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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Inverse Fourier transforms [migrated]
Find the inverse Fourier transform of:
$F_1 = -i\omega e^{-|2\omega|}$
I understand using the formula book to find that the IFT of the $ e^{-|2\omega|}$ is equal to $\frac{4}{\pi} \frac{1}{(4+t^2)}$ ...
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Can you use fourier analysis to identify recurring transactions?
My banking app provides some predictive analysis on what upcoming bills I have. This got me thinking on how the bank does this analysis. Is it possible to use fourier analysis with my transaction ...
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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
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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 ...
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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.
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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^...
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FFT: peak spectral power vs peak frequency
I am having my head confused about this paper I am reading, where they used FFT to create frequency domain features.
Specifically, they calculated ...
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how to calculate the Coefficient coherence and imaginary part of a coherence in R?
I performed a coherency analysis in R, using the wavelet comp library: analyze.coherency, but I need to know the Coherence Coefficient and the imaginary part of the coherence. Could someone help me ...
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Discovering alternative waveforms for Fourier transform (alternatives to sinusoidal waves)
I'm working on a classification algorithm that utilises a discrete Fourier transform on sensor data, as a means to detect the presence of an oscillating signal.
My issue is that the signal-generating ...
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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 ...
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Periodogram versus Fourier transform
The periodogram can be used to find signal frequencies by taking the Fourier transform of the signal's autocorrelation.
Why is the autocorrelation needed or desired?
Why don't we simply take the ...
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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 ...
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fourier signal in regression, statistical test
Based on my assumption, I want to see the effect of some variables on dependent variables through linear regression.
But I couldn't find some variables for my assumption, so I used Fourier transform(...
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interpreting correlation between a target varialbe and Fouriers coefficients
my dataset is composed by a set of time series, and a target variable.
So to analize the correlations between the time series and the target variable, i am extracting features from the time series, ...
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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.
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Bivariate normal CFD approximation using characteristic function
The normal distribution CFD can be approximated using
$$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$
where the characteristic ...
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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 ...
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Feature engineering in frequency-domain time series
I am solving a task of a real-time binary signal (electric current) classification. This is a project that I am continuing so I have been supplied with a feature extraction part already done. The ...
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Interpreting results from preforming Power spectral analysis on Hourly Temperature data
I'm analyzing hourly temperature data for a school project.
Analysis data set 1 | Hourly data from 01/01/2020-12/31/2020. So I have n=24*365 = 8760 data points.
I did the analysis using R.
Let x be ...
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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-
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Unknown factor when using the fftpack from scipy
The following code example from this site illustrates the use of the fftpack from scipy. I am aware that the Fourier transformation in the real-valued case is symmetrical and therefore only the values ...
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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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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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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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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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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 [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 ...
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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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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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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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