# Crude 18th century mathematician, Leonhard Euler, opened up

Topic: EconomicsFinancial Growth
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Last updated: June 5, 2019

Crude oil is one of the most financially liquid products that are being traded globally; therefore, an extensive literature about crude oil markets is available, and these researches are conducted by financial, academic, and government institutions. Fundamentals of crude oil, such as supply and consumption, are key indicators for local and global economies; therefore, the relation between price drivers of crude oil and macro variables was tackled in many researches. In this section, literature on crude oil markets is presented to elaborate on what has been done until today.Secondly, wavelet analysis is a relatively new concept that is used at various fields of science and engineering.

The literature review of wavelet analysis illustrates multi-disciplinary approach of academics, and it is evident that wavelet analysis has significant contribution to literature due to its unique properties.2.1 WAVELET ANALYSIS IN FINANCENowadays, finance world is interested in components of the price series rather than the price series themselves. Wavelet is the most recent and robust tool, which is able to decompose the price series into scales. In other words, wavelet analysis is an extended version of spectral analysis that examines frequency domain. Therefore, physicists, software engineers, and mathematicians have been involved in finance sector for several decades. Although wavelet transformation is recently developed, mathematical functions have been investigated in terms of sinusoidal components since Joseph Fourier. Before Joseph Fourier, 18th century mathematician, Leonhard Euler, opened up a new era in mathematics by discovering the concept of Euler’s identity.

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The concept is used to transform sinusoidal functions into exponential form, and the equation was later called Euler’s formula. The technical details of the formula are explained at the quantitative methods section of the thesis.Joseph Fourier transformed components of a time domain function into components at frequency domain in 19th century. This is known as Fourier transform, and the transformation process can identify sinusoidal components along with their amplitudes from the signal. However, this methodology assumes that all components of the signal6do not change over time.

In other words, jumps, seasonalities, and trends in the signal cannot be captured by the transformation. Time localization of sinusoidal components is required to accurately analyze non-stationary signals.Dennis Gabor (1946) inserted a rolling window into the model to localize the sinusoidal functions in time, which is called short term Fourier transformation (STFT). His solution is to apply the transformation within a rolling window. However, according to Heisenberg’s uncertainty principle; the window cannot achieve to localize time and frequency simultaneously.In the beginning of 20th century, Alfred Haar found first simple wavelet with finite energy in his study on orthogonal functions.

Then, Jean Morlet, who is an engineer working for an oil company, found a new way to localize the components in time and frequency domains, and named it as “wavelets of constant shape” (Mackenzie, 2001).