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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Spectral decomposition of the DC part of the signal

The title of the question might be misleading, but it shows how clueless I am about this problem. I have a time series of some quantity $X(t)$. I can calculate autocorrelation of this quantity $Y(\tau)...
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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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Is it appropriate to interpolate a signal for frequency analysis?

From an experiment, I have (somewhat) irregularly sampled data. The aim is to find the dominant frequency of the signal. As I understand it, most methods for frequency analysis require evenly spaced ...
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Difference of two independent lognormal random variables by Fourier transform methods?

Is it possible to calculate the difference $X_1-X_2$ of two independent lognormal variables $X_1$ and $X_2$ where $\log(X_1)\sim N(\mu_1,\sigma_1)$ and $\log(X_2)\sim N(\mu_2,\sigma_2)$? Could I ...
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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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Long term time-series prediction for non-stationary but regular water flow data

I am looking at a very interesting time series dataset for the volume of water flow through some groundwater sensors. Our goal is longer term prediction of water levels, such as 1 - 5 years in the ...
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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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Can you perform a Fourier transform of a Frequency / Time plot? (Number of phone taps (t))

I'm working on a thesis problem. I need to analyze tapping behaviour (number of phone taps). The raw data is the time stamp of every app, and the app in use. It was recommended to me that I create a ...
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Representing a time-series smoothed curve as a sinsoidal?

So attaches is an example of the kind of time series data I am working with. So far I have used Gaussian filters with sigma=3 and 6 to smooth the data, which has worked very well (especially sigma=3). ...
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Fourier transforms for noise reduction

Given a signal, which is regularly sampled over time and is noisy. The standard method is with a Fourier transform to reduce the noise and minimise the change to the signal. ...
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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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Exactly how are cyclical components computed?

So suppose we have some sort of a time series model. y_t = trend_t + cyclical_t + x_t + epsilon_t So, I'm interested in obtaining the seasonal component. Here "x_t" refers to other potential ...
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FFT for a binary time-series

I have a multivariate time-series of a binary values where 0 means that some state was not observed in the given second and 1 meaning that it was observed. I know that for a time-series consisting of ...
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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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Discrete Fourier transform (DFT) of 2 signals using a single DFT

Given two different(x,y) and independent signals, can one find dft of both of them using a single dft chip and using that that chip only once. I tried finding dft of x+jy but was not able to find ...
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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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Relationship between multivariate normal random variable and its Fourier transform

I was looking at the distribution of N dimensional real momenta $\pi$ where $\pi \sim \mathcal{N}(0,M)$. It turns out that $M$ is diagonal (and real) in Fourier space and so I wanted to transform to ...
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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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225 views

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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K in Fourier series - How to find value of K to use it in ARIMA?

I am using the forecast library for doing some time series forecasting. I need to forecast number of sold items. I am planning to add holidays as xreg in auto.arima. The holiday will be a 0/1 list, ...
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Continuous time Fourier representation

I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is: $$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad ...
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Inverse Fast Fourier Transform in R

plot(fft(fft(1:100),inverse = "true")/100) plot(1:100) In the above example, I was expecting the plots to be identical. The first, however, returns: while the ...
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Frequency analysis of categorical/binary data

I want to do frequency analysis with a data set of consecutive binary values, such as "Rainy - Sunny - Rainy - Rainy - Rainy - Sunny - ...". Using this data, I want to extract the frequency (= the ...
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341 views

Maximum number of Fourier terms in forecast package

I am using the forecast package in R to get some Fourier components - namely, function fourier(ts, K, ..). For a time series <...
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Adaptive knot selection for B-spline fitting

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
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Non-linear regression. Obtain B.spline coefficients using Fourier Transform?

I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way. Given that $$s(x)=\sum_kc(k)\...
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How to fit a model to both daily and weekly periodic data using linear regression and fourier transform

I have a variable, y, in 30 minute intervals, and is highly dependent both on the time of day and the day of the week - mainly Sundays. It is also dependent on bank holidays, which could by any day of ...
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464 views

R forecast multiple seasonality optimal model search using fourier and msts objects

Hi I have hourly data (one obs one hour) with multiple seasonality. I would like to fit an ARIMA model using forecast R package taking into account the multiple seasonality, maybe taking also in ...
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Fourier Output Meaning

I just ran a fourier series on weekly sales data for 3 years worth of data. I optimally chose the number of k-terms based on the AIC. First 6 lines of my data: ...
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RKHS norm and Fourier transform link

In the notes here, it is stated that norms of some reproducing kernel Hilbert spaces can be written in terms of Fourier transforms, and this is often used to argue that a higher RKHS norm implies a ...
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Is there a concept of weakly periodic in stats? If so, what is it called?

Ok, say we have a signal that reoccurs within every T+x time where T is a constant and x is a random number small in size compared to T. We know that a function is actually periodic if F(x) = F(x+T), ...
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Need help understanding output of a periodogram

In my effort to understand the output of a periodogram I created a series (s) where 1,1,1,1,1,1,1,1,1,10 is repeated 100 times and then created a periodogram of this series using the following R code: ...
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430 views

Discrete Fourier transform of an exponential decay

I have a vector with an exponential decay signal, using Numpy: t=np.arange(128) a=0.1 decay=np.exp(-a*t) I would like to compute the discrete Fourier transform (...