Questions tagged [functional-data-analysis]
Functional Data Analysis defines a framework where the fundamental units of analysis are functions.
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Is there a statistical test to compare two heatmaps/response surfaces or two curves?
I am conducting a series of experiments in which I analyse the effect of the order of treatment of bacterial cells with different concentrations of two compounds (Drug 1 and Drug 2). The values in the ...
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Dimension reduction for multivariate functional target that varies in length
I have target features consisting of about 70 different discrete time-series and I want to use this data to learn a model. Now the data is of course very high-dimensional, as I can also use a very ...
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Why does FPCA not use scaling as PCA?
Functional principal component analysis (FPCA), according to the original paper, does not use scaling before FPCA, as in PCA. Instead, it uses a covariance matrix to compute the eigen-components.
I ...
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Is the FPCA just a PCA on coefficients?
Is functional principal-components analysis (FPCA) on basis representations just a PCA on the component coefficients? The following seems to indicate that yes, it is:
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Error function of Functional Principal Component Regression is itself a curve. Why?
I have data indexed from 0 to 54. Using 6 principal components that explains 99% variation of data I was not expecting the error function to exhibit such a clear shape? By error function from ...
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Functional Data Analysis - minimum points per curve?
I am embarking on a project for which I'd like to use functional Data Analysis (FDA). I have several thousand discrete curves objects on which I'd like to fit continuous time curves. These discrete ...
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How can I use functional time series or functional principal component regression for forecasting this type of data?
Lets say I have this type of data. For one of the series, I observed data until day 300. Can I use functional principal component regression to forecast next 50 observations? What is the best way to ...
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Convergence of variational posterior
Let $q_\lambda(z)\in\mathcal Q$ be the variational posterior approximation of $p(z|y)$.
Note that the optimal $\lambda^*$ is approximated by the following recursive sequence
$$
\lambda^{(k+1)}=\lambda^...
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Simulating iid mean zero Matérn processes
I'm currently learning FDA (functional data analysis), following "Introduction to Functional Data Analysis" by Kokoszka & Reimherr. Here it is exercice 1.5:
1.5 The Matérn covariance ...
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Covariance of functional data object
This is a follow-up question on my earlier question Functional Regression in R: Functional Response Regressed on Scalar Covariate.
I have a functional response $Y(t)$ (i.e., a stochastic process) ...
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Empirical basis functions
Preliminary
Consider $n$ individuals each with observed data $ Z_i, i = 1, \ldots, n$. For each individual $i$, the longitudinal predictor $Z_i = \{Z_i(t_{i1}), \ldots, Z_i(t_{i,R_i})\}$ is measured ...
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Functional Principal Component Analysis - Explaining Functional Principal Component Scores
I was wondering if someone can help with explaining Functional Principal Component Scores?
I am working with a dataset which reflects participants in a weight loss management trial (longitudinal data)....
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Functional Regression in R: Functional Response Regressed on Scalar Covariate
I have a functional response $Y(t)$ (i.e., a stochastic process) which I regress on a set of scalar explanatory variables $X_1$ ($=1$, i.e., $X_1$ is a constant term), $X_2$, and $X_3$. The equation I ...
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Functional Data Analysis with Missing Data
I am trying to analyze a dataset containing several observations for n = 27. Unfortunately, not every participant answered the question every time, i.e., when plotting the graph it has holes in it. I ...
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Generate survival data with functional data using R
I am reading this article here and trying to regenerate their simulation study. Here is this scenario here, among others, but if I can figure out one, the rest follow. That is,
Simulation set-up
we ...
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Simulating models for longitudinal and survival data
This simulation study is taken from this [article] (https://pubmed.ncbi.nlm.nih.gov/35574725/). I am trying to generate this simulation
Theoretical Set up
Define the set of true basis functions,
\...
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What are the advantages of using Functional Data Analysis (FDA) over traditional Time Series or Stochastic Process approaches?
In the case where the curves (functions) are defined on a time interval.
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Does anyone know how to model selection for function on function linear model to find the best subset of functional covariate in R [closed]
Does anyone know how to model selection for function on function linear model to find the best subset of functional covariate in R
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Influence function of conditional quantile
I'm trying to derive the influence function of the estimand $\Psi$
$$\Psi(P) = P(Y > y | X = x)$$
Following tutorials for deriving the influence function of the average treatment effect here. Has ...
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Functional analysis in R with fda
I'm working with time series data for drug response. And, I wanted to as is there are some alternative ways to analyse it since the FDA package in R is not working in my case.
The type of my data is ...
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truncated Functional data analysis + identify subgroups?
I have intensive longitudinal data about heartbeat rates, movement activity measured by actiwatch, respiration rates etc... collected from about 100 patients at a terminal care.
For each patient, data ...
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Is there a substitute for an "average" which best inherently represents data, rather than reducing it to one or another aspect?
I was thinking about the arithmetic average as a type of metric or a representation of data. Maybe there is a mathematical argument for the non-arbitrariness of that function. I think something like ...
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Curve quantification
I have some longitudinal measurement data of 15,000. I smoothed that data with B-spline smoothing and got the following curve.
I then want to quantify this curve and extract features for clustering ...
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How to combine dissimilar curves into a single "average"?
With relevance to "functional data analysis", I am looking for some approach which can give an intermediate function among these five different function plots, where this "average ...
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Statistical method for finding homogeneous groups of curves
I need to divide a set of 100 or more response curves into groups.
These curves are formed by backscattering intensity along a range of frequencies. Basically, each curve represents the intensity in ...
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Forecasting using functional data analysis with parametric models for the functions
I have collected data at regularly spaced time intervals, and at each time, the observed data is commonly fit by a simple parametric functional form involving several parameters.
A naive approach to ...
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Reproducing kernel hilbert space norm as smoothness functional
Let $K:X \times X \rightarrow \mathbb{R}$ be a Mercer kernel with an associated RKHS $H$ then the norm $|f|_H^2$ can be used as a way to ensure that $f$ is smooth in $H$. If i understand correctly, ...
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Estimating the standard deviation
I have data of a function $f(t)$, where I always have just a single data point per $t$. It is known that the data is affected by noise, where the noise itself is also a function of $t$. Now I want to ...
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Efficient storage of functional data
I have access to a sample (size $N$) of functional data. Each observation corresponds to $C$ functions. Each function $f_{n,c}$ is represented by $T_n$ points for $1\geq n \geq N, 1\geq c \geq C$. All ...
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Methods for modelling distributions?
As predictor X I have particle size distributions
and I would like to run a model y ~ X.
I.e. each trial has a response ...
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Fitting an functional autoregressive model in R with mgcv
I'm working with the 'mgcv' package in R and I just want to know if I am going about the right way of fitting a functional autoregressive (FAR) model. The FAR model is represented as
$$Y=f_1(x.1)x.1+...
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Why is Functional Data Analysis (FDA) not as popular?
I am interested in FDA (data perceived as functions), as someone from a pure mathematics background and I think it can help provide solutions to some major challenges in data analysis (also data ...
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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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Two sample hypothesis tests for functional (curve) data
Suppose I have two samples of functional data or curves $x_i(t), i \in \{1,\dots,n_1\}$ and $y_i(t)\in\{1,\dots,n_2\}$ for $t\in\{1,\dots,T\}$. What tests are suitable for testing the null hypothesis ...
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Principles in experiment design for functional data analysis
The goal or objective of "standard"$^†$ experiment design is to design a data collection scheme (for a finite budget) such that linear models$^†$ can be applied and inference results can be ...
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Approximating RKHS norm with samples
I have a function $f'$, sampled from a Gaussian process prior with a known kernel $f' \sim N(0,K)$. Is it possible to find/approximate a solution for $min_w||f_w-f'||^2_H$ if I can calculate both $...
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Covariance kernel: understanding the term within groups and between groups terms of a two-way functional ANOVA
I was studying the paper "Multilevel Functional Principal Component Analysis" by Chong-Zhi Di et. al. To find the functional PCA for functional data with group factor, it is suggested to use ...
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Is panel data the same as functional data?
Panel data is defined in wikipedia as:
In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of ...
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What kind of machine learning models fit data of this kind?
I have been working with a manufacturing process. It would be very efficient to build a machine learning model for the kind of data that I have. So, my dataset has typically three inputs.
VAl_1
VAl_2
...
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Functional Autoregressive Process - Predictive Factors
Let $X_1,...,X_N$ be random functions on an Hilbert space $\mathbb{H}$, following a Functional Autoregressive process of order 1:
$$ X_{t+1} = \Psi X_t + \varepsilon_t $$
Where $\Psi$ is a linear ...
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What are the simplest examples of nonlinear statistical functionals?
I am reading Wasserman's book "All of Statistics" in which he defines a statistical functional as any function $T(F)$ of the cumulative distribution function $F(x)$ that outputs a real ...
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Functional time series forecasting
I am working in hourly multivariate time series forecasting, so due to big data, I made the forecasting through functional time series. The model that I am using is the functional autoregressive model ...
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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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Square root transformation of Poisson process. How $\small Var[\sqrt{P(\lambda)}] \approx \frac{1}{4}$
I am working on Kaggle Neural data challenge. I am trying to understand the transformation applied on the neural spiking data. A number of spikes given a stimulus are Poisson distributed as
$$Y_i \...
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Roughness penalty matrix for the Fourier basis functions
I am trying to understand how the roughness penalty matrix for the Fourier basis functions is calculated in R using the fda ...
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Prove that $K(x_{1},x_{2})=f(x_{1})K_{1}(x_{1},x_{2})f(x_{2})$ is symmetric positive definite if $K_{1}(x_{1},x_{2})$ is symmetric positive definite
I have been trying to prove this 5th proposition on the 25th slide
We can prove that $ K(x_{1},x_{2}) $ will be a valid kernel as:
\begin{align}
K(x_{1},x_{2}) = & \; f(x_{1})K_{1}(x_{1},x_{2})f(...
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Generalized additive models vs. functional data models
I am not sure I understand the difference between functional data analysis (FDA) and GAM.
Or in short, I was reading about GAM and I found the following model which seems like an FDA model (there is a ...
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Coefficients in Functional Regression
I would like to estimate the effect of DoE factors in my response which is a function.
For this reason, I tried to apply the Functional Data Analysis method.
As example,
I have a 12 treatment DoE,
<...
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Predict parameters of another model
I would like to make a model to predict the parameters of another model.
As example,
By using design of experiment I made some trials and I measured the temperature (response).
e.g.
...
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How to Generalize functional Autoregressive model
I am working with the multivariate hourly electricity price data, data consist of six years hourly electricity data in which the first five years are use as testing set and the last year as validation ...