The Sigma-MT Suite (includes Sigma-MT EA and the Sigma-MT Scanner) is available now for new VIP Cycle Analysts (founding members). Please contact me (on X or Substack chat) once you have joined or upgraded from standard membership to receive the software.

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Prerequisites: the full single-chart course, parts one through five, or at minimum the quick start, spectrogram and detection guides. The scanner assumes you can read a rank, a wavelength and a phase because it is about to hand you dozens of them at once!

Everything this course has taught you so far is deliberate, single-chart work: attach the EA, read the spectrogram, judge the rank, project the turn. The obvious question follows you the whole way through. Which chart? A clean, high-ranking, well-phased cycle is a needle, and the market is a very large haystack; the trader who checks ten instruments by hand has still not checked the other two hundred. The Sigma-MT Scanner, (currently on version 1.7 and part of the Sigma-MT portfolio) exists to remove that problem entirely. It runs the same wavelet convolution, the same CFAR detection and the same quality scoring you know from the EA across every symbol in your Market Watch, on up to five timeframes at once, and returns a ranked, sortable table of every tradeable rhythm it finds. This part covers the operational half, getting it installed, configured and scanning. The next part covers reading what it returns and the part after that the click-through from scanner to chart for a full deep dive.

Install and the one-time template

The Scanner is itself an Expert Advisor. Copy it into MQL5\Experts alongside Sigma-MT, refresh the navigator, and attach it to any chart; the symbol and timeframe of that host chart are irrelevant, since the scanner ignores it’s host and takes over the window as a dedicated panel. Like the EA it places no trades and requests no permissions.

Before your first scan, do one piece of setup you will never have to repeat because it transforms the whole workflow later. Open any chart, attach Sigma-MT to it, configure the inputs however you like your analysis, then right-click the chart and save it as a template named SigmaMT, matching the Scanner’s SigmaMTTemplate input (in Chart Launch section), which defaults to exactly that name. Note, you can also use another name, if you wish just ensure they line up.

When you eventually click a result row, the scanner opens the suggested chart and applies this template automatically, so the cycle analysis appears pre-loaded with the EA rather than as a bare chart, matching the characteristics found via the scanner which uses the same engine. The parameter handoff that accompanies this is genuinely clever and gets full treatment in part eight; for now, just save the template and forget about it. If the template is missing the scanner will still open charts, with a warning in the log, and setting the input blank disables template application deliberately.

Choosing the universe

The Scanner’s universe is simply your Market Watch: every symbol currently shown there is scanned, and nothing else is. This is a deliberate design choice, because it means curating your scan list requires no configuration dialog at all, you add and remove symbols in the Market Watch exactly as you always have, and the next scan reflects it. Keep a lean, liquid watchlist and scans complete quickly; show hundreds of exotic crosses and the scan will dutifully churn through all of them. My own habit is a working set of the majors, the key crosses, the indices and the metals, which keeps a full multi-timeframe pass comfortably fast while covering everything I would realistically trade.

Timeframes and the analysis settings

The Analysis Settings group offers up to five timeframe slots. Timeframe1 defaults to M1 and any slot left at current is disabled, so you enable exactly as many resolutions as you want swept, up to all five. Each symbol is analysed independently on each enabled timeframe, which is what later allows the Scanner to notice when one instrument is carrying related cycles at several resolutions at once, the nested-cycle grouping that part seven unpacks. Be aware the workload multiplies accordingly: five timeframes across a large watchlist is five full analyses per symbol, and the progress readout will show you the cost in time and processing accordingly. Scanning through my main market watch of around 60 instruments at N=500, across 5 timeframes, takes around 45 seconds! In practice one should aim for Market Watch lists around 30-50 instruments in order to avoid clutter if analysing multiple timeframes. I leave it up to the user to find their preferred balance of this powerful feature.

The rest of the analysis inputs you already know, because they are the same engine. N is the sample size per analysis, AnalyseOn the price it decomposes, and the Time Frequency group carries the identical LowestFreq, HighestFreq, DetectionMethod, SNRThreshold, CFAR cell and bias settings and RecencyBias from the EA, with the same defaults and the same meanings. This is worth a moment’s appreciation: the scanner is not a lightweight screener applying some crude proxy and hoping the EA later agrees. Every row in its results table earned its place through the full wavelet convolution and CFAR detection you studied in the detection guide, at the settings you chose. What the scanner calls rank is precisely the 0 to 1000 rank the EA shows you and so forth.

The three ways to scan

The LoopInterval input sets the scanning rhythm, and there are three modes of operation within it. Manual, the default, scans once when you press R and then waits for you; this is the mode I recommend while learning and for the session trader who wants a fresh sweep at moments of their choosing. Continuous begins a new scan the moment the previous one completes, keeping the table as live as your hardware allows, which suits a dedicated scanning terminal left running. Between the two sit the fixed intervals, one, five, fifteen, thirty or sixty minutes, where the Scanner rests between sweeps and refreshes on schedule; these are the sensible default for most people, frequent enough that the table stays current, quiet enough that your terminal is not permanently occupied. Whichever mode is set, R always triggers a scan on demand provided one is not already running, and the panel’s status line shows progress through the symbol list as it works.

Filtering the flood

A broad watchlist on several timeframes can return more rows than any human should triage raw, so the scanner filters at the source. IgnoreBelow sets the minimum acceptable rank for a cycle to appear at all, in tiers of 600, 700, 800 or 900, and I would encourage you to use it aggressively: the whole discipline of this course has been that detections below 700 deserve suspicion, and a scan-level floor simply enforces that discipline before the noise ever reaches your eyes. MinWavelengthMinutes discards cycles shorter than a chosen wavelength, expressed in minutes, which matters because a market-wide sweep of low timeframes will always find an abundance of very fast, technically valid but practically untradeable rhythms. A floor of an hour or two clears them out at a stroke. Set to zero it shows everything. Personally, if trading intraday, I set the minimum at 45 minutes - in my experience there is often great cycle features around an hour intraday but anything shorter is dicing with randomness just too much!

GroupInPhase, gathers rows of the same symbol across timeframes when their cycles are in phase, and it sits right on the border between operating the scanner and interpreting it. I will introduce it here and do it justice in part seven, where phase confluence and the harmonic validation behind that grouping are the entire subject.

Your first sweep

Put it together and the first session looks like this: Attach the scanner, confirm your Market Watch holds the instruments you actually care about, enable two or three timeframes to start, set IgnoreBelow to 700 and let it cook. Watch the progress line walk the symbol list, and when the table settles, simply read it with the eyes this course has given you: wavelengths in real units, ranks you know how to weigh, phases you know how to interpret. Click a column header to sort, shift-click to add a second sort, and scroll with the arrow keys. Press R to rerun the scan at a time of your choosing or set it to loop autonomously with the loop input described above.

Resist clicking through to charts for now; the next part is about extracting the full meaning of this table, the phase confluence between it’s rows and the harmonic relationships that mark a genuinely special result. The process after that completes the journey from a row in the table to a fully audited trade candidate.

Where to go next

Part seven, Reading the Results: Phase Confluence and Harmonic Validation, takes the table you can now produce and teaches you to read it the way the engine intends. What each column asserts, why in-phase cycles across timeframes are grouped together and the harmonic test that separates coincidence from genuine nested structure.

Licence: Sigma-MT is provided exclusively to founding members of Sigma-L and is licensed for your personal use only. Redistribution or copying, cloning or in any way reproducing the EA or Scanner files is prohibited without express prior permission.

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 Sigma-MT Suite is available now for new VIP Cycle Analysts (founding members). Please contact me (on X or Substack chat) once you have joined or upgraded from standard membership to receive the software.

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Prerequisites: the quick start, the spectrogram guide, detection guide, projection guide and the yield guide. You should be comfortable reading spectral power, trusting a score and understanding the bandpass as the dominant cycle extracted cleanly from price. No new keys are introduced in this part; the composite is configured entirely through the inputs.

Introducing composite analysis

Every part of this course so far has interrogated one instrument at a time. This part widens the aperture, and in doing so it reaches back to one of the oldest ideas in the cycle literature. Hurst called it the Principle of Commonality: the observation that broadly related instruments do not each dance to a private tune but share their dominant cyclic behaviour, with the differences between them being largely a matter of amplitude and noise rather than of rhythm. A basket of dollar pairs breathes with the dollar. A group of index futures inhales and exhales risk together. Hurst asserted this from painstaking manual study in the 1960s; the composite mode in Sigma-MT lets you test it numerically on any basket you care to assemble, in seconds, and then, crucially, profit from the answer.

The logic of the feature is simple to state. Each member of a related basket carries the shared cycle buried under its own idiosyncratic noise: the news spike particular to one pair, the earnings gap particular to one stock. Combine the members correctly and the common rhythm, being roughly in phase across the basket, reinforces, while the private noise, being uncorrelated, tends toward cancellation. The signal-to-noise ratio of the commonality cycle rises above anything a single member could offer. Then, having detected that shared cycle with a confidence no lone chart could give you, the software answers the practical question that follows: on which instrument in the basket is trading optimal for yield over time?

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The Sigma-MT Suite is available now for new VIP Cycle Analysts (founding members). Please contact me (on X or Substack chat) once you have joined or upgraded from standard membership to receive the software.

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Prerequisites: the quick start, the spectrogram guide, detection guide and the projection guide. You should be comfortable reading the spectral properties of a signal and a bandpass. You should understand that the bandpass is the dominant cycle extracted cleanly from the price action at the targeted bandwidth.

Every guide so far has pointed forward: detect the cycle, score it, project it, box the next turn. This one turns around and looks back historically, giving an estimation of past performance. Yield mode, on the Y key, asks a simple question of the cycle currently on your chart: if you had traded this rhythm across the visible window, buying every trough and selling every peak, what would it have returned? The answer is one of the most useful sanity checks in the software, and also one of the easiest to misread, so this guide is as much about what the number does not mean as what it does.

What yield mode actually does

Press Y and the engine walks the weighted bandpass from the oldest bar to now-time, finding every inflection point along the way. It then lines these up into a sequence of trades, which the software calls legs, always starting from a trough so that the first move is a buy. Trough to the next peak is a long; peak to the next trough is a short. The cycle is simply traded in both directions, turn by turn, across the whole window.

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The Sigma-MT Suite is available now for new VIP Cycle Analysts (founding members). Please contact me (on X or Substack chat) once you have joined or upgraded from standard membership to receive the software.

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Prerequisites: the quick start and the spectrogram guide. You should be comfortable reading a spectrogram, recognising a persistent band and understanding the bandpass origin.

The spectrogram guide showed you how to read power, but it left one question open: how does the engine itself decide that a particular band is worth your attention, rather than a passing flare of noise? That decision is the whole game. A projection is only as trustworthy as the detection beneath it, so this guide opens the bonnet (somewhat!) on detection, explains the score that grades it, and hands you the manual controls for when you want to override the engine’s choice.

Why a threshold has to adapt

Let’s begin with the naïve approach: The obvious way to call a peak significant is to set a fixed threshold of power: any frequency whose power rises a certain multiple above the average counts as a real cycle in this approach. This is the signal-to-noise method, and Sigma-MT offers it, although really it is a legacy of testing I thought I would leave in. You set SNRThreshold (default 3.0), the engine measures each frequency against the median power of the spectrum, and any local maximum that clears that ratio is flagged.

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The Sigma-MT Suite is available now for new VIP Cycle Analysts (founding members). Please contact me (on X or Substack chat) once you have joined or upgraded from standard membership to receive the software.

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Prerequisites: the quick start guide. You should already have Sigma-MT on a chart and recognise the spectrogram, bandpass panel and info panel by sight.

The quick start guide told you what the spectrogram is. This guide tells you what it means, and why a picture built from wavelet convolution is a fundamentally more honest way to look at market cycles than counting peaks on a chart. Everything that follows in the course (detection, projection, yield, composites, the Scanner) is built on the foundation laid here, so this is the one guide worth reading twice.

The problem with counting cycles..

Classical cycle analysis, the kind you find in most trading literature and some descending from the work of J.M. Hurst, begins by simple counting of visually apparent inflection points. You find two lows, measure the bars between them, and declare a cycle of that length. The trouble is that this is subjective and brittle. Two analysts looking at the same chart can pick different lows and arrive at different cycles, and a single noisy swing can throw the count off entirely. Indeed higher frequency features can often have bursts in power than exceed the power of the longer cycle, influencing price in a transitory and misleading (to the cycle counter) manner. Worse, it assumes a cycle has one fixed length, when real market rhythms drift in period (frequency modulation) and fade in and out of strength (amplitude modulation) over time. The real marker of a quality cycle is the minimisation of any modulation (or a high degree of stationarity), since modulation is a constant in financial market cycles.

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A reminder before the list. Several inputs have a keyboard twin, so anything you adjust live with the keys (sample size, frequency edges, panel toggles) only persists for that chart session. The inputs are where you set the values that load every time the EA attaches.

Time Series

SampleN (default 500) — The number of bars in the analysis window. Range 64 …

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Sigma-MT is driven almost entirely from the keyboard. There are no buttons cluttering the chart and no menus to dig through; once the key map is in your fingers, reshaping an entire analysis takes seconds. The keys fall into four groups, mirroring the in-app panel: analysis controls that change what the engine computes, cycle visibility toggles, display…

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What is Sigma-MT?

Sigma-MT is an advanced Expert Advisor for the trading platform Meta Trader 5, available exclusively to VIP cycle analyst members on Sigma-L (founding members).

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It is a direct port and enhancement of the python/numpy version I have developed over the last few years and has been seen previously in posts on Sigma-L. It is a bespoke, unique and powerful way to detect, analyse and act upon statistically significant periodic features in financial markets. It abstracts away many of the more technical aspects of this approach but retains a great degree of adaptability, alongside a streamlined interface and fast computation (processor dependent!). I hope you will enjoy using it as much as I do and discover some cyclic gems in the process.

Sigma-MT is not an indicator/expert advisor in the usual Meta Trader sense. It is a time-frequency analysis engine, drawn from the same family of mathematics used in radar and seismology, pointed at price. Rather than smoothing price and hoping a crossover means something, it decomposes the series into it’s component frequencies, watches how their power evolves through time, and tells you which cycle is dominant right now, where that cycle sits in it’s phase, and how much confidence the detection deserves.

This guide takes you from a fresh install to reading that information. Everything deeper, from the detection mathematics through projection, yield statistics, composites and the Sigma-MT Scanner has its own guide in this series. Here we simply get you on the chart.

For a complete reference to all of the controls for analysis navigation, see Sigma-MT keyboard reference.

For a complete reference to all in the input parameters, see Sigma-MT v1.0 Input Reference

1. Install

Copy Sigma-MT.ex5 into your MT5 data folder under MQL5\Experts (in MT5, go to File → Open Data Folder), then restart the terminal or right-click Expert Advisors in the Navigator and hit Refresh.

Sigma-MT places no trades. It is pure analysis, so there is nothing to configure on the trading side: no DLLs, no web requests, and no permissions beyond a standard EA attach.

2. Attach to a chart

Open a chart on anything liquid. A major FX pair or an index on H1 makes a good first run. Drag Sigma-MT onto it and the input dialog will appear. For your first analysis, change nothing. The defaults are the configuration I run daily: CFAR detection, a target range of 4 to 40 cycles across the sample window, trade boxes on, and the info panel on. Click OK.

One constraint is worth knowing up front. Sigma-MT pauses on the monthly timeframe, because there simply are not enough MN1 bars to support meaningful spectral analysis. Weekly and below is it’s territory.

Within a few seconds the engine completes its first wavelet convolution pass and the display builds at the bottom of the chart.

One aspect worth adjusting at the outset is the display resolution of the panels. This can be adjusted in several ways in the ‘Display Settings’ and ‘Information Panel’ settings. Play around with font sizes and width/height adjustments to fit your screen size as appropriate. Save as an EA preset or save the whole chart as a template named ‘default’ to automatically load this setup on every chart that is launched (not recommended!). Alternatively save as a named template, such as ‘SigmaMT’ so you can load in the settings quickly if required on future charts. This will also be needed when we come to setup the Sigma-MT Scanner workflow.

3. What you’re looking at

Three things appear, and together they carry the entire read. The spectrogram is the heart of the system. Time runs left to right, cycle wavelength runs bottom to top, and colour intensity shows spectral power. A bright horizontal band is a cycle that has persisted (it is stationary), and that persistence, rather than any single point, is what makes a cycle worth your attention for possible trading. To the right of the spectrogram, the summed spectrum shows where power concentrates across frequencies, with the detection threshold overlaid so you can see exactly what the engine considers significant.

Above/below it sits the bandpass panel (press F if you prefer it at the top of the chart). This is the dominant cycle extracted from the noise and plotted as a clean oscillation, with its projection extending past the current bar. It is the Hurst idea made explicit: isolate the dominant rhythm, then let its phase tell you where you are in the swing.

On the left you have the info panel, which summarises everything into a glance. It shows the detected wavelength in real units such as hours or days, the current phase status indicating whether the cycle is at a trough, rising, peaking or falling, and a score box grading detection quality from 0 to 1000 (1000 being a perfectly stationary sine wave and impossible in real markets!) over the recent window. Treat that score as your gatekeeper. Anything above 650 is worth attention, 800 and up is a solid detection, and 900 or more means the engine has found an unusually clean cycle. Below those levels, stand aside, and remember that the absence of a detection is also information.

4. Your first interactions

Press K and a shortcuts panel opens listing every keyboard control, grouped by function: analysis, cycle visibility, display options and utilities. You do not need to memorise anything, because K is always there. Press it again to close.

Now try the controls you will use constantly. The number keys 1, 2 and 3 toggle the short, medium and long-term cycle overlays (the medium and long-term layers activate once you enable their timeframes in the inputs, which Guide 2 covers properly). Press X to draw constant-width envelopes around price, calibrated so that price respects them roughly 95% of the time. Press B to toggle the trade boxes that mark projected inflection zones, and Z to switch on the Kalman extrapolation of the underlying trend. If you want a clean chart, G hides the spectrogram, and C captures a screenshot straight to disk.

There is one mouse interaction worth learning immediately. Click anywhere on the spectrogram and the summation area shows you the frequency spectrum at that exact point in time; click outside it and the view returns to the summed spectrum. Scrubbing through spectral history this way will teach you more about how cycles emerge and decay than any paragraph I can write.

5. Shaping the analysis window

Everything Sigma-MT tells you is computed over a window of bars, and that window is entirely under your control from the keyboard. Press W to extend the sample size and S to contract it. The current size is always visible in the small label beside the dotted line marking the start of the analysis, shown as N= along with S= for the shift and I= for the increment. The engine accepts anything from 64 bars up to 1500, and it will keep the count even for you automatically, since the wavelet transform requires it.

Why would you change it? Because the window defines what the engine can see. A short window resolves recent behaviour sharply but cannot detect a cycle longer than the data it holds, while a long window reveals the slow rhythms at the cost of diluting recent changes. If the spectrogram shows strong power pressed against its upper edge, that is usually a cycle bigger than your window, and a few presses of W will bring it fully into view. The frequency targets are defined per window, so resizing also rescales what each row of the spectrogram represents in real time units; the info panel always translates the detected wavelength back into hours or days for you.

Each press moves the size by the current increment, which you control with Q to halve it and E to double it, anywhere from 1 bar up to 100. Set a coarse increment to reposition quickly, then drop it down for fine adjustment. The I= value in the start line label always shows where it stands.

Shifting through history

The window does not have to end at the current bar. Press A to slide the entire analysis into the past and D to bring it back towards the present, again moving by the increment each press. The S= value shows how many bars into history you currently sit, and when it returns to zero the engine confirms you are back at the live edge. The only limits are the data itself, since the EA will stop you when there is no longer enough history to fill the window, and one display restriction: the price envelopes are unavailable while shifted into the past, because their calibration only has meaning at the live edge.

This is one of the most instructive things you can do with the software. Park the window over a past period, look at what the engine detected and projected with only the data available at that moment, then step forward with D and watch how the cycle evolved. It is the closest thing to a flight simulator for cycle analysis, and it costs nothing.

For larger jumps there is a faster route than holding D. Hold Ctrl and click anywhere on the price chart and the analysis window recentres on that bar, placing it at the middle of the sample. One click takes you from the present to a period of interest months back, and another Ctrl+Click near the right edge, or a few presses of D, returns you home.

Moving through the spectrogram with HILO keys!

The vertical extent of the spectrogram is set by the lowest and highest frequencies the engine targets, and four keys let you reshape that band live. L and O move the low-frequency edge, which governs the longest wavelengths at the top of the spectrogram: L pushes it down towards 2, admitting longer cycles, while O raises it towards 10, trimming them away. H and I move the high-frequency edge at the bottom: H lowers it towards 15, cutting out the fastest, noisiest wavelengths, and I restores them up to the maximum of 40.

In practice these keys are how you walk up and down the frequency axis. If the action you care about lives in the slower cycles, press H a few times to discard the high-frequency clutter and the engine re-spreads its full wavelet budget across the band that remains, so resolution improves exactly where you want it. If a fast intraday rhythm is your target, push the low edge up with O instead. Every adjustment triggers a full re-analysis, and the detection, score and projection all update against the new band, so you can watch in real time how the dominant cycle call depends on where you point the engine.

A useful habit: when a detection seems ambiguous, nudge the band edges and see whether the call survives. A robust cycle keeps being found as the band shifts around it. A fragile one vanishes the moment its supporting noise is excluded, and that fragility is exactly what you want to discover before committing to the projection.

Selecting a custom band

A very useful aspect of the app is the ability to select a custom band of 2 or more frequencies (wavelengths) to examine from the spectra. Simply shift+click once on the spectra to draw one boundary of the band and whilst still holding shift, click somewhere else in the band to establish the bandwidth and second boundary. The panel will also indicate this is a custom (not automatically detected) band, selected by the user.

Press the ‘0’ (zero) key to return to auto detection and remove the custom band of selected wavelengths.

6. Read the cycle

Here is the whole workflow in miniature. First check the score and ask whether the detection deserves your trust. Then check the wavelength and ask whether this is a rhythm you can actually trade on your timeframe. Finally check the phase. A cycle at its trough and rising argues for longs into the projected peak, while a cycle peaking and rolling over argues the reverse. The trade boxes translate that phase logic into a zone on the chart where the next inflection is expected, drawn with an honest uncertainty window rather than a single line of false precision.

That is a complete first analysis, and you have touched perhaps a few keys to help you get around. What you have not yet done is understand why the detection fired, or how to audit a cycle’s historical performance with yield mode on the Y key. That is deliberate. Each of those topics earns its own guide.

Where to go next

Guide 1, Reading the Spectrogram and the Time-Frequency Engine, is the natural next step. It explains what the engine is actually computing and how the short, medium and long-term layers stack into a genuine multi-timeframe read. From there the course runs through detection, projection, yield, composites and the market-wide Sigma-MT Scanner.

Licence: Sigma-MT is provided exclusively to founding members of Sigma-L and is licensed for your personal use only. Redistribution or copying, cloning or in any way reproducing the EA or Scanner files is prohibited without express prior permission.

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.

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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