Statistical description and dimension reduction of continuous time categorical trajectories with multivariate functional principal components
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915974564806656 |
|---|---|
| author | Cardot, Hervé Peltier, Caroline |
| author_facet | Cardot, Hervé Peltier, Caroline |
| contents | Getting tools that allow simple representations and comparisons of a set of categorical trajectories is of major interest for statisticians. Without loosing any information, we associate to each state a binary random indicator function, taking values in $\{0,1\}$, and turn the problem of statistical description of the categorical trajectories into a multivariate functional principal components analysis. This viewpoint encompasses experimental frameworks where two or more states can be observed simultaneously. The sample paths being piecewise constant, with a finite number of jumps, this a rare case in functional data analysis in which the trajectories are not supposed to be continuous and can be observed exhaustively. Under the weak hypothesis assuming only continuity in probability of the $0-1$ trajectories, the means and the (multivariate) covariance function are continuous and have interpretations in terms of departure from independence of the joint probabilities. Considering a functional data point of view, we show that the binary trajectories, which are right-continuous functions with left-hand limits, can be seen as random elements in the Hilbert space of square integrable functions. The multivariate functional principal components are simple to interpret and we show that we can get consistent estimators of the mean trajectories and the covariance functions under weak regularity assumptions. The ability of the approach to represent categorical trajectories in a small dimension space is illustrated on a data set of sensory perceptions, considering different gustometer-controlled stimuli experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical description and dimension reduction of continuous time categorical trajectories with multivariate functional principal components Cardot, Hervé Peltier, Caroline Methodology Machine Learning Getting tools that allow simple representations and comparisons of a set of categorical trajectories is of major interest for statisticians. Without loosing any information, we associate to each state a binary random indicator function, taking values in $\{0,1\}$, and turn the problem of statistical description of the categorical trajectories into a multivariate functional principal components analysis. This viewpoint encompasses experimental frameworks where two or more states can be observed simultaneously. The sample paths being piecewise constant, with a finite number of jumps, this a rare case in functional data analysis in which the trajectories are not supposed to be continuous and can be observed exhaustively. Under the weak hypothesis assuming only continuity in probability of the $0-1$ trajectories, the means and the (multivariate) covariance function are continuous and have interpretations in terms of departure from independence of the joint probabilities. Considering a functional data point of view, we show that the binary trajectories, which are right-continuous functions with left-hand limits, can be seen as random elements in the Hilbert space of square integrable functions. The multivariate functional principal components are simple to interpret and we show that we can get consistent estimators of the mean trajectories and the covariance functions under weak regularity assumptions. The ability of the approach to represent categorical trajectories in a small dimension space is illustrated on a data set of sensory perceptions, considering different gustometer-controlled stimuli experiments. |
| title | Statistical description and dimension reduction of continuous time categorical trajectories with multivariate functional principal components |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2502.09986 |