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 0410. 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 0650. Essential Statistics
A first course in probability and statistics emphasizing statistical reasoning and basic concepts. Topics include visual and numerical summaries of data, representative and non-representative samples, elementary discrete probability theory, the normal distribution, sampling variability, elementary statistical inference, measures of association. Examples and applications from the popular press and the life, social and physical sciences. Not calculus-based.
APMA 1070. Quantitative Models of Biological Systems
Quantitative dynamic models help understand problems in biology and there has been rapid progress in recent years. This course provides an introduction to the concepts and techniques, with applications to population dynamics, infectious diseases, enzyme kinetics, and cellular biology. Additional topics covered will vary. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: APMA 0350 or equivalent.
APMA 1080/2080 Inference in Genomics and Molecular Biology
Massive quantities of fundamental biological and geological sequence data have emerged. The goal of this course is to enable students to construct and apply probabilistic models to draw inferences from sequence data on problems novel to them. Statistical topics: Bayesian inferences; estimation; hypothesis testing and false discovery rates; statistical decision theory; change point algorithm; hidden Markov models; Kalman filters; and significances in high dimensions. Prerequisites: APMA 1650 or equivalent; APMA 0160 or CSCI 0111 or equivalent.
[(APMA 1650 or APMA 1655 or MATH 1610 or CSCI 1450) and (APMA 0160 or APMA 0200 or APMA 1160 or APMA 1170 or APMA 1180 or APMA 1210 or APMA 1690 or APMA 1740 or APMA 1860 or APMA 1720 or CSCI 0111 or CSCI 0150 or CSCI 0170 or CSCI 0190 or CLPS 0950)] or minimum score of WAIVE in ‘Graduate Student PreReq’
APMA 1200. Operational Analysis: Probabilistic models
Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisites: APMA 1650 or equivalent; MATH 520 or equivalent.
APMA 1210. Operational Analysis: Deterministic Methods
An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming. Prerequisites: MATH 0100 or equivalent; MATH 520 or equivalent; APMA 0160 or CSCI 0111 or equivalent.
[(MATH 0100 or MATH 0170 or MATH 0180 or MATH 0190 or MATH 0200 or MATH 0350 or minimum score of 4 in ‘AP Calculus BC’) and (MATH 0520 or MATH 0540) and (APMA 0160 or APMA 0200 or APMA 1160 or APMA 1170 or APMA 1180 or APMA 1210 or APMA 1690 or APMA 1740 or APMA 1860 or APMA 1720 or CSCI 0111 or CSCI 0150 or CSCI 0170 or CSCI 0190 or CLPS 0950)] or minimum score of WAIVE in ‘Graduate Student PreReq’
APMA 1650. Statistical Inference I
APMA 1650 is an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. Prerequisites: MATH 0100 or equivalent.
APMA 1660. Statistical Inference II
APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year’s course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisites: APMA 1650 or equivalent; MATH 0520 or equivalent.
[(APMA 1650 or APMA 1655 or MATH 1610 or CSCI 1450) and (MATH 0520 or MATH 0540)] or minimum score of WAIVE in ‘Graduate Student PreReq’
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 mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, dimensionality reduction. Prerequisites: APMA 1650 or equivalent; programming experience is recommended.
APMA 1710. Information Theory
Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information, lossless data compression, communication in the presence of noise, channel capacity, channel coding, source-channel separation, lossy data compression. Prerequisites: APMA 1650 or equivalent.
APMA 1650 or APMA 1655 or MATH 1610 or CSCI 1450 or minimum score of WAIVE in ‘Graduate Student PreReq’
APMA 1941. An introduction to Pattern Theory
This course is an introduction to some probabilistic models and numerical algorithms that model some aspects of human cognition. The class begins with stochastic models of language introduced by Shannon and develops related models for speech and vision. The classes stresses mathematical foundations, in particular the role of information theory in developing Bayesian models and the increasing importance of dynamics in several algorithms, especially in optimization and deep learning. Student assessment will be based on computational projects that implement the principles discussed in lecture.
APMA 2110. Real Analysis
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)
Part one of a two semester course that provides an introduction to probability theory based on measure theory. The first semester (APMA 2630) covers the following topics: countable state Markov chains, review of real analysis and metric spaces, probability spaces, random variables and measurable functions, Borel-Cantelli lemmas, weak and strong laws of large numbers, conditional expectation and beginning of discrete time martingale theory. Prerequisites—undergraduate probability and analysis, co-requisite—graduate real analysis. Enrollment is limited to Graduate Level students.
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
This course presents advanced statistical inference methods. Topics include: foundations of statistical inference and comparison of classical, Bayesian, and minimax approaches, point and set estimation, hypothesis testing, linear regression, linear classification and principal component analysis, MRF, consistency and asymptotic normality of Maximum Likelihood and estimators, statistical inference from noisy or degraded data, and computational methods (E-M Algorithm, Markov Chain Monte Carlo, Bootstrap). Prerequisite: APMA 2630 or equivalent.
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.