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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Why is Prophet faster than dynamic harmonic regression?
Section 12.2 of Forecasting: Principles and Practice (3rd edition) discusses Facebook's Prophet model. The authors write:
Prophet has the advantage of being much faster to estimate than the [dynamic ...
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Can we apply Fourier transform on non stationary data?
Hi, I'm trying to predict US inflation rate. The unit is in percentage change from a year ago. Would it be possible to use Fourier transform on the independent data to create a new feature, knowing ...
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Characteristic function of transformed random variable
Consider a random variable $X$ and a function $g(\cdot)$. Let $Y:=g(X)$, and let $\phi_X(\cdot), \phi_Y(\cdot)$ be the characteristic function (cf) of $X,Y$, respectively. Suppose that $\phi_X$ is non-...
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Curve-fitting audio decay harmonics
I'm examining the frequencies produced by pressing middle C on a piano.
I've used Fourier analysis to extract the amplitude of the middle C frequency per window of time,
as well as the integer ...
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Is it necessary to transform the tide variable using a truncated Fourier series for GLM (Gamma), and how should I interpret interactions? [closed]
I’m using glm to see whether there is an association between zooplankton biomass (response) with two variables: 1) hours from high tide (high tide is zero, hours before are negative at one hourly ...
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How do B-splines differ from Fourier transforms?
I read that B-splines can be used as activation functions in KAN neural networks, whereas Fourier transforms are not widely used. Can someone please explain the difference between the two in a simple ...
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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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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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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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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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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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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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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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 ...
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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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