## Related Courses

The following are some Applied Mathematics courses which involve concepts and principles related to Pattern Theory.

**APMA 0180. Modeling the World with Mathematics: An Introduction for Non-Mathematicians**

Mathematics is the foundation of our technological society and most of its powerful ideas are quite accessible. This course will explain some of these using historical texts and Excel. Topics include the predictive power of 'differential equations' from the planets to epidemics, oscillations and music, chaotic systems, randomness and the atomic bomb. Prerequisite: some knowledge of calculus.

**APMA 0**410. Mathematical Methods in the Brain Sciences

Basic mathematical methods commonly used in the cognitive and neural sciences. Topics include: introduction to differential equations, emphasizing qualitative behavior; introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; and some elementary information theory. Examples from biology, psychology, and linguistics. Prerequisite: a course in integral and differential calculus.

**APMA 0**650. Essential Statistics

A first course in statistics emphasizing statistical reasoning and basic concepts. Comprehensive treatment of most commonly used statistical methods through linear regression. Elementary probability and the role of randomness. Data analysis and statistical computing using Excel. Examples and applications from the popular press and the life, social and physical sciences. No mathematical prerequisites beyond high school algebra.

**APMA **1070. Quantitative Models of Biological Systems

An intermediate course between Biomed 1100 and Biomed 2120 (Applied Mathematics 2220). Quantitative modeling techniques useful in molecular biology, physiology and ecology. Topics range from the subcellular level through the cellular level and the organ systems level to the level of the whole organism and to population dynamics and ecosystem analyses.

**APMA **1200. Operational Analysis: Probabilistic models

Methods of problem formulation and solution. Introduction to the theory of Markov chains, the probabilistic analog of a difference or differential equation. Birth-death statistical processes and their applications. Queuing, probabilistic service and waiting line theory. Sequential decision theory via the methods of dynamic programming. Prerequisite: APMA 1650, or MATH 1610, or equivalent.

**APMA **1210. Operational Analysis: Deterministic Methods

An introduction to the basic mathematical ideas and computational methods of optimization. Linear programming, the theory of optimal decision making under linear constraints on resources. Applications include decision theory in economics, transportation theory, optimal assignments, production and operations scheduling. Network modeling and flows. Integer programming. Prerequisites: an introduction to matrix calculations, such as APMA 0340 or MATH 0520.

**APMA **1650. Statistical Inference I

APMA 1650/1660 constitute an integrated first course in mathematical statistics. The first third of APMA 1650 is probability theory, and its last two thirds are statistics. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, large and small sample techniques for confidence intervals and hypothesis testing. Prerequisite: MATH 0100 or equivalent.

**APMA **1660. Statistical Inference II

Simple and multiple regression. Analysis of variance and covariance. The general linear model. Introduction to categorical and nonparametric data analysis. Prerequisites: APMA 1650 and some linear algebra.

**APMA **1670. Statistical Analysis of Time Series

Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of a time-series, but time domain methods are also covered. Prerequisite: elementary probability and statistics on the level of APMA 1650-1660. Offered in alternate years.

**APMA **1680. Nonparametric Statistics

A systematic treatment of the distribution-free alternatives to classical statistical tests. These non-parametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 1650 or equivalent. Offered in alternate years.

**APMA **1690. Computational Probability and Statistics

Examination of probability theory and statistical inference from the point of view of modern computing. Random number generation, Monte Carlo methods, simulation, and other special topics. Prerequisites: calculus, linear algebra, APMA 1650, or equivalent. Some experience with programming desirable.

**APMA **1710. Information Theory

Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for beginning graduate students and advanced undergraduates, offers a broad introduction to information theory and its applications: Entropy and information; lossless data compression, communication in the presence of noise, capacity, channel coding; source-channel separation; Gaussian channels; lossy data compression.

**APMA **2110. Real Analysis

This course provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.

**APMA **2120. Hilbert Spaces and Their Applications

A continuation of APMA 2110: The theory of Lp spaces, the geometric theory of Hilbert spaces, spectral theory and bounded and unbounded operators in Hilbert spaces, and applications to integral and partial differential equations.

**APMA **2630, 2640. Theory of Probability (MATH 2630, 2640)

A two-semester course in probability theory. Semester I includes an introduction to probability spaces and random variables, the theory of countable state Markov chains and renewable processes, laws of large numbers and the central limit theorems. Measure theory is first used near the end of the first semester (APMA 2110 may be taken concurrently). Semester II provides a rigorous mathematical foundation to probability theory and covers conditional probabilities and expectations, limit theorems for sums of random variables. martingales, ergodic theory, Brownian motion and an introduction to stochastic process theory.

**APMA **2660. Stochastic Processes

Topics in the theory on continuous parameter stochastic processes. The precise content varies from year to year, but generally includes selections from the following topics: second order stationary processes; ergodic processes and their applications; Markov processes, including jump processes and diffusions; applications to noise and communication theory.

**APMA **2670. Mathematical Statistics I

Topics in classical statistical inference: unbiased, maximum likelihood, minimax and equivariant estimators; Cramer-Rao inequality; confidence sets; hypothesis testing; likelihood ratio tests; large sample theory; consistency and asymptotic normality; Bayesian efficiency, and super-efficiency.

**APMA **2680. Mathematical Statistics II

Introduction to decision and game theories; admissibility; complete class theorems; the Bayesian approach to statistics; subjective and prior information; posterior distribution; Bayesian methods for point estimation, hypothesis testing, and multiple decision problems; Bayesian sequential analysis; the sequential likelihood tests; applications to classification and learning problems. Prerequisite: APMA 2670.

**APMA **2690, 2700. Topics in Statistics and its Applications

Advanced topics varying from year to year, including: non-parametric methods for density estimation, regression and prediction in time-series; cross-validation and adaptive smoothing techniques; bootstrap; recursive partitioning, projection-pursuit, ACE algorithm; non-parametric classification and clustering; stochastic Metropolis-type simulation and global optimization algorithms; Markov random fields and statistical mechanics; applications to image processing, speech recognition and neural networks.

**APMA **2720. Information Theory II

Information theory and its relationship with probability, statistics, and data compression. Entropy. The Shannon-McMillan-Breiman theorem. Shannon's source coding theorems. Statistical inference; hypothesis testing; model selection criteria; the minimum description length principle. Information-theoretic proofs of limit theorems in probability: Law of large numbers, central limit theorem, large deviations, Markov chain convergence, Poisson approximation, the Hewitt-Savage 0-1 law.