Multivariate Analysis

The volume of statistical data continues to grow, particularly the richness of the descriptions of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions—that is, when the number of variables exceeds the number of statistical units. This course presents the most common techniques used to regularize linear models in high dimensions.

To provide training in high-dimensional univariate and multivariate linear modeling, that is, in various regularization techniques for classical linear modeling.

Course on Multidimensional Data Analysis (ACP & CA). Course on Euclidean geometry, normed vector spaces, and endomorphism reduction.

Recommended prerequisites: Courses in univariate and bivariate descriptive statistics. A solid understanding of matrix algebra.

Introduction

High-Dimensional Data. Dimension Reduction and Regularization.

I - Regularized linear modeling of a continuous variable.

  1. The classical linear model.

    a) Express reminders.

    b) Failures caused by collinearities.

  2. Principal component analysis.

    a) The method.

    b) Strengths and weaknesses.

  3. PLS regression.

    a) Tier 1 Criteria and Program.

    b) Criteria and schedule for subsequent ranks.

    c) Why PLS performs normalization.

    d) Selection of the number of components for the prediction.

    e) A metric for the continuum between OLS and PLS.

  4. Regularized linear regressions.

    a) Ridge regression.

    b) LASSO.

    c) Elastic net.

II - Regularized linear modeling of a set of continuous variables.

  1. The Multivariate Gaussian Linear Model

    a) The classic model.

    b) The penalized model.

    c) The MANOVA model.

  2. Multivariate PLS regression.

    a) A rank-1 criterion and program with arbitrary metrics.

    b) Special cases: canonical analysis, ACP using instrumental variables, PLS2 regression.

    (c) Criteria and schedule for subsequent ranks.

    d) Prediction: selection of the optimal number of components.

    e) Metrics of the continuum between Canonical Analysis, ACPVI, and PLS.

III - Linear Modeling of a Nominal Variable: Linear Discriminant Analysis.

  1. Discriminant Factor Analysis

    a) Criteria and program.

    b) Components and discriminant powers.

  2. PLS discriminant analysis.

    a) Criteria and program.

    b) Components and discriminant powers.

    c) Barycentric discriminant analysis.

  3. Decision-making aspects.

    a) Decision (classification), losses, decision rules (allocation), risks.

    b) Choosing the right number of factors for the decision.

Hourly volumes:

            CM: 21

            TD:

            Practical Work: 

            Lot: