Spring 2021

 

Introduction to Mathematical Methods for Modeling and Data Analysis

 

Lecturer: Eli Galanti 

eli.galanti@weizmann.ac.il

 

Monday 09:15-11:00 (Zoom 2020)

TA: Keren Duer

keren.duer@weizmann.ac.il

 

Wednesday 12:15-13:00 (Zoom)

Now happening!

Final assignment      Forum

 Submit by Wednesday, July 28, 5pm

Recorded sessions

Overview and goals

Mathematical models are present in all of the scientific disciplines, providing a quantitative framework for understanding and prediction of natural phenomena. The output from such models, as well as observations, often requires complex mathematical analysis. The course provides an introduction to mathematical modeling and data analysis through in-depth discussion of a series of real examples, with an emphasis on 'hands on' exercises. Upon successful completion of this course students should be able to: (1) Understand the principles of mathematical modeling and data analysis, (2) Solve analytically and numerically a wide range of problems.

 

Outline

1)  Introduction to modeling: ordinary differential equations

Topics: First order equations; Second order equations

 

2)  Linear equations – Eigen values and vectors

Topics: Over and under determined problems; System of linear ODEs

 

3)  Introduction to data analysis

Topics: Single time series analysis; Multiple time series analysis; Multivariate time series analysis; Regression and interpolation

 

4)   Mining of Big Data

Topics: Clustering; Classification

 

5)  Advanced modeling: partial differential equations

Topics: Diffusion-advection equations; Wave equations

 

6)  Combining models and data: optimization of model initial conditions and parameters

Topics: Optimization of unconstrained linear problems, Optimization of constrained nonlinear problems

 

 

 

Tutorials

o   Week 1 - matlab example

o   Week 2 - notes

o   Week 3 - notes, matlab_class, matlab_generate_sparse

o   Week 4 - notes, matlab_class

o   Week 5 - notes, matlab_class, data

o   Week 6 - notes, matlab_ex1, matlab_ex2, data

o   Week 7 - notes, matlab_class

o   Week 8 - notes, matlab_class

o   Week 9 - notes, examples

o   Week 10 - notes

o   Week 11 - notes, example1, example2

o   Week 12 - notes

o   Week 13 – matlab1, matlab2, solve

o   Week 14 - all

 

 

 

Homework

o   Homework 1  – submit by 14/04/2021. P53, P185, matlab

 

 

o   Homework 2  – submit by 28/04/2021.

o   Homework 3  – submit by 14/05/2021.

o   Homework 4  – submit by 02/06/2021. Image

o   Homework 5 - submit by 16/06/2021.

o   Homework 6 - submit by 30/06/2021.

o   Homework 7 - submit by 14/07/2021.

 

 

 

Reading

o   Strogarz, S.H: Nonlinear Dynamics and Chaos. Perseus books, 1994

o   Leskovec, J., Rajaraman, A., and Ullman, J. D: Mining of massive datasets. Cambridge University Press, 2014.

o   Gill, Murray, and Wright: practical optimization (1981)

 

 

 

 

Syllabus

1)     Introduction to modeling: ordinary differential equations

 

1st order equations (Class notes)

o   Motivating examples: Population dynamics, global earth temperature

o   Graphical solution method and stability analysis (Strogatz 21-26)

o   Numerical solutions - Euler and improved Euler (Strogatz 32-33, matlab ode45)

 

2nd order equations (Class notes)

o   Motivating examples: Love affairs (Strogatz 138-140), pendulum (Strogatz 168-173)

o   Phase space and stability analysis (Strogatz 145-154)

o   Rabbits versus sheep (Strogratz 155-159) (given in tutorials)

 

2)     Linear equations and eigen decomposition

 

Over and under determined problems (Class notes)

o   Motivating example: Medical tomography

o   Row interpretations (Matlab example)

o   Over-determined - least squares solution

o   Under-determined - SVD/pseudo inverse

o   Matlab backslash operator vs. using matrix inverse (Matlab example)

o   Sparse matrices (given in tutorials)

 

System of linear ODEs - eigenvalues and eigenvectors (Class notes)

o   Motivating examples: Google PageRank, Romeo & Juliet

o   Review of eigenvalues and eigenvectors (Strogatz 129-140)

o   Back to Romeo & Juliet

o   Non-linear equations - linearizing around fixed points (Strogatz 150-151)

o   Analysis of damped pendulum

 

3)     Data analysis

 

Single time series analysis (Class notes)

o   Motivating examples – Tropical Pacific sea surface temperature

o   Mean, variance, STD, auto-correlations

o   Composite analysis

o   Spectral analysis (FFT) (Matlab example)

 

Multiple time series analysis (Class notes)

o   Motivating examples: stock prices, ENSO

o   The covariance matrix

o   Principal Component Analysis (PCA) using the covariance matrix

o   Fraction of variance explained

o   Examples in 1D and 2D

 

Multivariate time series analysis (Class notes)

o   Motivating examples: stock prices from two different markets, SST+winds

o   Multivariate PCA

o   Singular Value Decomposition (SVD)

o   MPCA vs. SVD                                        

o   Image compression

 

Regression and interpolation (Class notes)

o   Correlation and regression

o   Quality of fit with linear regression

o   Connection between a time series and a time varying field

o   Interpolation via nonlinear polynomial fit (notes)

 

4)     Mining of Big Data

 

Unsupervised learning: Clustering (notes)

o   Motivating examples

o   Distances: Euclidian (L2), Manhattan (L1), maximum (L), Jaccard.

o   Hierarchical clustering, dendrogram, elbow plot (Matlab)

o   K-means (Matlab)

 

Supervised learning: Classification (notes)

o   Motivating examples

o   Introduction to machine learning

o   Perceptrons (Matlab)

o   Neural networks (Matlab)

 

5)     Advanced modeling: partial differential equations

 

Diffusion-advection equations (Class notes)

o   Motivating examples: HIV, Computer virus, WalMart

o   Derivation of the diffusion-advection equation

o   Setting the boundary condition

o   Numerical solution - Finite differences (Matlab example)

o   Steady state solutions using eigenvalues

o   Implicit solver – Euler backward

 

Wave equations (Class notes)

o   Motivating examples

o   Derivation of the Wave equation

o   Setting the initial and boundary conditions

o   Numerical solution

 

6)     Combining models and data: optimization of model initial conditions and parameters

o   Motivating example – Juno gravity measurements

 

Optimization of unconstrained linear problems (Class notes)

o   Regression – linear model fitting

o   A general linear case with no constraints

o   Steepest descent (Gill 4.3.1, 4.3.2.2)

o   Conjugate gradient (Gill 4.8.3)

o   Iterative optimization of an unconstrained linear model

o   Optimizing the strike of a pendulum

 

Optimization of constrained nonlinear problems (Class notes)

o   The effect of nonlinearity on optimization – nonlinear pendulum

o   Motivating example - diffusion-advection with a source

o   Matlab `fmincon` (webpage)

o   Solving a bounded and constrained problem (Matlab code)

  

 

Administrative

Prerequisites:

This is a basic introductory breadth course and should be accessible to all Weizmann graduate students, given that they took university level introductory courses in linear algebra and calculus.

 Requirements:

o   Attendance of at least 80% of the lectures.

o   Submission of all homework assignments by deadline.

o   Submission of final assignment by deadline.

 Grading:  Home assignments 70 points, final assignment 30 points.