Where Models Fail
A conceptual framework – three genuinely different ways a formalized generalization can miss reality A formalization can fail to match
M_Foundation, YEM
YEM
M_Foundation, YEM
Computation – Where the Model Becomes Real, and Where Machine Learning Breaks the Old Rules
A conceptual framework – the third leg, and the one place the earlier framework needs an extra clause Mathematics gives
M_Foundation, YEM
Statistics and Data – Uncertainty and Inference
A conceptual framework – where modeling ends and estimation begins With generalization and formalization established as modeling’s core act, statistics
M_Foundation, YEM
The Machinery – Mathematics as Structure and Logic
A conceptual framework – the vocabulary formalization needs before it can say anything precisely Generalization makes a claim. Formalization gives
M_Foundation, YEM
Generalization and Formalization
A conceptual framework – from a rough sense of a pattern to a structure you can compute with Modeling is
Approximations, YEM
Approximations using Newton’s method
First-Order Approximation A first-order approximation of a function $f(\mathbf({\text{x}}))$ at a point $\mathbf{x_0}$ uses the tangent line of the function
Approximations, YEM
Taylor Series
Derivation Using Taylor series:- 1-D Consider $f: R \to R$ and let us expand around a point $x_0$ (Quadratic) $$f(x)
Linear Algebra, YEM
Singular Value Decomposition (SVD)
Introduction Singular Value Decomposition (SVD) is a matrix factorization technique that is widely used in linear algebra, statistics, signal processing,
Linear Algebra, YEM
Matrices of derivatives
Gradient Consider a scalar valued function $f(\text{X})$ defined on $n$ variables; that is, $f: \mathbb{R}^n \to \mathbb{R}$, where $\text{X}=(x_1,x_2,\cdots, x_n)^T$
Linear Algebra, YEM
Eigen Values and Eigen Vectors
Eigen value An eigen value of a square matrix $A = [a_{ij}]$ is a scalar $\lambda$ such that there exists