The composition of portfolios is one of the most fundamental and important methods in financial engineering, used to control the risk of investments. This book provides a comprehensive overview of statistical inference for portfolios and their various applications. A variety of asset processes are introduced, including non-Gaussian stationary processes, nonlinear processes, non-stationary processes, and the book provides a framework for statistical inference using local asymptotic normality (LAN). The approach is generalized for portfolio estimation, so that many important problems can be covered. This book can primarily be used as a reference by researchers from statistics, mathematics, finance, econometrics, and genomics. It can also be used as a textbook by senior undergraduate and graduate students in these fields.

Despite the unobserved components model (UCM) having many advantages over more popular forecasting techniques based on regression analysis, exponential smoothing, and ARIMA, the UCM is not well known among practitioners outside the academic community. Time Series Modelling with Unobserved Components rectifies this deficiency by giving a practical overview of the UCM approach, covering some theoretical details, several applications, and the software for implementing UCMs. The book's first part discusses introductory time series and prediction theory. Unlike most other books on time series, this text includes a chapter on prediction at the beginning because the problem of predicting is not limited to the field of time series analysis. The second part introduces the UCM, the state space form, and related algorithms. It also provides practical modeling strategies to build and select the UCM that best fits the needs of time series analysts. The third part presents real-world applications, with a chapter focusing on business cycle analysis and the construction of band-pass filters using UCMs. The book also reviews software packages that offer ready-to-use procedures for UCMs as well as systems popular among statisticians and econometricians that allow general estimation of models in state space form. This book demonstrates the numerous benefits of using UCMs to model time series data. UCMs are simple to specify, their results are easy to visualize and communicate to non-specialists, and their forecasting performance is competitive. Moreover, various types of outliers can easily be identified, missing values are effortlessly managed, and working contemporaneously with time series observed at different frequencies poses no problem.

Principles of Copula Theory explores the state of the art on copulas and provides you with the foundation to use copulas in a variety of applications. Throughout the book, historical remarks and further readings highlight active research in the field, including new results, streamlined presentations, and new proofs of old results. After covering the essentials of copula theory, the book addresses the issue of modeling dependence among components of a random vector using copulas. It then presents copulas from the point of view of measure theory, compares methods for the approximation of copulas, and discusses the Markov product for 2-copulas. The authors also examine selected families of copulas that possess appealing features from both theoretical and applied viewpoints. The book concludes with in-depth discussions on two generalizations of copulas: quasi- and semi-copulas. Although copulas are not the solution to all stochastic problems, they are an indispensable tool for understanding several problems about stochastic dependence. This book gives you the solid and formal mathematical background to apply copulas to a range of mathematical areas, such as probability, real analysis, measure theory, and algebraic structures.

Winner of the 2017 De Groot Prize awarded by the International Society for Bayesian Analysis (ISBA) A relatively new area of research, adversarial risk analysis (ARA) informs decision making when there are intelligent opponents and uncertain outcomes. Adversarial Risk Analysis develops methods for allocating defensive or offensive resources against intelligent adversaries. Many examples throughout illustrate the application of the ARA approach to a variety of games and strategic situations. Focuses on the recent subfield of decision analysis, ARA Compares ideas from decision theory and game theory Uses multi-agent influence diagrams (MAIDs) throughout to help readers visualize complex information structures Applies the ARA approach to simultaneous games, auctions, sequential games, and defend-attack games Contains an extended case study based on a real application in railway security, which provides a blueprint for how to perform ARA in similar security situations Includes exercises at the end of most chapters, with selected solutions at the back of the book The book shows decision makers how to build Bayesian models for the strategic calculation of their opponents, enabling decision makers to maximize their expected utility or minimize their expected loss. This new approach to risk analysis asserts that analysts should use Bayesian thinking to describe their beliefs about an opponent's goals, resources, optimism, and type of strategic calculation, such as minimax and level-k thinking. Within that framework, analysts then solve the problem from the perspective of the opponent while placing subjective probability distributions on all unknown quantities. This produces a distribution over the actions of the opponent and enables analysts to maximize their expected utilities.

1. This book is applicable to courses across the social and behavioral science on a wide range of quantitative methods courses. 2. The book is based solely on Stata for EFA - one of the top statistics software packages used in behavioral and social sciences. 3. Clear step-by-step guidance combined with screen shots to show how to apply EFA to real data.

From the Foreword: "Big Data Management and Processing is [a] state-of-the-art book that deals with a wide range of topical themes in the field of Big Data. The book, which probes many issues related to this exciting and rapidly growing field, covers processing, management, analytics, and applications... [It] is a very valuable addition to the literature. It will serve as a source of up-to-date research in this continuously developing area. The book also provides an opportunity for researchers to explore the use of advanced computing technologies and their impact on enhancing our capabilities to conduct more sophisticated studies." ---Sartaj Sahni, University of Florida, USA "Big Data Management and Processing covers the latest Big Data research results in processing, analytics, management and applications. Both fundamental insights and representative applications are provided. This book is a timely and valuable resource for students, researchers and seasoned practitioners in Big Data fields. --Hai Jin, Huazhong University of Science and Technology, China Big Data Management and Processing explores a range of big data related issues and their impact on the design of new computing systems. The twenty-one chapters were carefully selected and feature contributions from several outstanding researchers. The book endeavors to strike a balance between theoretical and practical coverage of innovative problem solving techniques for a range of platforms. It serves as a repository of paradigms, technologies, and applications that target different facets of big data computing systems. The first part of the book explores energy and resource management issues, as well as legal compliance and quality management for Big Data. It covers In-Memory computing and In-Memory data grids, as well as co-scheduling for high performance computing applications. The second part of the book includes comprehensive coverage of Hadoop and Spark, along with security, privacy, and trust challenges and solutions. The latter part of the book covers mining and clustering in Big Data, and includes applications in genomics, hospital big data processing, and vehicular cloud computing. The book also analyzes funding for Big Data projects.

* A useful guide to financial product modeling and to minimizing business risk and uncertainty * Looks at wide range of financial assets and markets and correlates them with enterprises' profitability * Introduces advanced and novel machine learning techniques in finance such as Support Vector Machine, Neural Networks, Random Forest, K-Nearest Neighbors, Extreme Learning Machine, Deep Learning Approaches and applies them to analyze finance data sets * Real world applicable examples to further understanding

Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels in Renewal Risk Models is written for academics and practitioners who are concerned about potential weaknesses of the Solvency II regulatory system. It is also intended for readers who are interested in pure and applied probability, have a taste for classical and asymptotic analysis, and are motivated to delve into rather intensive calculations. The formal prerequisite for this book is a good background in analysis. The desired prerequisite is some degree of probability training, but someone with knowledge of the classical real-variable theory, including asymptotic methods, will also find this book interesting. For those who find the proofs too complicated, it may be reassuring that most results in this book are formulated in rather elementary terms. This book can also be used as reading material for basic courses in risk measures, insurance mathematics, and applied probability. The material of this book was partly used by the author for his courses in several universities in Moscow, Copenhagen University, and in the University of Montreal. Features Requires only minimal mathematical prerequisites in analysis and probability Suitable for researchers and postgraduate students in related fields Could be used as a supplement to courses in risk measures, insurance mathematics and applied probability.

The beginning of the age of artificial intelligence and machine learning has created new challenges and opportunities for data analysts, statisticians, mathematicians, econometricians, computer scientists and many others. At the root of these techniques are algorithms and methods for clustering and classifying different types of large datasets, including time series data. Time Series Clustering and Classification includes relevant developments on observation-based, feature-based and model-based traditional and fuzzy clustering methods, feature-based and model-based classification methods, and machine learning methods. It presents a broad and self-contained overview of techniques for both researchers and students. Features Provides an overview of the methods and applications of pattern recognition of time series Covers a wide range of techniques, including unsupervised and supervised approaches Includes a range of real examples from medicine, finance, environmental science, and more R and MATLAB code, and relevant data sets are available on a supplementary website

"Prof. Nitis Mukhopadhyay and Prof. Partha Pratim Sengupta, who edited this volume with great attention and rigor, have certainly carried out noteworthy activities." - Giovanni Maria Giorgi, University of Rome (Sapienza) "This book is an important contribution to the development of indices of disparity and dissatisfaction in the age of globalization and social strife." - Shelemyahu Zacks, SUNY-Binghamton "It will not be an overstatement when I say that the famous income inequality index or wealth inequality index, which is most widely accepted across the globe is named after Corrado Gini (1984-1965). ... I take this opportunity to heartily applaud the two co-editors for spending their valuable time and energy in putting together a wonderful collection of papers written by the acclaimed researchers on selected topics of interest today. I am very impressed, and I believe so will be its readers." - K.V. Mardia, University of Leeds Gini coefficient or Gini index was originally defined as a standardized measure of statistical dispersion intended to understand an income distribution. It has evolved into quantifying inequity in all kinds of distributions of wealth, gender parity, access to education and health services, environmental policies, and numerous other attributes of importance. Gini Inequality Index: Methods and Applications features original high-quality peer-reviewed chapters prepared by internationally acclaimed researchers. They provide innovative methodologies whether quantitative or qualitative, covering welfare economics, development economics, optimization/non-optimization, econometrics, air quality, statistical learning, inference, sample size determination, big data science, and some heuristics. Never before has such a wide dimension of leading research inspired by Gini's works and their applicability been collected in one edited volume. The volume also showcases modern approaches to the research of a number of very talented and upcoming younger contributors and collaborators. This feature will give readers a window with a distinct view of what emerging research in this field may entail in the near future.

Data Stewardship for Open Science: Implementing FAIR Principles has been written with the intention of making scientists, funders, and innovators in all disciplines and stages of their professional activities broadly aware of the need, complexity, and challenges associated with open science, modern science communication, and data stewardship. The FAIR principles are used as a guide throughout the text, and this book should leave experimentalists consciously incompetent about data stewardship and motivated to respect data stewards as representatives of a new profession, while possibly motivating others to consider a career in the field. The ebook, avalable for no additional cost when you buy the paperback, will be updated every 6 months on average (providing that significant updates are needed or avaialble). Readers will have the opportunity to contribute material towards these updates, and to develop their own data management plans, via the free Data Stewardship Wizard.

In many branches of science relevant observations are taken sequentially over time. Bayesian Analysis of Time Series discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters. Features Presents a comprehensive introduction to the Bayesian analysis of time series. Gives many examples over a wide variety of fields including biology, agriculture, business, economics, sociology, and astronomy. Contains numerous exercises at the end of each chapter many of which use R and WinBUGS. Can be used in graduate courses in statistics and biostatistics, but is also appropriate for researchers, practitioners and consulting statisticians. About the author Lyle D. Broemeling, Ph.D., is Director of Broemeling and Associates Inc., and is a consulting biostatistician. He has been involved with academic health science centers for about 20 years and has taught and been a consultant at the University of Texas Medical Branch in Galveston, The University of Texas MD Anderson Cancer Center and the University of Texas School of Public Health. His main interest is in developing Bayesian methods for use in medical and biological problems and in authoring textbooks in statistics. His previous books for Chapman & Hall/CRC include Bayesian Biostatistics and Diagnostic Medicine, and Bayesian Methods for Agreement.

This book provides in-depth analyses on accounting methods of GDP, statistic calibers and comparative perspectives on Chinese GDP. Beginning with an exploration of international comparisons of GDP, the book introduces the theoretical backgrounds, data sources, algorithms of the exchange rate method and the purchasing power parity method and discusses the advantages, disadvantages, and the latest developments in the two methods. This book further elaborates on the reasons for the imperfections of the Chinese GDP data including limitations of current statistical techniques and the accounting system, as well as the relatively confusing statistics for the service industry. The authors then make suggestions for improvement. Finally, the authors emphasize that evaluation of a country's economy and social development should not be solely limited to GDP, but should focus more on indicators of the comprehensive national power, national welfare, and the people's livelihood. This book will be of interest to economists, China-watchers, and scholars of geopolitics.

New statistical methods and future directions of research in time series

A Course in Time Series Analysis demonstrates how to build time series models for univariate and multivariate time series data. It brings together material previously available only in the professional literature and presents a unified view of the most advanced procedures available for time series model building. The authors begin with basic concepts in univariate time series, providing an up-to-date presentation of ARIMA models, including the Kalman filter, outlier analysis, automatic methods for building ARIMA models, and signal extraction. They then move on to advanced topics, focusing on heteroscedastic models, nonlinear time series models, Bayesian time series analysis, nonparametric time series analysis, and neural networks. Multivariate time series coverage includes presentations on vector ARMA models, cointegration, and multivariate linear systems. Special features include:

• Contributions from eleven of the world’s leading figures in time series
• Shared balance between theory and application
• Exercise series sets
• Many real data examples
• Consistent style and clear, common notation in all contributions
• 60 helpful graphs and tables

Requiring no previous knowledge of the subject, A Course in Time Series Analysis is an important reference and a highly useful resource for researchers and practitioners in statistics, economics, business, engineering, and environmental analysis.

The proliferation of financial derivatives over the past decades, options in particular, has underscored the increasing importance of derivative pricing literacy among students, researchers, and practitioners. Derivative Pricing: A Problem-Based Primer demystifies the essential derivative pricing theory by adopting a mathematically rigorous yet widely accessible pedagogical approach that will appeal to a wide variety of audience. Abandoning the traditional "black-box" approach or theorists' "pedantic" approach, this textbook provides readers with a solid understanding of the fundamental mechanism of derivative pricing methodologies and their underlying theory through a diversity of illustrative examples. The abundance of exercises and problems makes the book well-suited as a text for advanced undergraduates, beginning graduates as well as a reference for professionals and researchers who need a thorough understanding of not only "how," but also "why" derivative pricing works. It is especially ideal for students who need to prepare for the derivatives portion of the Society of Actuaries Investment and Financial Markets Exam. Features Lucid explanations of the theory and assumptions behind various derivative pricing models. Emphasis on intuitions, mnemonics as well as common fallacies. Interspersed with illustrative examples and end-of-chapter problems that aid a deep understanding of concepts in derivative pricing. Mathematical derivations, while not eschewed, are made maximally accessible. A solutions manual is available for qualified instructors. The Author Ambrose Lo is currently Assistant Professor of Actuarial Science at the Department of Statistics and Actuarial Science at the University of Iowa. He received his Ph.D. in Actuarial Science from the University of Hong Kong in 2014, with dependence structures, risk measures, and optimal reinsurance being his research interests. He is a Fellow of the Society of Actuaries (FSA) and a Chartered Enterprise Risk Analyst (CERA). His research papers have been published in top-tier actuarial journals, such as ASTIN Bulletin: The Journal of the International Actuarial Association, Insurance: Mathematics and Economics, and Scandinavian Actuarial Journal.

Solve the DVA/FVA Overlap Issue and Effectively Manage Portfolio Credit Risk Counterparty Risk and Funding: A Tale of Two Puzzles explains how to study risk embedded in financial transactions between the bank and its counterparty. The authors provide an analytical basis for the quantitative methodology of dynamic valuation, mitigation, and hedging of bilateral counterparty risk on over-the-counter (OTC) derivative contracts under funding constraints. They explore credit, debt, funding, liquidity, and rating valuation adjustment (CVA, DVA, FVA, LVA, and RVA) as well as replacement cost (RC), wrong-way risk, multiple funding curves, and collateral. The first part of the book assesses today's financial landscape, including the current multi-curve reality of financial markets. In mathematical but model-free terms, the second part describes all the basic elements of the pricing and hedging framework. Taking a more practical slant, the third part introduces a reduced-form modeling approach in which the risk of default of the two parties only shows up through their default intensities. The fourth part addresses counterparty risk on credit derivatives through dynamic copula models. In the fifth part, the authors present a credit migrations model that allows you to account for rating-dependent credit support annex (CSA) clauses. They also touch on nonlinear FVA computations in credit portfolio models. The final part covers classical tools from stochastic analysis and gives a brief introduction to the theory of Markov copulas. The credit crisis and ongoing European sovereign debt crisis have shown the importance of the proper assessment and management of counterparty risk. This book focuses on the interaction and possible overlap between DVA and FVA terms. It also explores the particularly challenging issue of counterparty risk in portfolio credit modeling. Primarily for researchers and graduate students in financial mathematics, the book is also suitable for financial quants, managers in banks, CVA desks, and members of supervisory bodies.

Statistics for Finance develops students' professional skills in statistics with applications in finance. Developed from the authors' courses at the Technical University of Denmark and Lund University, the text bridges the gap between classical, rigorous treatments of financial mathematics that rarely connect concepts to data and books on econometrics and time series analysis that do not cover specific problems related to option valuation. The book discusses applications of financial derivatives pertaining to risk assessment and elimination. The authors cover various statistical and mathematical techniques, including linear and nonlinear time series analysis, stochastic calculus models, stochastic differential equations, Ito's formula, the Black-Scholes model, the generalized method-of-moments, and the Kalman filter. They explain how these tools are used to price financial derivatives, identify interest rate models, value bonds, estimate parameters, and much more. This textbook will help students understand and manage empirical research in financial engineering. It includes examples of how the statistical tools can be used to improve value-at-risk calculations and other issues. In addition, end-of-chapter exercises develop students' financial reasoning skills.

This book is about learning from data using the Generalized Additive Models for Location, Scale and Shape (GAMLSS). GAMLSS extends the Generalized Linear Models (GLMs) and Generalized Additive Models (GAMs) to accommodate large complex datasets, which are increasingly prevalent. In particular, the GAMLSS statistical framework enables flexible regression and smoothing models to be fitted to the data. The GAMLSS model assumes that the response variable has any parametric (continuous, discrete or mixed) distribution which might be heavy- or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution (location, scale, shape) can be modelled as linear or smooth functions of explanatory variables. Key Features: Provides a broad overview of flexible regression and smoothing techniques to learn from data whilst also focusing on the practical application of methodology using GAMLSS software in R. Includes a comprehensive collection of real data examples, which reflect the range of problems addressed by GAMLSS models and provide a practical illustration of the process of using flexible GAMLSS models for statistical learning. R code integrated into the text for ease of understanding and replication. Supplemented by a website with code, data and extra materials. This book aims to help readers understand how to learn from data encountered in many fields. It will be useful for practitioners and researchers who wish to understand and use the GAMLSS models to learn from data and also for students who wish to learn GAMLSS through practical examples.

Machine learning (ML) is progressively reshaping the fields of quantitative finance and algorithmic trading. ML tools are increasingly adopted by hedge funds and asset managers, notably for alpha signal generation and stocks selection. The technicality of the subject can make it hard for non-specialists to join the bandwagon, as the jargon and coding requirements may seem out of reach. Machine Learning for Factor Investing: R Version bridges this gap. It provides a comprehensive tour of modern ML-based investment strategies that rely on firm characteristics. The book covers a wide array of subjects which range from economic rationales to rigorous portfolio back-testing and encompass both data processing and model interpretability. Common supervised learning algorithms such as tree models and neural networks are explained in the context of style investing and the reader can also dig into more complex techniques like autoencoder asset returns, Bayesian additive trees, and causal models. All topics are illustrated with self-contained R code samples and snippets that are applied to a large public dataset that contains over 90 predictors. The material, along with the content of the book, is available online so that readers can reproduce and enhance the examples at their convenience. If you have even a basic knowledge of quantitative finance, this combination of theoretical concepts and practical illustrations will help you learn quickly and deepen your financial and technical expertise.

Oil and gas industries apply several techniques for assessing and mitigating the risks that are inherent in its operations. In this context, the application of Bayesian Networks (BNs) to risk assessment offers a different probabilistic version of causal reasoning. Introducing probabilistic nature of hazards, conditional probability and Bayesian thinking, it discusses how cause and effect of process hazards can be modelled using BNs and development of large BNs from basic building blocks. Focus is on development of BNs for typical equipment in industry including accident case studies and its usage along with other conventional risk assessment methods. Aimed at professionals in oil and gas industry, safety engineering, risk assessment, this book Brings together basics of Bayesian theory, Bayesian Networks and applications of the same to process safety hazards and risk assessment in the oil and gas industry Presents sequence of steps for setting up the model, populating the model with data and simulating the model for practical cases in a systematic manner Includes a comprehensive list on sources of failure data and tips on modelling and simulation of large and complex networks Presents modelling and simulation of loss of containment of actual equipment in oil and gas industry such as Separator, Storage tanks, Pipeline, Compressor and risk assessments Discusses case studies to demonstrate the practicability of use of Bayesian Network in routine risk assessments

High-Performance Computing for Big Data: Methodologies and Applications explores emerging high-performance architectures for data-intensive applications, novel efficient analytical strategies to boost data processing, and cutting-edge applications in diverse fields, such as machine learning, life science, neural networks, and neuromorphic engineering. The book is organized into two main sections. The first section covers Big Data architectures, including cloud computing systems, and heterogeneous accelerators. It also covers emerging 3D IC design principles for memory architectures and devices. The second section of the book illustrates emerging and practical applications of Big Data across several domains, including bioinformatics, deep learning, and neuromorphic engineering. Features Covers a wide range of Big Data architectures, including distributed systems like Hadoop/Spark Includes accelerator-based approaches for big data applications such as GPU-based acceleration techniques, and hardware acceleration such as FPGA/CGRA/ASICs Presents emerging memory architectures and devices such as NVM, STT- RAM, 3D IC design principles Describes advanced algorithms for different big data application domains Illustrates novel analytics techniques for Big Data applications, scheduling, mapping, and partitioning methodologies Featuring contributions from leading experts, this book presents state-of-the-art research on the methodologies and applications of high-performance computing for big data applications. About the Editor Dr. Chao Wang is an Associate Professor in the School of Computer Science at the University of Science and Technology of China. He is the Associate Editor of ACM Transactions on Design Automations for Electronics Systems (TODAES), Applied Soft Computing, Microprocessors and Microsystems, IET Computers & Digital Techniques, and International Journal of Electronics. Dr. Chao Wang was the recipient of Youth Innovation Promotion Association, CAS, ACM China Rising Star Honorable Mention (2016), and best IP nomination of DATE 2015. He is now on the CCF Technical Committee on Computer Architecture, CCF Task Force on Formal Methods. He is a Senior Member of IEEE, Senior Member of CCF, and a Senior Member of ACM.

Praise for the first edition: [This book] reflects the extensive experience and significant contributions of the author to non-linear and non-Gaussian modeling. ... [It] is a valuable book, especially with its broad and accessible introduction of models in the state-space framework. -Statistics in Medicine What distinguishes this book from comparable introductory texts is the use of state-space modeling. Along with this come a number of valuable tools for recursive filtering and smoothing, including the Kalman filter, as well as non-Gaussian and sequential Monte Carlo filters. -MAA Reviews Introduction to Time Series Modeling with Applications in R, Second Edition covers numerous stationary and nonstationary time series models and tools for estimating and utilizing them. The goal of this book is to enable readers to build their own models to understand, predict and master time series. The second edition makes it possible for readers to reproduce examples in this book by using the freely available R package TSSS to perform computations for their own real-world time series problems. This book employs the state-space model as a generic tool for time series modeling and presents the Kalman filter, the non-Gaussian filter and the particle filter as convenient tools for recursive estimation for state-space models. Further, it also takes a unified approach based on the entropy maximization principle and employs various methods of parameter estimation and model selection, including the least squares method, the maximum likelihood method, recursive estimation for state-space models and model selection by AIC. Along with the standard stationary time series models, such as the AR and ARMA models, the book also introduces nonstationary time series models such as the locally stationary AR model, the trend model, the seasonal adjustment model, the time-varying coefficient AR model and nonlinear non-Gaussian state-space models. About the Author: Genshiro Kitagawa is a project professor at the University of Tokyo, the former Director-General of the Institute of Statistical Mathematics, and the former President of the Research Organization of Information and Systems.

This book provides an introduction to the use of statistical concepts and methods to model and analyze financial data. The ten chapters of the book fall naturally into three sections. Chapters 1 to 3 cover some basic concepts of finance, focusing on the properties of returns on an asset. Chapters 4 through 6 cover aspects of portfolio theory and the methods of estimation needed to implement that theory. The remainder of the book, Chapters 7 through 10, discusses several models for financial data, along with the implications of those models for portfolio theory and for understanding the properties of return data. The audience for the book is students majoring in Statistics and Economics as well as in quantitative fields such as Mathematics and Engineering. Readers are assumed to have some background in statistical methods along with courses in multivariate calculus and linear algebra.