{"id":90663,"date":"2021-06-07T16:00:29","date_gmt":"2021-06-07T20:00:29","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=90663"},"modified":"2022-11-21T09:47:35","modified_gmt":"2022-11-21T14:47:35","slug":"multiscale-analysis-for-financial-time-series","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/multiscale-analysis-for-financial-time-series\/","title":{"rendered":"Multiscale Analysis for Financial Time Series"},"content":{"rendered":"\n<p>Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to rapid fluctuations in the short term. This suggests that financial time series are potentially embedded with different timescales.&nbsp;<\/p>\n\n\n\n<p>On the other hand, nonstationary and behaviors and nonlinear\u202fdynamics are often observed in financial time series. These characteristics can hardly be captured by linear models and call for an adaptive and nonlinear approach for analysis. For decades, methods based on short-time Fourier transform have been developed and applied to nonstationary time series, but there are still challenges in capturing nonlinear dynamics, and the&nbsp;often prescribed&nbsp;assumptions make the methods not fully adaptive. This gives rise to the need for an adaptive and nonlinear approach for analysis.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-hilbert-huang-transform-hht\"><strong>Hilbert-Huang Transform (HHT)&nbsp;<\/strong><\/h2>\n\n\n\n<p>One alternative approach in adaptive time series analysis is the Hilbert-Huang transform (HHT). The HHT method can decompose any time series into oscillating components with nonstationary\u00a0amplitudes and frequencies using empirical mode decomposition (EMD). This fully adaptive method provides a multiscale decomposition for the original time series, which gives richer information about the time series. The instantaneous frequency and instantaneous amplitude of each component are later extracted using the Hilbert transform. The decomposition onto different timescales also allows for reconstruction up to different resolutions, providing a smoothing and filtering tool that is ideal for noisy financial time series.\u00a0<\/p>\n\n\n\n<p>EMD is the first step of our multistage procedure. For any given time series\u202f<em>x(t)<\/em>&nbsp;observed over a period of time [0,<em>T<\/em>], we decompose it in an iterative way into a finite sequence of oscillating components&nbsp;<em>c\u2c7c(t)<\/em>, for&nbsp;<em>j=1, \u2026, n,&nbsp;<\/em>plus a&nbsp;nonoscillatory&nbsp;trend called the residue term:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-25.png\" alt=\"\" class=\"wp-image-90681 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>To ensure that each\u202f<em>c\u2c7c(t)<\/em>\u202fhas the proper oscillatory properties, the concept of IMF is applied. The IMFs are real functions in time that admit well-behaved and physically meaningful Hilbert transform. Specifically, each IMF is defined by the following two criteria:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>No local oscillation: the number of extrema and the number of zero crossings must be equal or at most differ by one.&nbsp;<\/li><\/ul>\n\n\n\n<ul class=\"wp-block-list\"><li>Symmetric: the maxima of the function defined by the upper envelope and the minima defined by the lower envelope must sum up to zero at any time\u202f<em>t\u202f<\/em>\u2208&nbsp;[0,<em>T<\/em>].&nbsp;<\/li><\/ul>\n\n\n\n<p>In addition, we apply the method of complementary ensemble empirical mode decomposition (CEEMD) to nonstationary financial time series. This noise-assisted approach decomposes any time series into a number of intrinsic mode functions, along with the corresponding instantaneous amplitudes and instantaneous frequencies.&nbsp;<\/p>\n\n\n\n<p>Below we illustrate the intrinsic mode functions (IMFs) and residual terms from the decomposition for the S&amp;P500 and VIX.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1100\" height=\"1392\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26-1100x1392.png\" alt=\"\" class=\"wp-image-90686 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26-1100x1392.png 1100w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26-700x886.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26-300x380.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26-768x972.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-26.png 1124w\" data-sizes=\"(max-width: 1100px) 100vw, 1100px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/1392;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center\"><em>Intrinsic mode functions (IMFs) and residual terms extracted from complementary ensemble empirical mode decomposition of S&amp;P 500 (log-price) from April 1, 2010 to March 31, 2020. The top row shows the original time series. The second to last but one rows show the IMF modes of the corresponding time series. The bottom row of each plot shows the residual term of the time series.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1100\" height=\"1408\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27-1100x1408.png\" alt=\"\" class=\"wp-image-90688 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27-1100x1408.png 1100w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27-700x896.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27-300x384.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27-768x983.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-27.png 1114w\" data-sizes=\"(max-width: 1100px) 100vw, 1100px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/1408;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center\"><em>Intrinsic mode functions (IMFs) and residual terms extracted from complementary ensemble empirical mode decomposition of the volatility index (VIX) from April 1, 2010 to March 31, 2020. The top row shows the original time series. The second to last but one rows show the IMF modes of the corresponding time series. The bottom row of each&nbsp;plot shows the residual term of the time series.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<p>The modes correspond to different frequencies, from rapid fluctuations to long-term trends. The decomposition allows us to&nbsp;i) smooth\/filter any time series by excluding some higher-frequency components, and ii) reconstruct any time series using a subset set of components.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-28-1100x803.png\" alt=\"\" class=\"wp-image-90691 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/803;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center\"><em>Time series reconstruction using different sets of components.&nbsp;Top: S&amp;P 500. Bottom: VIX.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-energy-frequency-spectrum\">Energy-Frequency Spectrum<\/h2>\n\n\n\n<p>Note that each mode (IMF) also corresponds to a different frequency of fluctuations. The above decomposition allows us to compare any&nbsp;two time&nbsp;series on a mode-by-mode (frequency-by-frequency) basis.&nbsp;<\/p>\n\n\n\n<p>An oscillating real-valued function can be viewed as the projection of an orbit on the complex plane onto the real axis. For any function in time\u202fX(t), the\u202f<strong>Hilbert transform<\/strong>\u202fis given by&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-29.png\" alt=\"\" class=\"wp-image-90695 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>so\u202fY(t)\u202fprovides the complementary imaginary part of\u202fX(t)\u202fto form an analytic function in the upper half-plane defined by&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-30.png\" alt=\"\" class=\"wp-image-90696 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>where&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"674\" height=\"94\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-31.png\" alt=\"\" class=\"wp-image-90697 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-31.png 674w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-31-300x42.png 300w\" data-sizes=\"(max-width: 674px) 100vw, 674px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 674px; aspect-ratio: 674\/94;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-32.png\" alt=\"\" class=\"wp-image-90698 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Then, the instantaneous frequency is defined as the 2????&nbsp;-standardized rate of change of the phase function, that is,&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-33.png\" alt=\"\" class=\"wp-image-90701 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Applying Hilbert transform to each of the IMF components individually yields a sequence of analytic signals&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"754\" height=\"82\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-34.png\" alt=\"\" class=\"wp-image-90702 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-34.png 754w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-34-700x76.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-34-300x33.png 300w\" data-sizes=\"(max-width: 754px) 100vw, 754px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 754px; aspect-ratio: 754\/82;\" \/><\/figure>\n\n\n\n<p>In turn, the original time series can be represented as a sparse spectral representation of the time series with time-varying amplitude and frequency:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-35.png\" alt=\"\" class=\"wp-image-90704 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>In addition, the instantaneous energy of the\u202fjth&nbsp;component is defined as&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-36.png\" alt=\"\" class=\"wp-image-90707 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Hence, for each time series, we obtain a diagram for the\u202f<strong>energy-frequency spectrum.<\/strong>&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-37.png\" alt=\"\" class=\"wp-image-90749 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n\n\n<p class=\"has-text-align-center\"><em>S&amp;P500: instantaneous energy&nbsp;E(t)&nbsp;tends to decrease as the instantaneous frequency\u202ff(t)&nbsp;increases.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-38.png\" alt=\"\" class=\"wp-image-90756 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n\n\n<p class=\"has-text-align-center\"><em>For VIX, the instantaneous energy&nbsp;E(t)&nbsp;decreases at a much lower rate than S&amp;P500.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<p>S&amp;P500 and VIX share similar instantaneous frequencies but significantly different instantaneous energies.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/06\/image-39.png\" alt=\"\" class=\"wp-image-90761 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n\n\n<p class=\"has-text-align-center\"><em>S&amp;P 500 vs VIX: the mode-by-mode instantaneous frequencies are very similar, meaning they are both driven by similar fluctuation frequencies albeit of significantly different magnitudes.&nbsp;Source: Leung and Zhao (2021) available at&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2105.10871\">https:\/\/arxiv.org\/abs\/2105.10871<\/a>&nbsp;<\/em><\/p>\n\n\n\n<p>In summary, the key outputs of this method are the series of IMFs, along with the time-varying instantaneous amplitudes and instantaneous frequencies. Different combinations of modes allow us to reconstruct the time series using components of different timescales. Using Hilbert spectral analysis, we compute the&nbsp;associated instantaneous energy-frequency spectrum to illustrate the properties of various timescales embedded in the original time series.&nbsp;<\/p>\n\n\n\n<p>Multiscale signal processing is also very suitable for analyzing cryptocurrency prices (see this&nbsp;<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3833262\" target=\"_blank\" rel=\"noreferrer noopener\">paper<\/a>). For additional examples, such as gold (GLD) and Treasuries, along with machine learning applications, we refer the reader to the full\u202f<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3595914\" target=\"_blank\" rel=\"noreferrer noopener\">paper<\/a>.&nbsp;<\/p>\n\n\n\n<p><strong>Disclaimer:&nbsp;This article&nbsp;is not intended to be investment advice.&nbsp;<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-references\">References&nbsp;<\/h2>\n\n\n\n<p>Leung and Zhao (2021), Financial Time Series Analysis and Forecasting with HHT Feature Generation and Machine Learning [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3595914\" target=\"_blank\" rel=\"noreferrer noopener\">pdf<\/a>],\u202f<a href=\"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/asmb.2625\" target=\"_blank\" rel=\"noreferrer noopener\">Applied Stochastic Models in Business and Industry<\/a>&nbsp;<\/p>\n\n\n\n<p>Leung and Zhao (2021), Adaptive Complementary Ensemble EMD and Energy-Frequency Spectra of Cryptocurrency Prices [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3833262\" target=\"_blank\" rel=\"noreferrer noopener\">pdf<\/a>]&nbsp;<\/p>\n\n\n\n<p><a href=\"https:\/\/scholar.google.com\/citations?user=P40aOHIAAAAJ&amp;hl=en\" target=\"_blank\" rel=\"noreferrer noopener\">Google Scholar<\/a>\u202f\/\/\u202f<a href=\"https:\/\/www.linkedin.com\/in\/timstleung\/\" target=\"_blank\" rel=\"noreferrer noopener\">Linkedin&nbsp;Page<\/a>\u202f\/\/\u202f<a href=\"https:\/\/sites.google.com\/site\/timleungresearch\/research\" target=\"_blank\" rel=\"noreferrer noopener\">Homepage<\/a>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to rapid fluctuations in the short term. This suggests that financial time series are potentially embedded with different timescales. <\/p>\n","protected":false},"author":189,"featured_media":58432,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,352,344],"tags":[9844,4922,9842,9839,9840,9843,9841,9838,494,5926],"contributors-categories":[13668],"class_list":{"0":"post-90663","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-north-america","11":"category-quant-regions","12":"tag-applied-stochastic-models","13":"tag-econometrics","14":"tag-energy-frequency-spectrum","15":"tag-financial-time-series","16":"tag-hilbert-huang-transform-hht","17":"tag-instantaneous-energy-et","18":"tag-intrinsic-mode-functions-imfs","19":"tag-multiscale-analysis","20":"tag-quant","21":"tag-quantamental","22":"contributors-categories-computational-finance-risk-management-university-of-washington"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.3) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Multiscale Analysis for Financial Time Series | IBKR Quant<\/title>\n<meta name=\"description\" content=\"Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/90663\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Multiscale Analysis for Financial Time Series | IBKR Quant Blog\" \/>\n<meta property=\"og:description\" content=\"Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to rapid fluctuations in the short term. 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