# Questions tagged [fourier-transform]

The Fourier transform decomposes a signal (a function of time) into frequencies, giving the energy at each frequency.

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### The length of spectral density is longer than the data using spectrum() in R

I'm using spectrum(method = "pgram") in R to calculate the spectral density in my time series. spectrum() returns the spectral density for each frequency(from 1/n, 2/n to 1/2, n is the time ...
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### Using Fourier Coefficients to approximate a time series

I'm trying to understand approximating periodic functions with Fourier Transforms. I'm using R, but I suppose the question is language agnostic. Say I have this discrete periodic series. It is some ...
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### Is a Laplace Prior the same thing or related to a Laplace Transformation?

Context: I was watching this video https://youtu.be/pOYAXv15r3A?t=796 about Facebook Prophet and the speaker mentioned they use a Laplace Prior $$\delta \sim Laplace(\lambda)$$. What I have gleaned so ...
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### Forecasting time series with multiple seasonaliy by using auto_arima(SARIMAX) and Fourier terms

I am trying to forecast a time series in Python by using auto_arima and adding Fourier terms as exogenous features. The data come from kaggle's Store item demand forecasting challenge. It consists of ...
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### How can i get the phase difference between two frequencies?

I have two signals of the same frequency. Both have frequencies of 13.56Mhz, and I want to find the phase difference between these two frequencies. The function generator generates a sinusoidal ...
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### Seasonal ARIMA vs ARIMA with Fourier terms for monthly series

I have monthly data and my goal is to forecast values with an inclusion of exogenous variables. I've built SARIMA and Dynamic Harmonic Regression models, where the second one performs little bit ...
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### DFT of Multivariate Normal Random Vector

I have a real zero-mean multivariate rv $X \sim \mathcal{N}(0, \Sigma)$, with $N^d$ entries. $X \in \mathbb{R}^{N^d}$ is the flattened representation of a $d$-dimensional signal, of "side length&...
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### Looking for repeated patterns in time series data

I have spent the best part of the last few days searching forums and reading papers trying to solve the following question. I have thousands of time series arrays each of varying lengths containing a ...
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### Random Fourier Features approximating a kernel inverse?

There is a method I have been studying called Spectral Normalised Neural Gaussian Processes which leaves me with a question I cannot answer. In this method, they utilize Random Fourier Features but ...
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### Examples of Fourier Analysis in statistics?

My brother in law is studying Electrical Engineering (undergrad level.) I was helping him select some coursework, and noticed that "engineering analysis" includes Fourier analysis. I didn't ...
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### How to compute the minimum functional norm solution $h_{n,\infty}$ obtained from $\mathcal{H}_{\infty}$

I stumbled upon the following paper Reconciling modern machine learning practice and the bias-variance trade-off and do not completely understand how they produce the line for the minimum functional ...
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### Correlated residuals - white noise analysis

I want to check and analyze the goodness of my compressed sensing reconstruction algorithm (fit) using the autocorrelation of the residuals. Given my problem, I guess there are two ways to do this: ...
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### Relation between Fourier transforms and coherence of signals

My overall aim is to compare the edges of two images by comparing their Fourier Transforms (FFT) and to calculate one number as a key performance indicator that describes how much they are similar to ...
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### Identifying a single dominant number with high probability in a data point array of 4 to 20 numbers

EDIT 3: added below picture of raw readings of oscope from doppler data showing "periods" of sine waves before processed sine signal into square pulses. EDIT 1: These numbers on the left ...
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### How can I quantify the oscillation score of a complex wave?

My question is similar to this question, where I need to quantify a score that expresses the rate of oscillation, as shown in the question above. In my case, the wave may contain multiple frequencies ...
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### Why are Deep CNN's mesh dependent?(Fourier Neural Operators article)

I am reading the article named "Fourier Neural Operator for Parametric Partial Differential Equations" by Zongyi Li et al. In the very first page, there is a paragragh about Finite ...
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### How to identify a non-periodic time-series?

I am working on a problem where I have to first classify whether a time-series is periodic or not and then, if it is periodic identify its period(s) (the time-series could have multiple periodicities, ...
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### Correlation estimation by the Blackman-Tukey method

Problem statement: Assume that the spectral estimation for an unknown signal is given by the Blackman-Tukey method as follows: $$S_x(\omega) = 5+8\cos(\omega)-6\cos(2\omega)+2\cos(3\omega).$$ Assume ...
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### Functional PCA for stationary signal

I just came across this technique: https://en.wikipedia.org/wiki/Functional_principal_component_analysis As far as I understand, if projects a signal onto a functional subspace that describe the "...
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### Finding autocorrelation coefficients given PSD values at 2 frequencies

Assuming that $S_X(w)$ denotes powers spectral density function at frequency $w$, we are given $$S_X\left(\frac{\pi}{4}\right)=10+3\sqrt{2},\quad S_X\left(\frac{\pi}{6}\right)=11+3\sqrt{3}.$$ We also ...
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### Do I need to use the complex conjugate when convolving two functions with the FFT?

I know that, due to the convolution theorem, two densities $f$ and $g$ can be convolved by (i) applying the FFT to both of them, (ii) multiplying the results, (iii) applying an inverse FFT. Since I ...
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### How to do clustering of fft values of a time series dataset?

I have a time series dataset, I have computed its fft. But I want to know if there is any specific clustering technique for fft values or can I use clustering techniques such as kmeans,heirachial?
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### Uncertainty principle in probability theory

In probability theory, there is the covariance inequality $$\operatorname{Var}(Y) \geq \frac{\operatorname{Cov}(Y,X)^{2}}{\operatorname {Var} (X)}.$$ In signal processing, there is a similar ...
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### Model specification for seasonal ARMA-GARCH model using rugarch

TL;DR: I'm trying to find an adequate model for time series data that exhibits multiplicative seasonality and volatility clustering by identifying an ARMA-GARCH-model with Fourier terms using ...
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### Are people still researching the use of spectral decomposition on finite groups for data analysis?

In A GENERALIZATION OF SPECTRAL ANALYSIS WITH APPLICATION TO RANKED DATA (Diaconis 1989), the author discusses a dataset of election results. There were 5 candidates, and each voter was asked to rank ...
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### Using Cross Validation for Highly Seasonal Data with small sample

I'm having trouble getting good scores on cross validated metrics on time series regression models. Essentially, I am trying to model product purchases based on amount of money spent on different ...
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### A space of functions and their Fourier Transforms?

Conjugate variables and the Fourier transform are often used to analyze different states of a single object. For example in Quantum Mechanics it can be used to describe changing information about ...
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### Characteristic function and Fourier transform for a discrete random variable!

Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function If X is a continuous random variable, then: $\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
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### Method to calculate optimal nbasis (K) for fourier basis

Is there any method to calculate the optimal K for fourier basis transformation? For example if I were to use RMSE, it seems that as K increases RMSE keeps going lower. I'm thinking of the elbow-...
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### Estimating Fourier spectrum from multiple time series of a system

I have a set of N time series, each of length T, that describe separate realisations of a single physical system. For each series, I can compute an FFT to find the Fourier spectrum up to a period 2/T, ...
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### Spectral analysis in R, how to interpret periodogram

I'm relatively new to spectral analysis and have been working through some online tutorials. I have some time series data that I would like to examine for periodicity / repeating patterns. When I ...
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### Why SARIMA has better accuracy on weekly dataset than on daily one?

I am studying time series right now. So, I have this dataset. My aim is temperature prediction. I've found out that ARIMA can't work with long period seasonality. So, I've resampled daily dataset ...
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### Fourier analysis to retrieve components of individual spectra

I have a basic, simple question, I am a physics student, and searching internet gives me a lot of signal processing theory but couldn't find this basic answer, which I plan to implement in my speech ...
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### Log-likelihood function for a filtered Fourier spectrum

I have time series data from which I am trying to infer parameters using MCMC. I normally infer parameters about the data in the time domain, using a Normal log-likelihood. However, I now have to ...
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### Measuring weather impact on sales as a crossed random effect

I'm trying to model sales of a clothing brand having longitudinal data (unbalanced panel: 20 stores with 50 - 157 weeks of datapoints). There are many regressors in my analysis, but I believe that ...