Department of Applied Computational Mathematics and Statistics

For achieving a qualifying score on the appropriate Advanced Placement (AP) exam, students earn credit for this course as the exam credit equivalent of ACMS 10145. A conceptual introduction to the science of data for students of business. Descriptive statistics: graphical methods, measures of central tendency, spread, and association. Basic probability theory and probability models for random variables. Introduction to statistical inference: confidence intervals and hypothesis tests. Many examples will be based on real, current business and economics datasets. Calculations will be illustrated in Microsoft Excel.

A conceptual introduction to probability and statistics for students in the Glynn Honors program. The course will cover Probability: basic probability theory and probability models for random variables; Descriptive statistics: graphical and numerical summaries of data; and Statistical Inference: sampling distributions, confidence intervals, hypothesis tests and linear regression. Credit will not be given if the student takes both ACMS 10141 and either ACMS 10145 or 10140

A conceptual introduction to the science of data for students of business. Descriptive statistics: graphical methods, measures of central tendency, spread, and association. Basic probability theory and probability models for random variables. Introduction to statistical inference: confidence intervals and hypothesis tests. Many examples will be based on real, current business and economics datasets. Calculations will be illustrated in Microsoft Excel.

The goal of this course is to give students an introduction to a variety of the most commonly used statistical tools. A hands-on approach with real data gathered from many disciplines will be followed. Topics include inferences based on two samples, analysis of variance, simple linear regression, categorical data analysis, and non-parametric statistics. This course counts only as general elective credit for students in the College of Science.

Concepts and applications of limits, differentiation, optimization, introduction to integration, and the fundamental theorem of calculus. Concepts will be illustrated using visualizations and animations using a computer algebra system (CAS).

Concepts and applications of integration, geometric and power series, and introduction to complex numbers as time allows. Concepts will be illustrated using visualizations and animations using a computer algebra system (CAS).

Lab that supplements Applied Calculus I. Course is limited to students intending to major in ACMS

Lab that supplements Applied Calculus II. Course is limited to students intending to major in ACMS

This module will cover the following: 1. Data collection 2. Descriptive statistics 3. Probability 4. Discrete Probability Distributions 5. Normal Distribution 6. Statistical Estimation 7. Hypothesis Testing 8. Linear Regression 9. Linear Programming

This module provides an introduction to the statistical techniques used by economists to collect, present, and analyse numerical data to inform decision making in business and/or public policy.

An introduction to descriptive statistics, elementary probability theory and inferential statistics. Included are: mean, median, mode and standard deviation; probability distributions, binomial probabilities and the normal distribution; problems of estimation; hypothesis testing, and an introduction to simple linear regression.

This module introduces students to basic concepts and methods of statistics that will enable them to perform appropriate data analyses to uncover meaningful insights. The statistical software R is taught alongside the material to introduce statistical computing. Students will learn to load raw data, make numerical and graphical summaries of data, and conduct various estimation and testing procedures. Topics include programming in R, descriptive statistics, concepts of probability, random variables and probability distributions, sampling distribution, statistical estimation, hypothesis testing, linear regression, and applications to real-world problems.

Independent research with UWA Faculty writing Honors Student Paper. The topic is on observing Hidden Markov Models through simulations.

This module provides a basic introduction to the ideas of probability and how simple probability models can be applied in a number of contexts. The topics covered in the module are: Sources of data, sampling, experiments, random variation Exploring data - graphical and numerical summaries Basic notions of probability - sample spaces, events, combination of events, counting Conditional probability and independence, Bayes' Theorem Random variables and probability distributions Binomial and related probability distributions Poisson distribution for counts, events over time Expectation - mean and variance Bivariate distributions - marginal and conditional probabilities, correlation and independence Normal distribution - properties, use of tables, central limit theorem and approximations Confidence interval in one sample problems using classical and computational (i.e. bootstrap) approaches. Hypothesis testing - Introducing null and alternative hypotheses, type I and II errors and p-values. Simple linear regression model. Use of R programming language for data exploration and probability model calculations.

This module introduces the research process. Starting with the formulation of a research question, it covers completing a literature review, choosing an appropriate research design, data collection, data analysis and how to communicate research findings. Working in groups, students put this into practice through the design and implementation of research into the feasibility of a business idea. The final output is a research-based business plan for their chosen idea. Upon completing this course, students should have an understanding of the nature of the research process, drawing upon primary and secondary data sources; be able to locate, analyse and interpret quantitative and qualitative data; and to present the findings.

A course that teaches fundamentals in data analytics and SQL.

This course will examine current trends in data science, including those in big data analytics, and how it can be used to improve decision-making across different fields, such as business, economics, social and political sciences. We will investigate real-world examples and cases to place data science techniques in context and to develop data-analytic thinking. Students will be provided with a practical toolkit that will enable them to design and realize a data science project using statistical software Module Description: This module provides an introduction to modelling using differential equations, via the MATLAB software system. In addition to modelling, a core part of the course is to introduce good software design principals in MATLAB, through the use of functions. The course also covers exploratory data analysis, and initial data modelling tasks in MATLAB, to provide a contrast to the differential equations approach. Learning Outcomes Understand matrix manipulation using MATLAB Explain key MATLAB concepts: function design, element-wise operations, and logicals Design difference equations for modeling state changes Develop ODE models in MATLAB, including Newton's Law of Cooling Design simulation models using Simulink Introduce data science approaches in MATLAB..

This course provides academic credit for students pursuing unpaid internships external to Notre Dame and not directly supervised by ACMS faculty. To be eligible, internships must be meaningfully related to topics in Applied and Computational Mathematics and Statistics (ACMS) and must receive prior approval from the Director of Undergraduate Studies. Proposals must be submitted at least two weeks in advance of the internship by the student, with an offer letter from the internship employer submitted no later than 1 week into the internship. Students may enroll in this course multiple times for different approved internships. The requirements are that the student will be working 30 hours per week for at least 6 weeks and write a journal of activities, submitted regularly to the Director of Undergraduate Studies. A letter from the supervisor regarding satisfactory completion of work, and a 5-to-10-page reflection is also required at the completion of the internship.

This course provides academic credit for students engaged in research projects related to Applied and Computational Mathematics and Statistics (ACMS) that are not supervised by ACMS faculty. To be eligible, the research must be substantially connected to ACMS topics and approved in advance by the Director of Undergraduate Studies. This course may be repeated for credit with different approved research experiences, but a student can not earn more than 3 credits total for this course during the entire undergraduate period at ND.

This course will prepare students to understand call and put options, other financial derivatives and financial strategies such as bull spread, bear spread and others. Financial models such as the binomial model and Black Scholes will also be utilized and students will prepare their own models. Calculus will not be widely used, but an understanding is necessary. This course prepares students for the IFM actuarial exam and also uses present value concepts.

This course provides an engaging introduction to the fundamental principles behind managing financial uncertainty and making data-driven decisions in a world of risk. Students will be introduced to key actuarial and financial concepts, including the time value of money, probability theory, fundamental business principles, and the foundations of insurance and risk management. Emphasizing practical applications, the course introduces financial risk analysis and actuarial modeling using spreadsheet tools (eg MS Excel), and guest speakers from the industry will provide insights into real-world applications across various financial fields. Designed for students from diverse backgrounds, this course lays the groundwork for future study in actuarial science, finance, and data-driven decision-making, offering an interdisciplinary approach to understanding risk and financial uncertainty. While this course does not prepare the student directly for a specific actuarial credentialing exam it provides a strong grounding for later courses that will (eg Financial Math, Probability, etc)

In this course, you will learn the foundational skills necessary in R that will enable you to acquire and manipulate data, complete exploratory data analysis (EDA), and create visualizations to communicate your findings. Note: this course is delivered fully online. The course design combines required live weekly meetings online with self-scheduled lectures, problems, assignments, and interactive learning materials. To participate, students will need to have a computer with webcam, reliable internet connection, and a quiet place to participate in live sessions

In this course, you will learn the foundational skills necessary in Python that will enable you to acquire and manipulate data, model data for the purposes of scientific analysis, and create visualizations to communicate your findings. The course will introduce you to efficient scientific computing using NumPy. You will learn how to apply the pandas library to perform a variety of data manipulation tasks, including selecting, subsetting, combining, grouping, and aggregating data. You will also learn how to generate and customize visualizations with matplotlib. The course will give you the basic ideas and intuition behind modern data analysis methods and their applications, with a strong focus on a course project and weekly assignments. Note: this course is delivered fully online. The course design combines required live weekly meetings online with self-scheduled lectures, problems, assignments, and interactive learning materials. To participate, students will need to have a computer with webcam, reliable internet connection, and a quiet place to participate in live sessions.

This course is an introduction to computer programming using the Python programming language, with an emphasis on solving mathematical and statistical problems.

An introduction to the principles of statistical inference following a brief introduction to probability theory. This course does not count as a science or mathematics elective for mathematics majors. NOTE: Students may not take more than one of ACMS 20340, BIOS 40411 and MATH 20340. Not open to students who have taken MATH 30540.

Introduction to Numerical Analysis is an introductory course that integrates mathematics and scientific computing to address complex real-world problems. The course covers a wide range of topics, including root finding, solutions to linear systems of equations, polynomial interpolation, numerical differentiation and numerical integration, solutions to differential equations, eigenvalues and singular values, and numerical optimization.

An introduction to the methods of applied mathematics. Topics include: basic linear algebra, partial derivatives, Taylor and power series in multiple variables, Lagrange multipliers, multiple integrals, gradient and line integrals, Green's theorem, Stokes theorem and divergence, Fourier series and transforms, introduction to ordinary differential equations. Applications to real-world problems in science, engineering, the social sciences and business will be emphasized in this course and ACMS 20750. Computational methods will be taught. Credit is not given for both ACMS 20550 and PHYS 20451.

The objective of this class is to impart the fundamental knowledge in linear algebra and computational linear algebra that are needed to solve matrix algebra problems in application areas. Appropriate software packages will be used.

The fundamental methods of applied mathematics are continued in this course. Topics include: variational calculus, special functions, series solutions of ordinary differential equations (ODE), orthogonal functions in the solution of ODE, basic partial differential equations and modeling heat flow, vibrating string, and steady-state temperature. Topics in complex function theory include contour integrals, Laurent series and residue calculus, and conformal mapping. The course concludes with a basic introduction to probability and statistics. Credit is not given for both ACMS 20750 and PHYS 20452.

Lab for Scientific Computing with Python.

two Lab sessions for Statistics for Life Sciences (ACMS 20340)

Tutorial for Introduction to Applied Mathematics Methods I

Tutorial for Introduction to Applied Mathematics Methods II.

This course provides an engaging introduction to the fundamental principles behind managing financial uncertainty and making data-driven decisions in a world of risk. Students will be introduced to key actuarial and financial concepts, including the time value of money, probability theory, fundamental business principles, and the foundations of insurance and risk management. Emphasizing practical applications, the course introduces financial risk analysis and actuarial modeling using spreadsheet tools (eg MS Excel), and guest speakers from the industry will provide insights into real-world applications across various financial fields. Designed for students from diverse backgrounds, this course lays the groundwork for future study in actuarial science, finance, and data-driven decision-making, offering an interdisciplinary approach to understanding risk and financial uncertainty. While this course does not prepare the student directly for a specific actuarial credentialing exam it provides a strong grounding for later courses that will (eg Financial Math, Probability, etc)

In this course, you will learn the foundational skills necessary in R that will enable you to acquire and manipulate data, complete exploratory data analysis (EDA), and create visualizations to communicate your findings. This course is delivered fully online. The course design combines required live weekly meetings online with self-scheduled lectures, problems, assignments, and interactive learning materials. To participate, students will need to have a computer with webcam, reliable internet connection, and a quiet place to participate in live sessions. Students with other prerequisite courses or equivalent background preparation may enroll by permission of the instructor or permission of the Director of Undergraduate Studies.

This course is an introduction to graph theory. Graph theory is a field of mathematics that studies graphs. A graph is a way to represent relationships. For example, graphs can be used to represent a train map or a social network. Graphs and graph theory play an important role in computer science.

Theoretical-practical course focused on the fundamentals and techniques to collect, analyze and evaluate quantitative data from the health field necessary for evidence-based nursing practice and scientific research. Develop skills and abilities in the calculation, presentation and analysis of statistical data applied to health. The course uses student-centered methodologies and is assessed through written assessments.

This introductory module, which requires only elementary algebra, is designed to explain and illustrate the statistical ideas and techniques that are an essential skill for a biological scientist engaged in the conduct or interpretation of experimentation. You will discover different types of data distributions and the parameters that define them. You will see how statistics calculated from samples are related to true values in the population from which the sample was drawn. The basic idea of a significance test will be developed and used to adjudicate on the significance, or otherwise, of observed differences. You will also be introduced to the measurement and analysis of the association between variables.

Linear Algebra is an important tool commonly used in many fields, in not only mathematics but also natural sciences, engineering, etc. This course extends the contents in "Linear Algebra A/B" courses (provided majorly for 1st year students) and discusses advanced concepts of linear algebra, such as orthogonality, diagonalization, Singular Value Decomposition (SVD) of a matrix, Jordan canonical form, and their applications to real-world problems, etc.

This course is a continuation of the Financial Economics I material and is the second of a 2-course sequence that prepares students for the Society of Actuaries' Exam MFE (Models for Financial Economics). It is a core exam course for preparing students to become future actuaries. This course prepares students to apply mathematical models to financial assets and manage risk in an insurance setting. The second semester moves to corporate finance issues. This course counts as an ACMS elective

An introduction to the theory of probability and statistics, with applications to the computer sciences and engineering. Topics include discrete and continuous random variables, joint probability distributions, the central limit theorem, point and interval estimation and hypothesis testing.

An introduction to the theory of probability, with applications to the physical sciences and engineering. Topics include discrete and continuous random variables, conditional probability and independent events, generating functions, special discrete and continuous random variables, laws of large numbers and the central limit theorem. The course emphasizes computations with the standard distributions of probability theory and classical applications of them.

An introduction to mathematical statistics. Topics include distributions involved in random sampling, estimators and their properties, confidence intervals, hypothesis testing including the goodness-of-fit test and contingency tables, the general linear model and analysis of variance.

An introduction to mathematical statistics. Topics include distributions involved in convergence concepts, estimators and their properties, confidence intervals, hypothesis testing, and linear models and estimation by least squares.

Introduction to statistical methods with an emphasis on analysis of data. Estimation of central values. Parametric and nonparametric hypothesis tests. Categorical data analysis. Simple and multiple regression. Introduction to time series. The SOA has approved this course for VEE credit in Applied Statistics.

The course serves as a preparation for first actuarial exam in financial math- ematics, known as Exam FM or Exam 2. The first part of the course deals with pricing of fixed income securities, such as bonds and annuities. The second part of the course can serve as an introduction to deriva- tive securities such as options and futures. Although the amount of material for both parts is almost the same, Exam FM devotes usually about 2/3 of its questions to Part 1. Therefore, about 2/3 of the course is devoted to Part 1.Topics covered: interest rates, annuities, loans and bonds, forwards, options, hedging, and swaps.

This course will teach students how to use Structured Query Language (SQL) to access and manipulate data stored in databases. Students will learn fundamental commands for filtering records, selecting variables, and merging data tables. Students will also create and modify relational database schemas to store structured data. Students will apply these skills in the context of solving a research question, using SQL to obtain the appropriate data set, and then creating an appropriate analysis or visualization.

In this course, students will learn different methods for designing experiments, analyzing the data, and presenting the results. Topics will include ANOVA, Completely Randomized Designs, Block Designs, Factorial Designs, Split-Plot Designs, Nested Designs, fixed and random effects, contrasts, and covariates.

Lab for Statistical Methods & Data Analysis I

Mathematical Analysis is a large and important branch of modern mathematics with a long history. The core notion is that of a limit, which gives the "long-term" behaviour of some process, or computes what happens as one variable gets closer and closer to a given value. The limit notion is behind many other fundamental concepts, such as continuity, derivatives, integrals, and so on. Historically, such ideas were discussed and used since at least the 17th century (for example, in connection with mathematical physics or statistics). However, they were not precisely understood; the working definitions of the day typically involved somewhat vague ideas of "infinitely small" or "infinitely large" quantities, and these can lead to confusion. We will follow in the footsteps of the pioneering mathematicians of the 19th century and formally define limits of sequences of real numbers. We will then use these definitions to rigorously deduce important properties of sequences, series and so on, which make up the bedrock of Real Analysis. Topics investigated will include: The Completeness Axiom, Sequences, Series, Absolute and Conditional Convergence of Series, Power series, Countability of sets, Continuity and properties of continuous functions, the Boundedness Theorem and the Intermediate Value Theorem.

Taught at a host institution. STAT 20060 at UCD; This module introduces the basic concepts of statistical modelling, which particular emphasis on engineeringapplications.Strong emphasis is placed on using the material covered in problem-solvingscenarios.The main sections of the course are:1) Descriptive StatisticsMean, median, mode, range, standard deviation, interquartile range, percentiles.2) Graphical MethodsPie charts, bar graphs, histograms, stem-and-leaf plots, cumulative frequencycurves, Venn diagrams.3) Laws of ProbabilityLaw of total probability, additive rule, multiplicative rule, mutually exclusive events,dependent and independent events, conditional probability, combinations rule,permutations rule, mean and variance of functions of random variables.4) Discrete DistributionsDiscrete random variables, E(X) and Var(X) for X discrete, binomial distribution,poisson distribution, hypergeometric distribution.5) Continuous DistributionsContinuous random variables, density functions, cumulative density functions, E(X)and Var(X) for X continuous, uniform distribution, exponential distribution, normaldistribution, Z values, standard normal distribution, Student's t-distribution.6) Confidence Intervals and Hypothesis TestingSampling distributions, biased and unbiased estimators, significance level, CentralLimit Theorem, large sample confidence interval for a population mean, small sampleconfidence interval for a population mean, large sample confidence interval for apopulation proportion, sample size calculations.7) RegressionCorrelation coefficient, residuals, simple linear regression, correlation and causation,coefficient of determination, making predictions from the regression equation.In addition students are required to complete a sequence of computer laboratorysessions using an interactive package that allows them to simulate commonprobability problems; and use Microsoft Excel to analyse data and perform regressionanalysis. Also taken in Heidelberg. National University of Singapore: This course introduces students to basic probability theory and statistical inference. Topics include basic concepts of probability, conditional probability, independence, random variables, joint and marginal distributions, mean and variance, some common probability distributions, sampling distributions, estimation and hypothesis testing based on a normal population. Students will learn basic probability theory and statistics. Including conditional probability, random variables, mean and variance, and more. Sydney, Australia (SY): https://www.sydney.edu.au/units/STAT2011/2026-S1C-ND-CC This unit offers a foundational introduction to probability theory and statistical inference, focusing on key concepts such as random variables and widely used probability distributions¿including the Binomial, Hypergeometric, Poisson, Normal, Geometric, and Gamma distributions. Students will engage with univariate data analysis techniques and gain practical skills in modelling variability using real-world data. Core estimation methods, including the method of moments, maximum likelihood estimation, and associated inference procedures are introduced in a rigorous yet accessible manner. Weekly computer laboratory sessions provide hands-on experience with statistical software for simulation, distribution fitting, and computational methods such as the bootstrap. By the end of the unit, students will have developed essential statistical modelling competencies, equipping them for further study in advanced statistical analysis and data science.

1. Descriptive Statistics 2. Introduction to Probability Theory 3. Random discrete variables 4. Random continuous variables 5. Regression 6. Statistic Inference

Summarising and displaying statistical data in R; Introduction to probability: discrete sample spaces; axioms; addition and multiplication laws; conditional probability and independence; reliability of systems; Bayes theorem; Discrete Random Variables: Bernouilli, hypergeometric, binomial, geometric and Poisson distributions; expectation; Sampling Inspection Schemes: Single and double sampling; operating characteristic function; average outgoing quality; consumers and producers risks. Continuous Random Variables: Uniform, exponential and normal distributions; normal approximation to binomial. Tchebechevs and Markovs inequalities Aims: To introduce the basic probability concepts and their applications to computer disciples; To provide an understanding of discrete and continuous distributions; To cover the essentials of the statistical computing system R. To introduce the essentials of statistical analysis using R.

The primary objective of 3E3 is to provide a secure and accessible grounding for all sophister engineering students in probability and statistics. The module equips them with consistent methods for reasoning amid the uncertainties they encounter in their professional practice. In this way, the module supports decision-making in the uncertain contexts of real engineering practice.

The aim of this module is to give a thorough grounding in probability, statistics and calculus of several variables as required for the successful understanding and solution of problems in science. Students will learn how mathematics can be used as a tool for solving scientific problems and a language for communicating information. This is a know-how and skills module. Students will participate in the following learning activities: Lectures: Students will attend two one-hour lectures per week. These lectures are designed to introduce learners to the mathematical principles and problem solving techniques that underpin this module. Tutorials: Each student will attend one one-hour tutorial per week. Problem sheets based on lecture content are distributed to the students and they are strongly advised to attempt all tutorial questions in advance of the tutorial.Reading: Students are expected to fully utilise the textbooks recommended.

This unit introduces fundamental concepts of probability and probabilistic methods and provides tools for understanding, addressing and solving problems in a wide range of areas including science, engineering and finance. The basic concepts cover (pairs of) discrete and continuum random variables, their distributions and properties of random variables; independence and the conditional probability of multivariate distributions, sums of random variables, laws of large numbers and the Central Limit Theorem. This unit also covers discrete and continuous random processes and their properties that are particularly useful in science, engineering and finance. Statistical computing will form an essential part in testing the theoretical ideas, understanding and interpreting them in real and simulated scenarios.

University College Dublin: This module is an introduction to probability, with examples and applications. Topics covered will include: outcomes, events, and probability; independence; random variables and distributions; expected value, moments and variance; permutations and combinations; binomial, multinomial and Poisson distributions; conditional probability; continuous distributions; law of averages; central limit theorem. __________________________________________________________________________________________________________________ This unit develops mathematical methods essential in the study of probability together with the distribution theory required for a study of statistical inference. Topics include random variables and their distributions; joint and conditional distributions; a survey of common distributions and some of their applications; the Poisson process and related distributions; convergence of random variables and the central limit theorem; and an introduction to Markov chains

Taught at a host institution. STAT 20100 Inferential Statistics at UCD; Continuous bivariate and multivariate distributions. Covariance and correlation. Chebyshev inequality. Law of Large Numbers Theory of Estimation.Method of moments and maximum likelihood. Point and interval estimationHypothesis Testing. Simple and Composite Hypotheses. Neyman Pearson Lemma and applications. Likelihood ratio tests. Bayesian statistical inference. Loss functions Normal/Normal, Binomial/Beta and Exponential/Gamma models. Probability generating functions. Taught at Trinity - Dublin - ST 1252 Introductin to Statistics II at Trinity College Dublin.On successful completion of this module students should: have a strong grasp of the fundamental statistical ideas of significance tests and confidence intervals, which underpin statistical analysis, be able to apply simple statistical methods to practical problems, be able to explain why statistical methods are so widely applied in both the natural and social sciences, engineering and business, have a sound basis for developing their knowledge of more advanced statistical ideas and methods. Hong Kong University Course Description: Emphasis is on the two major areas of statistical analysis: estimation and hypothesis testing. Through the disciplines of statistical modelling, inference and decision making, students will be equipped with both quantitative skills and qualitative perceptions essential for making rigorous statistical analysis of real-life data.

An introduction to the ideas of statistical inference from a mathematical perspective. Topics covered include: populations and samples, properties of estimators, likelihood functions, hypothesis testing and construction of tests.

This course discusses about various statistical methods (e.g., regression analysis) for conducting empirical studies in management research. The course emphasizes the combination of theory and practice. Through the study of this course, students will master various commonly used statistical analysis methods in management research (for example, regression analysis).

This course is an introduction to statistical ideas and methods. The fundamental concepts are introduced in the context of a series of practical problems of varying complexity. The theory will be illustrated by examples from biology, engineering, industry, medicine and the social sciences. Topics covered by ST1252 will include: Statistical variation and parameter estimation; Statistical tests and their properties; Design and analysis of comparative studies for both binary and continuous variables; Introductions to Analysis of Variance (ANOVA), regression and contingency tables.

This course will teach students how to use Structured Query Language (SQL) to access and manipulate data stored in databases. Students will learn fundamental commands for filtering records, selecting variables, and merging data tables. These skills will be applied in the context of solving statistical problems in which students are presented with a research question, use SQL to obtain the appropriate data set, and then use the data to create an appropriate visualization and/or conduct a statistical inference to answer the question.

This course will attempt to give a sufficiently detailed explanation of at least a few of the most common AI techniques. We will focus on supervised machine learning in general and deep learning in particular.

The course covers and Introduction to Data Science and Study Design; Techniques in Exploratory Data Analysis; Two sample comparisons for continuous variables; Comparing three or more means using ANOVA. An introduction to inference in simple linear regression; Residual diagnostics for testing regression assumptions; Basic concepts in machine learning and an introduction to classification trees.

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Statistics and data analytics consulting provide an important framework for many individuals to seek assistant with statistics and data-driven problems. This unit of study will provide students with an opportunity to gain real-life experience in statistical consulting or work with collaborative (interdisciplinary) research. In this unit, you will have an opportunity to have practical experience in a consultation setting with real clients. You will also apply your statistical knowledge in a diverse collection of consulting projects while learning project and time management skills. In this unit you will need to identify and place the client's problem into an analytical framework, provide a solution within a given time frame and communicate your findings back to the client. All such skills are highly valued by employers. This unit will foster the expertise needed to work in a statistical consulting firm or data analytical team which will be essential for data-driven professional and research pathways in the future.

Course that covers the basics of programming from the lens of statistics/informatics, then introduces the R-programming language. Additionally, there are laboratory sessions for completing programming assignments in the R language conducting statistical analysis of online economic databases.

Course that covers the basics of programming from the lens of statistics/informatics, then introduces the R-programming language. Additionally, there are laboratory sessions for completing programming assignments in the R language conducting statistical analysis of online economic databases.

This course provides students with exposure to real world actuarial science projects, which involve substantial use of probability concepts and financial mathematics throughout. This course will be created in conjunction with an industry partner. Case studies and projects will vary by semester. This course counts for ACMS elective.

An introduction to the mathematical foundations of cryptography. Topics include: number theory and basic algebraic structures, select pre-quantum cryptosystems, lattice problems, lattice-based cryptosystems, reduction algorithms, signature schemes, and zero knowledge protocols. Python will be used for the implementation of cryptographic algorithms.

This course presents a variety of topics associated with programming for scientific computing. Students will be introduced to programming tools that are widely used in scientific computing and data science, as well as learn when and how to use these tools for data visualization, data analysis, and machine learning. The course will also teach students how to program with databases.

This course covers fundamental material necessary for using high performance computing in science and engineering. There is a special emphasis on algorithm development, computer implementation, and the application of these methods to specific problems in science and engineering.

Algorithms and Data Structures for Scientific Computing covers the essential theoretical background for reasoning about algorithms, efficiency and data structures in computation. Students will gain practical experience implementing and applying such algorithms and data structures, with an emphasis on problems that arise in mathematical and scientific contexts.

An introduction to the numerical solution of ordinary and partial differential equations. Topics include the finite difference method, projection methods, cubic splines, interpolation, numerical integration methods, analysis of numerical errors, numerical linear algebra and eigenvalue problems, and continuation methods.

Complex analysis is a core part of applied and computational mathematics. Asymptotic methods for evaluation of functions and integrals, special functions (Gamma, elliptic, Bessel, ...), and conformal mappings arise naturally in applications, e.g., in the solution of physical models from electromagnetism, optics, tumor growth, fluid flow... In this course, an introduction to complex analysis will be given with special regard to those topics occurring in modeling and computation.

This course provides a practical introduction to models, algorithms, and modern software for large-scale numerical optimization, especially for decision-making in engineering and business contexts. Topics include (nonconvex) nonlinear programming, deterministic global optimization, integer programming, dynamic optimization, and stochastic programming. Multi-objective optimization, optimization with embedded machine learning models as constraints, optimal experiment design, optimization for statistical inference, and mathematical programs with complementarity constraints may be covered based on time and student interests. The class is designed for advanced undergraduate/graduate engineering, science, mathematics, business, and statistics students who wish to incorporate computation optimization methods into their research. The course begins with an introduction to modeling and the Python-based Pyomo computational environment. Optimization theory and algorithms are emphasized throughout the semester.

An introduction to the finite element method with applications to structural analysis, heat flow, fluid mechanics, and coupled multiphysics problems. Basics of linear and nonlinear finite element technology (theory and implementation) for continuum problems and engineering structures (bar, beams, frames, plates). Students will build their own finite element code and learn to use commercial software.

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects - that it brings researchers from many disciplines together with a common language. A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical rule which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist trained only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. This course will be self-contained.

Artificial neural networks are a class of machine learning algorithms inspired by biological neural networks in the brain. In recent years, great strides in the theory and application of artificial neural networks have made them one of the most powerful and popular choices for many machine learning applications. This course will cover the underlying theory and practice of using neural networks for machine learning problems, beginning with simple networks for linear and logistic regression and building up to deep convolutional neural networks. Students will learn to build and train artificial neural networks in Python using the popular PyTorch software package. Students should be comfortable with linear algebra, calculus, and probability/statistics. Experience with Python will not be assumed, but some previous programming experience will be helpful.

Introductory course on applied mathematics and computational modeling with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class in the end of the course.

This course will introduce students to some of the most common computational and mathematical models used in neuroscience. In addition to developing a deeper understanding of some biological processes in the brain, students will learn mathematical and computational approaches to studying dynamical systems and modeling physical phenomena. The course is appropriate as an elective for Neuroscience or ACMS majors. The course assumes some experience with linear algebra and probability or statistics, but does not assume any background in biology or neuroscience. Some programming experience is helpful, but not necessary. After completing the course, students will be able to create mathematical models of neural systems, simulate these models in Python, and use mathematical techniques to study the models. Students will also obtain some limited experience with analyzing neural data.

An introduction to partial differential equations. Topics include Fourier series, solutions of boundary value problems for the heat equation, wave equation and Laplace's equation, Fourier transforms, and applications to solving heat, wave and Laplace's equations in unbounded domains.

Stochastic modeling is a technique of presenting data or predicting outcomes that takes into account a certain degree of randomness, or unpredictability. Topics include (i) Short Review of Probability - Major discrete and continuous distributions, properties of random variables. (ii) Conditional probability and conditional expectation, sums of random variables, martingales. (iii) Introduction to Discrete Markov Chains - Transition probability matrix of a Markov chain, some Markov chain models, first step analysis, the absorbing Markov chains, various types and classifications of Markov chains. (iv) Long Run (asymptotic) Behavior of Markov Chains: Limiting distribution, the classification of states, irreducible Markov chains, periodicity of Markov chains, recurrent and transient states, the basic limit theorem of Markov chains. (v) Poisson Processes - The Poisson distribution and the Poisson process, the law of rare events, distributions associated with the Poisson process, the Uniform distribution and Poisson processes. (vi) Continuous Time Markov Chains - Pure birth and death processes and it's limiting behavior. (vii) Introduction to Brownian Motion, Drift and Diffusion, Geometric Brownian motion, Ornstein-Uhlenbeck process and it's long run behavior. (viii) Monte Carlo Simulations for Diffusion.

This course will develop practical techniques for the simulation of stochastic systems. Stochastic models (as opposed to deterministic) do not produce the same outcome from identical input parameters. Extreme weather, the outcome of sporting events, or the fluctuation of asset prices observed in the stock market are examples of natural and human systems that we wish to understand, but that are governed largely by stochastic (or random) processes. This course will develop a suite of computational methods, collectively called Stochastic Simulation Algorithms (SSAs), for understanding the range, or distribution, of outcomes that can arise from stochastic models. The course will emphasize practical implementation and benchmarking of algorithms in Python.

Selected Topics in Applied and Computational Mathematics

This is an introductory and applied course in time series analysis. Popular time series models and computational techniques for model estimation, diagnostic and forecasting will be discussed. Although the book focuses on financial data sets, other data sets, such as climate data, earthquake data and biological data, will also be included and discussed within the same theoretical framework.

This course introduces advanced statistical methods used in biological and biomedical research. Topics include study designs commonly used in health research including case-control, cross-sectional, prospective and retrospective studies; statistical analysis of different data types arising from biological and health research including Gaussian data, categorical data, count data, survival data, correlated/clustered data models, and diagnostic tests. All statistical methods are illustrated with examples from the biology and health sciences. Students are expected to have basic knowledge in R programming, probabilities and distribution theory, descriptive statistics, statistical inferences including hypothesis testing and estimation, and working knowledge of linear regression, before they can register for the course. Upon completion of the course, students are able to recognize and give examples of different types of data arising in biological and health studies, and apply appropriate methods to analyze such data.

The course aims at providing the foundations of methods for spatio-temporal models for environmental Statistics. The main topic covered will be Gaussian processes in space and time and related notions of stationarity, co-variance functions and optimal interpolation (kriging). Exploratory analysis and inference, with particular emphasis on approximation methods for very large data sets, will be covered in the second part of the course. The last part of the course will be either dedicated to more methodological (e.g. asymptotics for spatial processes) or applied problems (e.g. climate model emulation, air pollution, visualization in Virtual Reality), depending on the class interests.

This course introduces a set of the most popular methods for addressing four central problems in machine learning: dimension reduction, regression, classification, and clustering, with the greatest emphasis on classification. Dimension reduction aims to reduce the dimensions of high-dimensional data to make it easier to visualize and analyze. Regression involves predicting a continuous variable, classification concerns predicting a categorical variable, and clustering seeks to divide data into useful or meaningful groups. The topics likely to be covered in the course include the following, although we may need to omit a few each year due to limited lecture time: principal component analysis, multidimensional scaling, tSNE, UMAP, k-means clustering, hierarchical clustering, nearest neighbor classifiers, linear/quadratic discriminant analysis, Naive Bayes, decision trees, and ensemble methods (bagging, random forest, boosting), as well as artificial neural networks. Deep learning will also be seriously discussed in this course, although, due to limited lecture time, we may focus only on a small, carefully selected set of topics, such as self-supervised learning, transformers, BERT, and GPT.

Given the growing volume and complexities of real-world data, successful deployment of data science pipelines into practice often require intangible factors, beyond the modeling, including careful formulation of the substantive problem of interest, non-trivial data pre-processing, powerful computational software, intuition into when and why models work or fail to work, effective visualization and communication of the results, awareness of the ethical consequences, and close collaborative efforts. Data Science in Practice: Tools and Applications explores these computational and critical thinking skills necessary to solve data science problems in real-world applications. To this end, this course will guide students through a series of hands-on learning projects based on real scientific datasets. Through these real-data projects, students will gain experience with data pre-processing, advanced visualization tools, unsupervised and supervised learning tasks, interpretability tools, and advanced computing tools that are commonly used in industry such as git, distributed computing, reproducible documentation, and open-source software packaging.

This course provides an overview of probabilistic graphical models and their applications to various scientific fields. Probabilistic graphical models allow us to illustrate the dependence structure of numerous random variables by means of a graph, where nodes represent the random variables and edges or arrows connect pairs of nodes to describe their dependence structure. Topics of study include: foundations of statistics, linear dependence, nonlinear dependence, conditional independence, information theory, undirected and directed graphical models, inference of graphical models, model selection, high-dimensionality, sparsity, optimization, and statistical reproducibility. These topics will be discussed alongside real scientific data applications, including neuroscience, genomics, astronomy, forensic science, psychology, and other fields.

This course introduces basic computing methods for statistics. Topics are organized into two major parts: optimization and integral approximation. Optimization techniques are commonly used in statistics for finding maximum likelihood estimators, minimizing risks in a Bayesian decision problem, solving nonlinear least square problems, and a wide variety of other tasks all involving optimizations. Approximation of integrals is frequently required for Bayesian inference, since a posterior distribution may not belong to a familiar distributional family. Integral approximation is also useful in some maximum likelihood inference problems when the likelihood itself is a function of one or more integrals.

Selected advanced topics in Statistics. Possible topics include, but are not limited to, applied logistic and ordinal regression modeling including fitting, building, and interpreting regression models for binary and ordinal response variables, various modeling strategies addressing different sampling and experimental designs such as case-control studies and longitudinal data, advanced experimental designs, survey research, big data analysis, Bayesian analysis, survival analysis, spatial and longitudinal analysis, commonly-used nonparametric statistics, basics of robust statistics, tests of association in contingency tables, permutation tests, the bootstrap, introduction to data mining techniques, etc. Applications in a variety of fields such as medical biology, psychology, global health, psychiatry, etc will be introduced. The topic of the course could vary from one semester to another depending on the interests of the faculty member and the students. The course could potentially involve a student project in the area of the interests of the faculty member and could change from one semester to another. The course will count for science credit, ACMS elective credit as well as STAT major elective credit.

Polynomial interpolation and its applications in numerical integration, numerical differentiation, splines, and finite element methods for ODEs. Implementation of methods. (Language of instruction: English) Learning Outcomes Construct Lagrange and Hermite interpolating polynomials to a function/set of points Bound the error in polynomial interpolation Derive Cauchy's theorem Construct piecewise linear and cubic splines Derive formulas for Newton-Cotes quadrature in low degrees Derive formulas for Gaussian quadrature in low degrees Bound the error in Newton-Cotes and Gaussian quadrature Use the FEM to approximately solve ODEs Derive the system of equations of the FEM solution with piecewise linear basis functions Implementation of methods in MATLAB and/or Octave

In this course, we will learn to analyze and overcome the challenges of constructing often uncomputable analytic quantities when constrained by limitations introduced by the practical world.

This module provides an introduction to the theory of dynamical systems leading up to the concept of chaos. The course starts by considering one-dimensional flow, identifying fixed points, classifying stability, and introducing the saddle-node, transcritical and pitchfork bifurcations. We then progress to two-dimensional flows, and discuss classification of linear systems, methods of plotting the phase plane, and limit cycles. We will consider the behaviour of conservative systems, reversible systems and Lienard systems and discuss the Poincare-Bedixson theorem and Hopf bifurcations. Finally, we will look at chaotic systems, and study one-dimensional maps, fractals and strange attractors.

IR- Dublin, Ireland, UCD: Waves and oscillations are present in the modelling of many physical systems. An understanding is gained using a variety of physical examples including mechanical oscillators, waves on a string, acoustic waves, surface water waves and electromagnetic waves. Key mathematical techniques include Laplace transforms, normal mode analysis, Fourier analysis, Fourier transforms, method of characteristics. Content will include topics drawn from: simple harmonic oscillators, coupled oscillators, transverse standing waves, Longitudinal standing waves, sound waves in a gas, Fourier analysis, travelling waves, energy conservation, transmission lines, reflection and transmission at boundaries, wave pulses, Fourier transforms, dispersive waves, surface waves in water. Nonlinear kinematic waves, traffic flow.

Students will have ability to discuss and model simple versions of the following processes in time: Everyday examples of stochastic processes Understand and apply the Markov property Describe long run properties of Markov processes Deal with simple Markov processes in discrete time, continuous time and space

Students will have developed a sound knowledge and appreciation of the ideas and concepts related to modelling biological and ecological systems using continuous-time non-spatial models.

This unit introduces basic notions and applications of random processes, i.e. randomly evolving dynamical systems which exhibit dependencies over time. The core material focuses on the simplest and most important class of such processes, namely, Markov-dependent processes which occupy discrete states and evolve in discrete or continuous time. Fundamental concepts and properties are introduced and explained (rather than proved). This unit emphasises modelling and numerical computation.

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. This unit will establish basic properties of discrete-time Markov chains including random walks and branching processes. This unit will derive key results of Poisson processes and simple continuous-time Markov chains. This unit will investigate simple queuing theory. This unit will also introduce basic concepts of Brownian motion and martingales. Throughout the unit, various illustrative examples are provided in modelling and analysing problems of practical interest. By completing this unit, you will develop an essential basis for further studies stochastic analysis, stochastic differential equations, stochastic control, financial mathematics and statistical inference.

This module revolves around mathematical and computational theory from the field of combinatorial optimisation. In addition to specific optimisation problems, we will study general problem solving strategies. The module also serves as an introduction to computational complexity theory. This theory will allow us to gauge the difficulty of problems and rank algorithms according to their asymptotic efficiency. National University of Singapore (SI): This course teaches us various methods to solve linear optimization problems.

This module introduces the fundamental principles of scientific computing, object oriented programming, and the development of mathematical software. Key ideas in object oriented programming, such as encapsulation, polymorphism and inheritance, and presented in the context of solving problems that arise in numerical and computational modelling.

This module will study the practical analysis of spatial data. It commences with a discussion on different types of spatial data. Spatial point processes, random fields and spatial models for lattice data are discussed. There is a strong focus on the practical and computational aspects of model fitting and modern, computationally efficient model fitting software is introduced.

The module introduces the concept of financial risk and discusses the importance of its regulation. The emphasis is laid on the popular risk measure Value at Risk (VaR). After a brief discussion on asset returns, various modeling techniques - ranging from the simple Historical Simulation to the more advanced ARMA and GARCH models - are presented and applied for the calculation of VaR using real financial data. The aim of this module is to provide a solid basis in risk management for those students considering a career in finance.

The aim of this course is to introduce Machine Learning from the point of view of modern optimisation and approximation theory.

Mission Statement of the Actuarial Profession: To develop the role and enhance the reputation of the actuarial profession in providing expert and relevant solutions to financial and business problems, especially those involving uncertain future events. This module will show how to solve financial and business problems in actuarial science. The module is divided into five units: 10. Investments 11. Simple Compound Interest Problems 12. Forward Contracts and the No Arbitrage Assumption 13. Term Structure of Interest Rates 14. Stochastic Interest Rate Models. Use of Excel spreadsheets and simple VBA programmes to carry out data analyses and financial modelling.

On successful completion of this module, students will be able to: LO1: Define and describe the different patterns that can be found in times series and propose algorithms and statistical models that are suitable for their analysis. LO2: Program, analyse and select the best model for forecasting. LO3: Interpret output of data analysis performed by a computer statistics package. LO4: Compute predictions with their confidence intervals using the selected model.

This course introduces fundamental concepts of Bayesian statistics and illustrates how to apply them to various areas of scientific research. Probabilistic programming languages (WinBugs, JAGS and/or Stan) are introduced, and their interfaces to the statistical computing and graphics environment R are discussed. These languages are used, either directly or via their R interface, to fit statistical models within a Bayesian framework to real-world examples from many disciplines such as engineering, science (e.g. agricultural, biological, environmental, medical and physical), social sciences, economics, finance and astronomy.

Statistical methods are used to analyse data in a wide variety of fields (e.g. engineering, medicine, agriculture, business, economics, psychology, genetics, criminology, the social sciences). While statistical theory can be helpful in analysing such data, its direct application may be limited by practical problems. For example, some of the data may be missing, some observations may be inconsistent with the rest of the data, the standard assumptions (e.g. normality) may fail, and the standard methods may not answer the important questions. The best way to learn how to deal with these practical problems is to gain experience in analysing real data. This unit provides that experience through case studies and projects. The emphasis is on applying statistical methods to interesting practical problems rather than on the theory behind the methods. The unit covers applications of a number of widely used statistical techniques selected from generalised linear models, nonlinear regression models, advanced regression topics, survival analysis, non-parametric statistics, multivariate analysis, and time series analysis. Furthermore, throughout the unit a large emphasis is placed on data visualisation techniques.

The normal linear model: use of matrices, least squares and maximum likelihood estimation, normal equations, distribution theory for the normal model, hypothesis tests and confidence intervals. Practical aspects of linear models and analysis of variance: multiple regression, categorical variables and interactions, blocks and treatments, orthogonality, model selection (including AIC, but not the derivation of AIC), fit criteria, use of residuals, outliers, leverage, model interpretation. Normal linear mixed models, hierarchical models. Generalised Linear Models: logistic regression, linear exponential families and generalized linear models, scale parameter, link functions, canonical link. Maximum likelihood fitting. Iteratively reweighted least squares. Asymptotic theory: statement and applications to inference, analysis of deviance, model checking, residuals.

The normal linear model: use of matrices, least squares and maximum likelihood estimation, normal equations, distribution theory for the normal model, hypothesis tests and confidence intervals. Practical aspects of linear models and analysis of variance: multiple regression, categorical variables and interactions, blocks and treatments, orthogonality, model selection (including AIC, but not the derivation of AIC), fit criteria, use of residuals, outliers, leverage, model interpretation. Normal linear mixed models, hierarchical models. Generalised Linear Models: logistic regression, linear exponential families and generalized linear models, scale parameter, link functions, canonical link. Maximum likelihood fitting. Iteratively reweighted least squares. Asymptotic theory: statement and applications to inference, analysis of deviance, model checking, residuals.

IT On successful completion of this module, students will be able to: LO1. Understand and put into practice merging and cleaning of datasets LO2. Understand and put into practice use of inbuilt and user written functions LO3. Understand the graphics capabilities of R and use these methods to visualise data and create reports LO4. Understand the use of clustering methods and their application to different data types LO5. Understand the use of Generalized linear models and their application to different data types LO6. Understand methods of Classification and their application to different data types LO7. Principles of effective report writing and how to present research and analysis. BP Statistics teaches us how to behave in the face of uncertainties, according to the famous mathematician, Abraham Wald. Theoretically, we will learn strategies of treating chances in everyday life, where our inference is based on a randomly selected sample from a large population, and hence, we intensively use concepts of probability (laws of large numbers, Bayes rule). Estimation theory and hypothesis testing are introduced on a theoretical basis, but applications are also discussed. Methods of supervised and unsupervised learning are outlined; former include regression and discriminant analysis, while latter ones factor and cluster analysis. The students are also made capable of selecting the methods and making inference on real-life data, while outputs of a program package for medical data are analyzed.

This course consists principally of an introduction to the theory and applications of partial differential equations. Topics covered include the heat equation, the wave equation, Laplace's equation, and a brief introduction to the special theory of relativity.

Synopsis Smoothing methods (local polynomials). Nonparametric inference (bandwidth and Generalised Cross Validation). Multivariate smoothers and Generalised Additive Models. Inference using simulation methods. Monte-Carlo Tests. Permutation tests. Rank statistics. Bootstrapping. Hidden Markov Models: specification. Forward-backward algorithm. Kalman filter.

Readings not covered in the curriculum which relate to the student's area of interest.

Research in collaboration with members of the faculty. Evaluation of performance will be accomplished through regular discussions with the faculty member in charge of the course.

To produce a thesis that describes work of an undergraduate research project. The undergraduate thesis must go beyond what is found in an undergraduate course, and present a novel approach to a subject.