Weighted Fourier Transform: Spectral Gating & Main Frequency🙏🏻 This drop has 2 purposes:
1) to inform every1 who'd ever see it that Weighted Fourier Tranform does exist, while being available nowhere online, not even in papers, yet there's nothing incredibly complicated about it, and it can/should be used in certain cases;
2) to show TradingView users how they can use it now in dem endevours, to show em what spectral filtering is, and what can they do with all of it in diy mode.
... so we gonna have 2 sections in the description
Section 1: Weighted Fourier Transform
It's quite easy to include weights in Fourier analysis: you just premultiply each datapoint by its corresponding weight -> feed to direct Fourier Transform, and then divide by weights after inverse Fourier transform. Alternatevely, in direct transform you just multiply contributions of each data point to the real and imaginary parts of the Fourier transform by corresponding weights (in accumulation phase), and in inverse transform you divide by weights instead during the accumulation phase. Everything else stays the same just like in non-weighted version.
If you're from the first target group let's say, you prolly know a thing or deux about how to code & about Fourier Transform, so you can just check lines of code to see the implementation of Weighted Discrete version of Fourier Transform, and port it to to any technology you desire. Pine Script is a developing technology that is incredibly comfortable in use for quant-related tasks and anything involving time series in general. While also using Python for research and C++ for development, every time I can do what I want in Pine Script, I reach for it and never touch matlab, python, R, or anything else.
Weighted version allows you to explicetly include order/time information into the operation, which is essential with every time series, although not widely used in mainstream just as many other obvious and right things. If you think deeply, you'll understand that you can apply a usual non-weighted Fourier to any 2d+ data you can (even if none of these dimensions represent time), because this is a geometric tool in essence. By applying linearly decaying weights inside Fourier transform, you're explicetly saying, "one of these dimensions is Time, and weights represent the order". And obviously you can combine multiple weightings, eg time and another characteristic of each datum, allows you to include another non-spatial dimension in your model.
By doing that, on properly processed (not only stationary but Also centered around zero data), you can get some interesting results that you won't be able to recreate without weights:
^^ A sine wave, centered around zero, period of 16. Gray line made by: DWFT (direct weighted Fourier transform) -> spectral gating -> IWFT (inverse weighted Fourier transform) -> plotting the last value of gated reconstructed data, all applied to expanding window. Look how precisely it follows the original data (the sine wave) with no lag at all. This can't be done by using non-weighted version of Fourier transform.
^^ spectral filtering applied to the whole dataset, calculated on the latest data update
And you should never forget about Fast Fourier Transform, tho it needs recursion...
Section 2: About use cases for quant trading, about this particular implementaion in Pine Script 6 (currently the latest version as of Friday 13, December 2k24).
Given the current state of things, we have certain limits on matrix size on TradingView (and we need big dope matrixes to calculate polynomial regression -> detrend & center our data before Fourier), and recursion is not yet available in Pine Script, so the script works on short datasets only, and requires some time.
A note on detrending. For quality results, Fourier Transform should be applied to not only stationary but also centered around zero data. The rightest way to do detrending of time series
is to fit Cumulative Weighted Moving Polynomial Regression (known as WLSMA in some narrow circles xD) and calculate the deltas between datapoint at time t and this wonderful fit at time t. That's exactly what you see on the main chart of script description: notice the distances between chart and WLSMA, now look lower and see how it matches the distances between zero and purple line in WFT study. Using residuals of one regression fit of the whole dataset makes less sense in time series context, we break some 'time' and order rules in a way, tho not many understand/cares abouit it in mainstream quant industry.
Two ways of using the script:
Spectral Gating aka Spectral filtering. Frequency domain filtering is quite responsive and for a greater computational cost does not introduce a lag the way it works with time-domain filtering. Works this way: direct Fourier transform your data to get frequency & phase info -> compute power spectrum out of it -> zero out all dem freqs that ain't hit your threshold -> inverse Fourier tranform what's left -> repeat at each datapoint plotting the very first value of reconstructed array*. With this you can watch for zero crossings to make appropriate trading decisions.
^^ plot Freq pass to use the script this way, use Level setting to control the intensity of gating. These 3 only available values: -1, 0 and 1, are the general & natural ones.
* if you turn on labels in script's style settings, you see the gray dots perfectly fitting your data. They get recalculated (for the whole dataset) at each update. You call it repainting, this is for analytical & aesthetic purposes. Included for demonstration only.
Finding main/dominant frequency & period. You can use it to set up Length for your other studies, and for analytical purposes simply to understand the periodicity of your data.
^^ plot main frequency/main period to use the script this way. On the screenshot, you can see the script applied to sine wave of period 16, notice how many datapoints it took the algo to figure out the signal's period quite good in expanding window mode
Now what's the next step? You can try applying signal windowing techniques to make it all less data-driven but your ego-driven, make a weighted periodogram or autocorrelogram (check Wiener-Khinchin Theorem ), and maybe whole shiny spectrogram?
... you decide, choice is yours,
The butterfly reflect the doors ...
∞
Dominant
Adaptive Zero Lag EMA v2This is my most successful strategy to date! Please enjoy and join the Open Source movement by sharing your code and ideas online!
OPERATING PRINCIPLE
The strategy is based on Ehlers idea that any indicator can be turned into a signal-producing trade system through smoothing and other filtering processes.
In fact, I'm using his Zero Lag EMA (ZLEMA) as a baseline indicator as well as some code snippets he has made public (1). God bless open source!
Next, I've provided the option to use an Instantaneous Frequency Measurement (IFM) method, which will adaptively choose the best period for the ZLEMA (2)
I've written other studies that use the differential calculus approximations for IFM, so it was only natural to include them in this strategy.
The primary two are Cosine IFM (3) and In-phase Quadrature IFM (4). You can also find an indicator with both plotted and the ability to average them together, as one IFM prefers long periods and the other short. (5)
BEFORE WE BEGIN
1. This strategy only runs on "normal" FX pairs (EURUSD, GBPJPY, AUDUSD ...) and will fail on Metals or Commodities.
Cryptos are largely untested.
2. Please run it on these time frames: M15 to D.
Anything outside this range will likely fail.
HOW TO USE AND SUCCEED
1. If the Default settings don't produce good results right off the bat, then lower gain limit to 1 or 2 and threshold to 0.01.
2. Test each setting under adaptive method . If you want to leave it Off , then I'd recommend using some kind of IFM (see my links below) to
discover the most efficient period to use.
3. Once you have the best adaptive method chosen, begin incrementing gain limit until you find a nice balance between profit factor (PF) and drawdown.
4. Now, begin incrementing threshold . The goal is to have PF above 2 and a drawdown as low as possible.
5. Finally, change the source ! Typically, close is the best option, but I have run into cases where high
yielded the highest returns and win rate.
6. Sit back, relax, and tweak the risk until you're happy with the return and drawdown amounts.
ADVANCED
You may need to adjust take profit (TP) points and stop loss (SL) points to create the best entry possible. Don't be greedy! You'll likely have poor
results if the TP is set to 300 and SL is 50.
If you are trading a pair that has a long Dominant Cycle Period , then you may increase Max Period to allow the IFM
to accept longer periods. Any period above the Max Period will be rejected. This may increase lag time!
Cheers and good luck trading!
-DasanC
PS - This code doesn't repaint or have future-leak, which was present in Pinescript v2.
PPS - Believe me! These returns are typical! Sometimes you must push aside the "if it's too good to be true..." mindset that society has ingrained in you.
Do you really believe the most successful pass up opportunities before investigating them? ;)
(1) Ehlers & Ric Zero Lag EMA
(2) Measuring Cycles by Ehlers
(3) Cosine IFM
(4) Inphase Quadrature IFM
(5) Averaging IFM
Ehlers Smoothed Adaptive MomentumEhlers Smoothed Adaptive Momentum script.
This indicator was developed and described by John F. Ehlers in his book "Cybernetic Analysis for Stocks and Futures" (2004, Chapter 12: Adapting to the Trend).
Ehlers Instantaneous TrendlineEhlers Instantaneous Trendline script.
This indicator was described by John F. Ehlers in his book "Rocket Science for Traders" (2001, Chapter 10: The Instantaneous Trendline).
Missile RSI (RSI of momentum w/ Dominant Cycle length + Fisher)This is a predictive indicator that looks for explosions in momentum of the cycles in price and large shifts in Momentum (Fisher turns the Bimodal PDF into Guassian like) as statistically unlikely events, showing points to exit or reverse positions.
You can adjust the lowpass frequency cuttoff (Aka what cycles you want to remove from the calculations through the super smoother filter).
To be honest you can monkey trade the direction of the Signal if you'd like but the Divergences and Maxing of the values is whats most useful.
Let me know if you guys want me to add anything else.
DFT - Dominant Cycle Period 8-50 bars - John EhlerThis is the translation of discret cosine tranform (DCT) usage by John Ehler for finding dominant cycle period (DC).
The price is first filtered to remove aliasing noise(bellow 8 bars) and trend informations(above 50 bars), then the power is computed.
The trick here is to use a normalisation against the maximum power in order to get a good frequency resolution.
Current limitation in tradingview does not allow to display all of the periods, still the DC period is plot after beeing computed based on the center of gravity algo.
The DC period can be used to tune all of the indicators based on the cycles of the markets. For instance one can use this (DC period)/2 as an input for RSI.
Hope you find this of some interrest.
Relative Vigor Index with Dominant Cycle Detection (RVI)Relative Vigor Index with Dominant Cycle Detection. As Ehler's mentioned, fixed length look back is inherently flawed when it is possible to extract a length from a dominant price cycle. may be less effective if signal to noise ratio is greater than 2, but that usually would not happen at >5m candles, and honestly shouldn't be looking at RV(igor)I when price is moving sideways.
Read just like an RVGI, but adjusted to the current time frame. To reduce noise, changing to heiken ashi will help with signals as well. Let me know if there are improvements!
Made for JD, the OG.
