Questions tagged [functional-data-analysis]

Functional Data Analysis defines a framework where the fundamental units of analysis are functions.

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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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Handling densely observed functional data with R package fda in reasonable time

I am trying to use the fda package to analyze functional data that is densely observed; for example, for one function I have ~25,000 samples of that function. (To be more precise, I observe $f(t_i), 1 ...
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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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Applications of Functional Data Analysis (FDA) [duplicate]

Are there any problems that are particularly well suited for models from the field of Functional Data Analysis (FDA)? I was reading the Wikipedia page for Functional Data Analysis (https://en....
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How to estimate scores in functional PCA in R

Given a dataset divided into train and test sets, I run the functional pca algorithm implemented in the fda package on the training dataset. How can I obtain then ...
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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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Robust methods to choose the location of the B-spline knots

I would like to know if there are some simple methods to choose the location of knots for a B-spline basis function. Either an optimal knots selection scheme or just some generally good strategies ...
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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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Updating in functional outlier detection using a sampling approach and functional depth

I came across a question while trying to find outliers in a functional data set of 350000 observations and was hoping you could help me. The method I'm using to detect outliers stems from the ...
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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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Retransform Functional data curve into discrete data points

I am working in functional time series forecasting using multivariate time series data. Due, to the large dimensionality of the data I transform the multivariate time data into a functional form for ...
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Intergrating product of multivariate normal and univariate normal to find marginal density [duplicate]

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $U\sim MN(0,I_p)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and ...
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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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7 votes
1 answer
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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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3 votes
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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 ...
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Asymptotic properties of functional models

When working in Functional Data Analysis, a classical "preprocessing" step is to represent the "observations" using a B-spline expansion: $$ X_i(t) \approx \sum_{j=1}^J \lambda_{ij}...
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functional principle components is retransform to original data

I am working with multivariate time series hourly data of five years, I use the first four years as testing set and last one year as validation set. My objective is to obtain one year hourly forecast, ...
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2 answers
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How to express descriptive statistics as statistical functionals - and why?

I'm reading through class notes explaining statistical functionals and came across the following expressions with little explanation how they were derived: Mean $=T(F)=\int xdF(x)$ Variance $=T(F)=\...
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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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Comparing curve fits

I have imaging data where I imaged different instances of the (somewhat) same phenomenon (2 different experimental conditions). I have already settled on the equations I use for curve fitting and ...
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Functional Linear Regression without Orthonormal Basis for Prediction

What do you do in functional linear regression when for whatever reason you don't want to use an orthonormal basis expansion? In functional linear regression for scalar on function regression, one may ...
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Random Fourier Features vs Eigenfunctions for Gaussian Process Kernel Approximations?

Say we define kernels in Gaussian processes. There are two approaches to approximating them: random fourier features and eigenfunctions of the kernel. What are the tradeoffs to using each? If we ...
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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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Optimize a Function with Respect to a Set

Suppose I have some set of points $X = \{x_1,...,x_N\}, x_i \in \mathbb{R}^d$ and a measure $\mu$ (i.e. a probability distribution) defined with respect to those points. Specifically, if $A \subset \...
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Control of type I error rate in functional ANOVA F-test

I am following how to compare curves via functional data analysis, specifically functional ANOVA. This video makes sense in describing it as being a lot like vanilla ANOVA...up to a point. When we do ...
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Kernel on two functions?

I am curious whether there are any literatures considering kernel functions whose inputs are two functions. For example I would like consider two 1-Liptchiz mappings $\pi, \sigma:\mathbb{R}^M\...
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Meaning of a notation regarding mean square derivative

I'm reading a paper (On Differentiable Functionals, Van der Vaart, 1991, Annals of Statistics), and I've got a question regarding a notation in the following part: My Question: Does $dP^{1/2}$ mean $...
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Reference request: Modulus of continuity of the map from parameter space to the space of probability measures

I am thinking of the following condition in my research: Let $\Theta \subset \mathbb R$ be the parameter space and $\mathbb P_\theta$ be the probability measure on sample space $\mathbb X$ ...
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How to prove properties about mixed (discrete and continuous inputs) function spaces?

I am using Gaussian Processes for data with mixed inputs (discrete and continuous input variables) i.e. $𝑥=[𝑥_𝑐,𝑥_𝑑]$ where $𝑥_𝑐 \in \Re$ and $x_d \in \mathbb{Z}$. There is a lot of work on ...
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How should I understand smoothing in functional data analysis from a modelling perspective (specifically for temperature data)?

The specifics of this question are that I am looking at daily maximum and minimum temperature from the GHCND data set obtained from NOAA's API and I view the temperatures observed at days in a year as ...
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Inference on the minimizing/maximizing value of $x$ in a regression

I have 30 pairs of observations $(x_i, y_i)$ and hypothesize that in the observed range of $x$ a non-monotone relationship $y_i = f(x_i) + \epsilon$ may hold. I am willing to assume that $\text{Var}(\...
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Right way to bin the data-Fitting Voigt profiles to spectroscopy data

I have some measurements of the rate of a physical process versus energy. For each energy I have a number of counts and a measurement time associated to it. However, the step (in energy) at which the ...
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1 vote
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How to handle a feature set with unordered members?

I ask a group of people for the age of their 10 best friends, and will try to predict some output variable on basis of that. I don't ask them to rank these friends in any way, thus each person I ask ...
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What is functional data? [closed]

Let's say I run a separate regression on different cross-sections at each point in time t. I would then have a time series for each regression coefficient over different periods in time. Would this ...
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6 votes
2 answers
452 views

Forcing smoothness of regression coefficients

I'm building regression models on spectral datasets: the predictors are the intensites of signal at the different frequencies. In this case the intensities at close frequency values are highly ...
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conversion of multivariate time series into functional data analysis

I am trying to generalize the functional auto-regressive model of order one to some order $p$. For this, I've calculated the functional principle components and choose a particular $pc$s which explain ...
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