Questions tagged [functional-data-analysis]
Functional Data Analysis defines a framework where the fundamental units of analysis are functions.
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Functional principal component analysis (FPCA): what is it all about?
Functional principal component analysis (FPCA) is something I have stumbled upon and never got to understand. What is it all about?
See "A survey of functional principal component
analysis" by Shang, ...
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3
answers
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From a statistical perspective: Fourier transform vs regression with Fourier basis
I am trying to understand whether discrete Fourier transform gives the same representation of a curve as a regression using Fourier basis. For example,
...
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2
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Is there an unbiased estimator of the Hellinger distance between two distributions?
In a setting where one observes $X_1,\ldots,X_n$ distributed from a distribution with density $f$, I wonder if there is an unbiased estimator (based on the $X_i$'s) of the Hellinger distance to ...
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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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1
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When/where to use functional data analysis?
I am very new to functional data analysis (FDA). I am reading:
Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data
Analysis, 2nd ed., Springer, New York.
However, I am still not ...
13
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2
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How to simulate functional data?
I'm trying to test various functional data analysis approaches.
Ideally, i'd like to test the panel of approaches i have on simulated functional data. I've tried to generate simulated FD using an ...
12
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1
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Dynamic Time Warping and normalization
I'm using Dynamic Time Warping to match a "query" and a "template" curve and having reasonable success thus far, but I have some basic questions:
I'm assessing a "match" by assessing whether the DTW ...
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What is the difference between functional data analysis and high dimensional data analysis
There are a lot of references in the statistic literature to "functional data" (i.e. data that are curves), and in parallel, to "high dimensional data" (i.e. when data are high dimensional vectors). ...
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Predicting response from new curves using fda package in R
Basically all I want to do is predict a scalar response using some curves.
I've got as far as doing a regression (using fRegress from the fda package) but have no idea how to apply the results to a ...
10
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3
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Forecasting of density function
I am doing some research about forecasting time series of probability density functions. We are aiming to forecast a PDF given historically observed (usually, estimated) PDF. The forecasting method we ...
9
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3
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How to map a trajectory to a vector?
I have a series of data points in this form (timestamp, lat, long) for a set of users. Each user has a trajectory when he travels from point A to point B. There might be any number of points from A to ...
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Illustration of functional gradient descent
I'm trying to understand boosting as gradient descent (GD) in functional space. I've followed the argument in the classic paper on the subject, but would characterize my understanding as tenuous at ...
7
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2
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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 ...
7
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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 ...
7
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3
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Learning from unordered tuples?
EDIT: I reduced my problem to a more specific question:
https://math.stackexchange.com/questions/26573/
But I am still interested in other ideas.
Let's say our data is generated by
$$Y_i = f(X_i) + \...
6
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2
answers
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How to express descriptive statistics as statistical functionals - and why?
I'm reading through class notes explaining statistical functionals and came across the following expressions with little explanation how they were derived:
Mean $=T(F)=\int xdF(x)$
Variance $=T(F)=\...
6
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1
answer
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Modelling longitudinal data
We have longitudinal data on children(n<20) in which we measure different quantities A,B,C,D (like distance walked, time spent in school etc.). These are all continuous variables. We measure these ...
6
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3
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Principal Component analysis (vector space or inner product space?)
(WARNING: This question might seem dumb)
I see that the optimization problem in PCA involves the notion of inner product. For example, to solve for the loadings in second principal component, the ...
6
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1
answer
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Estimating speed from position updates with uncertain time intervals
I have 2 alternative methods to solve a problem, and I was just wondering what people who know the math better than I think, and if there is a better method to use for this type of problem.
The ...
6
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0
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Asymptotic properties of functional models
When working in Functional Data Analysis, a classical "preprocessing" step is to represent the "observations" using a B-spline expansion:
$$
X_i(t) \approx \sum_{j=1}^J \lambda_{ij}...
5
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1
answer
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Functional Data Analysis and Splines Regression?
I'm new to Functional Data Analysis so my question will'be very simple for an expert in this topic. Operatively, when I fit a model like this, in R :
...
5
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1
answer
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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 ...
5
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1
answer
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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 ...
4
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2
answers
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Different number of time-points in functional data analysis
I have a data set that I wish to analyze using functional data analysis methods. Data consists of repeated measures of some characteristic on a number of inviduals. I have the time of the measurements ...
4
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1
answer
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Multinomial Count Models
Is it possible to model a dependent variable which is both multinomial and count? If so, how would one do so with a tool such as R?
For example, suppose that my dependent variable looks like this:
...
3
votes
1
answer
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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 ...
3
votes
1
answer
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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 ...
3
votes
1
answer
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How to calculate RKHS norm of a function under given kernel transformation
This was a question asked before in mathoverflow but not yet got answered.
I have the same problem when reading Srinivas et al (2010) [appendix B]'s paper.
Here are my problems:
Definitions: ...
3
votes
1
answer
279
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Test for difference between time-series measurements
I have an experiment where the response to a treatment is measured over time. I now want to test for if the response profile differ between the treated group and the control group. There are ...
3
votes
2
answers
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Alternate terms, or definition for functional logistic regression
I have recently come upon a paper discussing "functional logistic regression."
I could not find literature related to functional logistic regression. Is there a different name for this kind of ...
3
votes
1
answer
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Variability of a curve with 4 parameters
Say I have 10,000 data in 2-D and I want to fit a curve to them. There are many functional forms this curve could take -- polynomial, B-spline, trigonometric, and so on. I've decided that I only want ...
3
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1
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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 ...
3
votes
1
answer
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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 ...
3
votes
1
answer
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How to consider different samples in functional data clustering?
In the engineering context several data sources like different kinds of measurement signals (for example distances, angles and efficiencies) are very common. If it would be possible to observe these ...
3
votes
1
answer
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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 ...
3
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0
answers
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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 $...
3
votes
1
answer
617
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Interpretation of functional regression models for scalar response
I have an application scenario in which I want to determine a single outcome from the course of a series of measurements.
I decided to give functional regression a try, so I read and ran the example ...
3
votes
0
answers
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Which method for anomaly detection on curves of sensor data
I have data of measurements of a sensor. The sensor measures a numeric value at 20 fixed positions every few seconds. You can think of a camera traversing over a plate and measuring the amount of ...
3
votes
0
answers
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mgcv GAM - is it possible to include the 'simple' product of two 1-d smooths?
I am trying to test out various mgcv::gam models in a scalar-on-function regression analysis.
The following is the 'hierarchy' of models I would like to test:
$$
...
3
votes
0
answers
318
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What is the basic difference between Sieves, Series and Splines estimators?
As far as I know, Sieve Estimators consists in a broader class of estimators for a function g(x) lying in a space of functions G. The estimation basically consists in choosing the function that best ...
3
votes
0
answers
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Predicting curve registration parameters in functional data analysis with noisy data
I have some data I'm studying where functional data analysis seems
like a promising approach. But having never tackled FDA
before, I'm having trouble wrapping my head around it.
For background, I ...
2
votes
1
answer
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Square root transformation of Poisson process. How $\small Var[\sqrt{P(\lambda)}] \approx \frac{1}{4}$
I am working on Kaggle Neural data challenge. I am trying to understand the transformation applied on the neural spiking data. A number of spikes given a stimulus are Poisson distributed as
$$Y_i \...
2
votes
3
answers
1k
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Positive smoothing with the fda-package (Functional data analysis)
In the book Functional Data Analysis with R (Ramsay&Silverman) there is described the possibility to do the "positive smoothing" if it’s needed instead of the "normal smoothing"...
2
votes
1
answer
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Functional Principal Component Analysis (FPCA): Mean Function
How does one calculate a mean function in Functional Principal Component Analysis (FPCA) given a data set with an unknown distribution? (A theoretical approach if possible)
I am wanting to implement ...
2
votes
2
answers
752
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R: Regression between Two related Dependent Variables (not paired)
I have two dependent variables that are measured during an experiment, they both change in time (independent variable). The measurements for both variables do not always coincide in time (so they are ...
2
votes
2
answers
524
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Forecasting ranges for multiple observations with quarterly data
I have 90 markets with quarterly results, with data from 2014 Q1 to 2016 Q2. I'd like to predict 2016 Q4 results. With a time-series in R, as I understand, you need a single observation over multiple ...
2
votes
1
answer
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Multivariate multiple non linear regression
I know there are already two questions on this topic, but neither has an answer.
I have a set of $N$ experiments. For each experiment, denoted by a vector of predictors $\mathbf{x}$, I measure $m$ ...
2
votes
2
answers
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Limiting joint distribution of estimators; Functional Statistics; Influence curves;
Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
2
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1
answer
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What is the level of measurement of image data?
Question:
This is a bit of an abstract question, but bear with me. I am averaging images, to try and deduce what the average image of a specific subject looks like (just out of curiosity, it might ...
2
votes
2
answers
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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 ...