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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How to calculate harmonic equations from a complex wave
I’m working through an online example that explains spectral analysis in R. Let’s say you have a complex wave:
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Does the Discrete Fourier Transform have a factor of $n^{-1}$ or $n^{-1/2}$? [closed]
While studying, I have encountered two formulations of the Discrete Fourier Transform, and I am not sure whether the choice of having a factor of $n^{-1}$ or $n^{-1/2}$ in the DFT is arbitrary. The ...
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A question about linear inference in random Fourier feature kernels
In Ali Rahimi's and Ben Recht's paper "Random Features for Large-Scale Kernel Machines," there is a line near the bottom of the introduction which I can not reason about...
In addition to ...
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Does autocorrelation means harmonic analysis?
They have very different algorithms, the first easier I'd say.
Autocorellation shows no significant lag.
while the periodogram shows an entirely different story at "lag" 11.
So which shall ...
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Fourier Basis with even number of basis functions (R, FDA package)
For a project on scalar-on-function regression using a truncated basis expansion of the coefficient function, I am trying to understand why an even number of Fourier basis functions is only useful in ...
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Modelling seasonality by means of Fourier decomposition
I'm trying to model seasonality of a time-series by means of Fourier decomposition in order to model some yearly recurring patterns. The Fourier decomposition consists out of a few steps.
The time-...
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Phillips-Perron Test
I'd like to have a confirmation about the interpretation of the Philipps-Perron Test (PP-Test):
The PP-test is a unit root test for testing the non-stationarity of a time series.
It is constructed by ...
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Spectral Graph Convolutions: What are the spectral filters functions
I am trying to understand the mathematical meaning of one of the steps that appear in the Convolution Theorem (Step 4 here).
To give some context, this is related to applying the convolution theorem ...
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Taking the Z transform and the Fourier transform of a Discrete-Time Random Walk
I saw that we could apply two transforms to the propagator of the Continuous-time random walk (CTRW),
$$P(x,t)= \sum_{N=0}^\infty [\lambda^{*N}(x)w^{*N}(t)*\int_t^\infty d\tau w(\tau)] $$
where the ...
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Periods of High Activity Detection
So I have some sensor data (time series) of heart rate of some users. I want to detect the times they start and finish exercising.
The data is sensor readings of heart rate every second, it's ...
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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 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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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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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 ...
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Periodicity of random kitchen sink feature mappings
In various papers, e.g. Random Features for Large-Scale Kernel Machines, Rahimi and Recht introduce the now popular methodology wherein a "low rank" approximation to a stationary, PSD, kernel $K(x,y) =...
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Fourier series question I need help with
For the case of $a_n$ I am confused how my notes say we have 1 for the case n = 0. When calculating this integral for sin(nx) when n=0, we should also get zero as at x=0, sin(x) is 0. I must be ...
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How to efficiently find spikes in 2D data projected to 1 axis
I have the following points:
My goal is to find the be able to differentiate the yellow points from the purple ones.
We can assume that the yellow points are aligned on the vertical axis.
We cannot ...
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Multitaper F statistics
I'm having some problems interpreting the F-statistics output from multitaper analysis. To illustrate, the following code-snip in R performs multitaper analysis on the same sine-frequency but with ...
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How to discard the first spike after auto-correlation and handle sloping auto-correlation output [closed]
Disclaimer: I am not very mathematically inclined and am mostly looking to be pointed in the right direction.
I have various signals that I am putting through an auto-correlation function that uses ...
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How to search for irregular signals: Fourier, DWT or k-means?
See my notebook here
I want to search for irregular time signals in a data set of ~3 500 000 time signals. I can't give a clear definition of irregular signal, but it must fulfil the criteria of: not ...
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What is the difference between machine learning approaches and Fourier series to fit a curve to data graph?
As I know machine learning(at least in some problems) tries to fit a curve to data graph. And I think Fourier transform tried to do it. But machine learning use a hypothesis curve with the formula ...
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Fast Non-uniform DFT in R
So, if I want to compute discrete Fourier transform (DFT) in R, I can create my own functions like so:
...
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Why is spectral density only defined for stationary processes?
I read Brockwell and Davis(2016), Shumway and Stoffer(2016), and Stoica and Moses(2004). However, none of them laid out clearly the reasoning behind the presumption of stationarity when conducting ...
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Probability of sparse spectrum
Consider a vector $v$ such that $v \sim \mathrm{Unif}(\mathbb{S}^{d-1})$, the uniform distribution on the unit sphere in $d$ dimensions.
Question: is there an upper bound on the probability that $v$ ...
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