Statistics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=234
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 18 Sep 2014 23:54:41 GMTThu, 18 Sep 2014 23:54:41 GMTNew applications in the Statistics categoryhttp://www.mapleprimes.com/images/mapleapps.gifStatistics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=234
Generating random numbers efficiently
http://www.maplesoft.com/applications/view.aspx?SID=153662&ref=Feed
Generating (pseudo-)random values is a frequent task in simulations and other programs. For some situations, you want to generate some combinatorial or algebraic values, such as a list or a polynomial; in other situations, you need random numbers, from a distribution that is uniform or more complicated. In this article I'll talk about all of these situations.<img src="/view.aspx?si=153662/thumb.jpg" alt="Generating random numbers efficiently" align="left"/>Generating (pseudo-)random values is a frequent task in simulations and other programs. For some situations, you want to generate some combinatorial or algebraic values, such as a list or a polynomial; in other situations, you need random numbers, from a distribution that is uniform or more complicated. In this article I'll talk about all of these situations.153662Mon, 18 Aug 2014 04:00:00 ZDr. Erik PostmaDr. Erik PostmaSpectral k-statistics
http://www.maplesoft.com/applications/view.aspx?SID=153618&ref=Feed
<p>The algorithm constructs natural statistics of a spectral sample, by using convolutions on the symmetric group and the Weingarten function. These statistics are unbiased estimators of cumulants of traces.</p><img src="/view.aspx?si=153618/39882f96bb55a4970488a9bcf94fd60d.gif" alt="Spectral k-statistics" align="left"/><p>The algorithm constructs natural statistics of a spectral sample, by using convolutions on the symmetric group and the Weingarten function. These statistics are unbiased estimators of cumulants of traces.</p>153618Thu, 03 Jul 2014 04:00:00 ZDr. Giuseppe GuarinoDr. Giuseppe GuarinoPrincipal Component Analysis
http://www.maplesoft.com/applications/view.aspx?SID=153591&ref=Feed
<p>Principal Component Analysis transforms a multi-dimensional data set to a new set of perpendicular axes (or components) that describe decreasing amounts of variance. </p>
<p>This worksheet reduces the complexity of a data set using principal component analysis. Those components that have the least impact on the variance are discarded, and the simplified data reconstructed from the remaining components.</p><img src="/view.aspx?si=153591/PrincipalComponentAn.jpg" alt="Principal Component Analysis" align="left"/><p>Principal Component Analysis transforms a multi-dimensional data set to a new set of perpendicular axes (or components) that describe decreasing amounts of variance. </p>
<p>This worksheet reduces the complexity of a data set using principal component analysis. Those components that have the least impact on the variance are discarded, and the simplified data reconstructed from the remaining components.</p>153591Mon, 26 May 2014 04:00:00 ZSamir KhanSamir KhanBlutdruckwerte aus Langzeitmessung (Blood Pressure Values)
http://www.maplesoft.com/applications/view.aspx?SID=153556&ref=Feed
<p>During a period of 24 hours the blood pressure of a patient at the University Hospital Aachen has been measured. Thus, we have a lot of Systole-, Diastole-, and Pulse-Values important for a medical doctor treating sick patients. To analyse these “data” the Maple Program 16 (with stats) is very useful.</p>
<p>For graphical representation cubic splines within the Maple Curve Fitting program has been used. In German.</p><img src="/view.aspx?si=153556/16b4d27b4d08cd3278be0fadcf544abd.gif" alt="Blutdruckwerte aus Langzeitmessung (Blood Pressure Values)" align="left"/><p>During a period of 24 hours the blood pressure of a patient at the University Hospital Aachen has been measured. Thus, we have a lot of Systole-, Diastole-, and Pulse-Values important for a medical doctor treating sick patients. To analyse these “data” the Maple Program 16 (with stats) is very useful.</p>
<p>For graphical representation cubic splines within the Maple Curve Fitting program has been used. In German.</p>153556Fri, 25 Apr 2014 04:00:00 ZProf. Josef BettenProf. Josef BettenJump-diffusion stochastic processes with Maple
http://www.maplesoft.com/applications/view.aspx?SID=153516&ref=Feed
<p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p><img src="/view.aspx?si=153516/Jump_image1.jpg" alt="Jump-diffusion stochastic processes with Maple" align="left"/><p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p>153516Sat, 08 Mar 2014 05:00:00 ZIgor HlivkaIgor HlivkaIntroduction to Statistics with Maple
http://www.maplesoft.com/applications/view.aspx?SID=149942&ref=Feed
An introduction to statistics and data analysis in Maple including a general overview of statistics. Examples include:
<ul>
<li> Importing data and simple data analysis, including showing Excel connectivity.</li>
<li> Working with Matrix data sets</li>
<li> Working with Random Variables and predefined distributions</li>
<li> Sampling, Monte Carlo & Bootstrapping Techniques</li>
<li> Creating custom distributions</li>
<li> Visualizing data</li>
<li> Hypothesis Testing</li>
<li> Maximum likelihood estimation</li>
<li> And more…</li>
</ul>
These examples were used in a webinar, Maple: Introduction to Statistics. <a href="http://www.maplesoft.com/webinars/recorded/featured.aspx?id=487" target="_blank" >A recording of this webinar is available for viewing</a>.<img src="/view.aspx?si=149942/img.gif" alt="Introduction to Statistics with Maple" align="left"/>An introduction to statistics and data analysis in Maple including a general overview of statistics. Examples include:
<ul>
<li> Importing data and simple data analysis, including showing Excel connectivity.</li>
<li> Working with Matrix data sets</li>
<li> Working with Random Variables and predefined distributions</li>
<li> Sampling, Monte Carlo & Bootstrapping Techniques</li>
<li> Creating custom distributions</li>
<li> Visualizing data</li>
<li> Hypothesis Testing</li>
<li> Maximum likelihood estimation</li>
<li> And more…</li>
</ul>
These examples were used in a webinar, Maple: Introduction to Statistics. <a href="http://www.maplesoft.com/webinars/recorded/featured.aspx?id=487" target="_blank" >A recording of this webinar is available for viewing</a>.149942Mon, 29 Jul 2013 04:00:00 ZMaplesoftMaplesoftIndependenceModel package
http://www.maplesoft.com/applications/view.aspx?SID=148816&ref=Feed
<p>The main purpose of this work was to write a procedure to implement an algorithm based on the Diaconis Sturmfels algorithm to compute the Monte Carlo p-value of the independence model considered, but we present a package containing also some preliminary commands that can be useful to everyone studying an independence model.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="IndependenceModel package" align="left"/><p>The main purpose of this work was to write a procedure to implement an algorithm based on the Diaconis Sturmfels algorithm to compute the Monte Carlo p-value of the independence model considered, but we present a package containing also some preliminary commands that can be useful to everyone studying an independence model.</p>148816Tue, 25 Jun 2013 04:00:00 ZValentina TrioloValentina TrioloQuandl Library for Maple
http://www.maplesoft.com/applications/view.aspx?SID=146879&ref=Feed
<p>The Quandl library for Maple provides easy access to Quandl’s repository of over 5 million time-series datasets from directly inside Maple, allowing you to utilize Maple’s robust tools for mathematical statistics and data analysis on Quandl’s extensive collection of data. This library features a similar set of functionalities to the Quandl <a href="http://www.quandl.com/help/api">API</a>, making it possible to get data by date ranges and/or time frequencies, as well as sort and transform your dataset.</p>
<p>For more details on the Quandl library for Maple, see the included help documentation on Quandl.</p>
<p>For more information on Quandl, see <a href="http://www.quandl.com">www.quandl.com</a> .</p><img src="/view.aspx?si=146879/quandl.JPG" alt="Quandl Library for Maple" align="left"/><p>The Quandl library for Maple provides easy access to Quandl’s repository of over 5 million time-series datasets from directly inside Maple, allowing you to utilize Maple’s robust tools for mathematical statistics and data analysis on Quandl’s extensive collection of data. This library features a similar set of functionalities to the Quandl <a href="http://www.quandl.com/help/api">API</a>, making it possible to get data by date ranges and/or time frequencies, as well as sort and transform your dataset.</p>
<p>For more details on the Quandl library for Maple, see the included help documentation on Quandl.</p>
<p>For more information on Quandl, see <a href="http://www.quandl.com">www.quandl.com</a> .</p>146879Thu, 09 May 2013 04:00:00 ZMaplesoftMaplesoftClassroom Tips and Techniques: Least-Squares Fits
http://www.maplesoft.com/applications/view.aspx?SID=140942&ref=Feed
<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezInterpolation and Smoothing
http://www.maplesoft.com/applications/view.aspx?SID=132223&ref=Feed
These examples illustrate 3-D interpolation and smoothing. It shows the use of a smoothing algorithm to create a smooth surface that approximates your noisy data 3-D data, and interpolation methods that generate a surface that matches your data exactly, regardless of whether the data points lie on a uniform or non-uniform grid. Many of these techniques are new in Maple 16.<img src="/view.aspx?si=132223/thumb.jpg" alt="Interpolation and Smoothing" align="left"/>These examples illustrate 3-D interpolation and smoothing. It shows the use of a smoothing algorithm to create a smooth surface that approximates your noisy data 3-D data, and interpolation methods that generate a surface that matches your data exactly, regardless of whether the data points lie on a uniform or non-uniform grid. Many of these techniques are new in Maple 16.132223Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftStatistics Enhancements in Maple 16
http://www.maplesoft.com/applications/view.aspx?SID=132195&ref=Feed
Statistical computations in Maple combine the ease of working in a high-level, interactive environment with a very large and powerful set of algorithms. Large data sets can be handled efficiently with 35 built-in statistical distributions, sampling, estimations, data smoothing, hypothesis testing, and visualization algorithms. In addition, integration with the Maple symbolic engine means that you can easily specify custom distributions by combining existing distributions or simply by giving a formula for the probability or cumulative distribution function. These examples illustrate the use of the Statistics package, with emphasis on enhancements in Maple 16.<img src="/view.aspx?si=132195/thumb.jpg" alt="Statistics Enhancements in Maple 16" align="left"/>Statistical computations in Maple combine the ease of working in a high-level, interactive environment with a very large and powerful set of algorithms. Large data sets can be handled efficiently with 35 built-in statistical distributions, sampling, estimations, data smoothing, hypothesis testing, and visualization algorithms. In addition, integration with the Maple symbolic engine means that you can easily specify custom distributions by combining existing distributions or simply by giving a formula for the probability or cumulative distribution function. These examples illustrate the use of the Statistics package, with emphasis on enhancements in Maple 16.132195Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftMath Apps in Maple
http://www.maplesoft.com/applications/view.aspx?SID=132220&ref=Feed
Math Apps in Maple have give students and teachers the ability to explore and illustrate a wide variety of mathematical and scientific concepts. These fun and easy to use educational demonstrations are designed to illustrate various mathematical and physical concepts. This application contains a sampling of some of the many Math Apps available in Maple: drawing the graph of a quadratic, epicycloids, monte carlo approximations of pi, and throwing coconuts.<img src="/view.aspx?si=132220/mathapps_thumb.png" alt="Math Apps in Maple" align="left"/>Math Apps in Maple have give students and teachers the ability to explore and illustrate a wide variety of mathematical and scientific concepts. These fun and easy to use educational demonstrations are designed to illustrate various mathematical and physical concepts. This application contains a sampling of some of the many Math Apps available in Maple: drawing the graph of a quadratic, epicycloids, monte carlo approximations of pi, and throwing coconuts.132220Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftRegression in Maple
http://www.maplesoft.com/applications/view.aspx?SID=129021&ref=Feed
I have always thought that regressions has been too complicated in Maple. The Fit command is too fiddly ie you have to specify too many things and it is easy to get it wrong plus the statistical output you get is far from mainstream ie you dont get t-values, p-values, R, R^2, Adj R^2 etc etc.
I have therefore designed a new procedure called Reg() which only needs one input and that is a datamatrix.<img src="/view.aspx?si=129021/regression_sm.jpg" alt="Regression in Maple" align="left"/>I have always thought that regressions has been too complicated in Maple. The Fit command is too fiddly ie you have to specify too many things and it is easy to get it wrong plus the statistical output you get is far from mainstream ie you dont get t-values, p-values, R, R^2, Adj R^2 etc etc.
I have therefore designed a new procedure called Reg() which only needs one input and that is a datamatrix.129021Thu, 22 Dec 2011 05:00:00 ZMarcus DavidssonMarcus DavidssonGreat Expectations
http://www.maplesoft.com/applications/view.aspx?SID=127116&ref=Feed
<p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p><img src="/view.aspx?si=127116/expectation_thum.png" alt="Great Expectations" align="left"/><p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p>127116Thu, 27 Oct 2011 04:00:00 ZAn analysis of Kolmogorov-Smirnov statistics and their distributions
http://www.maplesoft.com/applications/view.aspx?SID=123987&ref=Feed
<p>This worksheet examines the use of Kolomogorov-Smirnov (K-S) statistics and presents MAPLE implementations of their distributions under different conditions and assumptions. The presentation aims at being representative rather than comprehensive, and so may serve as an introduction to the properties and use of K-S statistics. The worksheet draws on existing work listed in the References section, and it implements transcriptions (by the author of this worksheet) of certain algorithms given in a subset of those references. </p><img src="/view.aspx?si=123987/417499\KSstat.JPG" alt="An analysis of Kolmogorov-Smirnov statistics and their distributions" align="left"/><p>This worksheet examines the use of Kolomogorov-Smirnov (K-S) statistics and presents MAPLE implementations of their distributions under different conditions and assumptions. The presentation aims at being representative rather than comprehensive, and so may serve as an introduction to the properties and use of K-S statistics. The worksheet draws on existing work listed in the References section, and it implements transcriptions (by the author of this worksheet) of certain algorithms given in a subset of those references. </p>123987Tue, 19 Jul 2011 04:00:00 ZDr. Melvin BrownDr. Melvin BrownGeneralised Smirnov two-sample homogeneity tests
http://www.maplesoft.com/applications/view.aspx?SID=121124&ref=Feed
<p>The problem addressed by this worksheet is: Given two samples of data, which may contain ties, how may one test the hypothesis that they are drawn from the same distribution?</p>
<p>The worksheet demonstrates the use of a MAPLE implementation of an algorithm to perform two-sample homogeneity tests, based on any one of three Kolmogorov-Smirnov (K-S) test statistics.</p>
<p>The MAPLE package KSNstat, which is introduced in this worksheet, contains the MAPLE procedure gsmirn which implements the GSMIRN algorithm given in 1994 by Nikiforov [1] to calculate exact p-values for generalised (conditionally distribution-free) two-sample homogeneity tests based on two-sided and one-sided Kolomogorov-Smirnov statistics. Notably, the Nikiforov algorithm covers the range from discrete to continuous distributions; specifically, it handles tied data points.</p>
<p>[1] Exact Smirnov two-sample tests for arbitrary distributions, A. Nikiforov, Appl.Stat., vol.43, No. 1. pp.265-270, 1994. </p><img src="/view.aspx?si=121124/384833\3506f3b8019b414a0d703d121b569aa5.gif" alt="Generalised Smirnov two-sample homogeneity tests" align="left"/><p>The problem addressed by this worksheet is: Given two samples of data, which may contain ties, how may one test the hypothesis that they are drawn from the same distribution?</p>
<p>The worksheet demonstrates the use of a MAPLE implementation of an algorithm to perform two-sample homogeneity tests, based on any one of three Kolmogorov-Smirnov (K-S) test statistics.</p>
<p>The MAPLE package KSNstat, which is introduced in this worksheet, contains the MAPLE procedure gsmirn which implements the GSMIRN algorithm given in 1994 by Nikiforov [1] to calculate exact p-values for generalised (conditionally distribution-free) two-sample homogeneity tests based on two-sided and one-sided Kolomogorov-Smirnov statistics. Notably, the Nikiforov algorithm covers the range from discrete to continuous distributions; specifically, it handles tied data points.</p>
<p>[1] Exact Smirnov two-sample tests for arbitrary distributions, A. Nikiforov, Appl.Stat., vol.43, No. 1. pp.265-270, 1994. </p>121124Wed, 06 Jul 2011 04:00:00 ZDr. Melvin BrownDr. Melvin BrownThe Advanced Encryption Standard and its modes of operation
http://www.maplesoft.com/applications/view.aspx?SID=6618&ref=Feed
<p>This is an update, labeled version 1.1, to the existing application The Advanced Encryption Standard and its modes of operation.</p>
<p>Version 1.1: Key generation function and related functions updated to facilitate the use of externally generated seeds. Some minor changes to presentation.</p>
<p>Version 1.0: Implementation of encryption and authentication schemes that use the Advanced Encryption Standard (AES) as their underlying block cipher. These schemes are constructed by using all the modes of operation for block ciphers so far approved by NIST (the US National Institute of Standards of Technology), namely, the five confidentiality modes: ECB, CBC, CFB, OFB and CTR, the authentication mode CMAC, and the "authenticated encryption" modes CCM and GCM/GMAC. The implementation is able to encrypt/decrypt and/or authenticate messages in several formats, including binary files, and we use it to explore the basic properties of these schemes. The implementation contains also detailed explanations of all the procedures used, including the lower level ones, and discusses both the programming and the cryptographic aspects involved.</p><img src="/view.aspx?si=6618/AES_1608.gif" alt="The Advanced Encryption Standard and its modes of operation" align="left"/><p>This is an update, labeled version 1.1, to the existing application The Advanced Encryption Standard and its modes of operation.</p>
<p>Version 1.1: Key generation function and related functions updated to facilitate the use of externally generated seeds. Some minor changes to presentation.</p>
<p>Version 1.0: Implementation of encryption and authentication schemes that use the Advanced Encryption Standard (AES) as their underlying block cipher. These schemes are constructed by using all the modes of operation for block ciphers so far approved by NIST (the US National Institute of Standards of Technology), namely, the five confidentiality modes: ECB, CBC, CFB, OFB and CTR, the authentication mode CMAC, and the "authenticated encryption" modes CCM and GCM/GMAC. The implementation is able to encrypt/decrypt and/or authenticate messages in several formats, including binary files, and we use it to explore the basic properties of these schemes. The implementation contains also detailed explanations of all the procedures used, including the lower level ones, and discusses both the programming and the cryptographic aspects involved.</p>6618Mon, 20 Jun 2011 04:00:00 ZJosé Luis Gómez PardoJosé Luis Gómez PardoMonte Carlo Integration with Parallelism
http://www.maplesoft.com/applications/view.aspx?SID=103811&ref=Feed
This example implements a Monte-Carlo integrator, and then adds parallelism to the algorithm so that the computation is split over multiple processors when run on a multi-core computer.<img src="/view.aspx?si=103811/thumb.jpg" alt="Monte Carlo Integration with Parallelism" align="left"/>This example implements a Monte-Carlo integrator, and then adds parallelism to the algorithm so that the computation is split over multiple processors when run on a multi-core computer.103811Wed, 06 Apr 2011 04:00:00 ZMaplesoftMaplesoftA new algorithm for computing the multivariate Faà di Bruno’s formula
http://www.maplesoft.com/applications/view.aspx?SID=101396&ref=Feed
<p>We provide a new algorithm for computing the multivariate Faà di Bruno's formula. We follow a symbolic approach based on the classical umbral calculus that leads back the computation of the multivariate Faà di Bruno's formula to a suitable multinomial expansion. The resulting computational times are faster compared with procedures existing in the literature.</p><img src="/view.aspx?si=101396/319207\UMFB.JPG" alt="A new algorithm for computing the multivariate Faà di Bruno’s formula" align="left"/><p>We provide a new algorithm for computing the multivariate Faà di Bruno's formula. We follow a symbolic approach based on the classical umbral calculus that leads back the computation of the multivariate Faà di Bruno's formula to a suitable multinomial expansion. The resulting computational times are faster compared with procedures existing in the literature.</p>101396Thu, 03 Feb 2011 05:00:00 ZDr. Giuseppe GuarinoDr. Giuseppe GuarinoCopula function in multivariate dependency analysis
http://www.maplesoft.com/applications/view.aspx?SID=100528&ref=Feed
<p>Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions. In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions.</p>
<p>Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.</p>
<p>Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.</p><img src="/view.aspx?si=100528/maple_icon.jpg" alt="Copula function in multivariate dependency analysis" align="left"/><p>Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions. In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions.</p>
<p>Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.</p>
<p>Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.</p>100528Wed, 29 Dec 2010 05:00:00 ZI. HlivkaI. Hlivka