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# Fourier Analysis

## What Is Fourier Analysis?

Fourier analysis is a type of mathematical analysis that attempts to identify patterns or cycles in a time series data set which has already been normalized. In particular, it seeks to simplify complex or noisy data by decomposing it into a series of trigonometric or exponential functions, such as sine waves. Each of these sine waves would have a specific cycle length, amplitude, and phase relationship with the other sine waves, which then could be added back together to reconstruct the observed data.

By first identifying and removing any effects of spurious trends or other complicating factors from the data set, the effects of periodic cycles or patterns can be identified more accurately, leaving the analyst with a better estimate of the direction that the data under analysis will take in the future.

### Key Takeaways

• Fourier analysis is a mathematical technique that decomposes complex time series data into components that are simpler trigonometric functions.
• The idea is to be able to remove noise or confounding factors from the data set in order to identify true patterns or trends.
• Fourier analysis has been applied to stock trading, but research examining the technique has found little to no evidence that it is useful in practice.

## Understanding Fourier Analysis

Named after the nineteenth-century French mathematician and physicist Jean Baptiste Joseph Fourier (1768-1830), Fourier analysis may sound complex, but it actually makes good sense.﻿﻿ Essentially it theorizes that complicated time series data can be construed as the sum of simpler functions such as those described by trigonometry.

Numerous studies have explored Fourier analysis for practical value in forecasting stock market price. Because Fourier analysis seeks to break down repetitive waveforms into harmonic components and the stock market doesn't move in a well-defined and repetitive manner, results are mixed, as most similar strategies are.

Fourier analysis methods are frequently implemented in algorithmic trading as a technical analysis tool for forecasting market direction and trends. Recent research that has sought to vigorously examine the usefulness of Fourier analysis in predicting stock prices has, however, shown the method to be a failure.﻿﻿

## Conceptual Example

For example, suppose a manufacturing company wanted to know what stage of its price cycle its main raw material was in. Because inflation would constantly be increasing the dollar price of the commodity over time, an analyst would remove the effects of inflation from the commodity's historical prices first.

Inflation is typically maintained between specified rates and if inflation meets or exceeds a pre-set limit, interest rates will be adjusted by central bankers to either increase or decrease inflation so it is brought within a target range. Thus, as the rate of inflation increases, decreases, or stays the same, interest rates will oscillate up and down to control an undesired rate of inflation.

If our analyst thus believes that inflation rates are cyclical, they can subtract a sine wave that matches the inflation cycle from the time series. Once inflation has been controlled for, the analyst would then have a much more accurate picture of the true price cycles experienced by the commodity.

Article Sources
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1. Encyclopædia Britannica. "Joseph Fourier." Accessed Nov. 27, 2020.

2. ResearchGate. "Fourier Analysis for Stock Price Forecasting: Assumption and Evidence." Accessed Nov. 27, 2020.

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