Analyse de données fonctionnelles pour des problématiques sportives : spécificités, approches classiques et probabilistes

Arthur Leroy - Department of Computer Science, The University of Manchester

Séminaire ENSAI - 11/01/2024

Can you spot the differences?

Functional data is all about smoothness and tidiness

Functional data in sports ? Time, space and continua

Irregular time series (in number of observations and location),

Functional data in sports ? Time, space and continua

Irregular time series (in number of observations and location),
A few observations per swimmer,

Functional data in sports ? Time, space and continua

Irregular time series (in number of observations and location),
A few observations per swimmer,
Many different swimmers per category.

Reconstructing functions from series of points

A function can be expressed as a linear combination of basis functions:

\[f(t) = \sum\limits_{b=1}^{B}{\alpha_b \ \phi_b(t)}\]

If we observed the function at \(N\) instants, we can find coefficients \(\boldsymbol{\alpha}\) through least squares:

\[LS(\boldsymbol{\alpha}) = \sum_{i=1}^{N}\left[f(t_i)-\sum_{b = 1}^{B} \alpha_{b} \phi_{b}\left(t_{i}\right)\right]^{2}\]

Functional data analysis is the art of drawing nice curves

Depending on the context, we may expect different properties for our function.

Do we interpolate or smooth? Are the variations periodic (Fourier basis), multi-scale (wavelets) or polynomial (B-splines)? Answers are probably is this book.

Functional Principal Component Analysis (FPCA)

Like for multi-variate statistics, FCPA is central method when studying functions. From the Karhunen-Loève theorem we can express any centred stochastic process as an infinite linear combination of orthonormal eigenfunctions:

\[X(t)-\mathbb{E}[X(t)]=\sum_{q=1}^{\infty} \xi_{q} \varphi_{q}(t)\]

I must admit, it’s a bit trickier to interpret

It’s still used to represent the trajectories explaining the most variance among our functions. Eigenfunctions are uncorrelated, and they provide the most parsimonious decomposition in terms of basis function.

People are doing many (more or less crazy) things, trust me

An important strategy is non-parametric FDA that assumes no finite decomposition and instead defines specific metrics to measure a distance between functions directly.

Longitudinal mixed models are also worth a mention as including time as an input variable (also called fixed effect) in a mixed/hierarchical model proved to be efficient in many applications.

The curse of dimensionality? Don’t care, I’m smooth

One nice and surprising property of FDA methods is that they generally don’t suffer in high dimensions. Intuitively, what is the true dimension of this object?

What if I observed multiple functions? Let’s make groups

One classical problem in FDA (and statistics in general) is clustering: allocating functions that seem similar into groups according to some property. There are different strategies:

Yup, seems easy. No?

In our case, recall that we are interest in analysing those swimmers:

Don’t worry, I only displayed 100 of them, cause with 3456 swimmers that was a bit messy.

Modelling functions gives you derivatives to study dynamics

Clustering performance curves of 100m freestyle swimmers into 5 groups using derivatives.

Leroy et al. - Functional Data Analysis in Sport Science: Example of Swimmers’ Progression Curves Clustering - Applied Sciences - 2018

FDA is limited (and it’s kinda old to be fair)

While average trajectories of groups are reasonable, individual curves may easily diverge,
Really low predictive capacities,
No quantification of uncertainty.

Cumulated workload? Computing integrals without noticing it

Moussa, Leroy et al. - Robust Exponential Decreasing Index (REDI): adaptive and robust method for computing cumulated workload - BMJ SEM - 2019

Naive approaches can be useful but are often limited

EWMA (Exponential Weighted Moving Average) and ACWR (Acute Chronic Workload Ratio) have been proposed to compute cumulated workload, but mathematical problems (sensible to missing data, biased, fixed decreasing behaviour) prevent their direct use in practice.

Understanding properties of functions to develop adapted tools

REDI (Robust Exponential Decreasing Index) has been developed to overcome those issues by looking at this problem from a functional perspective. Computing (discrete) integrals from measurements allow us to track cumulated workload in a more robust and flexible way.

Feel free to use the R package REDI or the web app https://arthurleroy.shinyapps.io/REDI/.

Functional data appears in more areas that we would imagine

Zech et al. - Effects of barefoot and footwear conditions on learning of a dynamic balance task: a randomized controlled study - EJAP - 2018

Hollander et al. - Adaptation of Running Biomechanics to Repeated Barefoot Running: A Randomized Controlled Study - TAJSM - 2019

Like: what would have happened to results without Covid?

Now you know. Functional data > 2020

Probabilistic modelling and predictions: how do we really learn?

Moving from exploring to learning functions

\[y = \color{green}{f}(x) + \epsilon\]

où :

\(x\) is the input variable (age of the swimmer),
\(y\) is the output variable (performance on 100m),
\(\epsilon\) is the noise, a random error term,
\(\color{green}{f}\) is a random function encoding the relationship between input and output data.

All supervised learning problems require to retrieve the correct function \(\color{green}{f}\), using observed data \(\{(x_1, y_1), \dots, (x_n, y_n) \}\), to perform predictions when observing new input data \(x_{n+1}\).

Learning the simplest of functions : linear regression

In the simplest (though really common in practice) case of linear regression, we assume that:

\[\color{green}{f}(x) = a x + b\]

Finding the best function \(\color{green}{f}\) reduces to compute optimal parameters \(a\) and \(b\) from our dataset.

Learning is about updating our knowledge

This well-known probability formula has massive implications on learning strategies:

\[\mathbb{P}(\color{red}{T} \mid \color{blue}{D}) = \dfrac{\mathbb{P}(\color{blue}{D} \mid \color{red}{T}) \times \mathbb{P}(\color{red}{T})}{\mathbb{P}(\color{blue}{D})}\]

with:

\(\mathbb{P}(\color{red}{T})\), probability that some theory \(\color{red}{T}\) is true, our prior belief.
\(\mathbb{P}(\color{blue}{D} \mid \color{red}{T})\), probability to observe this data if theory \(\color{red}{T}\) is true, the likelihood.
\(\mathbb{P}(\color{blue}{D})\), probability to observe this data overall, a normalisation constant.

Bayes’ theorem indicates how to update our beliefs about \(\color{red}{T}\) when accounting for new data \(\color{blue}{D}\) :

\(\mathbb{P}(\color{red}{T} \mid \color{blue}{D})\), probability that theory \(\color{red}{T}\) is true considering data \(\color{blue}{D}\), our posterior belief.

Let’s exercise

Assume there exists some trouble or disease such that:

\(\mathbb{P}(\color{red}{T}) = 0.001,\) 1 person out of 1000 contracted the trouble on average,
\(\mathbb{P}(\color{blue}{D} \mid \color{red}{T}) = 0.99,\) a detection test is 99% reliable if you have the trouble,
\(\mathbb{P}(\color{blue}{\bar{D}} \mid \color{red}{\bar{T}}) = 0.99,\) this same detection test is 99% reliable if you don’t have the trouble,

From Bayes’ theorem, the probability to have contracted the trouble when the detection test was positive is:

\[\mathbb{P}(\color{red}{T} \mid \color{blue}{D}) = \dfrac{\mathbb{P}(\color{blue}{D} \mid \color{red}{T}) \times \mathbb{P}(\color{red}{T})}{\mathbb{P}(\color{blue}{D})} = \dfrac{0.99 \times 0.001}{0.99 \times 0.001 + (1-0.99) \times 0.999} \simeq 0.09\]

Hence, we only have 9% chance to actually be sick despite a positive result to the detection test.

A visual explanation of Bayes’ theorem

We generally use probability distributions to express our initial uncertainty about a quantity of interest (balance of a coin, average size of a human, …) and its posterior probable values.

Probabilistic estimation as an alternative to testing

Gaussian process: a prior distribution over functions

\[y = \color{orange}{f}(x) + \epsilon\]

No restrictions on \(\color{orange}{f}\) but a prior distribution on a functional space: \(\color{orange}{f} \sim \mathcal{GP}(m(\cdot),C(\cdot,\cdot))\)

We can think of a Gaussian process as the extension to infinity of multivariate Gaussians:
\(x \sim \mathcal{N}(m , \sigma^2)\) in \(\mathbb{R}\), \(\begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix} \sim \mathcal{N} \left( \ \begin{pmatrix} m_1 \\ m_2 \\ \end{pmatrix}, \begin{bmatrix} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{bmatrix} \right)\) in \(\mathbb{R}^2\)

Credits: Raghavendra Selvan

A GP is like a long cake and each slice is a Gaussian

Credits: Carl Henrik Ek

A GP is like an infinitely long cake and each slice is a Gaussian

Credits: Carl Henrik Ek

Covariance functions: Squared Exponential kernel

While \(m\) is often assumed to be \(0\), the covariance structure is critical and defined through tailored kernels. For instance, the Squared Exponential (or RBF) kernel is expressed as: \[C_{SE}(x, x^{\prime}) = s^2 \exp \Bigg(-\dfrac{(x - x^{\prime})^2}{2 \ell^2}\Bigg)\]

Covariance functions: Periodic kernel

To model phenomenon exhibiting repeting patterns, one can leverage the Periodic kernel: \[C_{perio}(x, x^{\prime}) = s^2 \exp \Bigg(- \dfrac{ 2 \sin^2 \Big(\pi \frac{\mid x - x^{\prime}\mid}{p} \Big)}{\ell^2}\Bigg)\]

Covariance functions: Linear kernel

We can even consider linear regression as a particular GP problem, by using the Linear kernel: \[C_{lin}(x, x^{\prime}) = s_a^2 + s_b^2 (x - c)(x^{\prime} - c )\]

We can learn optimal values of hyper-parameters from data through maximum likelihood.

Gaussian process: all you need is a posterior

The Gaussian property induces that unobserved points have no influence on inference:

\[ \int \underbrace{p(f_{\color{grey}{obs}}, f_{\color{purple}{mis}})}_{\mathcal{GP}(m, C)} \ \mathrm{d}f_{\color{purple}{mis}} = \underbrace{p(f_{\color{grey}{obs}})}_{\mathcal{N}(m_{\color{grey}{obs}}, C_{\color{grey}{obs}})} \]

This crucial trick allows us to learn function properties from finite sets of observations. More generally, Gaussian processes are closed under conditioning and marginalisation.

\[\begin{bmatrix} f_{\color{grey}{o}} \\ f_{\color{purple}{m}} \\ \end{bmatrix} \sim \mathcal{N} \left( \begin{bmatrix} m_{\color{grey}{o}} \\ m_{\color{purple}{m}} \\ \end{bmatrix}, \begin{pmatrix} C_{\color{grey}{o, o}} & C_{\color{grey}{o}, \color{purple}{m}} \\ C_{\color{purple}{m}, \color{grey}{o}} & C_{\color{purple}{m, m}} \end{pmatrix} \right)\]

While marginalisation serves for training, conditioning leads the key GP prediction formula:

\[f_{\color{purple}{m}} \mid f_{\color{grey}{o}} \sim \mathcal{N} \Big( m_{\color{purple}{m}} + C_{\color{purple}{m}, \color{grey}{o}} C_{\color{grey}{o, o}}^{-1} (f_{\color{grey}{o}} - m_{\color{grey}{o}}), \ \ C_{\color{purple}{m, m}} - C_{\color{purple}{m}, \color{grey}{o}} C_{\color{grey}{o, o}}^{-1} C_{\color{purple}{m, m}} C_{\color{grey}{o}, \color{purple}{m}} \Big)\]

Gaussian process: what about a visual summary?

Training hyper-parameters leads to a prior distribution adapted to capture the variation of data.

Gaussian process: what about a visual summary?

The uncertainty of probability distributions can be represented by sampling realisations (probable trajectories) or through Credible Intervals (95% probability to be in the pink area)

Gaussian process: what about a visual summary?

The uncertainty of probability distributions can be represented by sampling realisations (probable trajectories) or through Credible Intervals (95% probability to be in the pink area)

Gaussian process: what about a visual summary?

Powerful non parametric method offering probabilistic predictions,
Computational complexity in \(\mathcal{O}(N^3)\), with N the number of observations.

Forecasting with a unique GP

Great, let’s try with some data!

Scan the QR code or go to https://magmashiny.shinyapps.io/magma/ for us to collect data and play with Gaussian Processes.

Multi-task Gaussian processes (Magma)

\[y_i = \mu_0 + f_i + \epsilon_i\]

with:

\(\mu_0 \sim \mathcal{GP}(m_0, K_{\theta_0}),\)

\(f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \perp \!\!\! \perp_i,\)

\(\epsilon_i \sim \mathcal{GP}(0, \sigma_i^2), \ \perp \!\!\! \perp_i.\)

It follows that:

\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Sigma_{\theta_i} + \sigma_i^2 I), \ \perp \!\!\! \perp_i\]

\(\rightarrow\) Unified GP framework with a common mean process \(\mu_0\), and individual-specific process \(f_i\),

\(\rightarrow\) Naturaly handles irregular grids of input data.

Goal: Learn the hyper-parameters, (and \(\mu_0\)’s hyper-posterior).
Difficulty: The likelihood depends on \(\mu_0\), and individuals are not independent.

EM algorithm

E step: \[ \begin{align} p(\mu_0(\color{grey}{\mathbf{t}}) \mid \textbf{y}, \hat{\Theta}) &\propto \mathcal{N}(\mu_0(\color{grey}{\mathbf{t}}); m_0(\color{grey}{\textbf{t}}), \textbf{K}_{\hat{\theta}_0}^{\color{grey}{\textbf{t}}}) \times \prod\limits_{i =1}^M \mathcal{N}(\mathbf{y}_i; \mu_0( \color{purple}{\textbf{t}_i}), \boldsymbol{\Psi}_{\hat{\theta}_i, \hat{\sigma}_i^2}^{\color{purple}{\textbf{t}_i}}) \\ &= \mathcal{N}(\mu_0(\color{grey}{\mathbf{t}}); \hat{m}_0(\color{grey}{\textbf{t}}), \hat{\textbf{K}}^{\color{grey}{\textbf{t}}}), \end{align} \] M step:

\[ \begin{align*} \hat{\Theta} &= \underset{\Theta}{\arg\max} \ \ \log \mathcal{N} \left( \hat{m}_0(\color{grey}{\textbf{t}}); m_0(\color{grey}{\textbf{t}}), \mathbf{K}_{\theta_0}^{\color{grey}{\textbf{t}}} \right) - \dfrac{1}{2} Tr \left( \hat{\mathbf{K}}^{\color{grey}{\textbf{t}}} {\mathbf{K}_{\theta_0}^{\color{grey}{\textbf{t}}}}^{-1} \right) \\ & \ \ \ + \sum\limits_{i = 1}^{M}\left\{ \log \mathcal{N} \left( \mathbf{y}_i; \hat{m}_0(\color{purple}{\mathbf{t}_i}), \boldsymbol{\Psi}_{\theta_i, \sigma^2}^{\color{purple}{\mathbf{t}_i}} \right) - \dfrac{1}{2} Tr \left( \hat{\mathbf{K}}^{\color{purple}{\mathbf{t}_i}} {\boldsymbol{\Psi}_{\theta_i, \sigma^2}^{\color{purple}{\mathbf{t}_i}}}^{-1} \right) \right\}. \end{align*} \]

Prediction: the key idea

Defining a multi-task prior distribution by:

conditioning on training data,
integrating over \(\mu_0\)’s hyper-posterior distribution.

\[\begin{align} p(y_* (\textbf{t}_*^{p}) \mid \textbf{y}) &= \int p\left(y_* (\textbf{t}_*^{p}) \mid \textbf{y}, \mu_0(\textbf{t}_*^{p})\right) p(\mu_0 (\textbf{t}_*^{p}) \mid \textbf{y}) \ d \mu_0(\mathbf{t}^{p}_{*}) \\ &= \int \underbrace{ p \left(y_* (\textbf{t}_*^{p}) \mid \mu_0 (\textbf{t}_*^{p}) \right)}_{\mathcal{N}(y_*; \mu_0, \Psi_*)} \ \underbrace{p(\mu_0 (\textbf{t}_*^{p}) \mid \textbf{y})}_{\mathcal{N}(\mu_0; \hat{m}_0, \hat{K})} \ d \mu_0(\mathbf{t}^{p}_{*}) \\ &= \mathcal{N}( \hat{m}_0 (\mathbf{t}^{p}_{*}), \underbrace{\Psi_* + \hat{K}}_{\Gamma}) \end{align}\]

Prediction: additional steps

Multi-task prior:

\[p \left( \begin{bmatrix} y_*(\color{grey}{\mathbf{t}_{*}}) \\ y_*(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix} \mid \textbf{y} \right) = \mathcal{N} \left( \begin{bmatrix} y_*(\color{grey}{\mathbf{t}_{*}}) \\ y_*(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix}; \ \begin{bmatrix} \hat{m}_0(\color{grey}{\mathbf{t}_{*}}) \\ \hat{m}_0(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix}, \begin{pmatrix} \Gamma_{\color{grey}{**}} & \Gamma_{\color{grey}{*}\color{purple}{p}} \\ \Gamma_{\color{purple}{p}\color{grey}{*}} & \Gamma_{\color{purple}{pp}} \end{pmatrix} \right)\]

Multi-task posterior:

\[p(y_*(\color{purple}{\mathbf{t}^{p}}) \mid y_*(\color{grey}{\mathbf{t}_{*}}), \textbf{y}) = \mathcal{N} \Big( y_*(\color{purple}{\mathbf{t}^{p}}); \ \hat{\mu}_{*}(\color{purple}{\mathbf{t}^{p}}) , \hat{\Gamma}_{\color{purple}{pp}} \Big)\]

with:

\(\hat{\mu}_{*}(\color{purple}{\mathbf{t}^{p}}) = \hat{m}_0(\color{purple}{\mathbf{t}^{p}}) + \Gamma_{\color{purple}{p}\color{grey}{*}}\Gamma_{\color{grey}{**}}^{-1} (y_*(\color{grey}{\mathbf{t}_{*}}) - \hat{m}_0 (\color{grey}{\mathbf{t}_{*}}))\)
\(\hat{\Gamma}_{\color{purple}{pp}} = \Gamma_{\color{purple}{pp}} - \Gamma_{\color{purple}{p}\color{grey}{*}}\Gamma_{\color{grey}{**}}^{-1} \Gamma_{\color{grey}{*}\color{purple}{p}}\)

Leroy et al. - MAGMA: Inference and Prediction using Multi-Task Gaussian Processes with Common Mean - Machine Learning - 2022

A GIF is worth a thousand words

An image is still worth many words

Magma + Clustering = MagmaClust

Leroy et al. - Cluster-Specific Predictions with Multi-Task Gaussian Processes - Journal of Machine Learning Research - 2023

A unique underlying mean process might be too restrictive.

\(\rightarrow\) Mixture of multi-task GPs:

\[y_i = \mu_0 + f_i + \epsilon_i\]

with:

\(\color{green}{Z_{i}} \sim \mathcal{M}(1, \color{green}{\boldsymbol{\pi}}), \ \perp \!\!\! \perp_i,\)

\(\mu_0 \sim \mathcal{GP}(m_0, K_{\theta_0}), \ \perp \!\!\! \perp_k,\)
\(f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \perp \!\!\! \perp_i,\)
\(\epsilon_i \sim \mathcal{GP}(0, \sigma_i^2), \ \perp \!\!\! \perp_i.\)

It follows that:

\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Psi_i), \ \perp \!\!\! \perp_i\]

Magma + Clustering = MagmaClust

Leroy et al. - Cluster-Specific Predictions with Multi-Task Gaussian Processes - Journal of Machine Learning Research - 2023

A unique underlying mean process might be too restrictive.

\(\rightarrow\) Mixture of multi-task GPs:

\[y_i \mid \{\color{green}{Z_{ik}} = 1 \} = \mu_{\color{green}{k}} + f_i + \epsilon_i\]

with:

\(\color{green}{Z_{i}} \sim \mathcal{M}(1, \color{green}{\boldsymbol{\pi}}), \ \perp \!\!\! \perp_i,\)
\(\mu_{\color{green}{k}} \sim \mathcal{GP}(m_{\color{green}{k}}, \color{green}{C_{\gamma_{k}}})\ \perp \!\!\! \perp_{\color{green}{k}},\)
\(f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \perp \!\!\! \perp_i,\)
\(\epsilon_i \sim \mathcal{GP}(0, \sigma_i^2), \ \perp \!\!\! \perp_i.\)

It follows that:

\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Psi_i), \ \perp \!\!\! \perp_i\]

Magma + Clustering = MagmaClust

A unique underlying mean process might be too restrictive.

\(\rightarrow\) Mixture of multi-task GPs:

\[y_i \mid \{\color{green}{Z_{ik}} = 1 \} = \mu_{\color{green}{k}} + f_i + \epsilon_i\]

with:

\(\color{green}{Z_{i}} \sim \mathcal{M}(1, \color{green}{\boldsymbol{\pi}}), \ \perp \!\!\! \perp_i,\)
\(\mu_{\color{green}{k}} \sim \mathcal{GP}(m_{\color{green}{k}}, \color{green}{C_{\gamma_{k}}})\ \perp \!\!\! \perp_{\color{green}{k}},\)
\(f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \perp \!\!\! \perp_i,\)
\(\epsilon_i \sim \mathcal{GP}(0, \sigma_i^2), \ \perp \!\!\! \perp_i.\)

It follows that:

\[y_i \mid \{ \boldsymbol{\mu} , \color{green}{\boldsymbol{\pi}} \} \sim \sum\limits_{k=1}^K{ \color{green}{\pi_k} \ \mathcal{GP}\Big(\mu_{\color{green}{k}}, \Psi_i^\color{green}{k} \Big)}, \ \perp \!\!\! \perp_i\]

Covariance structure assumption: 4 sub-models

Sharing the covariance structures offers a compromise between flexibility and parsimony:

\(\mathcal{H}_{oo}:\) common mean process - common individual process - \(2\) HPs,
\(\mathcal{H}_{\color{green}{k}o}:\) specific mean process - common individual process - \(\color{green}{K} + 1\) HPs,
\(\mathcal{H}_{o\color{blue}{i}}:\) common mean process - specific individual process - \(1 + \color{blue}{M}\) HPs,
\(\mathcal{H}_{\color{green}{k}\color{blue}{i}}:\) specific mean process - specific individual process - \(\color{green}{K} + \color{blue}{M}\) HP.

Prediction

Multi-task posterior for each cluster:

\[ p(y_*(\mathbf{t}^{p}) \mid \color{green}{Z_{*k}} = 1, y_*(\mathbf{t}_{*}), \textbf{y}) = \mathcal{N} \Big( y_*(\mathbf{t}^{p}); \ \hat{\mu}_{*}^\color{green}{k}(\mathbf{t}^{p}) , \hat{\Gamma}_{pp}^\color{green}{k} \Big), \forall \color{green}{k}, \]

\(\hat{\mu}_{*}^\color{green}{k}(\mathbf{t}^{p}) = \hat{m}_\color{green}{k}(\mathbf{t}^{p}) + \Gamma^\color{green}{k}_{p*} {\Gamma^\color{green}{k}_{**}}^{-1} (y_*(\mathbf{t}_{*}) - \hat{m}_\color{green}{k} (\mathbf{t}_{*}))\)
\(\hat{\Gamma}_{pp}^\color{green}{k} = \Gamma_{pp}^\color{green}{k} - \Gamma_{p*}^\color{green}{k} {\Gamma^{\color{green}{k}}_{**}}^{-1} \Gamma^{\color{green}{k}}_{*p}\)
Predictive multi-task GPs mixture:

\[p(y_*(\textbf{t}^p) \mid y_*(\textbf{t}_*), \textbf{y}) = \color{green}{\sum\limits_{k = 1}^{K} \tau_{*k}} \ \mathcal{N} \big( y_*(\mathbf{t}^{p}); \ \hat{\mu}_{*}^\color{green}{k}(\textbf{t}^p) , \hat{\Gamma}_{pp}^\color{green}{k}(\textbf{t}^p) \big).\]

Quick recap: we started from this…

Classical Gaussian processes

… then improved to that…

Multi-task Gaussian processes

… to arrive here. Quite a journey!

MagmaClust

Cluster-specific predictions

Each cluster-specific prediction is weighted by its membership probability \(\color{green}{\tau_{*k}}\).

Clustering and prediction performances

So what about female swimmers after all?

And male swimmers?

BMI evolution patterns and overweight predictions

Weight follow-up of children between birth and 6 years

You know what? It also works with multi-dimensional inputs

It’s your time to shine

Remember the data you entered a few minutes ago? Let’s have a look at your predictions!

Scan the QR code or go to https://magmashiny.shinyapps.io/magma/.

And more importantly, it’s your turn to work, I’m done!

You can install the R package MagmaClustR, the dataset named swimmers is directly available (feel free to use data from your favourite sport instead) and run your own analysis.

For those who don’t have R installed, or to help you understand the package, you can access notebooks at https://arthurleroy.github.io/MagmaClustR/ or by scanning the QR code.