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Harmonic Mean

Harmonic Mean

The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.
Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed...

Properties of Fourier series II

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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
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Discrete Fourier Transform

Discrete Fourier Transform

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Discrete-Time Fourier Series

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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
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Aliasing

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Properties of Fourier series I

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A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

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Harmonic analysis of environmental time series with missing data or irregular sample spacing.

Shabnam Dilmaghani¹, Isaac C Henry, Puripus Soonthornnonda

¹Department of Civil & Environmental Engineering, University of Southern California, 3620 South Vermont Avenue, Los Angeles, California 90089-2541, USA.

Environmental Science & Technology

|November 13, 2007

Summary

The Lomb periodogram and fast Fourier transform (FFT) reveal weekly and monthly cycles in air and water quality data. These methods are effective for analyzing environmental time series, even with irregular sampling and missing values.

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Area of Science:

Environmental Science
Data Analysis
Time Series Analysis

Background:

Environmental monitoring generates complex time series data.
Irregular sampling and missing data pose challenges for traditional spectral analysis.
Harmonic analysis is crucial for identifying periodic patterns in environmental systems.

Purpose of the Study:

To apply and compare the Lomb periodogram and fast Fourier transform (FFT) for harmonic analysis of environmental time series.
To identify anthropogenic periodicities in air and water quality data.
To provide guidance on selecting the appropriate method for irregularly sampled data.

Main Methods:

Application of the Lomb periodogram to analyze environmental time series.
Utilizing the discrete Fourier transform and fast Fourier transform (FFT) for harmonic analysis.
Demonstration of FFT on irregularly spaced air quality data with missing values.

Main Results:

Identification of a 7-day weekday/weekend effect in Washington D.C. particulate elemental carbon data.
Discovery of a 1-month cycle in several constituents of Milwaukee stormwater.
Quantification of random noise effects in Lomb periodogram analysis.

Conclusions:

The Lomb periodogram and FFT are valuable tools for detecting periodicities in environmental data.
Specific methods are recommended for handling irregularly sampled time series with missing data.
Anthropogenic cycles were identified in both air and water quality datasets.