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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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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Why assumption of negligible cross-correlation between missing points is important for PSD estimate

In this paper they propose a method for estimating the Fourier transform (FT), and therefore the power spectral density of a time series with one or more missing points. They do this by inserting ...
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19 views

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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16 views

applying fast Fourier transform without restriction on N

I want to apply fast Fourier transform for a sequence of complex numbers with length N. N can be anything (not necessarily a power of 2). It seems that Cooley–Tukey algorithm only works for the ...
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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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extracting seasonality - using fourier transform vs. learning the coefficients of fourier terms

I'm following hyndman's advice for using fourier terms when fitting a linear regression model to the taylor time-series (with the very long seasonality of 336). My ...
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36 views

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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27 views

Capturing cycles in time series with Fourier

In time series regression with Fourier, Fourier terms are limited to maximum half of the frequency of time series to capture seasonality. my question is: Can we extend Fourier terms beyond half of ...
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Marginalisation when conditioning on the Fourier transform of a random variable

I am looking to sample from a distribution $p(y)$ defined by the following expectation: $$p(y) = \mathbb{E}_{p(u)} \left[ p(y|u) \right]$$ Both $p(u)$ and $p(y|u)$ are multivariate Gaussians: $$p(u)...
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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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28 views

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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81 views

Fast Fourier on time series To decompose

I am new to fast Fourier transform. The data is a weekly sales data and has trend and yearly seasonal components. I want to Decompose the time series signals to identify certain unique signal pattern ...
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Analyzing Accelerometer and Gyroscope Data on a Drone

I am using an MPU-6050 Gyroscope and Accelerometer for a drone flight controller. I have been able to get raw data from the sensor, and account for bias and use the included Digital Low pass Filter at ...
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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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Gil-Pelaez condition

I'm struggling to understand some passages of proof of Gil-Pelaez condition here described (pag. 1,2), i.e. ** $P_j=\mathbb{Q}(S_T>K)=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{-iuK}...
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Why should we demean data?

In my case, I'm performing a vibration analysis using FFT and all that, and in this algorithm that I've been using, the author demeans the amplitude data before applying the FFT to it. Is it a thing ...
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66 views

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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Comparison of two groups of time-series

I have time series data collected from two sensors. Suppose that they are microphone recordings from a man and a woman. Each signal, e.g. the one below, is 5000 samples. They don't share some ...
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spatial correlation function, power spectral density function and homogeneity condition

Suppose that the spatial correlation function $R_{VV}(\mathbf{x})$ of a zero-mean random field $V(\mathbf{x})$ ($\mathbf{x} \in \mathbb{R}^3$) and its power spectral density function $\widehat{R_{VV}}(...
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146 views

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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186 views

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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199 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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279 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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330 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 (...
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Forecasting with ARIMA ( Training and Test Data split)

I have an hourly time series of the average parking occupancy with data available from September 2017 up until June 2018. I would like to use the ARIMA model with external regressors to produce a ...
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Fourier transform of a Gaussian process

I would like to discuss and ask a question regarding the Fourier transform of a Gaussian process, if it makes sense. For that purpose, let me describe the following situation. Let $z(s)$ be a ...
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265 views

Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
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104 views

Fourier Transform based imputation

Fourier Transform based imputation Can any body please assist me in understanding the Fourier Transform based imputation algorithm shown in Figure. I am struggling to understand ts and te. Link of ...
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Linear regression on circulant matrices

I am reading the paper on High-Speed Tracking with Kernelized Correlation Filters and I am a bit stuck on the equivalence of Ridge regression in the frequency domain. Minimizing the typical equation ...
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Time series/ ARMA Simulation

Given: I have a question, given a continuous real spectral density f(w), -infinity my idea: I would folding, discretize and truncate the spectral density to get a real (one-sided) discrete spectral ...
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308 views

Calculating right values of Periodogram using Fourier Analysis

In the book, Economic Cycles: There Law and Cause By Henry Ludwell Moore, he plots Periodogram of rainfall of Ohio valley. He uses 72 years data (1839-1910) and tries to find the most dominant cycle ...
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662 views

tbats() model not capturing seasonality (weekly data)

I have below 4 years of weekly data which has complex seasonality of varying seasonal length. I have assigned the first 160 data points as training and the rest as test with frequency=52. It seems ...
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580 views

Proof of Herglotz Theorem in Time Series Analysis

I am studying time series, following Time Series Theory and Methods by Brockwell and Davis. I am reading the proof of the Herglotz Theorem, I have the following questions. First, the proof given in ...
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Kernel density estimation with FFT for a univariate non-parametric regression

The non-parametric regression model to be estimated looks like the following x_t = b(x_t-1) + epsilon_t Forfinding the optimal bandwith h in the kernel ...
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108 views

Spectral density and Riemann Stieltjes Integral

I am confused with a part about spectral densities. I found it in Time Series Theory and Methods by Brockwell and Davis. I don´t understand how is applied the Riemann Stieltjes Integral in this ...
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Is this a well-studied problem? Problem: Optimally unlagging multiple time-series

Is the problem of optimally lagging/unlagging multiple time-series with integer lags to maximize a sum of pairs of cross correlations or coherence an already well-studied problem? If so, references? ...
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225 views

Power spectral density of the output of a linear time invariant system with a weakly-stationary process

Please note that I have no background whatsoever in Fourier analysis and very little in time series. This is exercise 2.10 in Theodoridis' Machine Learning: Show that the autocorrelation of the ...