###### Forcasting and dimension reduction

Objectives

• Invite the student to use some useful mathematical methods in data prediction and analysis.

• To familiarize the student with the mathematical language used in the fundamental methods of Data Science and Machine Learning.

• Formalize some supervised and unsupervised learning algorithms.

• Provide the student with the restrictions of use and the practical benefits that the mathematical formalization implies.

Syllabus

Block one: Prediction

1. Linear regressions

to. Examples

b. Advantages and disadvantages

c. Convex optimization

d. Stochastic noise

and. Stochastic attributes of the algorithm

F. Algebraic solution: matrix inversion

g. Geometric interpretation

h. Analytical solution: Gradient Descent

i. Stochastic Gradient Descent stochastic solution

j. Learning capacity

k. Logistic regressions

2. Generalizations

to. Polynomial regressions

b. Splines

c. Unvistazoaloskernels

d. A look at neural networks

3. The curse of dimension

to. K-nearest neighbors

b. Some concrete calculations

c. Solutions: regularizers or dimension reduction

4. Linear regressions with regularizers

to. Tychonoff's regularizer as a stabilizer

b. Ridge linear regression

i. Strongly convex optimization

ii. Algebraic solution

iii. Analytical solution

iv. Stochastic solution

v. Determination of the lambda parameter

saw. K-fold cross validation

c. Lasso linear regression

d. Elastic net linear regression

Block two: numerical linear algebra

1. Matrix algebra

to. Basic concepts

b. Relationships with linear regressions

c. Tensor products

2. Decomposition of matrices

to. Motivation: curve interpolation

b. Gaussian decomposition

c. Singular value decomposition

d. Singular value stochastic decomposition

and. Non-negative matrix factorization

F. Cholensky decomposition

3. Linear dimension reduction

to. PCA: Euclidean interpretation

b. PCA stochastic interpretation

c. Cut-off

d. Robust PCA