This manuscript is divided in two parts: the first one consists in the establishment of a coherent framework for the analysis of manifold-valued longitudinal data; the second one focuses on the algorithmic aspect by proposing a new class of stochastic approximations of the EM algorithm. More precisely, the chapters are organized as follows.
Thereafter, the chapter with symbol ♦ correspond to our contribution.
Chapter 3. ♦The first chapter is dedicated to the establishment of a coherent framework for the statistical analysis of manifold-valued longitudinal data. For this purpose, we propose a generalization of the model introduced bySchiratti et al.(2015,2017) in order to encompass the case of longitudinal data undergoing multivariate evolution dynamics. This model relies on the distinction of temporal deformations, linked with the acquisition process of the data and the pace of evolution of the observed phenomenon, versus space deformations, linked with intrinsic geometry of observed shapes.
To do so, we propose a nonlinear mixed effects model whose individual evolution trajectories are seen as spatio-temporal deformations of an average path representative of the evolution at the macroscopic population scale. We present this model under fairly generic hypotheses in order to encompass a wide class of more specific models.
Estimation of the parameters of the geometric model is performed through a maxi- mum a posteriori whose existence and consistency are proved; in other words, we prove that the larger the sample, the closer the estimator of the maximum a posteriori is from the real values of the parameters. This last result is all the more important given that the proposed model include the model of Schiratti et al. (2015, 2017) as a particular case. The later being applied to early detection of Alzheimer’s disease, we thus provide theoretical guarantees on ongoing pre-clinic studies.
This chapter is derived fromChevallier et al. (2017) andChevallier et al. (2019). Chapter 4. In order to allow for models dealing with 3-dimensional anatomical shapes,
this chapter provides the mathematical foundations of two shape spaces widely used in this context: the currents (Vaillant and Glaunès, 2005) and the varifolds (Charon and
Trouvé,2013).
Chapter 5. ♦ Then, the model proposed in Chapter 3 is able to quantify the pace of progression of a continuous process. We apply this model to chemotherapy monitor- ing and especially to follow-up of metastatic kidney cancer. Indeed, in this context, understanding the pace of progression of the cancer is at the heart of medical care.
The first application consists in monitoring RECIST scores. These scores being scalar data, we realize an instantiation of the generic model for real bounded data on segment [0, 1] endowed with logistic metric. This model was elaborated in collaboration with oncologists and radiologists of the Georges Pompidou European hospital (HEGP for hôpital européen Gorges Pompidou in French). Numerical experiments on synthetic and real data validate its relevance.
The second application consists in follow-up of 3D anatomical shapes, again for evaluation of tumors response. This model relies on the notion of large deformations we discussed in the introduction and can be applied to currents (Vaillant and Glaunès, 2005) and varifolds (Charon and Trouvé, 2013), which are standard shape spaces for the analysis of anatomical data and whose mathematical background was recalled at Chapter4. We also propose numerical experiments on synthetic data.
This chapter is partly derived fromChevallier et al. (2019).
Chapter 6. Numerically, estimation of the parameters is performed through a stochastic approximation of the EM algorithm, namely the SAEM algorithm. Before studying in depht this algorithm, we present in this chapter the classical literature on the EM algorithm and its usual variants. We pay particular attention to the seminal paper of Delyon et al.(1999).
Chapter 7. ♦ Despite numerical performance of the SAEM algorithm, and because of the complexity of our model, our numerical experiments suffered from limitations. In particular, the SAEM algorithm is very sensitive to its initial condition, and, even with the additional stochasticity associated to its approximation feature, it can be trapped in local minima. Moreover, the SAEM assumes that we are able to draw from the con- ditional distribution of latent variables given the observations (with the latent variable models terminology), possibly with MCMC-type method, which is not always the case. We propose in this chapter a new class of SAEM algorithms: the approximated- SAEM algorithms, for which we prove convergence toward local maxima under standard assumptions. This class relies on sampling from an approximation, in a sense to be specified, of the real conditional law in the simulation step. In particular, we encom- pass pre-existent algorithms like the ABC-SAEM (Picchini and Samson, 2018) whose numerical efficiency was empirically established but whose theoretical convergence was not proved.
the SAEM algorithm in order to favor convergence toward global minima. In this version, we approximate the conditional law by tempering it following a damped oscillatory temperature scheme. This method is applied to the estimation of parameters in Gaussian mixture models and we illustrate its numerical superiority over SAEM algorithm. This algorithm is also applied to blind source separation through independent factor analysis.
This chapter is derived fromAllassonnière and Chevallier (2019).