Price labels toggled ON:

BenfordsLawLibrary "BenfordsLaw"
Methods to deal with Benford's law which states that a distribution of first and higher order digits
of numerical strings has a characteristic pattern.
"Benford's law is an observation about the leading digits of the numbers found in real-world data sets.
Intuitively, one might expect that the leading digits of these numbers would be uniformly distributed so that
each of the digits from 1 to 9 is equally likely to appear. In fact, it is often the case that 1 occurs more
frequently than 2, 2 more frequently than 3, and so on. This observation is a simplified version of Benford's law.
More precisely, the law gives a prediction of the frequency of leading digits using base-10 logarithms that
predicts specific frequencies which decrease as the digits increase from 1 to 9." ~(2)
---
reference:
- 1: en.wikipedia.org
- 2: brilliant.org
- 4: github.com
cumsum_difference(a, b)
Calculate the cumulative sum difference of two arrays of same size.
Parameters:
a (float ) : `array` List of values.
b (float ) : `array` List of values.
Returns: List with CumSum Difference between arrays.
fractional_int(number)
Transform a floating number including its fractional part to integer form ex:. `1.2345 -> 12345`.
Parameters:
number (float) : `float` The number to transform.
Returns: Transformed number.
split_to_digits(number, reverse)
Transforms a integer number into a list of its digits.
Parameters:
number (int) : `int` Number to transform.
reverse (bool) : `bool` `default=true`, Reverse the order of the digits, if true, last will be first.
Returns: Transformed number digits list.
digit_in(number, digit)
Digit at index.
Parameters:
number (int) : `int` Number to parse.
digit (int) : `int` `default=0`, Index of digit.
Returns: Digit found at the index.
digits_from(data, dindex)
Process a list of `int` values and get the list of digits.
Parameters:
data (int ) : `array` List of numbers.
dindex (int) : `int` `default=0`, Index of digit.
Returns: List of digits at the index.
digit_counters(digits)
Score digits.
Parameters:
digits (int ) : `array` List of digits.
Returns: List of counters per digit (1-9).
digit_distribution(counters)
Calculates the frequency distribution based on counters provided.
Parameters:
counters (int ) : `array` List of counters, must have size(9).
Returns: Distribution of the frequency of the digits.
digit_p(digit)
Expected probability for digit according to Benford.
Parameters:
digit (int) : `int` Digit number reference in range `1 -> 9`.
Returns: Probability of digit according to Benford's law.
benfords_distribution()
Calculated Expected distribution per digit according to Benford's Law.
Returns: List with the expected distribution.
benfords_distribution_aprox()
Aproximate Expected distribution per digit according to Benford's Law.
Returns: List with the expected distribution.
test_benfords(digits, calculate_benfords)
Tests Benford's Law on provided list of digits.
Parameters:
digits (int ) : `array` List of digits.
calculate_benfords (bool)
Returns: Tuple with:
- Counters: Score of each digit.
- Sample distribution: Frequency for each digit.
- Expected distribution: Expected frequency according to Benford's.
- Cumulative Sum of difference:
to_table(digits, _text_color, _border_color, _frame_color)
Parameters:
digits (int )
_text_color (color)
_border_color (color)
_frame_color (color)

Profitable Supertrend v0.1 - AlphaThis a script to try detect the best combination of supertrend parameters in a space of time. Sadly the script is slow. Evaluate all possibilities params is hard for a pinescript and my knowledge too. In some cases, when you want evaluate many time could be the script fails for timeout. Perhaps with time I could enhance. For this problem of speed the calculate of combinatios it's not complete: In factor use a increment of 0.2 in each param (0.1, 0.3, 0.5 ...) in period the increment for each value is 3. The range for factor it's from 3.0 to 12.0. The range of period it's from 10 to 43
My knowledge don't let me go more far. Perhaps with time I can enhance the script.

Reinforced RSI - The Quant Science This strategy was designed and written with the goal of showing and motivating the community how to integrate our 'Probabilities' module with their own script.
We have recreated one of the simplest strategies used by many traders. The strategy only trades long and uses the overbought and oversold levels on the RSI indicator.
We added stop losses and take profits to offer more dynamism to the strategy. Then the 'Probabilities' module was integrated to create a probabilistic reinforcement on each trade.
Specifically, each trade is executed, only if the past probabilities of making a profitable trade is greater than or equal to 51%. This greatly increased the performance of the strategy by avoiding possible bad trades.
The backtesting was calculated on the NASDAQ:TSLA , on 15 minutes timeframe.
The strategy works on Tesla using the following parameters:
1. Lenght: 13
2. Oversold: 40
3. Overbought: 70
4. Lookback: 50
5. Take profit: 3%
6. Stop loss: 3%
Time period: January 2021 to date.
Our Probabilities Module, used in the strategy example:

Probabilities Module - The Quant Science This module can be integrate in your code strategy or indicator and will help you to calculate the percentage probability on specific event inside your strategy. The main goal is improve and simplify the workflow if you are trying to build a quantitative strategy or indicator based on statistics or reinforcement model.
Logic
The script made a simulation inside your code based on a single event. For single event mean a trading logic composed by three different objects: entry, take profit, stop loss.
The script scrape in the past through a look back function and return the positive percentage probability about the positive event inside the data sample. In this way you are able to understand and calculate how many time (in percentage term) the conditions inside the single event are positive, helping to create your statistical edge.
You can adjust the look back period in you user interface.
How can set up the module for your use case
At the top of the script you can find:
1. entry_condition : replace the default condition with your specific entry condition.
2. TPcondition_exit : replace the default condition with your specific take profit condition.
3. SLcondition_exit : replace the default condition with your specific stop loss condition.

Student's T-Distribution Bollinger BandsThis study shows the prediction interval as Bollinger Bands using Student's T-distribution. This means that the bands will be wider when the data features higher variation, as well as when the sample size (in the form of length) is smaller. The bands will also be wider when the confidence level is lower. The opposite is also true. Assuming we set a confidence level of 0.99 and a source set to the close price, we could reasonably expect that 99% of the time the close price would fall between the upper and lower bounds. Because this is a general statistical method which requires a lot of math, the script has a tendency to be relatively slow, but should be eligible to be used in a wide variety of situations.

Saty ATR LevelsThis indicator uses the previous period close and +/- 1 ATR to display significant day, multiday, swing, and position trading levels including:
- Trigger clouds for possibly going long/short @ 23.6 fib
- Mid-range level at 61.8 fib
- Full range level at +/- 1 ATR (from previous close)
- Extension level at 161.8 fib
Additionally, a convenient info table is provided that shows trend, range utilization, and numerical long/short values.
This indicator is most beneficial when you combine it with price, volume, and trend analysis. For educational content please check out the indicator website at atrlevels.com.
I am constantly improving this indicator, please use this one if you want to continue to get new features, bug fixes, and support.

SIMPLE CANDLESTICK PATTERN ALGO BACKTESTING - TESLA 4HMany traders spend a lot of time to create algorithms full of unrealistic and far from reality indicators and market conditions. With this script I want to help traders understand the advantage of the Pine language. Using indicators with no statistical foundation and creating algorithms with technical indicators and thousands of conditions is not always the right way to create an efficient tool.
With this script that we have called "SimpleBarPattern_LongOnly" we analyse the market through a simple condition, using bars or candles.
How it works
The condition is constructed as follows. You go long with 100% of the established capital and 0.03% commission. The first condition is that the minimum of the period under analysis falls below the opening level. The second condition is that the low of the period is below the low of the previous period. The third condition is that the close of the period is above the opening level. The final condition wants the current close to be higher than the previous open and higher than the previous close. We used a statistical approach in the creation of this script, some candlestick patterns that reflect these conditions are: Bullish Engulfing, Bullish Hammer and Morning Star .
This strategy aims to help traders make more accurate decisions while using candlesticks for their trading and scientifically demonstrates that candlesticks are valid statistical tools for financial analysis.
"SimpleBarPattern_LongOnly" is a very lightweight script created with Pine v5. We developed a user interface that can adjust the analysis period from a few days to several years.
The initial capital set is €1,000 (You can change this from the "Properties" section of the user interface).
Each individual trade uses 100% of the set capital, in this case €1,000.
The default commission per trade is 0.03% (You can change this in the "Properties" section of the user interface).
User Interface
1) General backtest time settings: Set the history period to be analysed
StartDate: backtest start date
StartMonth: backtest start month
StartYear: backtest start year
EndDate: backtest end day
EndMonth: backtest end month
EndYear: backtest end year
3) Stop Loss
4) Take Profit
Please do not hesitate to contact us for any questions or information.
Disclaimer
Be careful, the past is not a guarantee of future performance, so remember to use the script as a pure analysis tool. The developer takes no responsibility for any use other than research and analysis and can in no way be held liable for damages resulting from wrong use of this code.

Bar StatisticsThis script calculates and displays some bar statistics.
For the bar length statistics, it takes every length of upper or lower movements and calculates their average (with SD), median, and max. That way, you can see whether there is a bias in the market or not.
Eg.: If for 10 bars, the market moved 2 up, then 1 down, then 3 up, then 2 down, and 2 up, the average up bars length would be at 2.33, while the average for the down length would be at 1.5, showing that upper movements last longer than down movements.
For the range statistics, it takes the true range of each bar and calculates where the close of the bar is in relation to the true low of it. So if the closing of the bar is at 10.0, the low is at 9.0, and the high is at 10.2, the candle closed in the upper third of the bar. This process is calculated for every bar and for both closing prices and open prices. It is very useful to locate biasses, and they can you a better view of the market, since for most of the time a bar will open on an extreme and close on another extreme.
Eg.: Here on the DJI, we can see that for most of the time, a month opens at the lower third (near the low) and closes at the upper third (near the high). We can also see that it is very difficult for a month to open or close on the middle of the candle, showing how important the first and the last day are for determining the trend of the rest of the month.

Inferential Statistics And Quick Metrics For Strategy Analysis.Part of this script is used to calculate inferential statistics and metrics not available through the built in variables in the strategy tester.
A label will be created on the last bar displaying important strategy results, so you can test and analyze strategies quicker.
The built in strategy itself is just an example. You can copy and paste the metrics into any existing version 4 strategy and instantly use it**
**Just be sure all the variable names are unique in your target script.
I am looking for critique and would appreciate input on the statistical functions. I am aware that some of these functions are based on the assumption that the data is normally distributed. It's not meant to be perfect, but it is meant to be helpful. So if you think I can add or improve something to make it more helpful, let me know.