DOCSLIB.ORG

Mathematica Tutorial: Core Language

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematica Tutorial: Core Language Wolfram M athematica ® Tutorial Collection CORE LANGUAGE For use with Wolfram Mathematica® 7.0 and later. For the latest updates and corrections to this manual: visit reference.wolfram.com For information on additional copies of this documentation: visit the Customer Service website at www.wolfram.com/services/customerservice or email Customer Service at [email protected] Comments on this manual are welcomed at: [email protected] Printed in the United States of America. 15 14 13 12 11 10 9 8 7 6 5 4 3 2 ©2008 Wolfram Research, Inc. All rights reserved. No part of this document may be reproduced or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright holder. Wolfram Research is the holder of the copyright to the Wolfram Mathematica software system ("Software") described in this document, including without limitation such aspects of the system as its code, structure, sequence, organization, “look and feel,” programming language, and compilation of command names. Use of the Software unless pursuant to the terms of a license granted by Wolfram Research or as otherwise authorized by law is an infringement of the copyright. Wolfram Research, Inc. and Wolfram Media, Inc. ("Wolfram") make no representations, express, statutory, or implied, with respect to the Software (or any aspect thereof), including, without limitation, any implied warranties of merchantability, interoperability, or fitness for a particular purpose, all of which are expressly disclaimed. Wolfram does not warrant that the functions of the Software will meet your requirements or that the operation of the Software will be uninterrupted or error free. As such, Wolfram does not recommend the use of the software described in this document for applications in which errors or omissions could threaten life, injury or significant loss. Mathematica, MathLink, and MathSource are registered trademarks of Wolfram Research, Inc. J/Link, MathLM, .NET/Link, and webMathematica are trademarks of Wolfram Research, Inc. Windows is a registered trademark of Microsoft Corporation in the United States and other countries. Macintosh is a registered trademark of Apple Computer, Inc. All other trademarks used herein are the property of their respective owners. Mathematica is not associated with Mathematica Policy Research, Inc. Contents Building Up Calculations Using Previous Results . 1 Defining Variables . 2 Values for Symbols . 4 The Four Kinds of Bracketing in Mathematica . 8 Sequences of Operations . 8 Lists Making Lists of Objects . 10 Collecting Objects Together . 11 Making Tables of Values . 12 Manipulating Elements of Lists . 16 Vectors and Matrices . 17 Getting Pieces of Lists . 23 Testing and Searching List Elements . 25 Adding, Removing and Modifying List Elements . 26 Combining Lists . 28 Lists as Sets . 29 Rearranging Lists . 30 Grouping and Combining Elements of Lists . 31 Ordering in Lists . 32 Rearranging Nested Lists . 33 Manipulating Lists Constructing Lists . 35 Manipulating Lists by Their Indices . 37 Nested Lists . 44 Partitioning and Padding Lists . 47 Sparse Arrays . 51 Expressions Everything Is an Expression . 57 The Meaning of Expressions . 59 Special Ways to Input Expressions . 60 Parts of Expressions . 62 Manipulating Expressions like Lists . 65 Expressions as Trees . 66 Levels in Expressions . 68 Patterns Introduction to Patterns . 72 Finding Expressions That Match a Pattern . 75 Naming Pieces of Patterns . 77 Specifying Types of Expression in Patterns . 79 Putting Constraints on Patterns . 80 Patterns Involving Alternatives . 86 Pattern Sequences . 87 Flat and Orderless Functions . 88 Functions with Variable Numbers of Arguments . 92 Optional and Default Arguments . 94 Setting Up Functions with Optional Arguments . 96 Repeated Patterns . 99 Verbatim Patterns . 101 Patterns for Some Common Types of Expression . 101 An Example: Defining Your Own Integration Function . 104 Transformation Rules and Definitions Applying Transformation Rules . 107 Manipulating Sets of Transformation Rules . 111 Making Definitions . 112 Special Forms of Assignment . 114 Making Definitions for Indexed Objects . 118 Making Definitions For Functions . 120 The Ordering of Definitions . 122 Immediate and Delayed Definitions . 124 Functions That Remember Values They Have Found . 129 Associating Definitions with Different Symbols . 132 Defining Numerical Values . 136 Modifying Built-in Functions . 138 Manipulating Value.
Load more

Recommended publications
  • Stan: a Probabilistic Programming Language
    JSS Journal of Statistical Software January 2017, Volume 76, Issue 1. doi: 10.18637/jss.v076.i01 Stan: A Probabilistic Programming Language Bob Carpenter Andrew Gelman Matthew D. Hoffman Columbia University Columbia University Adobe Creative Technologies Lab Daniel Lee Ben Goodrich Michael Betancourt Columbia University Columbia University Columbia University Marcus A. Brubaker Jiqiang Guo Peter Li York University NPD Group Columbia University Allen Riddell Indiana University Abstract Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propa- gation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces sup- port sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.
    [Show full text]
  • Stan: a Probabilistic Programming Language
    JSS Journal of Statistical Software MMMMMM YYYY, Volume VV, Issue II. http://www.jstatsoft.org/ Stan: A Probabilistic Programming Language Bob Carpenter Andrew Gelman Matt Hoffman Columbia University Columbia University Adobe Research Daniel Lee Ben Goodrich Michael Betancourt Columbia University Columbia University University of Warwick Marcus A. Brubaker Jiqiang Guo Peter Li University of Toronto, NPD Group Columbia University Scarborough Allen Riddell Dartmouth College Abstract Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.2.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propa- gation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line, through R using the RStan package, or through Python using the PyStan package. All three interfaces support sampling and optimization-based inference. RStan and PyStan also provide access to log probabilities, gradients, Hessians, and data I/O. Keywords: probabilistic program, Bayesian inference, algorithmic differentiation, Stan.
    [Show full text]
  • Advancedhmc.Jl: a Robust, Modular and Efficient Implementation of Advanced HMC Algorithms
    2nd Symposium on Advances in Approximate Bayesian Inference, 20191{10 AdvancedHMC.jl: A robust, modular and efficient implementation of advanced HMC algorithms Kai Xu [email protected] University of Edinburgh Hong Ge [email protected] University of Cambridge Will Tebbutt [email protected] University of Cambridge Mohamed Tarek [email protected] UNSW Canberra Martin Trapp [email protected] Graz University of Technology Zoubin Ghahramani [email protected] University of Cambridge & Uber AI Labs Abstract Stan's Hamilton Monte Carlo (HMC) has demonstrated remarkable sampling robustness and efficiency in a wide range of Bayesian inference problems through carefully crafted adaption schemes to the celebrated No-U-Turn sampler (NUTS) algorithm. It is challeng- ing to implement these adaption schemes robustly in practice, hindering wider adoption amongst practitioners who are not directly working with the Stan modelling language. AdvancedHMC.jl (AHMC) contributes a modular, well-tested, standalone implementation of NUTS that recovers and extends Stan's NUTS algorithm. AHMC is written in Julia, a modern high-level language for scientific computing, benefiting from optional hardware acceleration and interoperability with a wealth of existing software written in both Julia and other languages, such as Python. Efficacy is demonstrated empirically by comparison with Stan through a third-party Markov chain Monte Carlo benchmarking suite. 1. Introduction Hamiltonian Monte Carlo (HMC) is an efficient Markov chain Monte Carlo (MCMC) algo- rithm which avoids random walks by simulating Hamiltonian dynamics to make proposals (Duane et al., 1987; Neal et al., 2011). Due to the statistical efficiency of HMC, it has been widely applied to fields including physics (Duane et al., 1987), differential equations (Kramer et al., 2014), social science (Jackman, 2009) and Bayesian inference (e.g.
    [Show full text]
  • Maple Advanced Programming Guide
    Maple 9 Advanced Programming Guide M. B. Monagan K. O. Geddes K. M. Heal G. Labahn S. M. Vorkoetter J. McCarron P. DeMarco c Maplesoft, a division of Waterloo Maple Inc. 2003. ­ ii ¯ Maple, Maplesoft, Maplet, and OpenMaple are trademarks of Water- loo Maple Inc. c Maplesoft, a division of Waterloo Maple Inc. 2003. All rights re- ­ served. The electronic version (PDF) of this book may be downloaded and printed for personal use or stored as a copy on a personal machine. The electronic version (PDF) of this book may not be distributed. Information in this document is subject to change without notice and does not rep- resent a commitment on the part of the vendor. The software described in this document is furnished under a license agreement and may be used or copied only in accordance with the agreement. It is against the law to copy the software on any medium as specifically allowed in the agreement. Windows is a registered trademark of Microsoft Corporation. Java and all Java based marks are trademarks or registered trade- marks of Sun Microsystems, Inc. in the United States and other countries. Maplesoft is independent of Sun Microsystems, Inc. All other trademarks are the property of their respective owners. This document was produced using a special version of Maple that reads and updates LATEX files. Printed in Canada ISBN 1-894511-44-1 Contents Preface 1 Audience . 1 Worksheet Graphical Interface . 2 Manual Set . 2 Conventions . 3 Customer Feedback . 3 1 Procedures, Variables, and Extending Maple 5 Prerequisite Knowledge . 5 In This Chapter .
    [Show full text]
  • End-User Probabilistic Programming
    End-User Probabilistic Programming Judith Borghouts, Andrew D. Gordon, Advait Sarkar, and Neil Toronto Microsoft Research Abstract. Probabilistic programming aims to help users make deci- sions under uncertainty. The user writes code representing a probabilistic model, and receives outcomes as distributions or summary statistics. We consider probabilistic programming for end-users, in particular spread- sheet users, estimated to number in tens to hundreds of millions. We examine the sources of uncertainty actually encountered by spreadsheet users, and their coping mechanisms, via an interview study. We examine spreadsheet-based interfaces and technology to help reason under uncer- tainty, via probabilistic and other means. We show how uncertain values can propagate uncertainty through spreadsheets, and how sheet-defined functions can be applied to handle uncertainty. Hence, we draw conclu- sions about the promise and limitations of probabilistic programming for end-users. 1 Introduction In this paper, we discuss the potential of bringing together two rather distinct approaches to decision making under uncertainty: spreadsheets and probabilistic programming. We start by introducing these two approaches. 1.1 Background: Spreadsheets and End-User Programming The spreadsheet is the first \killer app" of the personal computer era, starting in 1979 with Dan Bricklin and Bob Frankston's VisiCalc for the Apple II [15]. The primary interface of a spreadsheet|then and now, four decades later|is the grid, a two-dimensional array of cells. Each cell may hold a literal data value, or a formula that computes a data value (and may depend on the data in other cells, which may themselves be computed by formulas).
    [Show full text]
  • An Introduction to Data Analysis Using the Pymc3 Probabilistic Programming Framework: a Case Study with Gaussian Mixture Modeling
    An introduction to data analysis using the PyMC3 probabilistic programming framework: A case study with Gaussian Mixture Modeling Shi Xian Liewa, Mohsen Afrasiabia, and Joseph L. Austerweila aDepartment of Psychology, University of Wisconsin-Madison, Madison, WI, USA August 28, 2019 Author Note Correspondence concerning this article should be addressed to: Shi Xian Liew, 1202 West Johnson Street, Madison, WI 53706. E-mail: [email protected]. This work was funded by a Vilas Life Cycle Professorship and the VCRGE at University of Wisconsin-Madison with funding from the WARF 1 RUNNING HEAD: Introduction to PyMC3 with Gaussian Mixture Models 2 Abstract Recent developments in modern probabilistic programming have offered users many practical tools of Bayesian data analysis. However, the adoption of such techniques by the general psychology community is still fairly limited. This tutorial aims to provide non-technicians with an accessible guide to PyMC3, a robust probabilistic programming language that allows for straightforward Bayesian data analysis. We focus on a series of increasingly complex Gaussian mixture models – building up from fitting basic univariate models to more complex multivariate models fit to real-world data. We also explore how PyMC3 can be configured to obtain significant increases in computational speed by taking advantage of a machine’s GPU, in addition to the conditions under which such acceleration can be expected. All example analyses are detailed with step-by-step instructions and corresponding Python code. Keywords: probabilistic programming; Bayesian data analysis; Markov chain Monte Carlo; computational modeling; Gaussian mixture modeling RUNNING HEAD: Introduction to PyMC3 with Gaussian Mixture Models 3 1 Introduction Over the last decade, there has been a shift in the norms of what counts as rigorous research in psychological science.
    [Show full text]
  • GNU Octave a High-Level Interactive Language for Numerical Computations Edition 3 for Octave Version 2.0.13 February 1997
    GNU Octave A high-level interactive language for numerical computations Edition 3 for Octave version 2.0.13 February 1997 John W. Eaton Published by Network Theory Limited. 15 Royal Park Clifton Bristol BS8 3AL United Kingdom Email: [email protected] ISBN 0-9541617-2-6 Cover design by David Nicholls. Errata for this book will be available from http://www.network-theory.co.uk/octave/manual/ Copyright c 1996, 1997John W. Eaton. This is the third edition of the Octave documentation, and is consistent with version 2.0.13 of Octave. Permission is granted to make and distribute verbatim copies of this man- ual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the en- tire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the same conditions as for modified versions. Portions of this document have been adapted from the gawk, readline, gcc, and C library manuals, published by the Free Software Foundation, 59 Temple Place—Suite 330, Boston, MA 02111–1307, USA. i Table of Contents Publisher’s Preface ...................... 1 Author’s Preface ........................ 3 Acknowledgements ........................................ 3 How You Can Contribute to Octave ........................ 5 Distribution .............................................. 6 1 A Brief Introduction to Octave ....... 7 1.1 Running Octave...................................... 7 1.2 Simple Examples ..................................... 7 Creating a Matrix ................................. 7 Matrix Arithmetic ................................. 8 Solving Linear Equations..........................
    [Show full text]
  • Stan Modeling Language
    Stan Modeling Language User’s Guide and Reference Manual Stan Development Team Stan Version 2.12.0 Tuesday 6th September, 2016 mc-stan.org Stan Development Team (2016) Stan Modeling Language: User’s Guide and Reference Manual. Version 2.12.0. Copyright © 2011–2016, Stan Development Team. This document is distributed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). For full details, see https://creativecommons.org/licenses/by/4.0/legalcode The Stan logo is distributed under the Creative Commons Attribution- NoDerivatives 4.0 International License (CC BY-ND 4.0). For full details, see https://creativecommons.org/licenses/by-nd/4.0/legalcode Stan Development Team Currently Active Developers This is the list of current developers in order of joining the development team (see the next section for former development team members). • Andrew Gelman (Columbia University) chief of staff, chief of marketing, chief of fundraising, chief of modeling, max marginal likelihood, expectation propagation, posterior analysis, RStan, RStanARM, Stan • Bob Carpenter (Columbia University) language design, parsing, code generation, autodiff, templating, ODEs, probability functions, con- straint transforms, manual, web design / maintenance, fundraising, support, training, Stan, Stan Math, CmdStan • Matt Hoffman (Adobe Creative Technologies Lab) NUTS, adaptation, autodiff, memory management, (re)parameterization, optimization C++ • Daniel Lee (Columbia University) chief of engineering, CmdStan, builds, continuous integration, testing, templates,
    [Show full text]
  • Maple 7 Programming Guide
    Maple 7 Programming Guide M. B. Monagan K. O. Geddes K. M. Heal G. Labahn S. M. Vorkoetter J. McCarron P. DeMarco c 2001 by Waterloo Maple Inc. ii • Waterloo Maple Inc. 57 Erb Street West Waterloo, ON N2L 6C2 Canada Maple and Maple V are registered trademarks of Waterloo Maple Inc. c 2001, 2000, 1998, 1996 by Waterloo Maple Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the copyright holder, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Contents 1 Introduction 1 1.1 Getting Started ........................ 2 Locals and Globals ...................... 7 Inputs, Parameters, Arguments ............... 8 1.2 Basic Programming Constructs ............... 11 The Assignment Statement ................. 11 The for Loop......................... 13 The Conditional Statement ................. 15 The while Loop....................... 19 Modularization ........................ 20 Recursive Procedures ..................... 22 Exercise ............................ 24 1.3 Basic Data Structures .................... 25 Exercise ............................ 26 Exercise ............................ 28 A MEMBER Procedure ..................... 28 Exercise ............................ 29 Binary Search ......................... 29 Exercises ........................... 30 Plotting the Roots of a Polynomial ............. 31 1.4 Computing with Formulæ .................. 33 The Height of a Polynomial ................. 34 Exercise ...........................
    [Show full text]
  • Bayesian Linear Mixed Models with Polygenic Effects
    JSS Journal of Statistical Software June 2018, Volume 85, Issue 6. doi: 10.18637/jss.v085.i06 Bayesian Linear Mixed Models with Polygenic Effects Jing Hua Zhao Jian’an Luan Peter Congdon University of Cambridge University of Cambridge University of London Abstract We considered Bayesian estimation of polygenic effects, in particular heritability in relation to a class of linear mixed models implemented in R (R Core Team 2018). Our ap- proach is applicable to both family-based and population-based studies in human genetics with which a genetic relationship matrix can be derived either from family structure or genome-wide data. Using a simulated and a real data, we demonstrate our implementa- tion of the models in the generic statistical software systems JAGS (Plummer 2017) and Stan (Carpenter et al. 2017) as well as several R packages. In doing so, we have not only provided facilities in R linking standalone programs such as GCTA (Yang, Lee, Goddard, and Visscher 2011) and other packages in R but also addressed some technical issues in the analysis. Our experience with a host of general and special software systems will facilitate investigation into more complex models for both human and nonhuman genetics. Keywords: Bayesian linear mixed models, heritability, polygenic effects, relationship matrix, family-based design, genomewide association study. 1. Introduction The genetic basis of quantitative phenotypes has been a long-standing research problem as- sociated with a large and growing literature, and one of the earliest was by Fisher(1918) on additive effects of genetic variants (the polygenic effects). In human genetics it is common to estimate heritability, the proportion of polygenic variance to the total phenotypic variance, through twin and family studies.
    [Show full text]
  • Bayesian Data Analysis – Assignment 6
    Bayesian data analysis { Assignment 6 General information • The recommended tool in this course is R (with the IDE R-Studio). You can download R here and R-Studio here. There are tons of tutorials, videos and introductions to R and R-Studio online. You can find some initial hints from RStudio Education pages. • Instead of installing R and RStudio on you own computer, see how to use R and RStudio remotely. • When working with R, we recommend writing the report using R markdown and the provided R markdown template. The remplate includes the formatting instructions and how to include code and figures. • Instead of R markdown, you can use other software to make the PDF report, but the the same instructions for formatting should be used. These instructions are available also in the PDF produced from the R markdown template. • Report all results in a single, anonymous *.pdf -file and return it to peergrade.io. • The course has its own R package aaltobda with data and functionality to simplify coding. To install the package just run the following (upgrade="never" skips question about updating other packages): 1. install.packages("remotes") 2. remotes::install_github("avehtari/BDA_course_Aalto", subdir = "rpackage", upgrade="never") • Many of the exercises can be checked automatically using the R package markmyassignment. Information on how to install and use the package can be found here. There is no need to include markmyassignment results in the report. • Recommended additional self study exercises for each chapter in BDA3 are listed in the course web page. • Common questions and answers regarding installation and technical problems can be found in Frequently Asked Questions (FAQ).
    [Show full text]
  • Probabilistic Programming in Python Using Pymc3
    Probabilistic programming in Python using PyMC3 John Salvatier1, Thomas V. Wiecki2 and Christopher Fonnesbeck3 1 AI Impacts, Berkeley, CA, United States 2 Quantopian Inc, Boston, MA, United States 3 Department of Biostatistics, Vanderbilt University, Nashville, TN, United States ABSTRACT Probabilistic programming allows for automatic Bayesian inference on user-defined probabilistic models. Recent advances in Markov chain Monte Carlo (MCMC) sampling allow inference on increasingly complex models. This class of MCMC, known as Hamiltonian Monte Carlo, requires gradient information which is often not readily available. PyMC3 is a new open source probabilistic programming framework written in Python that uses Theano to compute gradients via automatic differentiation as well as compile probabilistic programs on-the-fly to C for increased speed. Contrary to other probabilistic programming languages, PyMC3 allows model specification directly in Python code. The lack of a domain specific language allows for great flexibility and direct interaction with the model. This paper is a tutorial-style introduction to this software package. Subjects Data Mining and Machine Learning, Data Science, Scientific Computing and Simulation Keywords Bayesian statistic, Probabilistic Programming, Python, Markov chain Monte Carlo, Statistical modeling INTRODUCTION Probabilistic programming (PP) allows for flexible specification and fitting of Bayesian Submitted 9 September 2015 statistical models. PyMC3 is a new, open-source PP framework with an intuitive and Accepted 8 March 2016 readable, yet powerful, syntax that is close to the natural syntax statisticians use to Published 6 April 2016 describe models. It features next-generation Markov chain Monte Carlo (MCMC) Corresponding author Thomas V. Wiecki, sampling algorithms such as the No-U-Turn Sampler (NUTS) (Hoffman & Gelman, [email protected] 2014), a self-tuning variant of Hamiltonian Monte Carlo (HMC) (Duane et al., 1987).
    [Show full text]