Interestingly, there is little to few information on the usefulness of Fourier Transform<\/strong> (FT<\/strong>) applied to financial time-series<\/strong>. No wonder why. In discrete edition of the FT, the incoming time-series, x(t)<\/em><\/strong> where t<\/strong><\/em> denotes time, is transformed into frequency domain of information, P(f)<\/em><\/strong> where f<\/em><\/strong> is frequency and P<\/em><\/strong> represents effective power of transformed signal, e.g.<\/p>\n\n\n\n where In real life applications of FT, very rarely trading signals<\/strong> are dominated by distinct periodicities. Due to unlimited market factors<\/strong>, the power spectrum may display broader \u201cpeaks\u201d, suggesting the existence of a latent dumping process or defined by a centroid frequency, f0<\/sub><\/em><\/strong>, a full width at half maximum (FWHM) of \u0394f<\/em><\/strong> and amplitude. A ratio of the QPO frequency to Lorentzian FWHM is known as a quality factor, Q = f0<\/sub>\/\u0394f<\/em><\/strong>. The distinguishing feature of the QPO as opposed to the noise components is that its quality factor can be large. The higher Q<\/em><\/strong> the wider QPO. All these parameters are correlated in the low-frequency QPO, with the amplitude and quality factor increasing along with the increase in centroid frequency.<\/p>\n\n\n\n It is obviously important to know what sets the (lack of) coherence of the QPO. Are QPOs composed of a set of longer-lasting continuous modulations or are they rather fragmented in time with frequencies concentrated around the peak QPO frequency? Is there any correlation between observed oscillations or they are preferably excited in a random manner? To answer these questions we need to resolve and understand the internal structure of detected QPOs i.e. to study the behaviour of the QPO on timescales which are short<\/em> compared to its observed broadening. A continuous modulation would then show up as a continuous drift in QPO frequency with time, while a series of short timescale oscillations will show up as disjoint sections where the QPO is on and off.<\/p>\n\n\n\n One way to do this is using dynamical power spectra<\/strong> (spectrograms<\/strong>; hereafter also referred to as Short-Fourier Transform<\/strong> or STFT<\/strong> for simplicity). While the actual techniques for this can be quite sophisticated in essence, an observation of length T<\/em><\/strong> sampled every \u0394t <\/strong><\/em>(giving a single power spectrum spanning frequencies 1\/T<\/em><\/strong> to 1\/(2\u0394t)<\/em><\/strong> in steps of 1\/T<\/em><\/strong>) is spilt into N<\/em><\/strong> segments of length T\/N<\/strong><\/em>. This gives N<\/em><\/strong> independent power spectra spanning frequencies N<\/em><\/strong>\/T<\/em><\/strong> to 1\/(2\u0394t)<\/em><\/strong> but crucially, the resolution is now lower, at N\/T<\/strong><\/em>. While this is useful in tracing the evolution of the QPO, it introduces \u201cinstrumental\u201d frequency broadening from the windowing of the data which prevents us following the detailed behaviour of the QPO on the required timescales.<\/p>\n\n\n\n The real problem with such Fourier analysis techniques is that they decompose the signal onto a basis set of sinusoid functions. These have frequency f<\/strong><\/em>, with resolution \u0394f=N\/T<\/strong><\/em>, but exist everywhere in time across the duration of the observation Tdur<\/sub>=T\/N<\/strong><\/em>. In the same way that the Heisenberg uncertainty principle sets a limit to the measurement of momentum and location of a particle, namely \u0394px<\/sub>\u0394x\u2265\u210e\/2\u213c<\/strong><\/em> where \u210e is a Planck constant, there is a Heisenberg-Gabor uncertainty principle<\/strong> setting the limiting frequency resolution for time-series analysis. This states that we cannot determine both the frequency and time location of a power spectral feature with infinite accuracy.<\/p>\n\n\n\n 2. Wavelet Transform<\/strong><\/p>\n\n\n\n Instead, if the QPO is really a short-lived signal, we will gain in resolution and get closer to the theoretical limit by using a set of basis functions which match the underlying physical shape of the QPO. This is the idea behind wavelet analysis<\/strong>. For example, one particular basis function shape is the Morlet wavelet, which is a sinusoid of frequency f<\/strong><\/em>, modulated in amplitude by a gaussian envelope such that it lasts only for a duration Tdur<\/sub><\/strong><\/em>. The product f \u00d7<\/em><\/strong> Tdur<\/sub><\/strong><\/em> is set at a constant<\/em>, fixing the number of cycles seen in the basis function. The lightcurve is then decomposed on these basis functions, calculated over a set of frequencies so that the basis function shape is maintained (i.e. that the oscillation consists of 4 cycles, so low frequencies have longer Tdur<\/sub><\/strong><\/em> than high frequencies). The resolution adjusts with the frequency, making this a more sensitive technique to follow short duration signals.<\/p>\n\n\n\n![\"\"](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-1-1.png\")
<\/figure>\n\n\n\n![](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-2.png\")
denotes mean values and N<\/strong><\/em> number of data points in x(t)<\/em><\/strong> and fj<\/sub><\/em><\/strong> is frequency index. The quantity of P(f)<\/em><\/strong>, defined above, often is referred to as Fourier Power Spectrum. The key information here is what it does. In plain English, the FT scans a wide range of frequencies and matches a degree of similarity between a sine wave of frequency fj<\/sub><\/em><\/strong> and the signal x(tj<\/sub><\/em><\/strong>) itself. If such \u201ccorrelation\u201d exists, the P<\/em><\/strong> (or the inner product between the signal and base sine function) will deliver a large value compared to the other frequencies. In other words, FT is best if we look for purely periodic oscillations in our time-series. The best scenario would be if such periodic pattern is active along the entire record of the examined signal; it loses it power if it lasts shorter under additional condition, i.e. if signal-to-noise ratio allows for its detection. Signal that is a realisation of white noise has a flat power spectrum while time-series with long up\/down-trends will reveal lots of colour noise in low frequency section. In a log-log representation of P(f)<\/em><\/strong>, such colour noise is modelled by a power-law model<\/strong>, ![](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-3.png\")
where the higher ![](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-5.png\")
the steeper slope or stronger trend in time-domain of the signal.<\/p>\n\n\n\n![](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-6.png\")
where ![](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-7.png\")
denotes a damping time-scale. In science, we talk about not periodic but a bunch of periodic<\/em> or localised periodic<\/em> or quasi-periodic oscillations<\/strong> (QPOs<\/strong>) present in the signal. A Lorentzian has<\/p>\n\n\n\n![\"\"](\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/06\/quant-at-risk-matching-pursuit-algo-4.png\")
<\/figure>\n\n\n\n