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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). The dataset consists of 1139 participants and four observations of weight (sparse data). The observations of weight were collated on four different days. The objective of my work is to apply Functional Principal Component Analysis to the dataset to uncover distinct trajectories of change in weight (trajectory modelling). The four measurements of weight are considered functional data.

In modelling, I have uncovered four distinct trajectories. FPC1 accounts for 96.59% of explained variance. FPC2, 2.25% and FPC3/ FPC4 account for a combined 1.15% of variance. FPC1 exhibits an upward trajectory, this demonstrating weight gain over the trial. The mean function reflects initial weight loss in the trial followed by consistent weight regain thereafter. The image below is a plot of the Eigen Function for the dominant component, FPC1.

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Observing mean, correlation coefficients and FPC Loadings, I can see that mean weight increases over time. The correlation coefficients become tighter towards the end of the trial and in FPC1, the FPC loadings increase over time. Therefore, we can determine that later in the trial, weight change accounts for the greater degree of variation. Therefore suggesting that greater weight is gained later in the trial.

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Going beyond this, I have produced participant level FPC Scores for FPC1. These scores demonstrate participant level contribution to the variance or pattern in FPC1. In this, we see evidence of strong positive and strong negative scores per participants, thus suggesting that participants strongly, positively contribute to the pattern in the component.

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Taking participant 1180 as an example, they scored strong positive FPC score in FPC1. This suggests that they strongly correlate or contribute to the pattern uncovered in FPC1. However, observing their functional data, their pattern exhibits differently to that of the component.

My comment on this is that participants may score strong positive or negative in a given component, however, where their functional data exhibits differently to the pattern of the component, this suggests Individual Variability (uniqueness) not captured by variation, outlier, possible sample size issues and others. Is my understanding of FPC Scores correct or is there a better way in which the difference in patterns between a strong positive participant and the pattern of the component can be explained?

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Given the nature of your data, I'm not sure I understand the point of functional modelling (apart from the fact that the data are linked temporally with a certain smoothness, given that the weight of the participants won't vary too much). Indeed, you could also assume that you have $n$ vectors of observations in $R^4$ and perform a standard multivariate PCA.

What are FPCs

In either case (whether you decide to treat things with functional or multivariate modelling) the FPCs represent the scalar products of the observations (seen as a function in a Hilbert space of functions or a vector in $R^4$) with the eigenfunctions/vectors of the covariance operator/matrix.
For a formal definition of this quantity in the functional framework see equation (7.24) on page 193 of Tailen Hsing, Randall Eubank (2015) (for the vectorial PCA case any lecture on the subject should contain this information).

Interpreting FPCs

As a result, to interpret the FPCs correctly, you need to have interpreted the eigenfunctions/vectors correctly.
I can't go into too much detail on this point, as I'm clueless on the subject of weighloss. Interpretation of the eigenfunctions/vectors is generally a job for people who know the field in question (for example, in the case of data on monitoring weight loss of human beings, I would think of doctors or biologists).

Nevertheless, in view of the information you give us on the data, a fairly plausible explanation of your observations can be given, keeping in mind that the FPCs are the results of the scalar products between the observations and the eigenfunctions/vectors.

Probable interpretation of your results

A fortiori, if the norm of an observation is large it will tend to have a large FPC. Here, participant number 1180 is probably one of the heaviest participants. Also, bearing in mind that prior to a principal component analysis we have centred the data, negative FPCs (for example participant 1679) probably often indicate that this is a person whose weight is below average (in the code the FPC1 vector only has positive entries).

I don't have access to the data in question, so I'll leave comments about interpretations concerning participants as mere hypotheses.
They shouldn't be too difficult to corroborate/infirm by looking at the participants' data more closely: for example, to your 'strong positive / strong negative' table you could add a 'person's average weight' column (which gives a good idea of the norm of the observation vectors and the influence of centring on the sign.

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