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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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Help with PSD graph interpretation and Time Series Analysis

I am pretty new to time series analysis and I am currently studying it. My latest work requires me to work with a time series that was generate by VASP AIMD. I tried power spectrum density analysis ...
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Product of RVs of Which Distribution Approximates Normal Well?

Suppose that I have $N$ i.i.d. random variables with a distribution $Q$, which has mean around 1. That is $R_1, R_2,\ldots,R_N \sim Q$. I would like $\prod_{i=1}^N R_i$ to approximate a normal random ...
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Fourier transform in information transfer in biological neural network

Principles of Neural Design by Peter Sterling and Simon Laughlin describes a usage of information theory in calculating the rate of information transfer in the brain. ...when successive signal states ...
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What is the spectral density matrix, in the context of vector stochastic processes

I cannot seem to find any good/easy to read resources on the spectral theory, and in particular for multivariate stochastic processes. I want to know: Any resources explaining spectral theory ...
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Forecasting Square Waves

I am involved in a social experiment with other college students. The experiment involves simulating current price of the market at which a given asset can be bought or sold for immediate delivery. We,...
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Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis

Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle ...
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Approximation for a correlation matrix

I have a cross-correlation matrix of some parameter for each time period. E.g. expected economy growth for each months in the future, i.e. growth for Apr 2014, May 2014, ...., Dec 2018, and ...
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The sampling frequency or sampling period for financial time series in doing DFT

The discrete Fourier transform (DFT) is widely utilized in computer engineering, and its formula is as follows: $$X(k)=\sum_{n=0}^{N-1}x[n]e^{-i\frac{2\pi}{N}nk},$$ where the result $X(k)$ refers to ...
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How do I approximate a timeseries with a sum of sin and cos functions?

I have a time series and I am trying to approximate it using an equation of the type: y = A + B*sin(2*pi*x/n) + C*cos(2*pi*x/m) + D*sin(2*pi*x/q) + ... The ...
point618's user avatar
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Fourier coeficients as feature to classification model

I have to solve a timeseries multiclass classification problem. Some of these classes (Class 0, Class 1 and Class 4) have timeseries that are really similar to each other, as follows: ...
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How to test if a particular frequency is significant or not in a fourier frequency spectrum

I have got a timeseries data with which I plotted a frequency spectrum and I could eyeball all the significant peaks (significant frequencies) in the plot. But I want to do this for a spatial data (...
arya lakshmi's user avatar
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How to dectect sudden change in signal frequency? (analysis of EEG signals)

I am trying to analyze data from EEG electrodes to understand how brain activity changes in different coginitive states (for simplicity, assume that there are only 2 states: baseline (A) and chanelled ...
Brzoskwinia's user avatar
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170 views

Interpretation of fast fourier transform in a simple time series

I am struggling with understanding the result of FFT in R. I have 729 observations, two years data. In the data I simulated there were more visitors every sunday. I divided 729/7 which is ...
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Are moduli of components of Fourier transformed Gaussian random vector still independent?

Suppose $X=[X_1,\ldots, X_n]^T$ is a random (column) vector such that: $X_i \stackrel{i.i.d}{\sim} \mathcal{N}(0,\sigma^2), \ 1 \le i \le n$ $\mathcal{F} \in \mathbb{C}^{n \times n}$ is the discrete ...
Adam O. G.'s user avatar
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Rescaling matrix W in Random Fourier Features

I came across this beautiful idea of Random Fourier Features by Rahimi and Recht while working on optimising my GP model using Predictive Entropy Search. I understand the overall idea of approximating ...
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How should I interpret parameters of the SARIMA model in time series analysis?

I am a bit confused as to why the SARIMA model requires four parameters beyond the ARIMA model just to remove the seasonal component from a time series. Obviously $m$ is required to specify the ...
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Calculating convolution in R [closed]

I am struggling to get the correct answer for the simple calculation of convolution in R. The convolution of $f(t) = e^{-t}$ and $g(t) = \sin(t)$ is: $$ (f * g)(t) = 1/2 \left( e^{-t} + \sin(t) - \cos(...
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How does a fast Fourier transform yield the probabilities for a multinomial random variable?

Let $(A,B,C)$ follow a multinomial distribution: $$(A,B,C) \sim \text{MultiNom}\left(n=100,p_1=p_2=p_3=\frac13\right).$$ Define $X$, a discrete random variable as $$X = f(A,C) = 2A + 3B + 4C = 2A + 3(...
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Energy of a signal contained within a given band

Suppose I wish to compute the energy contained in a given frequency range using FFT. So, first, I compute FFT and take the squares of the absolute values of $x_k$. Can I then use discrete integration ...
Dmitry's user avatar
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FFT: zero-padding and mean subtraction

When computing FFT of a finite-time discrete sample with a non-zero mean, I get two different results depending on the order of operations: whether I zero-pad and then subtract the mean or subtract ...
Dmitry's user avatar
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2 votes
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Why can weekends cause harmony?

The following plot shows power spectra (periodograms) of a sample from $X_t \sim \operatorname{Poisson}(1)$ along with that same sample where: Weekends were set to zero Sundays were set to zero ...
Galen's user avatar
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Non-stationary Random Fourier Features

Random Fourier Features (RFFs) were introduced by A. Rahimi and B. Recht in their 2007 publication Random Features for Large-Scale Kernel Machines. RFFs are based on Bochner's theorem, which applies ...
LoveRKHS's user avatar
12 votes
3 answers
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Is it possible to have seasonality at 24, 12, 8 periods in hourly based wind power data?

I have taken hourly-based wind power data, by taking its periodogram after making it stationary in R. It gives me four seasonal patterns at periods of 24, 12, 08, and 06, as shown in the figure below. ...
Mastoi's user avatar
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CLT for sums of Fourier transform of white noises r.v

Define $I_n(\lambda_j) = \frac{1}{2\pi n} |\sum_{t = 1}^n Z_t e^{it\lambda_j}|^2 = \frac{1}{2 \pi} \sum_{h = - \infty}^{\infty} \hat{\gamma}_n(h) e^{ih\lambda_j}$ where $Z_t$ is a $WN \sim (0, \sigma^...
Eryna's user avatar
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4 votes
1 answer
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Can regression forecasts of univariate time series be independent (of one another)

Suppose I have short-term forecasts from two univarite regression models of the same time series. I am choosing the models to be as different as possible in structure and assumptions. For instance, ...
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Stan: modulated distribution

I am modeling a process where events happen every X days with X following a gamma distribution. I already have a model for that ...
salva's user avatar
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Can you decompose a wave approximately?

I have data which looks like composition of sine waves. I need to decompose it to fewest possible sine waves that would give me tolerable error. The picture is of a half-period. Each half-period ...
Boppity Bop's user avatar
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How to calculate the transalation and/or rotation of two images using fourier transaform?

I need find the translation and/or rotation of an image and himself translated and/or rotated (x0, y0) px and/or J degrees. Given the two images I need to find N.
Sebastian Jose's user avatar
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Distribution of a constrained Gaussian distribution in frequency domain

We know that a Gaussian Distribution is not limited and it spans from $-\infty$ to $+\infty$ . However, practically if we sample the Gaussian with a finite sampling frequency, the maximum frequency is ...
CfourPiO's user avatar
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1 answer
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How does the plot of the output of librosa.stft work?

So, I have been analyzing audio data. I used the librosa package and used its .stft() function to calculate the Fourier transform of my audio data. I did the following- ...
dhananjaya's user avatar
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1 answer
433 views

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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A question about linear inference in random Fourier feature kernels [duplicate]

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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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 ...
Jakob J's user avatar
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2 votes
1 answer
203 views

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 ...
Gonzalo Polo's user avatar
1 vote
2 answers
39 views

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 ...
Mohamed Gamal Hamed's user avatar
1 vote
0 answers
76 views

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 ...
Fuhan YANG's user avatar
2 votes
1 answer
755 views

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 ...
invictus's user avatar
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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 ...
user2529589's user avatar
1 vote
1 answer
1k views

Forecasting time series with multiple seasonaliy by using auto_arima(SARIMAX) and Fourier terms [closed]

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 ...
Downforu's user avatar
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0 votes
1 answer
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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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2 votes
1 answer
367 views

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 ...
Tom's user avatar
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2 votes
1 answer
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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 ...
Dexter Jeffery's user avatar
1 vote
0 answers
337 views

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 ...
Joff's user avatar
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1 vote
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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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1 vote
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31 views

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: ...
Miguel Cárcamo's user avatar
1 vote
0 answers
209 views

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 ...
MolineraNegra's user avatar
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35 views

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 ...
TommyS's user avatar
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4 votes
2 answers
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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, ...
Hari's user avatar
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1 vote
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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 "...
Thomas's user avatar
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0 votes
1 answer
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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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