## Data assimilation. A mathematical introduction.(English)Zbl 1353.60002

Texts in Applied Mathematics 62. Cham: Springer (ISBN 978-3-319-20324-9/hbk; 978-3-319-20325-6/ebook). xviii, 242 p. (2015).
The text in the back cover of the book summarises adequately the content of the book, and is therefore not reproduced here.
The mathematical style of the book is accessible to post-graduate students and combines formal mathematics with intuitive arguments and summaries of higher level results. However, in order to read this book, I would recommend to have some experience with applied problems involving (linear) dynamic systems, time-series and likelihood based statistical inference. In these conditions the book is a good guide on dynamic data assimilation.
The limitations of the book derive from the fact that the whole presentation of the subject of data assimilation is restricted to random variables in (multivariate) real space, while it does not mention that most data and real phenomena have a constrained sample space, like the sphere for directional data or the simplex for compositional data. Nevertheless, this makes the book suitable as a reference book for modelling on coordinates, whenever the sample space has a Euclidean vector space structure. A typical example of such a sample space is the sample space of compositional data, i.e., the simplex. The simplex, endowed with the Aitchison geometry, has a Euclidean space structure [the reviewer and J. J. Egozcue, Stoch. Environ. Res. Risk Assess. 15, No. 5, 384–398 (2001; Zbl 0987.62001)], and could be well used as an example of working on coordinates. Furthermore, the structure of the finite dimensional simplex has been extended to infinite dimensions, leading to a Hilbert space structure of densities or, more generally, for measures [K. G. Van Den Boogart et al., SORT 34, No. 2, 201–222 (2010; Zbl 1208.62003); Aust. N. Z. J. Stat. 56, No. 2, 171–194 (2014; Zbl 1335.62025)]. It would be interesting to see how the proposed methodology works with such a structure.

### MSC:

 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 62-07 Data analysis (statistics) (MSC2010) 60G35 Signal detection and filtering (aspects of stochastic processes) 62M20 Inference from stochastic processes and prediction 93E11 Filtering in stochastic control theory

### Citations:

Zbl 0987.62001; Zbl 1208.62003; Zbl 1335.62025

Matlab
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