# 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 have Ramsay and Silverman's "Functional Data Analysis" and Ramsay, Hooker, and Graves "Functional Data Analysis with R and Matlab" but only got them recently, and am still very much the newbie with FDA. I am using R with package fda as the analysis software.

My data are a sample of hundreds of young adult subjects, each of whom were measured annually by performing a task with a binary outcome over some number of trials. The outcome of interest is the success rate on the task. I'll include a short sample of data at the end of my question.

Several parameters were collected on each subject, including height, weight, education level, amount of prior training on the task, and the results of an aptitude pre-test.

The year-to-year change in success rate over time is the curve I'd like to model with FDA. There is significant variation across subjects in overall performance, but I am mostly interested in how a subject's success rate evolves over time rather than a subject's actual performance.

The subjects generally follow a pattern of starting off with a low success rate, improving yearly until reaching a personal maximum, then declining with age. The typical subject enters the study at age 20 with a low success rate, improves until age 25, then declines until age 30, at which point no further data is collected. One of the goals is to characterize the "canonical" development curve for the population. FDA seems like a good fit for this aspect of the problem.

Some of my challenges are:

• Subjects may reach their maximum success rate at different ages, and may improve/decline faster than others, so both phase variation and amplitude variation are potentially present. So it seems like curve registration will be an important part of the analysis.

• The number of trials in the test is relatively small each year, so the outcome data is noisy, and registering the data properly seems daunting since the noise in one subject can obscure the landmarks in the data.

• I am also interested in whether the other variables predict the registration characteristics of the subject's curve. i.e. Do college-educated subjects reach their maximum performance at an earlier age than others (phase), or do particularly tall subjects sustain their peak performance longer than short subjects do (amplitude)?

• We would like to be able to produce a customized curve for a subject. e.g. If we observed the tests through age 23, we would be able to predict how much improvement to expect, at what age the maximum performance would be reached, and how sharp the decline phase would be through age 30.

My questions include:

1. Is FDA an appropriate method for this problem?

2. Are there techniques in FDA for predicting a subject's phase and amplitude variation using other variables? What I've read so far seems to treat registration as a correction for noise rather than as containing features worth examination on its own.

3. What are some good ways to handle registration where the individual curves have a lot of noise?

4. Should I be able to produce individualized forecast curves that include subject-specific predicted phase/amplitude variation? What difficulties should I anticipate?

Thank you for any and all suggestions.

Sample data

Subject  Education    Ht   Wt   Training   Aptitude

A        College      72  200   0          Medium
B        High School  77  250   100        High
C        High School  68  160   50         Low

Subject Age Trials  Success Success%
A   20  15  3   20%
A   21  18  5   28%
A   22  30  7   23%
A   23  28  8   29%
A   24  32  13  41%
A   25  8   2   25%
A   26  20  8   40%
A   27  40  11  28%
A   28  33  10  30%
A   29  18  5   28%
A   30  10  2   20%

B   20  24  4   17%
B   21  27  5   19%
B   22  30  8   27%
B   23  33  2   6%
B   24  41  8   20%
B   25  39  5   13%
B   26  39  5   13%

C   24  13  4   31%
C   25  19  6   32%
C   26  18  5   28%
C   27  23  6   26%
C   28  16  6   38%
C   29  9   3   33%

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