Exponential Grid [Phi, Pi, Euler]If you disagree with one of the EMH principles that price is too random, then by definition you must agree that historic price has deterministic function to a scenario ahead.
I personally believe that constants like phi, pi and e can mimic exponential growth of the price.
In this script, first grid is based on the Lowest price multiplied with self fraction of the constant.
For example:
If you are familiar with fib ratio 1.272, then you must know that it is 1.618 to the power of 0.5.
With default settings of exponent step 0.25
First grid = Lowest price x phi^0.25
Second grid = Lowest price x phi^0.25x2
Third grid = Lowest price x phi^0.25x3 and so on
The script will automatically find the lowest price and update the grid values.
Or you can set up your custom Lowest price manually if you feel like the All Time Low level loses its relevance value after long period.
There are 64 grids including Lowest price level. And it wasn't by a chance. Pine Script has a limitation of max 64 plots. Number of grids shown in the chart depends on the highest price. Once price breaks above ATH a couple of next grids will be plotted automatically. In most cases if everything is plotted, the chart appears squeezed and you'll need to zoom in to see it. Therefore, I adjusted it relatively to the scale of the chart for the comfort.
In some cases 64 plots aren't enough to cover the whole chart. For example, let's take a look at NVIDIA chart:
Since the price has started with 0.0333, it is way too small to cover all with default settings.
We are left with 2 choices:
Either Enable "Round"
OR increase Exponent Step (from 0.25 to 0.5 in the particular example below)
If you set constant to pi or e which is a bigger number than phi, expect the gaps to be bigger. To reduce it to a more gradual way of expansion you can decrease Exponent Step.

# Euler

Fourier Smoothed Hybrid Volume Spread AnalysisIndicator id:
USER;91bdff47320b4284a375f428f683b21e
(only relevant to those that use API requests)
MEANINGFUL DESCRIPTION:
The Fourier Smoothed Hybrid Volume Spread Analysis (FSHVSA) indicator is an innovative trading tool designed to fuse volume analysis with trend detection capabilities, offering traders a comprehensive view of market dynamics.
This indicator stands apart by integrating the principles of the Discrete Fourier Transform (DFT) and volume spread analysis, enhanced with a layer of Fourier smoothing to distill market noise and highlight trend directions with unprecedented clarity.
This smoothing process allows traders to discern the true underlying patterns in volume and price action, stripped of the distractions of short-term fluctuations and noise.
The core functionality of the FSHVSA revolves around the innovative combination of volume change analysis, spread determination (calculated from the open and close price difference), and the strategic use of the EMA (default 10) to fine-tune the analysis of spread by incorporating volume changes.
Trend direction is validated through a moving average (MA) of the histogram, which acts analogously to the Volume MA found in traditional volume indicators. This MA serves as a pivotal reference point, enabling traders to confidently engage with the market when the histogram's movement concurs with the trend direction, particularly when it crosses the Trend MA line, signalling optimal entry points.
It returns 0 when MA of the histogram and EMA of the Price Spread are not align.
HOW TO USE THE INDICATOR:
The FSHVSA plots a positive trend when a positive Volume smoothed Spread and EMA of Volume smoothed price is above 0, and a negative when negative Volume smoothed Spread and EMA of Volume smoothed price is below 0. When this conditions are not met it plots 0.
ORIGINALITY & USEFULNESS:
The FSHVSA is unique because it applies DFT for data smoothing, effectively filtering out the minor fluctuations and leaving traders with a clear picture of the market's true movements. The DFT's ability to break down market signals into constituent frequencies offers a granular view of market dynamics, highlighting the amplitude and phase of each frequency component. This, combined with the strategic application of Ehler's Universal Oscillator principles via a histogram, furnishes traders with a nuanced understanding of market volatility and noise levels, thereby facilitating more informed trading decisions.
DETAILED DESCRIPTION:
My detailed description of the indicator and use cases which I find very valuable.
What is the meaning of price spread?
In finance, a spread refers to the difference between two prices, rates, or yields. One of the most common types is the bid-ask spread, which refers to the gap between the bid (from buyers) and the ask (from sellers) prices of a security or asset.
We are going to use Open-Close spread.
What is Volume spread analysis?
Volume spread analysis (VSA) is a method of technical analysis that compares the volume per candle, range spread, and closing price to determine price direction.
What does this mean?
We need to have a positive Volume Price Spread and a positive Moving average of Volume price spread for a positive trend. OR via versa a negative Volume Price Spread and a negative Moving average of Volume price spread for a negative trend.
What if we have a positive Volume Price Spread and a negative Moving average of Volume Price Spread ?
It results in a neutral, not trending price action.
Thus the indicator returns 0.
In the next Image you can see that trend is negative on 4h, neutral on 12h and neutral on 1D. That means trend is negative .
I am sorry, the chart is a bit messy. The idea is to use the indicator over more than 1 Timeframe.
What is approximation and smoothing?
They are mathematical concepts for making a discrete set of numbers a
continuous curved line.
Fourier and Euler approximation of a spread are taken from aprox library.
Key Features:
Noise Reduction leverages Euler's White noise capabilities for effective Volume smoothing, providing a cleaner and more accurate representation of market dynamics.
Choose between the innovative Double Discrete Fourier Transform (DTF32) and Regular Open & Close price series.
Mathematical equations presented in Pinescript:
Fourier of the real (x axis) discrete:
x_0 = array.get(x, 0) + array.get(x, 1) + array.get(x, 2)
x_1 = array.get(x, 0) + array.get(x, 1) * math.cos( -2 * math.pi * _dir / 3 ) - array.get(y, 1) * math.sin( -2 * math.pi * _dir / 3 ) + array.get(x, 2) * math.cos( -4 * math.pi * _dir / 3 ) - array.get(y, 2) * math.sin( -4 * math.pi * _dir / 3 )
x_2 = array.get(x, 0) + array.get(x, 1) * math.cos( -4 * math.pi * _dir / 3 ) - array.get(y, 1) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(x, 2) * math.cos( -8 * math.pi * _dir / 3 ) - array.get(y, 2) * math.sin( -8 * math.pi * _dir / 3 )
Fourier of the imaginary (y axis) discrete:
y_0 = array.get(x, 0) + array.get(x, 1) + array.get(x, 2)
y_1 = array.get(x, 0) + array.get(x, 1) * math.sin( -2 * math.pi * _dir / 3 ) + array.get(y, 1) * math.cos( -2 * math.pi * _dir / 3 ) + array.get(x, 2) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(y, 2) * math.cos( -4 * math.pi * _dir / 3 )
y_2 = array.get(x, 0) + array.get(x, 1) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(y, 1) * math.cos( -4 * math.pi * _dir / 3 ) + array.get(x, 2) * math.sin( -8 * math.pi * _dir / 3 ) + array.get(y, 2) * math.cos( -8 * math.pi * _dir / 3 )
Euler's Smooth with Discrete Furrier approximated Volume.
a = math.sqrt(2) * math.pi / _devided
b = math.cos(math.sqrt(2) * 180 / _devided)
c2 = 2 * math.pow(a, 2) * b
c3 = math.pow(a, 4)
c1 = 1 - 2 * math.pow(a, 2) * math.cos(b) + math.pow(a, 4)
filt := na(filt ) ? 0 : c1 * (w + nz(w )) / 2.0 + c2 * nz(filt ) + c3 * nz(filt )
Usecase:
First option:
Leverage the script to identify Bullish and Bearish trends, shown with green and red triangle.
Combine Different Timeframes to accurately determine market trend.
Second option:
Pull the data with API sockets to automate your trading journey.
plot(close, title="ClosePrice", display=display.status_line)
plot(open, title="OpenPrice", display=display.status_line)
plot(greencon ? 1 : redcon ? -1 : 0, title="position", display=display.status_line)
Use ClosePrice, OpenPrice and "position" titles to easily read and backtest your strategy utilising more than 1 Time Frame.
Indicator id:
USER;91bdff47320b4284a375f428f683b21e
(only relevant to those that use API requests)

Prime, E & PI Superiority CyclesIf you have been studying the markets long enough you will probably have noticed a certain pattern. Whichever trade entry/exit logic you try to use, it will go through phases of working really well and phases where it doesn't work at all. This is the markets way of ensuring anyone who sticks to an oversimplified, one-dimensional strategy will not profit. Superiority cycles are a method I devised by which code interrogates the nature of where price has been pivoting in relation to three key structures, the Prime Frame, E Frame and Pi Frame which are plotted as horizontal lines at these values:
* Use script on 1 minute chart ONLY
prime numbers up to 100: 2.0,3.0,5.0,7.0,11.0,13.0,17.0,19.0,23.0,27.0,29.0,31.0,37.0,41.0,43.0,47.0,53.0,59.0,61.0,67.0,71.0,73.0,79.0,83.0,89.0,97.0
multiples of e up to 100: 2.71828, 5.43656, 8.15484, 10.87312, 13.5914, 16.30968, 19.02796, 21.74624, 24.46452, 27.1828, 29.90108, 32.61936, 35.33764,
38.05592, 40.7742, 43.49248, 46.21076, 48.92904, 51.64732, 54.3656, 57.08388, 59.80216, 62.52044, 65.23872, 67.957, 70.67528, 73.39356000000001, 76.11184,
78.83012, 81.5484, 84.26668000000001, 86.98496, 89.70324, 92.42152, 95.13980000000001, 97.85808
multiples of pi up to 100: 3.14159, 6.28318, 9.424769999999999, 12.56636, 15.70795, 18.849539999999998, 21.99113, 25.13272, 28.27431, 31.4159, 34.55749,
37.699079999999995, 40.840669999999996, 43.98226, 47.12385, 50.26544, 53.40703, 56.54862, 59.69021, 62.8318, 65.97339, 69.11498, 72.25657, 75.39815999999999,
78.53975, 81.68133999999999, 84.82293, 87.96452, 91.10611, 94.2477, 97.38929
These values are iterated up the chart as seen below:
The script sums the distance of pivots to each of the respective frames (olive lines for Prime Frame, green lines for E Frame and maroon lines for Pi Frame) and determines which frame price has been reacting to in the least significant way. The worst performing frame is the next frame we target reversals at. The table in the bottom right will light up a color that corresponds to the frame color we should target.
Here is an example of Prime Superiority, where we prioritize trading from prime levels:
The table and the background color are both olive which means target prime levels. In an ideal world strong moves should start and finish where the white flags are placed i.e. in this case $17k and $19k. The reason these levels are 17,000 and 19,000 and not just 17 and 19 like in the original prime number sequence is due to the scaling code in the get_scale_func() which allows the code to operate on all assets.
This is E Superiority where we would hope to see major reversals at green lines:
This is Pi Superiority where we would hope to see major reversals at maroon lines:
And finally I would like to show you a market moving from one superiority to another. This can be observed by the bgcolor which tells us what the superiority was at every historical minute
Pi Frame Superiority into E Frame Superiority example:
Prime Frame Superiority into E Frame Superiority example:
Prime Frame Superiority into Pi Frame Superiority example:
By rotating the analysis we use to enter trades in this way we hope to hide our strategy better from market makers and artificial intelligence, and overall make greater profits.

Euler Frame (Dynamic)Euler frame posted as lines instead of plots, allowing for complete colour change and extension.
Be warned, the entire frame moves in real time with price so do not look at the corresponding lines for scale factors below (i.e. it will show the current levels in this graph for also under 10k).
To see the correct levels for the <10k range you can use the TV replay function or refer to my "Euler Price Levels" script, which plots accordingly with scale factor.

Confluence Zones & MidpointsConfluence zones between tight Prime / Euler / Pi levels, and their midpoints.
Colour and extend options included.

Pythagorean Means of Moving AveragesDESCRIPTION
Pythagorean Means of Moving Averages
1. Calculates a set of moving averages for high, low, close, open and typical prices, each at multiple periods.
Period values follow the Fibonacci sequence.
The "short" set includes moving average having the following periods: 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.
The "mid" set includes moving average having the following periods: 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597.
The "long" set includes moving average having the following periods: 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
2. User selects the type of moving average: SMA, EMA, HMA, RMA, WMA, VWMA.
3. Calculates the mean of each set of moving averages.
4. User selects the type of mean to be calculated: 1) arithmetic, 2) geometric, 3) harmonic, 4) quadratic, 5) cubic. Multiple mean calculations may be displayed simultaneously, allowing for comparison.
5. Plots the mean for high, low, close, open, and typical prices.
6. User selects which plots to display: 1) high and low prices, 2) close prices, 3) open prices, and/or 4) typical prices.
7. Calculates and plots a vertical deviation from an origin mean--the mean from which the deviation is measured.
8. Deviation = origin mean x a x b^(x/y)/c.
9. User selects the deviation origin mean: 1) high and low prices plot, 2) close prices plot, or 3) typical prices plot.
10. User defines deviation variables a, b, c, x and y.
Examples of deviation:
a) Percent of the mean = 1.414213562 = 2^(1/2) = Pythagoras's constant (default).
b) Percent of the mean = 0.7071067812 = = = sin 45˚ = cos 45˚.
11. Displaces the plots horizontally +/- by a user defined number of periods.
PURPOSE
1. Identify price trends and potential levels of support and resistance.
CREDITS
1. "Fibonacci Moving Average" by Sofien Kaabar: two plots, each an arithmetic mean of EMAs of 1) high prices and 2) low prices, with periods 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
2. "Solarized" color scheme by Ethan Schoonover.

Eliza S&R FrameSource code for the e (euler based) S&R frame as requested by a few people. Yellow line are major levels, blue lines are minor levels but all are important. Use in conjunction with 27 minute chart or 271 minute chart if you want to go full Eulerian in your trading approach. For an asset over $10,000 the lines are $2718 apart starting from 0. For an asset over $1,000 the lines are $271.8 apart starting from 0. This pattern continues through the decimal point scale. Remember the first real attempt on one of these lines by the price is the most important. If you see a wick through price will often need to move to the next line in the frame as strong support or resistance will turn price away without ever being broken.

Euler Cubes - CubᵋI give you the "Euler Cubes", inspired by the mathematical number 'e' (Euler's number).
It is suggested (fibonacci ratios analogy) that price/e ratio can give Support/Resistance area's.
The first cube is made by a low/high of choice, for example:
You set the 'source low'/'source high' in position:
Then you choose the 'e ratio' (x times 'e')
This multiplies the distance 'high-low' times '0.271828' times 'the set number' .
For example, choosing 5 gives 5 x 0.271828 = 1.35914, the distance 'high-low' hereby multiplied by 1.35914, the following cubes multiply the previous distance by 1.35914.
(Settings below 5 will give cubes smaller than the 'high-low' distance)
In the case of x times 'e' = 5:
You can extend the lines:
Now you can give it an angle:
Do mind, using it over very little bars and using an angle can cause some lines to not align as intended, because for now, it is not possible to plot in between bars.
There are also 'Euler' SMA and EMA available with following length's:
27, 54, 82,109, 136, 163, 190 and 217
Cheers!

[RS][V4]ZigZag Percent Reversal - Helper - Retrace LevelsA helper script with multiple retrace level options.