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Data science & ML Video Tutorials Part II - Group & Set Theory (Groups, Rings & Fields)

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We continue building the knowledge required for basic probability theory and statistics (needed for data science & ML). The videos in this post are about set algebra and algebraic structures like groups, rings & fields. It includes an introduction to identity, inverse and idempotent elements. We'll see that Boolean conjunction and disjunction can be seen as multiplication and addition and investigate possible additive and multiplicative identities and inverses. This is followed by power sets and basic operations on elements of power sets (union, intersection, complement, difference). All concepts are then combined in a discussion of algebraic structures including groups, rings and fields. We end with Boolean rings and describe set algebra as an example Boolean ring. In the review session we complete an example proof from CH1 of W. Rudin's Principles Of Mathematical Analysis using the axioms of a field. Introduction to part II. The main subject of the sessions of Par
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Data science & ML Video Tutorials Part I - Sets, Tuples, Relations & Functions, Basic Boolean algebra

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What follows is part I of a series of short instructional videos in mathematics. The videos cover concepts needed for analysis & probability theory. The part I video sessions are embbeded in order below: Intro: Introduction to Part I of a course in mathematics for probability, statistics, data science and machine learning. 1A: The basic concepts of set algebra. Set builder notation, common sets of numbers (natural numbers, whole numbers, rational number, real numbers), subset, superset, universal set & complement. This session is part of a series of math tutorials for statistics, data science & machine learning( ML ). 1B: Introduce tuples, the set Cartesian product, Cartesian triple product and set cardinality. 2A: Define a mathematical relation as a subset of the Cartesian product. Define the inverse relation. Example relations. Position and function input. Continuation of session 1A. 2B: Definition of a function as a special case of a relation. Compare relati
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Finite Difference Simulation of a Helmholtz Resonator - 2D Acoustic Wave Equation

In this project we wish to simulate the acoustics of a Helmholtz Resonator (HR) as may be found in a vented loudspeaker enclosure design. The vent functions as a Helmholtz resonator and has a low frequency attenuation effect. The enclosure is compartmentalized due to the vent design. Therefore, in order to carry out the simulation we must implement reflecting boundary conditions in the interior of the simulation region. In addition, a mixture of absorbing and reflecting boundary conditions are needed at the edges of the simulation region. Absorbing boundary conditions for the wave equation require additional attention and are implemented as described by [Alam & Mohiuddin, 2021]. Read a detailed explanation report can be found here here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/acoustic_simulation .
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Combination Finite Difference & Spectral Solution to Advection-Diffusion PDE

Vorticity $\omega$ is a measure of local rotation (in a fluid). It is related to the angular velocity of a solid. Vorticity is generally a vector (valued function). However, it can be viewed as a scalar in 2D because it is always perpendicular to the plane. If fluid velocity $\vec{v}$ (in 2D) has components $u$ and $v$ \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} \end{equation} then these vector components are related to a scalar quantity called the \textbf{stream function} $\psi$ as follows \begin{align} u = -\psi_y \\ v = \psi_x \end{align} so that \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = \begin{pmatrix} -\psi_y \\ \psi_x \end{pmatrix} \end{equation} So $\psi$ is a scalar valued function while the components of the fluid velocity vector can be derived from it. \begin{equation} \label{eq:hw4_laplacian_eqn} \nabla^2 \psi = \omega \end{equation} Vorticity-Streamfunction PDE \begin{align} \omega_t + [\psi,\omega] &am
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MNIST Classification LDA SVM CART & Generalized Eigenvalue Problems

In this UW assignment report we analyze the MNIST handwritten digit dataset. We start with Principal Component Analysis (PCA) for dimensionality reduction and visualization. Visualization of the digits is accomplished by projection on 3 principal components such that each image is represented by a single 3D point. We next build a few supervised classifiers and compare their accuracy. Classification methods explored in this report include Linear Discriminant Analysis (LDA), Support Vector Machine (SVM) and Classification And Regression Tree (CART) used in the classification mode. Read a detailed explanation of MNIST classification with LDA, SVM & CART in terms of eigenvalue problems here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/MNIST_classifier_LDA_SVM_CART_582HW4 . Python implementation of DMD:
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Video Background Subtraction With Dynamic Mode Decomposition

In this UW AMATH582 assignment we will subtract stationary background from videos of moving objects using Dynamic Mode Decomposition (DMD). We approximate the unknown dynamics with a linear system by organizing the image sequence into two sets related by the linear difference equation allowing us to solve for its matrix by least squares. We then discard the low frequency mode corresponding to the static background. Read a detailed explanation of Dynamic Mode Decomposition and its application to video background subtraction here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/video_background_dynamic_mode_decomposition_582HW5 . Python implementation of DMD:
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Music Scoring With STFT-Gabor Transform

In this UW AMATH582 assignment report we analyze samples of music and attempt to reproduce guitar and bass music scores from them. In order to accomplish this we carry out a time-frequency decomposition of the signal. For the latter, we make use of the Short Time Fourier Transform (STFT) also called Gabor transform. We use low, high or bandpass filters to isolate the bass or guitar from other sound. Read a detailed explanation of the short time Fourier (Gabor) transform and its application to music scoring here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/stft_gabor_music_scoring_582HW2 . Example source code:
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