Extrapolation

One of the most requested and often well discussed features is now live. Sigma-MT uses a sinusoidal tuned Kalman filter to decompose the bandpass-filtered cycle into trend and oscillatory components, then projects forward. The extrapolation respects the detected wavelength and current phase, producing a natural continuation rather than a naive repetition. A short cross-fade at the boundary prevents discontinuities where historical data meets the forecast.

Each enabled timeframe — short, medium, and long-term — extrapolates independently, with the projection length configurable in terms of forward cycles.

Inflection Points and Uncertainty

Building on extrapolation, the system identifies upcoming inflection points — peaks and troughs in the projected cycle — and marks them on the chart with estimated timing. The information panel displays a countdown to the next expected inflection alongside phase status.

Since any projection carries uncertainty, the inflection markers include a time window rather than a single bar, giving a realistic range for when the turn is expected. UTC and broker time display are both supported.

Envelopes

Amplitude envelopes now wrap each cycle component, calibrated so that approximately 95% of price action falls within the bands. The envelope width is driven by the instantaneous magnitude from the wavelet transform rather than a fixed offset, so the bands widen and narrow as cycle strength fluctuates. These extend through the extrapolation region, providing a projected price range alongside the cycle forecast. This concept, first espoused by JM Hurst in ‘Profit Magic for Stock Transaction Timing’, is a useful way to visualise the volatility in an instrument defined by the cycle feature driving it

Trade Boxes

Trade zones highlight regions around upcoming inflection points where the cycle is expected to reverse. Each box spans the uncertainty window of the inflection and uses the envelope amplitude to set price bounds. Red boxes mark projected peaks, green boxes mark projected troughs. These provide a quick visual for where the cycle analysis suggests actionable zones may develop.

Yield Mode

Pressing Y activates yield mode, which walks through historical cycle inflections and calculates what a systematic approach of buying troughs and selling peaks (and vice versa) would have produced over the analysis window. The chart displays individual trade legs with percentage gains, a running total, win rate and expectancy. Each enabled timeframe calculates its own yield independently, letting you compare how different cycle horizons have performed on the same instrument.

Composite Instruments

A new cross-instrument analysis mode examines multiple related markets simultaneously. Define a basket of instruments — for example the major USD pairs — and Sigma-MT z-score normalises each price series independently, then averages them in the time domain to produce a single composite series. This averaged series is then fed through the standard wavelet convolution pipeline — the same CFAR detection, bandpass filtering, and phase analysis that runs on any individual chart. By convolving the average rather than averaging individual convolutions, the process identifies periodicity that is genuinely shared across the basket rather than artefacts of summing unrelated cycles.

The composite average line overlays the primary chart, rescaled back into the chart instrument’s price space.

Once the dominant cycle is detected from the composite, Sigma-MT calculates a Cycle Amplitude Ratio (CAR) for each instrument in the basket. CAR measures how much of an instrument's actual price movement falls within the detected cycle band, expressed as bandpass amplitude relative to mean price. The instruments are then ranked by CAR in a volatility ranking displayed alongside the composite line. Higher CAR means a larger proportion of that instrument's price action is cyclical at the shared frequency — making it the better candidate to trade if the wavelength range around the composite cycle is your target signal.

Cycle Band Lock and Manual Selection

The detected dominant frequency band can now be locked, preventing it from jumping to any transient higher frequency power as time progresses. A lock indicator in the info panel shows the current state, toggled by clicking the box or pressing 0.

Beyond locking the auto-detected band, you can now manually define your own frequency range by shift-clicking twice on the spectrogram — once for each boundary. A dashed marker line provides visual feedback after the first click. The selection auto-locks immediately, and an Auto/User indicator next to the lock box shows whether the current band was detected algorithmically or chosen manually. A minimum bandwidth of 5% of total frequencies is enforced to ensure meaningful results

Expanded Keyboard Interface

The beta introduces a comprehensive keyboard-driven workflow. A dedicated shortcuts panel (K key) documents everything, organised into sections:

Interface: F flips the spectrogram position, G toggles spectrogram visibility, P toggles the info panel, < > adjusts panel width, C takes a screenshot.

Analysis Control: W/S adjusts sample size, A/D shifts through history, Q/E changes the increment, L/O/H/I adjusts frequency bounds, Ctrl+Click centres on a bar, and shift+click selects a custom band.

Cycle Visibility: 1/2/3 toggle short/medium/long-term cycles, 0 toggles band lock, X toggles envelopes, B toggles trade boxes.

Modes: Y activates yield mode.

Score Threshold Filtering

A configurable score threshold now filters longer-term cycles, suppressing medium and long-term overlays when their detection score falls below a chosen level (700, 800, or 900). This keeps the chart clean when only the short-term cycle has strong evidence, avoiding visual clutter from marginal detections on higher timeframes.

What’s Next

The beta is currently with founding members for testing. As we work toward a stable release, the focus is on refinement — edge cases, performance, and ensuring everything behaves predictably across instruments and timeframes. Sigma-MT will be listed on the MetaTrader market at release, and for future founding members of Sigma-L.

More to follow.

David

Since the initial alpha announcement, development on Sigma-MT, a Meta-Trader 5 expert advisor, has progressed steadily and is nearing beta. This update covers some improvements to the underlying time-frequency analysis, visual refinements, and how the tool integrates with the recently announced scanner.

Time-Frequency Engine Updates

The multi-timeframe analysis system performs independent wavelet convolutions for each enabled timeframe - short, medium, and long-term. Each process transforms the price series into the frequency domain, applies a rapid convolution process and returns a time-frequency power matrix showing how spectral energy evolves across the analysis window.

When analysing higher timeframes, the system retrieves the appropriate bar data, performs the full convolution at that resolution, then interpolates the resulting bandpass signal back to the chart timeframe for overlay. This interpolation maps the higher timeframe’s filtered output onto the short-term bar structure, accounting for the timeframe ratio. A cycle detected on H4 while viewing an M15 chart will display correctly aligned with the underlying price bars.

Dominant cycle detection begins by using CFAR (Constant False Alarm Rate) as the default method - the same adaptive thresholding technique used in radar systems. Rather than applying a fixed SNR threshold across all frequencies, CFAR calculates a local threshold for each frequency bin based on the surrounding spectral energy. Reference cells on either side of each bin establish the local noise floor, with guard cells preventing the signal itself from contaminating the estimate. The detection threshold adapts to the spectral shape, maintaining consistent false alarm probability regardless of whether you’re examining a flat spectrum or one with significant slope.

The ranking score combines multiple metrics: peak prominence above the CFAR threshold, spectral isolation from neighbouring frequencies, amplitude stability across the recency window, and consistency of the instantaneous frequency estimate. A high score indicates a cycle that stands clearly above the noise, maintains stable characteristics, and persists through the analysis period rather than appearing transiently.

Wavelength display throughout the interface now uses adaptive unit conversion. The base unit is determined by your chart timeframe - minutes for M1-M30, hours for H1-H12, days for daily charts. Values then convert upward when they exceed natural thresholds: minutes to hours beyond 120, hours to days beyond 24, days to weeks beyond 100. This ensures the Y-axis labels, information panel, and cycle readouts all express wavelengths in immediately meaningful terms.

Workflow with Sister Scanner App

Sigma-MT and the Sigma-MT Scanner serve different analytical needs within the same workflow.

The scanner processes your entire market watch, applying the same wavelet convolution engine across every instrument and ranking results by cycle strength. It answers the question of where to focus attention. Which markets are exhibiting periodicity now?

Sigma-MT provides the detailed examination. The full spectrogram reveals how cycle strength evolves through time, the bandpass filter isolates the dominant component and the information panel displays phase status, wavelength, and detection score. A cycle flagged by the scanner will appear with identical characteristics when examined in Sigma-MT - the underlying detection will be consistent across both tools.

Keyboard Configuration

So far we have added some very useful keyboard shortcuts to help the user navigate the spectra in an analysis.

Analysis Window:

• A / D - Shift analysis left/right through price history

• W / S - Increase/decrease sample size (N)

• Q / E - Decrease/increase increment value

• Ctrl + Click - Centre analysis on clicked bar

Frequency Bounds:

• L / O - Decrease/increase lowest frequency of analysis

• I / H - Increase/decrease highest frequency of analysis

Display:

• F - Flip spectrogram/bandpass to top or bottom of chart

• P - Toggle information panel

• Left click on spectrogram - show spectral summation and detection threshold at the clicked timepoint

• < / > - Increase/decrease panel width

Cycles:

• 1 / 2 / 3 - Toggle short/medium/long-term cycle visibility

• 4 - Toggle composite cycle overlay
  

Availability

Sigma-MT remains available to a small group of founding members for testing and I thank them for their help. As we approach a stable release, it will also be listed on the Meta-Trader market for all, in addition to being available for all new founding members of Sigma-L at release.

More updates to follow and exciting features to add, not least extrapolation, which everyone seems to love!

David

The scanner applies the same robust wavelet convolution engine that powers Sigma-MT across every instrument in your market watch, simultaneously analysing multiple timeframes and ranking results by cycle strength. In seconds, you have a prioritised list of instruments exhibiting the most statistically significant cyclical behaviour.

Under the Hood: Time-Frequency

Under the hood, the scanner leverages the same frequency-domain convolution that makes Sigma-MT so fast in MT5. Each instrument undergoes a full wavelet transform, with automatic cycle detection using either a simple SNR (Signal-to-Noise Ratio) and filter or CFAR (Constant False Alarm Rate) algorithms - the same detection methods used in radar systems for picking signals out of noise.

The cycle ranking score then combines multiple factors: peak prominence above the noise floor, spectral isolation, amplitude stability, frequency consistency, and persistence across the analysis window. To be clear this isn’t just finding any cycle - it’s finding cycles that exhibit the characteristics of genuine periodic phenomena rather than random fluctuations.

A scan of several hundred instruments across multiple timeframes completes in seconds.

Intelligent Phase Grouping

This is where the scanner becomes more than just a ranking tool. When analysing multiple timeframes, the scanner can automatically identify phase confluence - instruments showing aligned cycle phases across different time horizons.

When the same symbol exhibits similar phase, these groups lock to the top of the results, immediately drawing your attention to potential multi-timeframe confluence setups.

However, matching phases alone isn’t enough. To ensure meaningful alignment rather than the same cycle detected at different resolutions, the scanner validates that each higher timeframe shows a wavelength greater than at least 1.5 times the previous timeframes’ detected cycle. This filters for genuine nested cycle structures - the kind of fractal patterns that Hurst and others identified as characteristic of market behaviour.

Multi-Timeframe Analysis

The scanner has capacity for up to five timeframes running at simultaneous analysis. The scanner will evaluate every market watch instrument against each timeframe, giving you a comprehensive view of cyclical activity from intraday through to longer-term structures, if required.

Results display the detected wavelength (automatically formatted in minutes, hours, days, or weeks), the cycle rank score, timeframe, and current phase status. Click any row to instantly open a chart with that symbol and timeframe - ready for deeper analysis with the main Sigma-MT EA.

Filtering & Configuration

Focus your attention with built-in filtering (more features to come):

• Minimum Rank Threshold - Hide weak cycles below a specified score

• Minimum Wavelength - Filter out very short cycles that may represent noise rather than tradeable features

• Recency Bias - Weight the analysis toward recent price action

All active parameters display clearly at the top of the interface, so you always know exactly what you’re looking at:

Bars: 500 | Frequency Range: 5-40 | Detection: CFAR | Recency Bias: 50% | Analysis on Median Price | Min Wavelength: 30 min | Timeframes: M1, M15, H1

The Workflow

The scanner and Sigma-MT EA are designed to work together although one can use the scanner standalone if you have a system already setup to identify periodicities in markets.

1. Scan - Run the Scanner to identify instruments with strong cyclical behaviour. Set the scan to loop continuously after completion, manually by pressing ‘R’ or at set intervals.

2. Filter - Use phase grouping and rank thresholds to surface the most interesting opportunities

3. Click - Open a chart directly from the scanner results

4. Analyse - Apply Sigma-MT for detailed time-frequency analysis, phase projection, extrapolation and cycle visualisation. Automatic trading will also be in place further along the development process.

This workflow transforms cycle analysis from a manual, instrument-by-instrument process into a systematic scan of the entire market.

The scanner is currently in development alongside the main Sigma-MT EA and will be available to new founding members on Sigma-L in addition to being directly available on the MQL store. I believe this combination of tools - broad market scanning paired with deep individual analysis - represents something genuinely new for retail cycle analysts. I am excited to finally be able to share the insight I have been using for the past few years with others.

More updates to follow as development progresses. Thank you for your continued support.

David

So many people have asked me over the last few years whether they can use my software and I am always flattered by this. I developed it initially for myself to examine the claims by Hurst and others of the existence of periodic features in financial markets. I never really intended it to be used by others although we successfully built a multi instrument live scanner which was somewhat useful! Since then I am grateful to have shared many analyses with subscribers here and on X for those who are interested in this rather niche but fascinating area.

I am pleased to say, therefore, over the holiday period and just before, I began development work on porting over the familiar time frequency analysis readers have seen over many years on Sigma-L, to Metatrader’s MQL language. The main question was one of speed. I wanted to ensure that this new version would be fast enough to use intraday (it updates on every new bar) and also flexible enough to perform the many computations required for a robust time frequency analysis. The underlying process is already highly optimised, performing the calculations and wavelet formation all in the frequency domain.

Fortunately, with MQL’s C based language the vectorisation present in the python/numpy version was fairly easily transferred and has produced excellent results. A convolution over 1000 points and 100 frequencies takes less than a quarter of a second.

Enhancements

Over the last few years I have collated a large list of enhancements I want to bring to this kind of market analysis, both from a user experience point of view and from a signal processing point of view. Some of the expectations for features on this expert advisor, as we move toward release, will be (non exhaustive):

• Ability to shift analysis anywhere on the chart (currently live in alpha 1.0)

• Ability to quickly move up and down the frequency spectra to inspect any features of interest (currently live in alpha 1.0)

• Automatic intraday updating (currently live in alpha 1.0, updates on each new bar)

• Time frequency analysis on return series

• Multi length analysis and composite lines, combining multiple cycles of influence and true, numerically evidenced based compositing / extrapolation

• Auto detection at all timeframes

• Tuned Kalman filter extrapolation

• Composite analysis of many instruments for phase correlation

• Aesthetic configuration (colours, spectrogram etc)

• High pass filtering at the touch of a button

• Future time phase estimation for all detected features

• Social media sharing

Currently the alpha is available for testing to founding members only but as we draw closer to something I feel is ready, we will likely list it on the Metatrader shop and also make it available to future founding members here. The exact format remains to be decided.

I can’t wait to see lots of very cool cycle charts coming up on my X feed over 2026 as you guys get your hands on one of the most powerful analysis techniques out there!

Do continue to watch this space for updates, thank you all for your continued support.

David

Although Tradingview’s pinescript language is, unfortunately, limited for the purpose of performing frequency domain analysis on market data, one can implement time domain filters to a decent degree, given some verbose coding (compared to optimal python/numpy/C and vectorisation).

Below is a fairly simple implementation of a Butterworth bandpass filter (on close prices) for your analysis pleasure. The price series is first detrended prior to filtering using a quadratic (order 2) or cubic (order 3) polynomial fit.

Copy and paste the code into the Tradingview code area to use on your charts, remembering to configure the upper and lower bounds of the filter (the cutoffs) according to your desired cycle target.

It is important to consider when configuring the cutoffs that Tradingview does not interpolate for weekends / holidays as we do in the main time frequency analysis. Therefore cycles identified in the main time frequency analysis will be slightly longer than the cutoffs needed for this Butterworth filter. I recommened experimenting yourself to discover the best parameters for optimal correlation to the time frequency result.

//@version=6
indicator(”Sigma-L BW Sweep”, overlay=false)

// Inputs
low_per = input.int(40, “Low Cutoff Period”, minval=10)
high_per = input.int(80, “High Cutoff Period”, minval=10)
filter_length = input.int(100, “Number of Bars for Filter/Detrend”, minval=20)
detrend_order = input.int(2, “Detrend Order (2 or 3)”, minval=1, maxval=3)
src = input.source(close, “Source”)

// Polynomial Detrend Function (least squares fit for order 2 or 3)
poly_detrend(src_in, len, order) =>
    if len <= order
        src_in
    else
        n = len
        x_mean = (n - 1) / 2.0
        sum_y = 0.0
        sum_z_y = 0.0
        sum_z2_y = 0.0
        sum_z3_y = 0.0
        sum_z2 = 0.0
        sum_z4 = 0.0
        sum_z6 = 0.0
        for i = 0 to n - 1
            xi = float(n - 1 - i)
            zi = xi - x_mean
            yi = src_in[i]
            sum_y += yi
            sum_z_y += zi * yi
            sum_z2_y += zi * zi * yi
            sum_z2 += zi * zi
            sum_z4 += zi * zi * zi * zi
            sum_z3_y += zi * zi * zi * yi
            sum_z6 += zi * zi * zi * zi * zi * zi
        y_mean = sum_y / n
        denom_even = sum_z4 - sum_z2 * sum_z2 / n
        b2 = (sum_z2_y - y_mean * sum_z2) / denom_even
        b0 = y_mean - (sum_z2 / n) * b2
        b1 = 0.0
        b3 = 0.0
        z_current = float(n - 1) - x_mean
        if order == 2
            b1 := sum_z_y / sum_z2
        else
            det = sum_z2 * sum_z6 - sum_z4 * sum_z4
            b1 := (sum_z6 * sum_z_y - sum_z4 * sum_z3_y) / det
            b3 := (sum_z2 * sum_z3_y - sum_z4 * sum_z_y) / det
        trend = b0 + b1 * z_current + b2 * z_current * z_current + b3 * z_current * z_current * z_current
        src_in - trend

// Detrended source
detrended_src = poly_detrend(src, filter_length, detrend_order)

// 3-Pole Butterworth Low-Pass Function
butterLP(src_in, len_in) =>
    cf = 2 * math.tan(math.pi / len_in)
    cf2 = cf * cf
    cf3 = cf2 * cf
    a0 = 8 + 8 * cf + 4 * cf2 + cf3
    a1 = -24 - 8 * cf + 4 * cf2 + 3 * cf3
    a2 = 24 - 8 * cf - 4 * cf2 + 3 * cf3
    a3 = -8 + 8 * cf - 4 * cf2 + cf3
    c = cf3 / a0
    d0 = -a1 / a0
    d1 = -a2 / a0
    d2 = -a3 / a0
    out = 0.0
    out := nz(c * (src_in + src_in[3]) + 3 * c * (src_in[1] + src_in[2]) + d0 * out[1] + d1 * out[2] + d2 * out[3], src_in)
    out

// Band-Pass on detrended series: High-Pass (detrended - LP(high_per)) then LP(low_per)
lp_high = butterLP(detrended_src, high_per)
hp = detrended_src - lp_high
bp_full = butterLP(hp, low_per)

// Plot in indicator window
plot(bp_full, “Butterworth Band-Pass (Detrended)”, color=color.blue, linewidth=2)
hline(0, “Zero Line”, color=color.gray, linestyle=hline.style_dashed)

The Cycle Hunt is On

If, in the course of using this indicator, you discover what you believe to be a metronomic signal (mostly stationary amplitude and frequency!), please let me know so I can run the TFA across it and share it to subscribers. Many pairs of eyes are better than one!

Thanks in advance.

David

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Firstly, if you have not already studied the original Understanding Charts on Sigma-L post, please do so at your convenience, the underlying mechanisms are largely unchanged.

Main Analysis

Using the example shown above of US Stock Market indices, via SPY, IWM and QQQ, there are several changes over and above the aesthetic shifts for controls and input that are immediately obvious…

Composite Constituents

For a composite analysis (an analysis of more than one instrument) we have added various colours for the constituents in the time domain. This makes for easier viewing of variance between each and the overall time-domain average (white). Importantly, the time-frequency analysis is performed only on the time domain average of the close prices of each instrument. This allows us to view only those periodic features common to all markets in the composite, improving, in most cases, the signal to noise ratio when compared to a detection of the same component for an individual market.

Cycle Amplitude Metrics

In addition to the amplitude modulation (AM) and maximum amplitude (Max) for the band-passed area of the spectra (white box on TFA), we have added a couple of new measures. The Max4 is simply the maximum amplitude of the detected cycle (band-pass region) multiplied by 4. We use this as a good level for a stop loss in trading, where we do not expect price to increase in volatility (amplitude) beyond 4 times the maximum amplitude of the cycle detected. In the rare case it does it is generally due to news/macro events. The average leg (Avg.Leg) is the average of the actual price legs measured from the inflection points of the band-pass over the whole sample. This gives a broad sense of how much actual price movement is being powered by the detected dominant cycle, per leg.

Recency Marker and Spectra

Frequency and amplitude modulation is constant in financial markets and a major challenge in trading. This motivates the use of a time frequency analysis method to capture both temporal and frequency information from the signal, as opposed to simply just frequency information via a static spectral analysis (like a FFT). As traders we are interested in cycles that are influencing price movements at now-time, with a nod to the recent past, replete with the knowledge that frequency modulation tends to change slowly (amplitude modulation more randomly and less predictably). In order to augment our signal detection algorithm we take two samples of the spectra, one over the whole sample and one from the recency marker point, to now-time. We can then feed these two useful snapshots back into our calculations which further informs our calculation of amplitude, frequency and phase. The two spectra are also shown at the end of the time frequency plot, the brighter being the more recent sample.

Phase and Average Phase

We show the now-time phase of each frequency within the detected dominant cycle areas. These are represented within the recency spectra at the end of the time frequency plot by varying colours, red for peak, moving to green for troughs. As the overall cycle captured by the detected band progresses, you will see these individual colours change, each frequency within the band moving through it’s own phase at now-time. Trivially, these will all be of similar phase but can give a good indication of higher frequencies peaking or troughing first, for example. The average phase is the weighted by amplitude average of the phase values at now-time of all the frequencies within the detected band, denoted by the dotted lines. In the above example the two components at 177.03 days and 270.16 days have phases of hard up and trough respectively. As an aside, these two components are likely the Hurst 20 and 40 week nominal waves.

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Cycle Scanner

In order to identify potentially profitable cycles of significance we needed a method to perform the same time frequency analysis across a wide range of instruments, quickly. This is where the new cycle scanner comes in! Our bespoke technique is fast, utilising a stripped back convolution process that will typically analyse a 4000 point market in less than half a second. Under the hood we still use the same time-frequency process that is on full display in the main analysis above but in the scanner’s case we only retain information pertinent to quick trading decisions.

Of course the scanner can also be put to good use on a range of other markets at different timescales. Below the scanner is used on 40 US stocks at a daily resolution, identifying signals of significance (the rank score) and presenting the top 9, in this case. We can isolate a specific bandwidth of interest to use, in the below case it is between ~ 40 and 200 days (that is, the scanner will attempt to detect cycles of significance between ~ 40 and 200 days wavelength).

Let’s now look at a couple of examples more closely to explain the concise information presented for each market.

Rank

Our main metric for differentiating significant signals from our analysis is the Sigma-L rank. This score, an integer from 0-1000, relies on several criteria to arrive at the final measure. Importantly, a score of 1000 would only ever be achieved by a signal represented by a sinusoidal of perfect amplitude and frequency stationarity. So you will never see it in financial markets but it is an important calibration! A non exhaustive list of ranking criteria are: level of noise, distance from noise of the peaks, width of the broader peak (denoting more or less frequency modulation), spectral isolation (how quickly the signal drops off to noise) and how far other significant peaks are from the main peak. A visual representation of rank is shown in the top left. The ‘fire’ emoji represents signals great than or equal to a 900 rank. An ‘ok’ emoji represents signals ranging from 800 to a less than 900 rank. A ‘fingers crossed’ emoji represents signals with rank ranging from 700 to less than 800. A ‘warning’ emoji represents all other signals below 700.

Cycle Wavelength

This one is straightforward. It is the wavelength of the detected dominant cycle in this instrument, within the bandwidth analysed.

Phase Information

This is the real juicy stuff for trading. We have three boxes: firstly the cycle status denotes the current position of the cycle. Trade information is only shown when the phase is either peak, peaking, troughing or trough. Other phases represent mid points of the cycle. To the immediate right of the cycle status is the inflection point countdown and expected range. These show when the next peak or trough is due and within what range it can be anticipated. The range is defined by the frequency modulation calculated from the time frequency analysis output. A tighter range is better, but not too tight! The future inflection point is also shown visually on the plot as a dotted line (the absolute inflection point) and shaded area surrounding, which represents the modulation and uncertainty.

Trade Information

When the cycle phase is appropriate, trade information is presented. From left to right we have: current price (yellow), max4 metric (see main analysis above), max4 stop loss, take profit 1 (max4 1:1) and take profit 2 (max4*2 2:1) trade levels. Finally we have the most important trading metric, that of the time target denoted as TT. This gives the expected future time to exit the trade and is the basis of most trade exits when using this primarily time based method.

Band Pass and Underlying Trend

The yellow band-pass on price is the same as previously shown on all Sigma-L charts. It is the result of a weighted by amplitude band-pass region of the time frequency analysis, related to the detected area of interest. The blue dotted line is simply the first few lower frequency outputs from the convolution and is a fair representation of longer cycle power.

Meta Information

Finally, some useful meta information is shown at the top of each scan result, relating to the scan parameters and some metrics. These include the current scan and the number of markets scanned, the last scan time and elapsed time for the scan to complete. The time-frame, recency bias (see main analysis above) and the bandwidth to target cycles within.

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DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Price Action

Time Series

Starting with the basics and using USDCAD as an example here, price action per unit time is displayed using ‘candlesticks’ of high, low, open and close price, on individual instruments. For the actual wavelet analysis only the close price is used. The name of the instrument being analysed is displayed on the y-axis.

A composite analysis is any analysis comprising two or more instruments. We typically use this to increase the signal to noise ratio amongst commonality instruments which are highly correlated. The concept is used widely in neuroscience (ERP via an EEG analysis) and uses the simple process of averaging to emphasise only those parts of the signal that are phase locked over time. The surviving signal is more likely to be the dominant component across all instruments in the composite. The price action displayed here in white is the averaged close value of the mean centered time series per instrument in the composite. Instruments within the composite are displayed in dark grey and will typically show variation via amplitude modulation.

Low and High Pass Filtering

Within the main price action we often show a representation of smoothed price, in yellow. The low pass filter is formed by allowing all frequencies lower than the lower bound of the bandpassed region (in white, see below, time frequency) to pass. This is typically useful to view the underlying trend and how the signal that is being investigated is influencing overall price action.

Highpassed price is very useful because it approaches a mean reverting time series, usually around zero. We apply a high pass filter to price action, retaining frequencies higher than the upper bound of the bandpass region, in white. The passband of the highpass filter is shown by the yellow arrows on the time frequency plot. The red and green envelopes on the price action plot represent +- 2 standard deviations of the amplitude envelope. Excursions by price above and below these zones is a high probability reversal area for the component being analysed.

It is an important technical note to mention that the cutoff in both the low and high pass filters used is not as abrupt as shown here and features some gentle rolloff around the bounds of the bandpassed region. We use a Butterworth filter for both approaches on Sigma-L.

Entry Zones

We typically will display time and price based zones for entry based upon the component’s spectral attributes in the bandpassed region. Buy zones, in green, and sell zones, in red, show both typical frequency modulation across the sample (in unit time) and amplitude modulation (in price units) around the entry zone. This is also influenced by the frequency modulation calculation in the time frequency plot (FM). As a rule of thumb the smaller and tighter the zones the better! There will always be some kind of modulation involved in financial markets, cycle lows and highs based around absolute price peaks and troughs are a fallacy.

Time Frequency

Overview and Frequency Resolution

We use a mathematical process entitled ‘complex morlet wavelet convolution’ on Sigma-L to produce the time frequency plots shown in each analysis. For a simple conceptual analysis imagine a sine wave mixed with a gaussian (the ‘wavelet’), as a function of time, being ‘dragged’ across price action.

At each time point the ‘dot product’ is taken, (multiplication and sum) between the function and price. The result of this is a measure of correlation between the function and price. Now imagine that process performed 1000’s of times at different frequencies and timepoints. The end result is an array of dot products - the time frequency plot.

For a more detailed look at time frequency and it’s features in general, please take some time to read the article: Hacking the Uncertainty Principle: Time Frequency.

On the time frequency plot we display wavelength (frequency) on the y-axis and time on the x-axis. The colour scale (or heatmap) within the plot relates to the correlation between each per frequency wavelet and price action. In this case lighter yellow represents a higher correlation, moving down to black for lower correlations. The wavelets can also can be thought of as individual precise bandpass filters, each occupying a small portion of the overall bandwidth in the analysis. A strong signal will show as a band of yellow across time. Often signals will modulate in amplitude and frequency. The more reliable signals have minimal modulation over time.

The number of wavelets used in each analysis is defined in the filter control section. Increasing the number raises the frequency resolution as shown above with 20 and 100 wavelets respectively. This in turn allows better spectral separation and observation of frequency modulation over time. Typically on Sigma-L reports we will always use the maximum number of wavelets (100).

Static Spectra

At the end of each time frequency plot there is a representation of the overall magnitude at each frequency. This gives a quick visual summary of the periodic activity across the overall bandwidth defined. Prominent peaks are shown with their associated peak wavelengths. This representation removes any temporal information (frequency and amplitude modulation) from the analysis and is analogous to a simple Fast Fourier Transform (FFT).

Bandpass Region

Upon inspection of the time frequency plot, we can isolate frequency bands of interest and use them to speculate on future price action. The frequencies included in the bandpass region (white box on time frequency plot) are defined in the filter control. We can also automatically identify these regions with our auto detection algorithm and dominant signal selection. The auto detected regions are shown in green and assigned a number dependent on significance.

The bandpass plot is the result of a weighted average of wavelet convolution outputs at the frequencies specified. The frequencies used controlled by either the manual filter control or the dominant signal detection lock, as above. Weights in the average are assigned based upon the magnitude of the output of each filter over time, per frequency.

At the bounds of the bandpass region box we show, on the left, the wavelengths at the margins and, on the right, the average frequency modulation (FM). At the bottom right the overall average wavelength of the identified signal within the bandwidth specified (height of white box) is shown.

This overall average wavelength is the wavelength used to describe the component in all our reports and is suitably indicative of it’s dominant frequency characteristics.

The white line within the bounding box is instantaneous frequency and shows how frequency changes over time within the bandwidth specified. A flatter line is favourable for a stationary and reliable signal.

Extrapolation

Using the spectral properties of the weighted bandpass filter we can extrapolate beyond nowtime. The regions around future peaks and troughs are flanked by bounding arrows, these can be thought of the error terms based upon the modulation in the sample. As time goes on, the uncertainty grows as to the position of the peaks or troughs of the component. The amplitude of the bandpass filter extrapolation is simply assumed to modulate back to the average amplitude from the sample.

Ready for more? Take a look at Part 2 of Understanding Charts on Sigma-L

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Time frequency analysis and the fast Fourier transform (FFT) are two approaches used to analyse the frequency content of a signal. Financial markets can be thought of as noisy signals, a time series that is assumed to be composed of a number of periodic functions and random components. Time frequency analysis allows for the identification of how a signal's frequency content varies over time, while the static FFT provides a snapshot of a signal's frequency content at a single point in time. The FFT squashes any temporal information out of the final analysis (how a signal is changing over time) and is solely based in the frequency domain.

At the time of writing Profit Magic for Stock Transaction Timing there was huge excitement over the newly established FFT algorithm and it’s application across a great many disciplines. Areas of modern life we take for granted now - the internet, mobile communication, streaming video and much much more, were made possible by this algorithm

Prior to it’s invention, performing the discrete Fourier transform was a computationally intensive task that required a lot of time and resources, even more so with early computing power. The FFT greatly improved the efficiency of this computation, making it possible to quickly analyse signals and extract useful information related to periodic components. Today, the FFT is used in a wide variety of applications, from audio and image processing to scientific data analysis and communication systems.

The FFT is a powerful tool for analysing the frequency content of a signal, but it has some limitations. One of the more notable limitations is that the FFT only provides information about the frequency content of a signal at a single point in time. This means that it cannot capture how the frequency content of a signal varies over time. In contrast, time frequency analysis with wavelets allows for a more detailed analysis of a signal's frequency content over time. This is quite important when we look at financial markets as the signals that we assume comprise price action are modulated over time. Hurst referred to this modulation as ‘the principle of variation’.

In the 60s and despite the new algorithm, computing the FFT at scale was in the domain of supercomputers and those with the resources to hire them. So when a certain engineer was asked, well, the rest is history…

Wavelets and the Time Frequency Trade Off

A common method of time frequency analysis is wavelet convolution, which uses wavelets to decompose a signal into it’s frequency components. Wavelets are mathematical functions that can be used to break a signal down into different frequency components. The wavelet convolution process involves convolving a wavelet with a signal to extract its frequency components. This is achieved by taking the inner product of the signal and the wavelet at each point in time. The wavelet coefficients can then be used to reconstruct the signal's frequency content at each point. This is quite distinct from the static FFT which assumes all frequencies occur across all time and are crystallised into one point. Said more succinctly, the process of wavelet convolution allows for the identification of how the frequency content of a signal varies over time.

One type of wavelet that is commonly used in time frequency analysis is the Morlet wavelet. The Morlet wavelet is a sinusoidal function multiplied with a Gaussian. This makes it well-suited for analysing signals featuring changes in frequency and amplitude, such as those in financial markets. To perform wavelet convolution, a signal is convolved (multiplication in the frequency domain) with the wavelet to obtain the wavelet coefficients.

One advantage of using wavelets for time frequency analysis is that the parameters of the wavelet can be adjusted to better suit the characteristics of the signal being analysed. This can help to overcome the general uncertainty principle, which states that it is impossible to simultaneously measure both the time and frequency of a signal with perfect accuracy. The static FFT only caters for the frequency side of that uncertainty.

By adjusting the parameters of the wavelet, such as the central frequency and the width of the Gaussian envelope, it is possible to tune the wavelet to the characteristics of the signal. This can help to reduce the uncertainty in the measurements of the signal, allowing for a more accurate analysis of its frequency content over time. We do this with our own analysis, changing the width of the Gaussian over frequencies to capture components in the most coherent way.

The video below from ubermath youtuber 3blue1brown gives an excellent insight into the process of convolution.

If that video piques your interest then the following tour de force of the wavelet transform as a mathematical microscope is awesome, from Artem Kirsanov:

Time Frequency in the Wild

Time frequency analysis is a powerful tool that can be used in a variety of applications:

• Speech recognition: In speech recognition systems, time frequency analysis can be used to analyse the frequency content of speech signals. By identifying how the frequency content of a speech signal varies over time, it is possible to differentiate between different words and sounds. This can be useful for improving the accuracy of speech recognition systems.

• Structural health monitoring: Time frequency analysis can be used to analyse the frequency content of signals from vibrating structures, such as bridges or buildings. By identifying changes in the frequency content of these signals over time, it is possible to detect potential problems with the structure. This can be useful for ensuring the safety of structures and preventing accidents.

• Electroencephalography (EEG): In neuroscience, time frequency analysis can be used to study brain activity. Electroencephalography (EEG) involves recording the electrical activity of the brain using electrodes placed on the scalp. Time frequency analysis can be used to analyse the frequency content of these signals, which can provide insights into brain function and help to diagnose neurological disorders. Some of the techniques employed on Sigma-L use insight from this field.

• Seismology: In seismology, time frequency analysis is used to study earthquakes and other seismic events. By analysing the frequency content of seismic waves over time, it is possible to identify the location and magnitude of earthquakes, as well as the properties of the materials that the waves are traveling through. This can be useful for predicting and mitigating the effects of earthquakes.

• Music: Time frequency analysis can also be used in music. By analysing the frequency content of a musical signal over time, it is possible to identify the different notes and chords that make up a piece of music. This can be useful for music transcription and other applications in music analysis.

Of course on Sigma-L we also use this technique quite widely to compliment the more established methods. It is more objective than other pattern based techniques and allows us to really see any significant periodic activity. Below is the S&P 500 and the wonderful 20 week component, a familiar sight for subscribers.

Time frequency analysis with wavelets provides a more detailed and flexible approach to analysing the frequency content of a signal than the FFT. By adjusting the parameters of the wavelet, it is possible to overcome the uncertainty principle somewhat and obtain more accurate measurements of a signal's frequency content over time. Whilst the FFT is a powerful tool for quickly computing the frequency content of a signal, time frequency analysis with wavelets allows for a more insight into any modulation. This modulation motivated Hurst to apply his comb filter technique, shown in the appendix of Profit Magic for Transaction Timing and equivalent to a low resolution wavelet convolution. We have compute power orders of magnitude above what Hurst could leverage in the 60s, it makes sense to use it.

In Summary

• Time frequency analysis and the fast Fourier transform (FFT) are two approaches used to analyse the frequency content of a signal.

• Time frequency analysis allows for the identification of how a signal's frequency content varies over time, while the FFT provides a snapshot of a signal's frequency content at a single point in time.

• One common method of time frequency analysis is wavelet convolution, which uses wavelets to decompose a signal into its frequency components. The parameters of the wavelet can be adjusted to better suit the characteristics of the signal being analysed, which can help to overcome the uncertainty principle.

• The FFT is a powerful tool for quickly computing the frequency content of a signal, but it has some limitations. It only provides information about the frequency content of a signal at a single point in time, and it assumes that the signal being analysed is wholly periodic.

• Time frequency analysis with wavelets provides a more detailed and flexible approach to analysing the frequency content of a signal than the FFT. By adjusting the parameters of the wavelet, it is possible to overcome the uncertainty principle and obtain more accurate measurements of a signal's frequency content over time.

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

What is a Band Pass Filter?

In signal processing and the related topic of time-series filtering we often hear the terms ‘low pass filter’, ‘high-pass filter’, ‘band pass filter’ or ‘band stop filter’. They each refer to the process of creating a new signal with part of the original signal removed, or ‘filtered’. In the case of the high pass filter we remove the lower frequency components from a signal and retain the higher frequencies; we allow the higher frequencies to pass. This was achieved in the previous article with the help of a moving average - a particular type of low pass filter when plotted correctly to account for the digital filter lag. When the filtered (low passed) signal was subtracted from the original signal all that remained of the original signal were components above the cutoff (the span, or period) of the low pass filter. The resulting signal had therefore been highpassed. Highpass filters are useful to detrend time series or examine lower magnitude, higher frequency components of a signal.

We can take a similar approach to construct a bandpass filter. Instead of leaving one end of the frequency spectrum open as is the case for the low and high pass filters we can combine two low pass filters of different cutoffs (in the moving average case, periods or span). The distance between these two cutoffs is the passband of the filter. Any components with wavelengths in this passband will be passed by the filter and retained in the output signal.

A Recipe for Extraction

For the moving average bandpass filter recipe we will need three ingredients: a signal and two moving averages (low pass filters). The latter will form the bandpass filter. One of the moving averages should be set to a period lower than the other or vice versa. The exact specification of the moving average periods will vary depending on the signal component we wish to isolate and extract.

The bandpass filtered signal output is calculated by subtracting the lower frequency moving average from the higher frequency moving average.

It is paramount both moving averages are correctly lagged in time ( (odd period-1)/2) ) prior to the subtraction for the output to be accurate! The subsequent bandpass output is then plotted with a lag equal to the lower frequency moving average lag period.

As with most time series calculation it is often best to see the results visually. Below follows several examples of using this technique in action on a composite, simulated signal.

Example 1

In this first example we have a signal of 1000 data points in length. The signal is a composite of white noise and sinusoidal components of wavelength: 500, 100 and 50, each of diminishing amplitude. A linear trend is also applied to the signal to simulate a smooth underlying motion.

In this first example we wish to isolate the component of 100 wavelength. Therefore we will set our low pass filters accordingly and construct two moving averages of period 101 and 51. The longer period moving average is set at the closest odd number available to attenuate the 100 wavelength component almost completely. Therefore when this signal is subtracted from the higher low pass filter the component of 100 will be retained. The higher frequency low pass filter is set around the closest odd number to the wavelength of the component at 50. Therefore the signal at a wavelength of 50 is almost completely attenuated and will not come through in the bandpass calculation.

We then perform the calculation to form the bandpass by subtracting the lower frequency moving average from the higher frequency moving average, leaving a band of data inbetween cleanly passed and retaining the signal of 100 wavelength. Importantly the bandpass is plotted so it is aligned with the lower frequency moving average lag.

The extracted bandpass signal has virtually no trace of high frequency noise and the higher frequency component at 50 is non-existent.

Example 2

For the next example we will compare the outputs of the high pass filter and the bandpass filter. Below is the same signal with the bandpass filter applied as above and a highpass filter applied to retain all components of 100 wavelength and above.

Clearly the obvious difference between these two outputs is that the high pass retains both the noisy part of the signal and the component at a wavelength of 50. The latter can be seen in the slight undulation of the signal between the 100 wavelength component troughs. Both outputs also contain a tiny part of the component at 500 wavelength, an unavoidable feature of using a moving average as a time series filter due to it’s frequency response.

Now you may ask why we can’t just filter the high pass output with a low pass filter to achieve the same result as the bandpass? Well the answer is we can but the additional operation introduces a further filter lag, as demonstrated below.

Example 3

In this example we will attempt to extract the component at a wavelength of 50. The amplitude of this component is relatively (and deliberately!) low at 0.1 so we can expect a rather rougher result due to the similar amplitude of the noise component. Indeed the component of 50 wavelength is only slightly visually apparent in the signal.

Here the result is somewhat more what we might expect from real markets and amplitude modulated components. The periodic signal of interest is buried deeply in the overall signal and the amplitude is only just above that of the noise. Regardless, a bandpass filter set at 51/31 gives a reasonable result, maintaining the amplitude of around 0.1 and revealing the periodic component. Note how the much higher amplitude component of 100 wavelength is bleeding through. The highpass filter is also shown for comparison with the noise retained in the output.

In Summary

We hope these articles on the humble moving average have been insightful and prompted some of you to look at this staple of technical analysis in a different (and perhaps more correct?) light. All that is required in order to take advantage of the filter characteristics of this indicator is to lag it correctly! New trading strategies and indicators can also be constructed - for example a correctly plotted MACD or RSI. The ubiquitous nature of the incorrectly plotted moving average operation in many indicators implies a ripe ground for experimentation.

The obvious path now is to take this knowledge and apply some moving average style filtering on real markets! Keep an eye out for that article, coming in the next few weeks on Sigma-L.

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Inversion Excursion

We saw in previous articles of this series how only a correctly plotted moving average can be used as a low pass filter for time series extraction of periodic components. Now, let’s take it to the next level by inverting that moving average - throwing away the low passed data and retaining only components shorter in period than the cutoff of the moving average (period). Since in our work we have both the low pass filtered time series (via the application of the centred moving average) and the original signal time series (be that a test signal as above or financial time series) we can perform the operation easily.

Below is how it is done, via the help of some pseudo code. Note carefully that the subtraction performed to reveal the higher frequency components not present in the low pass filter can only be done if the original signal and low pass filtered signal are correctly aligned. The low passed signal must be shifted, as demonstrated in the previous article, by the (moving average period - 1)/2. This is essential and dictates that the moving average period chosen must really be an odd number!

// Our original signal
signal = [4,5,6,7,8,9,8,7,8] 

// MA period (lp filter cutoff)
ma_period = 3  

// Filtered signal (apply moving average)              
lp_signal = [NaN,NaN,5,6,7,8,8.3,8,7.6] 

// Filter lag correction by: (ma_period-1)/2 = 1 in this case
lp_shift = [NaN,5,6,7,8,8.3,8,7.6,NaN] 

// Subtract the filtered signal from the original signal
hp_signal = signal - lp_shift

High Pass Output, Visualised

Now, let’s build a very simple signal with only two sinusoidal components to demonstrate a clean extraction of the higher frequency component. Below, a 1000 point time series containing components at wavelengths of 500 and 100 respectively. The higher frequency component has an amplitude of 0.5, this will be an important number to remember as we progress. The whole plot is vertically offset for visual clarity.

We now apply a moving average of period 101 (must be odd!) to effectively remove the component of period 100 wavelength. This is shown below via the white dashed line, a clean sinusoid of period 500, the only component left in the low passed signal. The plot is correctly centred with lags of 50 points each side using the equation (ma_period-1)/2.

We now have the ingredients to perform our high pass filter extraction. We subtract the lagged low pass filter from the corresponding points in time of the original signal to reveal only what is left, namely components of wavelength equal to and less than 100. This is shown in green at the foot of the chart and approaches the correct amplitude of 0.5.

With the unavoidable digital filter lag we are still left with gaps at nowtime (and the tail) that are problematic for any trading purposes when we switch to real signals. We can solve this to some extent by using the previously discussed technique of signal reflection:

Drawbacks of a Moving Average LPF

Although the moving average is a useful tool and available to many traders across common platforms there is a subtle drawback readers should be aware of. The frequency response of a moving average is tainted by it’s sidelobes. This means, in practice, that extraction of periodic components is minimally spectrally leaky compared to more sophisticated filtering techniques. It is not that bad for our purposes and certainly the ubiquity and ease of use of the moving average outweighs this disadvantage. It is something to be aware of, regardless. Hurst outlined the frequency response of the moving average as a low pass filter in the appendix of ‘Profit Magic for Stock Transaction Timing’, shown below:

Below, the effect of the leaky spectral profile of the moving average is seen by the slight oscillation of the high pass filter output when the exact frequency is approached by the low pass filter operation on the original signal. In order to correctly return the accurate amplitude of the component (in this case 0.5) one has to accept a slight leaking through of lower frequency components. In the below case the component with a wavelength of 500 is very slightly affecting the high pass output (in green). Note how the oscillation is more pronounced as the filter cutoff approaches the wavelength of the target component at 101.

More Extractions!

Let’s now add a third sinusoid to our signal, of even higher frequency, amplitude 0.25 and some small amount of white noise to taste:

Now we will attempt to extract the very highest frequency component at a wavelength of 50 via our high pass filter technique. But first let’s see if we can isolate just the noise, for demonstration:

For the high frequency component we sweep across a range of filter cutoffs from 11 to 51, odd numbers only. The result is interesting:

What we see here is as we approach the low pass filter cutoff of 51, which would remove the component of that wavelength in the low pass filter (and therefore return it in the high pass) is once again, spectral leakage. The component of 100 wavelength is corrupting the signal. However this is marginally only the case if we wish to return the original amplitude of 0.25. It turns out that the leakage from larger components will begin to affect the high pass signal after around halfway to the wavelength of the component we are trying to target. If we are only really interested in the frequency of a component we can compromise in this high pass filter approach by only filtering up to around half of the target component wavelength. This is shown below:

Next, Bandpass

The final installment of this moving average series will take the next natural step and detail construction of a bandpass filter. This is possible by computing two low pass filters at different cutoffs (periods), the frequencies between those two filters acting as the ‘band’ to ‘pass’. A bandpass filter is possibly the most powerful of the techniques and allows us to construct ‘comb filters’ that can sweep through signals, isolating specific bands for further examination. It requires careful thought about the cutoff frequencies to employ and how to correctly align signals when we account for filter lag. It is also an easy way to remove noise from a noisy high pass signal, as is the case above.

After that we will take these techniques and apply them to real markets. That is where the fun really begins!

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

In our first look at the humble moving average we discussed how to correctly plot this popular trend indicator and how the vast majority of chartists today do not. We discussed it’s true operation, that of a low pass filter, or data smoother and the inherent digital filter lag resulting from the mathematical operation of averaging over a sliding window.

The filter lag is not unique to the moving average however, it is present in every filtering operation, at least in the time domain with a sliding window. The amount of digital filter lag can vary depending on the type of filter but the general rule is that it will always be present to some extent. The result of the digital filter lag are ‘edge effects’ at the start and tail of the data. As traders we are generally interested in the situation at ‘nowtime’, not least for timely entries.

Consider the above signal, a simple composite of 3 components of wavelength 500, 100 and 50 respectively and of diminishing amplitude - with some noise added. A moving average (low pass filter) is set to 200, removing all components above the 200 point wavelength, leaving just the 500 point component. The result is a plot, when correctly lagged, that loses 100 points each side. Now the tail loss is not so much of an issue for traders but the nowtime loss is something that needs to be addressed, ideally.

There are a few appropriate solutions. In this article we will look at signal reflection.

Mirror, Mirror

Reflection is the process of adding a mirrored version of the signal to the end(s) of the real signal and then applying filtering to the signal and reflection together.

The advantage of reflection for our purposes of periodic function extraction is quite compelling. We expect periodic signals to continue with similar frequency and fairly similar amplitude through nowtime and beyond. The accuracy diminishing with higher lag periods. Therefore a moving average applied to a reflected signal is a valuable tool that has minimal drawbacks, especially at negligible lags relative to the overall signal length. The fact we can use the correctly lagged moving average as a band or high pass filter also means a method to help remove the filter lag is of even greater benefit.

Let’s write some pseudo code to visualise a double sided, reflected signal:

signal = [2,3,4,7,5,3,7,8] // The original signal
ref_left = [8,7,3,5,7,4,3] // Reflected signal at the start
ref_right = [7,3,5,7,4,3,2] // Reflected signal at nowtime

reflected_signal = concatenate(ref_left, signal, ref_right)

// Reflected signal result, spaces added for clarity
>> [8,7,3,5,7,4,3, 2,3,4,7,5,3,7,8, 7,3,5,7,4,3,2]

Note that we omit reflecting the start and end points of the original signal to avoid repeating data points. Thus a signal consisting of 8 data points will have two 7 point arrays either side when implementing a fully reflected, double sided approach.

In reality it is only necessary to reflect the original signal at the ‘nowtime’ edge because we aren’t really interested in the tail. In addition, the amount of reflection can be limited to the amount of digital filter lag expected via the application of the moving average. So for a moving average of period 200, reflection of only 100 data points from nowtime is required for the moving average to be extrapolated to nowtime.

However in this example, for the sake of completeness we will reflect each side in it’s entirety.

Apply and Extrapolate

Let’s take another composite signal and demonstrate double side reflection:

When we apply the moving average to this entire signal, which includes the original signal plus the reflection, we are in effect extrapolating the low pass filter to nowtime (and beyond if required). The accuracy of the extrapolation falls the further from the original moving average lag point. As a general rule of thumb, no significance should be given to an extrapolation performed in this manner beyond a distance of nowtime plus the filter lag.

Now let’s target the component of 100 wavelength. We apply a moving average of period 50 to remove the component of wavelength 50 and the high frequency noise. This leaves just the 100 and 500 period components in the signal, which we can extrapolate through nowtime and beyond:

Signal reflection is a powerful and established tool in signal processing to use when trying to battle inherent digital filter lag. It is relatively easy to implement in most trading platforms with a touch of coding and we hope this article inspires a fresh look at the humble moving average once again.

Try it in your strategies today!

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

The above statement is apt to describe the humble moving average and the common use of it as an indicator by technical analysts of financial markets across the world. However, in the vast majority of cases (and certainly by a brief look at some charts on TA twitter), it is being used almost entirely incorrectly! This is also compounded by the fact many indicators employed by chartists to form trading algorithms or signals use moving averages incorrectly in their own composition.

Two such examples are the relative strength index (RSI) and the moving average convergence divergence (MACD).

The Inescapable Digital Filter Lag

To understand why moving averages are being plotted incorrectly in most cases we need to understand what a moving average actually is. It is a data smoother, or, in signal processing parlance, a low pass filter. It is most often utilised to emphasise or ‘uncover’ some kind of price trend, composed of lower frequency components, via the attenuation of higher frequency components. It does this via the mathematical operation of averaging over all data points in a certain window of data points, the length of which is stipulated by the user as the ‘period’ or ‘span’ of the average.

As an example, below is some pseudo code detailing a moving average of span 3, passed over a vector of 5 points, analogous to prices in a financial market over a period of 5 sample points:

data = [2,3,4,8,6] // A 5 point vector of data
ma_span = 3 // The 'span or period' of the moving average

// Step by step calculation:
> 2+3+4 / 3 = 3 // 1st MA data point
> 3+4+8 / 3 = 5 // 2nd MA data point
> 4+8+6 / 3 = 6 // 3rd MA data point
> 8+6+NaN // Not enough points for window!

// Original data and MA vector, incorrectly aligned for time 
data = [2,3,4,8,6]
MA =   [3,5,6,-,-]

// Original data and MA vector, correctly aligned for time
data = [2,3,4,8,6]
MA =   [-,3,5,6,-]

The process of averaging via a window, as shown above, results in unavoidable data loss. This is because at the end of the series we simply run out of data points to satisfy the number required for the window of the moving average. In this particular case we lose 2 points at the end of the series. In general terms the equation for the number of points lost due to digital filter lag is simply:

ma_span-1

So in the above case where the moving average span is 3, we lose 2 data points.

Because of this loss we must adjust the placement in time on our charts to reflect the loss of data. We are dealing with timeseries for the majority of our charts in financial markets so must adjust accordingly if we wish to judge trend (or other metrics) correctly. A moving average that is plotted at nowtime and not lagged to account for the inherent data loss is attempting to estimate trend via data that reflects that trend incorrectly! This is negligible for a short MA such as the 3MA above but if averages based upon longer spans are employed, as they so often are to smooth data for any use in trend estimation, the larger error inherent renders such lines as useless.

In order to correctly align moving averages with price to account for digital filter lag we must use the following equation to center the average in regard to price and time. This is seen above in the ‘correctly aligned’ data and MA vector:

shift = (ma_span-1)/2

Due to the discrete sampling of timeseries in financial markets it is often best to choose an odd span for your moving average when accounting for digital filter lag. This is so the average can be accurately centred using the calculation above. This is not so important for very large moving average calculations.

For example, a moving average of period 11 should be shifted back in time by 5 points. If the moving average was 10, the midpoint would be 4.5, or halfway between two of our discrete sample points.

The fact we have to shift the moving average back in time like this reveals the reason perhaps that many chartists have ignored (knowlingly or not) the correct placement: We lose the moving average at nowtime!

Of course trading decisions are made often on the basis of price interactions at nowtime and a moving average is somewhat compelling for the naive trader/investor. It is though, nonsense, in the majority of cases to infer anything from a non-lagged moving average.

Moving Average = LPF

All is not lost however. What we lose in nowtime clarity we gain back several fold in the ability to utilise the humble moving average as a low pass filter when correctly plotted to account for digital filter lag.

Let’s build a composite timeseries that will consist of simple sine waves and some added gaussian noise before the addition of a moving average to demonstrate a low pass filter. We will see when the average is correctly shifted and span carefully chosen we can begin to utilise the benefits.

A feature of MAs used as low pass filters is the ability to attenuate completely ALL instances of cyclic components in a signal equal to the span of the moving average. This is a wonderful property and leads onto all kinds of possibilities to construct highpass, bandpass and bandstop filters via the humble moving average, available to almost all traders on all platforms.

Below, we add 3 moving averages (correctly lagged) that approach the wavelength of the 1000 point component. Note how as the span approaches the wavelength in the signal, it is removed, leaving only the other two components and some tiny residual noise:

Now, let’s try removing the larger wavelength component of 2500, leaving us with a signal consisting only of the component of wavelength 5000.

Finally, we attempt to remove the very longest component in this signal with the low pass filter characteristics of the moving average. Since there is no more signal components available after this stage of filtering, we should get a flat line. We use one moving average here with a span of 5000 to match the target wavelength:

In this way we can use the moving average in a useful way to isolate cyclical components of interest in financial markets. It can only be done if the average is correctly lagged though and we pay the price via the necessary shifting of the timeseries.

Solutions to Filter Lag

There are a few solutions available when we need to extrapolate the moving average to nowtime due to the lag inherent in this approach. We will look into one of these in a future article, namely polynomial regression. It is very well suited to approximating such smooth curves and with good benefits at the medium term wavelengths (think 80 day, 20 week, 40 week components in the nominal model).

Flipping the Average

We can also exploit another benefit in using MAs as low pass filters, the ability to ‘flip’ the average and arrive at a highpass filter easily. Hurst called these ‘inverse’ moving averages. It is simply the process of subtracting the low pass filtered timeseries from the original signal. This is where things get really interesting as we can isolate components away from the modulating effects of the very long period trends.

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The term ‘nominal’, in relation to Hurst Cycles, refers to the distinct naming convention of groups in Hurst’s nominal model. These are a collection of groups that coalesce around similar frequencies, named for ease of use and convention. In reality, periodic functions in financial markets are plagued by all kinds of variation, excursions from the nominal model groups are described by the principle of variation. These variations are comprised of both frequency and amplitude modulation. Hurst referred to this variation as ‘magnitude-duration-fluctuation’. Catchy!

Importantly, Hurst found in his research that the grouping of component frequencies exhibited in stockmarkets (and commodities later in the Cycles Course) show distinct spectral spacing. Another way to describe the spacing of frequencies is that of a non normal distribution. The latter might indicate that markets are infact entirely explained by ‘pink noise’ - the type of noise that displays a distinct ‘1/f’ magnitude spectrum in the frequency domain. The spectral spacing gave rise to the grouping described above and the resulting nominal model. The groups within the nominal model can also be thought of as ‘carrier’ frequencies, each one consisting of finer frequencies closely related to the carrier and suitably amplitude modulated.

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Twitter is very useful for traders because we can go back and examine the performance of an analysis over time, which lends itself to this kind of helpful insight for readers of Sigma-L. In this first edition of a series where we look at entering, managing and exiting trades using Hurst Cycles, USDCAD will be our initial candidate.

Price had moved from a low of at least 40 week magnitude in June 2021, which itself was a great swing trade targeting the 18 month FLD resistance (prior to the start of the Sigma-L site!). In mid September price began the move down to a 20 week nominal low, expected mid October and driven by resistance at the 18 month FLD (yellow).

The setup was ready in mid-late October with a cascade of FLDs forming a bullish setup and a choice of entry points. At this stage a cross of the 10 day FLD would confirm the 80 day nominal low and by the principle of synchronicity the 20 week nominal low too. It was important to be patient at this point, entering only on valid cyclical signals. For entries we will usually place a stop order at the level of the chosen entry FLD so it will automatically trigger when price reaches that level. Each day we adjust the stop level if it has not been triggered to reflect the new FLD value.

The 20 week FLD and 18 month FLD was at a clear resistance level above price around 1.28 and was chosen as the trade target. As price moves through the bullish FLD cascade FLD cross targets are noted and extrapolated.

Entry Criteria and Targets

Entry: 10 Day VTL / 10 Day FLD / 20 Day FLD

Target: 20 Week FLD / 18 Month FLD resistance around 1.28

Stop: Below confirmed 20 week nominal low

Trade

October 15th

Price continued to move down and the placement of the 20 day component was clarified as occurring on the 28th of September. This assisted us further as we could measure the number of days elapsed since that low and compare it to the sample average of 17.6 days. The 20 week low was imminent but we still await our correct entry point. We now also add the 10 day downward VTL (pink) as a valid entry point and have a choice between that and the 10 day FLD, both being good entries. We note that price also has found resistance at the 5 day FLD (white). This is noted as a very risk on entry but still valid.

October 21st

Price is now extremely close to the 20 week low with the nest of lows suggesting it is overdue - but we still do not have a point of entry. This is now shaping up to be a wonderful low risk / high reward trade as the 10 day downward VTL forms obvious resistance to price. A daily median price break of that VTL is our entry point.

Subsequent to the below image being posted on the day, median price broke the VTL and we entered our trade.

November 18th

Due to the low being a 20 week nominal trough we knew price should reach up to the 18 month FLD easily within the first 80 day component. We noted the position of FLD crosses and found in the rise price exceeded both the 20 day FLD cross target and the 40 day FLD cross. It also crossed past the 80 day FLD and projected a target to 1.278. This point was chosen as our exit area, being close to the 20 week FLD resistance around 1.28.

We also noted that should price meet and exceed the 80 day target then we should possibly assess whether the 40 week low in June was also infact of 18 month magnitude. A median price cross of the 18 month FLD itself would also add further evidence.

November 26th

With the 20 week low confirmed and price marching upward toward our 80 day FLD cross target we could have a look at the weekly view. We noted that price had crossed the 18 month FLD, providing evidence that the low in June 2021 may have been the 18 month / 9 year nominal low. This is vital information going forward and in anticipation of the 40 week nominal low due in march 2022.

November 30th

Price reached our target of 1.278 and the general area of resistance around the 20 week FLD at 1.28, previously mentioned at the start of the trade. Another option for a cyclic exit may have been the 40 day upward VTL, a cross of which confirms the peak of the 80 day component. As Sentient Trader had not yet assigned the 40 day low it is not drawn on the charts here but it would have been a valid exit point, were it not for our main 80 day FLD cross target to be reached already.

Summary

This was a good ‘swing’ style trade with an excellent risk:reward ratio. The entry point required patience and a trust in the technique to overcome psychological barriers. The principle of synchronicity was important here and defined our ability to enter at a cycle entry degree much lower (10 day) than the cycle target degree signal (20 week). We selected valid cyclical entry points and chose that which optimised our risk:reward ratio (10 day VTL/FLD).

An estimation of the underlying trend (sigma-l) was also required to anticipate what kind of a move we might expect.

DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

A simple concept, but one which must be considered for our purposes as Hurstonians, is underlying trend. Named by Hurst as ‘sigma-l’ to simply mean ‘the sum of all cycles’, it is inherent in a system which assumes cycles form an infinite series of increasing period. This underlying trend (which can be thought of as a sum of sinusoidal functions) is never static. It may appear to be flat when shorter cycles form sideways consolidations but this is just a result of a much larger period cycle, relative to the cycle in focus, turning up or down.

Traditional technical analysis has come to love the simple moving average as an indicator of trend, usually set at a arbitrary period of 50, 100 or 200 to smooth higher frequencies. This is a blunt tool and flawed. Firstly by the fact moving averages must be time shifted by the (period/2) to account for the digital filter lag and, secondly, the moving average makes no attempt to decompose the signal, save for it’s low pass characteristics.

In the Cyclitec cycles course Hurst tabulated sigma-l by assessing the phase of each component in question and arriving at an estimation for sigma-l. Below is an example of this in Sentient Trader using the daily phasing of Bitcoin.

In Practice

We will take our advanced FLD trading strategy which uses three discrete components of the nominal model and their respective signal cycle FLDs to demonstrate a rudimentary application of sigma-l. In this example we will assign a linear trend to the model and observe the effect on price, interactions and the summary status of ‘M-Sigma’ - our subset of underlying trend only applicable to this 40 week sample.

In real markets the trend is not linear and is the sum of what can be thought of as discrete, sinusoidal functions of increasing wavelength.

Trend is Up

Here we assign sigma-l a value of 1 and the dotted line below price is an indicator of it’s function. The effect on the composite price line is obvious with the familiar ‘higher low’ forming at the end of the chart and price skewed upwards.

Trend is Down

Here we assign sigma-l a value of -1 and the inverse effect of a positive sigma-l is observed. The effect on the composite price line a ‘lower low’ forming at the end of the chart and price skewed downwards.

We can now look at what effect the addition of this simple linear trend has on our overall summary at each 20 day FLD interaction and the strength of the trade associated with it.

The tabulation above shows the summary of each 20 day FLD interaction in the idealised 40 week period, via the model described previously. M-Sigma shows interaction trade direction and strength when the underlying trend is assumed to be 0, essentially a localised subset of underlying trend.

Subsequent summary columns show the resulting differences when underlying trend is up and down, respectively, as a true reflection of Sigma-L. This more closely resembles real price action in financial markets.

Looking at the summary columns with underlying trend accounted for we can see that in the bullish scenario, those trends previously described as ‘risk buy’ are now standard ‘buys’, some ‘buys’ become ‘strong buys’ and so on. The underlying trend has the effect of amplifying bullish signals and attenuating bearish signals. Of course the inverse is true when the underlying trend is bearish.

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DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Expanding and Combining

The FLD trading strategy, explored piece by piece in the basic FLD trading strategy article, is a wonderful addition to our Hurst based kit of trading tools. The breakdown of trades into relatively easy to recognise and categorise interactions with an FLD provides an insight into both the underlying status of a market and what is …

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Strategy Fundamentals

The first aspect to note when examining the strategy outlined here is that it is in no way related to anything Hurst wrote himself in any of his literature (aside from the FLD)! It is instead the result of many years diligent work by David Hickson (creator of the excellent Sentient Trader software) into the nature of the FLD and it’s relation to price action.

The FLD itself is explained in some detail in this article so please give that a read for the very basics of what the FLD actually is. Briefly recalled, it is price projected forward by half the period of the cycle component in question. The FLD is therefore 180 degrees out of phase with the price action of that component. This allows the analyst to gain robust insight into certain periodic components of interest within the instrument. For example, whether a peak or trough is left or right translated, informing the likely directional bias of the underlying trend (sigma-l). The FLDs available to use are based upon identified periodic components present in price action, derived from a formal phasing analysis of the instrument.

For this basic explanation of the FLD trading strategy we will use the concept of the trading cycle as being the 80 day nominal FLD and the FLD used as the signal cycle being the 20 day FLD.

A crucial point to note is the relationship of the signal cycle to the trading cycle. It is always at 2 degrees lower in the cyclic model and the components within the trading cycle must follow a 2:1 harmonic ratio.

Examples: If the 20 week cycle was selected as the trading cycle for analysis the associated signal cycle, upon which the FLD is based, would be the 40 day cycle. If the 40 week cycle was selected as the trading cycle, the associated signal cycle upon which the FLD is based would be the 80 day cycle, and so on.

Fortunately for us in Hurst’s nominal model the majority of cycles are proposed to be arranged in a 2:1 harmonic ratio (principle of harmonicity). Be cautious if using the 18 month component in your analysis along with associated FLDs for this strategy - it has a 3:1 harmonic ratio with the 54 month component.

Interaction Series A - H

Once an trading cycle has been decided upon and it’s associated signal cycle FLD plotted, we can begin to assess the phase of the trading cycle (and future opportunities) by looking at the characteristics of each interaction. Each interaction is assigned a letter, from A to H, representing 8 individual interactions with the signal cycle FLD. There are 3 bullish interactions, 3 bearish interactions and 2 neutral, all are far from equal and depend to a large extent on the phase of the trading cycle.

Let’s now examine each interaction in turn, with the help of real world examples and utilising the 80 day trading cycle / 20 day signal cycle.

A Category Interaction (Bullish)

This interaction occurs at the very start of the trading cycle and is often strong, prolonged and will, in the majority of cases, hit the 20 day FLD cross target. Sounds great, right? The inevitable downside, unfortunately, is that phasing analysis error can lead to the G/H interaction being wrongly identified as the A category! Traders risk then being caught in a bull trap. It is therefore a good trade to enter but carries more transaction risk than a B to C category interaction or E category interaction. It can be especially profitable if identified in combination with the A category interaction of higher degree FLDs, a concept we will explore more in the advanced article.

B Category Interaction (Neutral)

The ‘B’ category interaction occurs, in this case, at the first 20 day cycle low in the 80 day trading cycle. At this point sigma-l is up but price is forced sideways in a neutral scenario to meet or ‘touch’ the 20 day FLD. This is the definition of what Hurst described as the ‘mid channel pause’ in his early work, where price reaches the middle of a curvelinear channel drawn around the trading cycle.

In general, ‘B’ category interactions should not be traded but that does not imply by any means they are useless to the skilled Hurstonian. The manner in which the ‘B’ category interaction occurs can tell us more about the possibility of a profitable, bullish, ‘C’ category move and the underlying trend of the market at this point in the trading cycle. For example, if price finds significant support at the 20 day FLD (price tracks / straddles the FLD) this is a bullish sign for sigma-l. If price, on the other hand, were to break below the 20 day FLD in a significant manner (several days below the FLD, or approaching the FLD cross target), this is bearish for sigma-l and we should take note that bullish trades are perhaps not the best strategy going forward.

C Category Interaction (Bullish)

The ‘C’ category interaction with the 20 day FLD is sometimes a tricky one to find an entry point to but presents less transaction risk than an ‘A’ category interaction as the underlying trend is already established. Assuming that the result of the ‘B’ category interaction is deemed to be bullish there are a couple of tactics to use.

Firstly a stop entry above the high of the 20 day nominal peak can be used, a measured move from the accompanying lows perhaps serving as a good target, or potential resistance above at a higher degree FLD. Secondly, a lower degree FLD can be plotted above price to gain a more objective target. The 10 day FLD is a good choice in our example here using the 80 day trading cycle / 20 day signal cycle.

D Category Interaction (Bearish)

The ‘D’ category interaction is the first short position any trader using this strategy should consider entering into during the trading cycle. It’s low co-incides with the trough of the 40 day component in our idealised example and therefore represents the mid point of the 80 day trading cycle. The FLD cross target in both the ‘D’ and ‘E’ category interactions are very important to inform us of the underlying trend. If there has been weakness (bearish signs) noted at ‘A’, ‘B’ or ‘C’ it is more likely the FLD cross target will be overshot to the downside (bearish sigma-l). On the contrary, if price misses the FLD target here sigma-l is likely to be more bullish and the ‘E’ category bullish trade should exceed it’s target. Quite often support is found at the 40 day FLD in this move.

E Category Interaction (Bullish)

The ‘E’ category interaction with the 20 day FLD is the inverse of the ‘D’ category interaction and is a bullish trade from the 40 day nominal low in our idealised example. Expectations are that price should hit the FLD cross target in a neutral shaped trading cycle, any large deviations from a simple meeting of the target indicating either a bullish (overshot target) or bearish (undershot target) sigma-l.

F Category Interaction (Bearish)

The ‘F’ category interaction is the optimal trade to enter into given a bearish sigma-l and represents the start of the falling side of our idealised 80 day trading cycle. If the ‘D’ category interaction has overshot it’s target and price has subsequently struggled in the ‘E’ category interaction, Hurstonian traders should be on alert for a profitable bearish trade ahead. If sigma-l is bearish the ‘F’ category trade should exceed it’s target in most cases easily and can often be held longer, given higher degree FLD confirmations.

G Category Interaction (Neutral)

The ‘G’ category interaction is the inverse of the ‘B’ category interaction and generally will move price toward the 20 day FLD (in our idealised example), either tracking the FLD or breaking above it in a sideways consolidation. In a bearish scenario the ‘G’ category interaction can be expected to find resistance at the 20 day FLD and is therefore sometimes a risk on fade short if sigma-l is indicated to be heavily bearish. The risk here is that of phasing error and more specifically a 20 day nominal cycle margin of error. This may result in the ‘G’ category interaction infact being the ‘A’ category interaction of a new 80 day cycle, so be careful! A less risky strategy for a bearish scenario is to enter the ‘H’ category trade on stops below the 20 day nominal low that initialised the ‘G’ category interaction.

H Category Interaction (Bearish)

The ‘H’ category interaction is the most bearish of the trading cycle, being the final 20 day component to bottom. This does not mean it is the best bearish trade. Given a robust assessment of sigma-l, however, it can be very good. It is somewhat the inverse of the ‘A’ category interaction and if previous bullish trades have all struggled to meet targets (or just met them), the ‘H’ category is likely to be profitable. It can either be held from the ‘F’ category interaction or entry can be made via a stop sell order at the level of the prior 20 day low. Alternatively the 10 day FLD might be used in our idealised scenario to gain confirmation of a 10 day peak occurring previously, giving more confidence to the trade.

Combining FLD Interactions

We can take the FLD strategy applied to one single signal cycle FLD and combine it with other signal cycle FLDs for an even more powerful approach. This is explored in depth via the advanced FLD trading strategy article.

This incredibly powerful technique allows us to pinpoint optimal entry areas for maximising yield over time, bullish or bearish. By combining higher and lower degree cycles we can take advantage of both smaller risk entries and higher rewards.

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DISCLAIMER: This website/newsletter and the charts/projections contained within it are intended for educational purposes only. Results and projections are hypothetical. We accept no liability for any losses incurred as a result of assertions made due to the information contained within Sigma-L. This report is not intended to instruct investment or purchase of any financial instrument, derivative or asset connected to the information conveyed in the report. Trade and invest at your own risk.

Throughout Sigma-L readers will find references to ‘nominal’ cycles. What does this mean? Hurst coalesced his spectral work in ‘Profit Magic for Stock Transaction Timing’ into a concise and easily referenced ‘model’ for average wavelengths of components in financial markets, most notably stockmarkets, but research and experience shows valid utility in other markets.

This model consists of several entries; from the 18 year component down to the 5 day component. Although the model was somewhat more populated in Hurst’s original work the classic version we use frequently today was introduced in the Cycles Course and attempted to simplify the lineup for practical trading purposes. Let’s take a look at it below.

JM Hurst Nominal Model

Name: 18 year nominal cycle | Average wavelength: 17.93 years

Name: 9 year nominal cycle | Average wavelength: 8.96 years

Name: 54 month cycle | Average wavelength: 53.77 years

Name: 18 month cycle | Average wavelength: 17.93 months

Name: 40 week cycle | Average wavelength: 38.97 weeks

Name: 20 week cycle | Average wavelength: 19.48 weeks

Name: 80 day cycle | Average wavelength: 68.2 days

Name: 40 day cycle | Average wavelength: 34.1 days

Name: 20 day cycle | Average wavelength: 17 days

Name: 10 day cycle | Average wavelength: 8.5 days

Name: 5 day cycle | Average wavelength: 4.3 days

The important aspect to grasp from this model is the fact that wavelengths specified in the model are a guide because they are average wavelengths found to be present in the sample set Hurst originally researched upon in the 1960s and 1970s.

Of course today we can bring to bear the power of modern software to any market and it is striking to see this model work extremely well for most financial instruments, cryptocurrency being the most recent and compelling.

The principle of variation dictates that the wavelengths of components in this model will inherently oscillate around the average wavelengths. We see this as frequency modulation visually. For example, stockmarkets are in general running faster than nominal globally. This may well continue or may indeed revert to longer cycles to compensate in the coming years. Clearly there is also the chance that the average wavelengths are in need of adjustment - the more samples we have the closer we come to converging on the most cogent, consistent period for each component.

What is really wonderful is that over the last 80 years the components listed above have changed very little. This is compelling evidence that the model above is a true synopsis of financial market motion. The 18 month component in particular has stayed very close to the nominal average in stockmarkets, only really decreasing in wavelength in recent years.

Hurst Nominal Model - Visualised

Below is a visual representation of Hurst’s nominal model using simple sinusoids and spanning one 18 month nominal cycle (545.6 days). Cycles here obey principles of proportionality (longer cycles = larger amplitude) and the principle of harmonicity (cycles are related by a simple integer ratio, usually 2:1 or 3:1).

The model here assumes a flat sigma-l and so is an entirely neutral cycle.

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Whilst it is true the majority of phasing analysis performed on Sigma-L is assuming synchronised troughs it is always possible to choose a peak phasing if the instrument looks suitable. The general school of thought is that in stockmarkets one should phase the troughs and in some metals like gold and silver one should phase assuming peaks are synchronised.

This thought process is based around some of Hurst’s work (although Hurst phased copper at the troughs in the Cycles Course) and some of it is just based on what everyone else is doing! The proper approach should be to assess the chart and look carefully at the shape of troughs and peaks. Are the troughs generally sharp and precise and the peaks rounded? Then a trough phasing is appropriate. How about sharp peaks and rounded troughs? Then of course a peak phasing is appropriate. How about sharp peaks and troughs? Or rounded peaks and troughs?

This is where we should use our software and human knowledge to assess which is the better option. How do we know which is the better option? The answer is that one of the analysis will reveal a much better phasing, with minimal frequency modulation amongst the components and the FLD of each component will provide support and resistance at various points of the price action. These are two solid signs of an acceptable phasing analysis.

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The FLD

What is the FLD? Proposed by JM Hurst in the Cycles Course and other work, this simple tool informs traders of several important cyclical factors at work in the market. The FLD, or Future Line of Demarcation, is a humble tool with no complicated setup or need for intricate indicator configuration! However it is one of the most powerful tools we can use in…

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5 years ago · 1 like · David F

A bad phasing analysis on the other hand will require a lot of licence in isolating the synchronised peaks or troughs. Frequency modulation will be high and price highs or lows will more often not be aligned to proposed cycle highs and lows. In Sentient Trader perhaps a lot of ‘pinning’ will be required to influence the analysis - not a good sign in our experience.

So why can we only choose either peak or trough synchronisation according to Hurst’s cyclic principles? It is because of the way the individual components of the overall price motion are summed. Let’s look at an idealised example below with sinusoids.

Idealised Price Motion Model Example

We can simply illustrate why it is not possible to have both synchronised peaks and synchronised troughs with the above chart. Here we have 4 sinusoidal components, each obeying a simple principle of harmonicity (2:1) and proportionality as espoused by Hurst in his nominal model. You can imagine each component in an analogous fashion to the periodic components in financial markets. So the purple sinusoid might be the 40 week component, the red the 20 week component and so on. The blue thicker line is the price motion composite, a result of summing the individual components.

When these discrete components are summed, as they are in financial markets, the resulting composite function has peaks that are phase shifted! The vertical lines mark the peaks in the individual sinusoids and you will see that none of the individual peaks are exact peaks in the composite model of price action! Infact the peak of the largest periodic component (purple) coincides with a fairly large low in the composite price motion.

The ‘peaks’ in the composite are rounded as each cycle component adds it’s influence into the summation and then wanes in strength. This is why trying to perform a phasing analysis on a financial market via a peak analysis and then struggling to do so will reveal it requires a trough based analysis only.

Of course the chart would be reversed for an analysis assuming synchronised peaks - the troughs would be rounded and lows out of phase with the discrete low of each periodic component.

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One of the most useful tools in any toolkit for a Hurst Cycles devotee is the humble FLD. The simplicity combined with the powerful insight it can offer are extremely useful once a full phasing analysis has been performed and wavelengths derived.

The trading signals it can produce are good, robust and various strategies around crosses are available of course. However, the nature of market motion is composed of periodic components and totally random components. This random noise tends to increase as we move to lower timeframes and in our experience is highest at timeframes of less than 1 hour per bar. This makes sense as volume per bar is relatively lower and there is more chance (especially in an illiquid or overnight market) of rogue participants pushing price outside the range of what we expect the smoother, robust and periodic components to produce. This random nature of markets is something we accept as Hurst Cycles traders and is detailed by JM Hurst himself in ‘Profit Magic for Stock Transaction Timing’. A section where he also describes the periodic components rather awesomely as ‘x-motivation’!

Removing the Noise (some of it)

One of the benefits to studying Hurst Cycles on a deep level is an increased knowledge of signal processing techniques to improve trading strategies. We know in our pure price signal there is random, high frequency noise which can produce spurious spikes in price and even skew median price as a marker for FLD crosses. However, we can employ one of the most simple low pass filters to the FLD itself and attenuate a good proportion of the noisy components. Let’s see how.

Fire up the MA

The most simple low pass filter available to the majority of traders is the moving average. Of course we can employ all manner of digital filters to price action and we will be exploring many such approaches in future articles but to simply attenuate high frequencies on an FLD signal the moving average is perfect for our purpose.

If we take the example in the original humble FLD post, applied to the S&P 500 we can start to show how we approach smoothing a noisy FLD to taste.

1. Plot your FLD with a moving average of period 1, applied to median price and projected forwards by the wavelength of the cycle component of interest, divided by 2. This should look similar to below where we have plotted the 20 day FLD for the 20 day nominal component, sampled to to be running at 16.5 days from the phasing analysis.

   

2. To apply low pass filtering to the FLD signal increase the period but only by an odd number, we will start at 3. You must now also account for digital filter lag which is an unavoidable tradeoff when using digital filters. Therefore we must shift the FLD back in time by the (period-1)/2. In the below case that is (3-1)/2 = 1. So the period is 3 and the horizontal shift becomes (8-1)=7. The FLD is smoothed and components with frequencies equal to and higher (2 day, 1 day etc..) than those of 3 day frequency are attenuated! Importantly the correct position of the FLD is maintained as we accounted for the filter lag inherent in the moving average calculation.

   

3. Let’s smooth further with periods of 5 and 7 respectively to further attenuate high frequency components.

   

   

When we start to approach the period being equal to the digital filter lag the benefits of applying the filter diminish. This is because the component we are applying the FLD for begins to be attenuated itself. So in the above example we would probably not want to go above 7 for the period. However try it yourself and watch how the periodic components are slowly removed from the signal. It is important to always account for the digital filter lag however!

Benefits of FLD 2.0

In practice this application of a low pass filter to the FLD allows a slightly better signal for FLD crosses. Where it really shines is when we look for dynamic support and resistance at FLD levels, a feature of Hurst Cycles that is really useful. Median price will so often track or hover around FLDs and to have the line smoothed (not too much, just to taste!) is a very good tactic to inform decisions.

I encourage you to try this approach for yourself and judge the results.

Hold on a Minute…

Keen eyed readers will have spotted the following: having correctly applied the digital filter lag to a moving average we have the correct position of the FLD, but what about moving averages applied to price as a matter of course on charts throughout the financial world? Yes, these must be lagged too and the vast majority are using moving averages incorrectly!

This will be the subject of a future newsletter/post and will likely be controversial as we challenge the arbitrary and ubiquitous application of these tools in technical analysis everywhere.

Thank you for reading and your support.

DISCLAIMER: This site is intended for educational purposes only. All results and projections hypothetical. We accept no liability for any losses incurred as a result of trading with real money based upon assertions made on this website.