The integrated course (Mod A + Mod B) deals with a variety of predicting techniques applied to geospatial data and available in a variety of software suites like those devoted to Earth Observation data analysis or Geographic Information System data manipulation and visualization tools.

In the first module the basics of probability, statistics and linear algebra are revised and different numerical aspects arising in the solution of prediction problems are discussed.

The polynomial exact and least square interpolation problem and the exact cubic spline interpolation one are treated in the laboratory sessions.

The second module deals with data gridding and classification techniques, interpolation with combination of known base functions (including the discrete fourier transform), and finally with the stochastic interpolation.

The goal of the whole course is to make students aware of the mathematical background needed to properly apply those techniques. They will be asked to implement the geospatial data manipulation tools under study and apply them to simulated and real data.

Lectures proposed  (Mod. A)

An introduction to stochastic modeling: from random variables and random vectors to random processes and fields.

Review of basics: Random variables and random vectors. Mean and covariance propagation laws. Sample variables. Estimators. Tests. Observation error modeling and error covariance propagation within the least squares interpolation problem. Geometric interpretation of the least squares solution and the orthogonalization of the normal matrix. Outlier detection. 

Practice and laboratory sessions proposed  (Mod. A)

Practice lessons with numerical examples will be given and exercises will be proposed to make students more familiar with the proposed techniques. Laboratory sessions will be devoted to the implementation of the following algorithm:

- Exact and least squares polynomial interpolation. Test on least squares nested models. Test on least squares parameters.

Lectures proposed  (Mod. B)

Data gridding: nearest neighbor, distance weighted interpolation. Commission and omission error in data interpolation and error evaluation with the leave one out technique.

Data classification: hierarchical and optimization techniques. Stochastic techniques. Clasification quality evaluation.

Discrete Fourier Transform.

The stochastic modeling of deterministic interpolation residuals. The concepts of stationary signals and homogeneous and isotropic random fields, empirical variogram and covariance function estimation and the linear prediction with kriging techniques.

Practice and laboratory sessions proposed  (Mod. B)

Practice lessons with numerical examples will be given and exercises will be proposed to make students more familiar with the proposed techniques. Laboratory sessions will be devoted to the implementation of the following algorithm:

- Hierarchical Classification: divisive and agglomerative clustering algorithm implementation.  Partitioning around medoids by minimization of a target function. Maximum likelihood classification.

- DFT

- Stochastic prediction: empirical covariance/variogram function estimation and modeling. Simple kriging and collocation.

- Gram-Schmidt orthogonalization. Cholesky decompostition.

- Least squares interpolation with linear and cubic splines. Tichonov regularization.

Mathematics and probability theory.

A unique exam will be held for the whole course (Mod A + Mod B)

To assess students’ ‘knowledge and understanding’  written test sessions will be held.

To assess their ability to ‘apply the knowledge’, to ‘make judgements’ as well as to assess their ‘communication skills’ students are asked to do a final oral presentation, reporting on an experimental activity carried out on a real dataset by using the software developed during the laboratory sessions.

During the oral presentation questions will be asked to better evaluate student’s awareness of the course topics and on the their general understanding of the data modeling problem.