Comparison of Mathematical and Statistical Functionality of Machine Learning Tools for Data Analysis Research

DOI : 10.17577/IJERTCONV4IS21010

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Comparison of Mathematical and Statistical Functionality of Machine Learning Tools for Data Analysis Research

Shamitha S. K

Research Scholar, VTU, Belgaum Bangalore, India

Nikitha Pai

Guest Lecturer, NMKRV College for Women, Jayanagar, Bangalore, India

B. Nithya

Asst.Professor,

New Horizon College of Engineering, Marathahalli

Dr. V. Ilango

Professor,

New Horizon College of Engg Marathahalli

Abstract – Over the last three decades many general- purpose machine learning frameworks and libraries have emerged from both academia and industry. The aim of this paper is to compare mathematical and statistical programming languages on a fair level by showing only facts about the tested programs and attempts have been made to avoid subjective remarks. This could be used as base information to make own decision. The paper takes a closer look on mathematical and statistical programming, data analysis and simulation functionality for huge and very huge data sets. The following machine learning tools have been tested: Mathematica from Wolfram Research Inc., MATHLAB from The Mathworks Inc. This type of functionality is of great interest for econometrics, the financial sector in general, biology, chemistry, physics and have immense usage in other areas as well, where the numerical analysis of data is very important.

Keywords: Machine Learning frameworks, Mathematica, Mat lab

  1. INTRODUCTION

    Given the enormous growth of collected and available data in companies, industry and science, techniques for analyzing such data are becoming ever more important. [1] Research in machine learning (ML) combines classical questions of computer science (efficient algorithms, software systems, databases) with elements from artificial intelligence and statistics up to user oriented issues (visualization, interactive mining, user assistance and smart recommendations).[2] Over the last three decades, many general purpose machine learning frameworks, as well as special purpose machine learning libraries, such as for phishing detection or speech processing [1], has emerged from both academia and industry. In this survey, we will only consider the general purpose frameworks. It is good to have a look around to see what languages and platforms are popular in

    self-selected communities of data analysis and machine learning professionals.[3] The study consists of tables which lists the availability of functions for each program [6][7]. It is divided in functional sections of mathematical, graphical functionality and programming environment, a data import/export interface section, the availability for several operating systems, a speed comparison and finally a summary of the whole information. To rate all these information, a simple scoring system has been used and following machine learning tools have been tested: Mathematica from Wolfram Research Inc, MATHLAB from The Mathworks Inc. A recent poll is titled What programming/statistics languages you used for an analytics work in 2013 [6]. The results suggest heavy use of R and Python and SQL for data access.

    FIG 1: Popular Data Analytic Tools

  2. COMPARISON OF THE MATHEMATICAL FUNCTIONALITY

    Actually there are a lot of different mathematical and statistical programs in the market which are covering a huge amount of functions. The above figure(FIG 1) discusses the popular Analytical tools. The following

    tables should give an overview about the functionality for analyzing data in numerical ways and should mark out which functions are supported by which program and whether these functions are already implemented in the base program or whether you need an additional module. The functions are sorted by the categories: Standard mathematics, Linear algebra, Numerical mathematics, Stochastic, Statistics, Other mathematics.

    1. Standard Mathematics

      Standard mathematics functions are an essential part of any kind of mathematical work. Not necessary to mention that these type of functions should be available in all programs. Therefore the following results are not very surprising.[6].Comparison of Mathematica and MATHLAB on standard mathematics are discussed in the table below

      Table1.1: Comparison of Mathematica and MATHLAB with respect to standard mathematics

    2. Algebra

      Algebra and especially linear algebra offers a basic functionality for any kind of matrix oriented work.

      i.e. Optimization routines are widely used in the financial sector but also very useful for logistic problems (remember the traveling salesman problem).Most simulation and analyzing routines are relying on decomposition equations solving and other routines from algebra.Table 1.2 discusses the comparison between Mathematica and MATHLAB with respect to Algebra

      Functions (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008a

      )

      Eigenvalues

      Eigenvalues

      Eigenvectors

      Matrix

      Characteristic polynom

      Determinant

      Hadamard matrix

      M

      Hankel matrix

      Hilbert matrix

      Householder matrix

      Inverse matrix

      Kronecker product

      Pascal matrix

      Toeplitz matrix

      Upper Hessenberg form

      Decomposition

      Cholesky decomposition

      Crout decomposition

      Dulmage-Mendelsohn decomposition

      LU decomposition

      QR decomposition

      Schur form of quadratic matrix

      Smith normal form

      Functions (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008a

      )

      Eigenvalues

      Eigenvalues

      Eigenvectors

      Matrix

      Characteristic polynom

      Determinant

      Hadamard matrix

      M

      Hankel matrix

      Hilbert matrix

      Householder matrix

      Inverse matrix

      Kronecker product

      Pascal matrix

      Toeplitz matrix

      Upper Hessenberg form

      Decomposition

      Cholesky decomposition

      Crout decomposition

      Dulmage-Mendelsohn decompsition

      LU decomposition

      QR decomposition

      Schur form of quadratic matrix

      Smith normal form

      Table 1.2: Comparison of Mathematica and MATHLAB tools with respect to Algebra

      Functions (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008a)

      BesselI

      Bessel J

      BesselK

      Bessel Y

      Beta function

      Binomial

      Factorial

      FresnelC

      FresnelS

      Gamma function

      Hyperbolic trig.

      Function

      Incomplete Gammafunc.

      Log / Ln / Exp

      / /

      / /

      Log-Gammafunc.

      Poly gamma

      Square root

      Sum / Product

      /

      /

      Trig. / arg trig.

      Functions

      /

      /

      Singular value decomposition

      Optimization

      Optimization – linear models

      (Unconstr. / Constr.)

      /

      /

      Optimization – nonlinear models

      (Unconstr. / Constr.)

      /

      /

      Optimization – quadratic models (QP)

      (Unconstr. / Constr.)

      /

      /

      Equation solver

      Linear equation solver

      Non-linear equation solver

      Ordinary Differential Equation solver

      Partial Differential Equation solver

      Miscellaneous

      Moore-Penrose pseudo- inverse

      Sparse matrices handling

      Singular value decomposition

      Optimization

      Optimization – linear models

      (Unconstr. / Constr.)

      /

      /

      Optimization – nonlinear models

      (Unconstr. / Constr.)

      /

      /

      Optimization – quadratic models (QP)

      (Unconstr. / Constr.)

      /

      /

      Equation solver

      Linear equation solver

      Non-linear equation solver

      Ordinary Differential Equation solver

      Partial Differential Equation solver

      Miscellaneous

      Moore-Penrose pseudo- inverse

      Sparse matrices handling

      Most simulation and analyzing routines are relying on decompositions, equation solving and other routines from algebra.

    3. Numerical Mathematics

    Numerical mathematics offers fundamental algorithms for several appliances. It is marked out that especially any kind of interpolation algorithms are commonly used in technical and non-technical businesses. Without really recognizing, interpolation routines are used in nearly any kind of graphical representation. .[6][4]Table 1.3 explains the comparisons of Mathematica and MathLab tools with respect to Numerical functions.

    Table 1.3: Comparison of Mathematica and MATHLAB tools with respect to Numerical mathematics

  3. COMPARISON OF THE STATISTICAL FUNCTIONALITY

    Pade interpolation

    Piecewise cubic hermite polynomial interpolation

    Piecewise polynomial interpolation

    Other functions

    Bisection

    Newton method for finding roots

    Runge Kutta method for solving ODE

    Pade interpolation

    Piecewise cubic hermite polynomial interpolation

    Piecewise polynomial interpolation

    Other functions

    Bisection

    Newton method for finding roots

    Runge Kutta method for solving ODE

    1. Descriptive statistic and Distribution functions

      Very important to get familiar with data and to understand samples of data are stochastic and descriptive statistic routines. Distribution functions, their CDF- Cumulative Distribution Function and PDF- Probability Distribution Functions function are commonly used to figure out what are representative samples and what are outliers. A typical simple but common usage might be in a productive area to take samples of the manufactured product and to see whether the faulty parts in a sample are within a normal range. More complex usages might be in load balancing simulations of telecommunication hardware. However it might be possible to mention example usages for nearly all kind of business.[6].Comparison of Mathematica and MathLab tools to distribution function is explained in Table 2.1

      Functions (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008a)

      General Function

      Contingency tables

      Correlation

      Cross tabulation

      Deviation

      Kurtosis

      Markov models

      Mean /geometric Mean / Mode

      / /

      / /-

      Functions (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008a)

      General Function

      Contingency tables

      Correlation

      Cross tabulation

      Deviation

      Kurtosis

      Markov models

      Mean /geometric Mean / Mode

      / /

      / /-

      Table 2.1 Comparison of Mathematica and MATHLAB tools with respect to distribution function

      Function (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008)

      Interpolaton

      B-Spline interpolation

      Classical interpolation (1D/2D/3D/n D)

      /

      /

      /

      /

      /

      /

      k-Spline interpolation

      Function (Version)

      Mathematica

      MATHLAB

      (6.0)

      (2008)

      Interpolation

      B-Spline interpolation

      Classical interpolation (1D/2D/3D/n D)

      /

      /

      /

      /

      /

      /

      k-Spline interpolation

      Log-normal

      / / /

      / / /

      Log-normal (multivariate)

      / / /

      -/-/-/-

      Negative binomial

      / / /

      / / /

      Normal

      / / /

      / / /

      Normal (bivariate)

      -/-/-/-

      -/-/-/-

      Normal (multivariate)

      / / /

      / / /

      Pareto

      / / /

      -/-/-/-

      Poisson

      / / /

      / / /

      Rayleigh

      / / /

      / / /

      S

      / / /

      -/-/-/-

      Student's t

      / / /

      / / /

      Student's t (non- central)

      / / /

      / / /

      Student's t (multivariate)

      / / /

      / / /

      Uniform

      / / /

      / / /

      Von Mises

      / / /

      -/-/-/-

      Weibull

      / / /

      / / /

      Wishart

      / / /

      -/-/-/

      Min / Max

      /

      /

      Quantile / Percentile

      /

      /

      Skewness

      Variance

      Variance-covariance matrix

      Distribution Functions

      (PDF / CDF / iCDF/ random number)

      Bernoulli

      / / /

      / / /

      Beta

      / / /

      / / /

      Binomial

      / / /

      / / /

      Brownian motion

      -/-/-/

      -/-/-/-

      Cauchy

      / / /

      -/-/-/-

      Chi-squared

      / / /

      / / /

      Chi-squared (non- central)

      / / /

      / / /

      Dirichlet

      / / /

      -/-/-/-

      Erlang

      / / /

      -/-/-/-

      Exponential

      / / /

      / / /

      Extreme value

      / / /

      / / /

      F

      / / /

      / / /

      F (non-central)

      / / /

      / / /

      Gamma

      / / /

      / / /

      Geometric

      / / /

      / / /

      Gumbel

      / / /

      -/-/-/-

      Half-normal

      / / /

      -/-/-/-

      Hotelling T2

      / / /

      -/-/-/-

      Hyper-exponential

      / / /

      -/-/-/-

      Hypergeometric

      / / /

      / / /

      Kernel

      / / /

      -/-/-/-

      Laplace

      / / /

      -/-/-/-

      Logarithmic

      / / /

      -/-/-/-

      Logistic

      / / /

      -/-/-/-

      Min / Max

      /

      /

      Quantile / Percentile

      /

      /

      Skewness

      Variance

      Variance-covariance matrix

      Distribution Functions

      (PDF / CDF / iCDF/ random number)

      Bernoulli

      / / /

      / / /

      Beta

      / / /

      / / /

      Binomial

      / / /

      / / /

      Brownian motion

      -/-/-/

      -/-/-/-

      Cauchy

      / / /

      -/-/-/-

      Chi-squared

      / / /

      / / /

      Chi-squared (non- central)

      / / /

      / / /

      Dirichlet

      / / /

      -/-/-/-

      Erlang

      / / /

      -/-/-/-

      Exponential

      / / /

      / / /

      Extreme value

      / / /

      / / /

      F

      / / /

      / / /

      F (non-central)

      / / /

      / / /

      Gamma

      / / /

      / / /

      Geometric

      / / /

      / / /

      Gumbel

      / / /

      -/-/-/-

      Half-normal

      / / /

      -/-/-/-

      Hotelling T2

      / / /

      -/-/-/-

      Hyper-exponential

      / / /

      -/-/-/-

      Hypergeometric

      / / /

      / / /

      Kernel

      / / /

      -/-/-/-

      Laplace

      / / /

      -/-/-/-

      Logarithmic

      / / /

      -/-/-/-

      Logistic

      / / /

      -/-/-/-

      1. Statistics

    Statistical functions are fundamental for any kind of data analysis. Routines like regression or time series

    are commonly used to find out trends or to predict future values i.e. for stock market courses. Filter routines are

    used to smooth or filter effects in data acquisition.

    Multivariate statistics are used to find patterns or common characteristics in data i.e. for market basket

    analysis by clustering routines.[9].Comparison of Mathematica and MAthLab tools with respect to

    statistical functions are explained in the table 2.2.

    Table 2.2 Comparison of Mathematica and MATHLAB tools with respect to statistical functions

    Functions (Version)

    Mathematica

    MATHL AB

    (6.0)

    (2008a)

    Regression models

    Linear

    Loess

    Logistic Regression

    LOGIT / PROBIT

    m- / m

    /

    Nonlinear / Polynomial

    /

    /

    PSN

    Tobit models

    Test statistics

    Ansari-Bradley test

    Bartlett multiple- sample test

    Besley test

    Breusch-Pagan test for homoscedasticity

    Chow Test for stability

    CUSUM test for stability

    Davidson-MacKinnon J-Test

    Dickey Fuller test

    Durbin-Watson test

    Engles LM test

    Friedmans test

    F-Test

    Goodness of fit test

    Goldfeld-Quandt test for homoscedasticity

    Grangers causality test

    Hausmans specification test

    Kolmogorov-Smirnov test

    Kruskal-Wallis test

    Kuh test

    Lagrange multiplier test

    Lilliefors test

    Ljung-Box Q-Test

    Mann-Whitney U test

    Sign test

    T-Test

    Wald test

    Walsh test

    Wilcoxon rank sum / sign test

    – / –

    /

    Z-Test

    Filter / smoothing models

    Bandpass / Lowpass / Highpass / Multiband

    / /

    /-/

    / / /

    / Bandstop

    /

    Battle-Lemarie

    Bessel

    Butterworth

    Chebyshev

    Coiflet

    Daubechies

    Elliptic

    Haar

    Hodrick-Prescott

    IIR / FIR

    /

    /

    Kernel

    Linear

    Meyer

    Pollen

    Riccati

    Shannon

    Savitzky-Golay

    Time series models

    ARMA / ARIMA /

    /

    /-/

    ARFIMA / ARMAX

    /-/-

    GARCH / ARCH / AGARCH / EGARCH / FIGARCH / IGARCH

    /

    /-

    /-/-/-/-/-

    //-/

    /

    -/-/-/-/-

    Logistic Regression

    LOGIT / PROBIT

    m- / m

    /

    Nonlinear / Polynomial

    /

    /

    PSN

    Tobit models

    Test statistics

    Ansari-Bradley test

    Bartlett multiple- sample test

    Besley test

    Breusch-Pagan test for homoscedasticity

    Chow Test for stability

    CUSUM test for stability

    Davidson-MacKinnon J-Test

    Dickey Fuller test

    Durbin-Watson test

    Engles LM test

    Friedmans test

    F-Test

    Goodness of fit test

    Goldfeld-Quandt test for homoscedasticity

    Grangers causality test

    Hausmans specification test

    Kolmogorov-Smirnov test

    Kruskal-Wallis test

    Kuh test

    Lagrange multiplier test

    Lilliefors test

    Ljung-Box Q-Test

    Mann-Whitney U test

    Sign test

    T-Test

    Wald test

    Walsh test

    Wilcoxon rank sum / sign test

    – / –

    /

    Z-Test

    Filter / smoothing models

    Bandpass / Lowpass / Highpass / Multiband

    / /

    /-/

    / / /

    / Bandstop

    /

    Battle-Lemarie

    Bessel

    Butterworth

    Chebyshev

    Coiflet

    Daubechies

    Elliptic

    Haar

    Hodrick-Prescott

    IIR / FIR

    /

    /

    Kernel

    Linear

    Meyer

    Pollen

    Riccati

    Shannon

    Savitzky-Golay

    Time series models

    ARMA / ARIMA /

    /

    /-/

    ARFIMA / ARMAX

    /-/-

    GARCH / ARCH / AGARCH / EGARCH / FIGARCH / IGARCH

    /

    /-

    /-/-/-/-/-

    //-/

    /

    -/-/-/-/-

    / MGARCH / PGARCH /

    TGARCH models

    Holts Winter additive / multiplicative

    – / –

    – / –

    Multivariate GARCH models

    (Diagonal VEC / BEKK / Matrix Diagonal / Vector Diagonal)

    -/-/-/-

    -/-/-/-

    Partial autocorrelation

    Spectral analysis

    State space models

    Time series analysis (Stationary / Non- stat.)

    /

    /

    Wavelets

    Multivariate statistics

    ANOVA / MANOVA

    / –

    /

    Cluster analysis (hierarchical/k-means)

    /

    /

    Discriminant analysis

    Factor analysis

    Fuzzy clustering

    Procrustes analysis

    Principal component analysis

    Principal coordinate analysis

    Survival analysis

    Design of Experiments

    Box-Behnken design

    Central composite design

    D-Optimal design

    Full / Fractional factorial design

    – / –

    /

    Hadamard design

    / MGARCH / PGARCH /

    TGARCH models

    Holts Winter additive / multiplicative

    – / –

    – / –

    Multivariate GARCH models

    (Diagonal VEC / BEKK / Matrix Diagonal / Vector Diagonal)

    -/-/-/-

    -/-/-/-

    Partial autocorrelation

    Spectral analysis

    State space models

    Time series analysis (Stationary / Non- stat.)

    /

    /

    Wavelets

    Multivariate statistics

    ANOVA / MANOVA

    / –

    /

    Cluster analysis (hierarchical/k-means)

    /

    /

    Discriminant analysis

    Factor analysis

    Fuzzy clustering

    Procrustes analysis

    Principal component analysis

    Principal coordinate analysis

    Survival analysis

    Design of Experiments

    Box-Behnken design

    Central composite design

    D-Optimal design

    Full / Fractional factorial design

    – / –

    /

    Hadamard design

    Response surface design

    Other statistical functions & models

    Bootstrapping

    Duration models

    Entropy models

    Event count models

    Heckman two step estimation

    Heteroscedasticity

    Jacknife estimation

    Lagrange multiplier test

    Markowitz efficient frontier

    Maximum Likelihood (Unconstr. / Constr.)

    /

    / –

    Monte Carlo simulation

    Response surface design

    Other statistical functions & models

    Bootstrapping

    Duration models

    Entropy models

    Event count models

    Heckman two step estimation

    Heteroscedasticity

    Jacknife estimation

    Lagrange multiplier test

    Markowitz efficient frontier

    Maximum Likelihood (Unconstr. / Constr.)

    /

    / –

    Monte Carlo simulation

  4. OTHER MACHINE LEARNING TOOLS

Table 3: Machine Learning tools and Libraries

Name

HLD

OS

Language

Aleph

No

Win/Unix

Yap Prolog

C4.5/C5/See5

Yes

Win/Unix

C/C++

Encog

Yes

Win/Unix

Java/.NET

FuzzyML

Yes

Win/Unix

ADA

IBM Cognos

Yes

Web

PowerHouse

IBM SPSS

Modeler

Yes

Win/Unix/OSX

Java

JavaML

Yes

Win/Unix

Java

JHepWork

Yes

Win/Unix

Java/Jython/Jru by/BeanShell

Joone

No

Win/Unix

Java

KNIME

Yes

Win/Unix

Java/Python/Per l

LIONsolver

Yes

Win/Unix

C/C++

MLC++

No

Win/Unix

C++

Mlpy

No

Win/Unix

Python

p>MS SQL Server

Yes

Win

.NET

Neuroph

No

Win/Unix

Java

Oracle Data Miner

Yes

Win/Unix

Java

Orange

No

Win/Unix

C++/Python

PCP

Yes

Win/Unix

C/C++/Fortran

Pyml

No

Win/Unix

Python

R

Yes

Win/Unix

C/Fortran/R

RapidMiner

Yes

Win/Unix/OSX

Java/Groovy

Salford Systems

Yes

Win

C/C++/.NET?

SAS Enterprise Miner

Yes

Win/Unix

C

scikit-learn

Yes

Win/Unix/OSX

C/C++/Python/ Cython

Shogun

Yes

Win/Unix

C/C++/Python/ R/MATHLAB

Statistica

Yes

Win

.NET/R

Mlpy

No

Win/Unix

Python

MS SQL Server

Yes

Win

.NET

Neuroph

No

Win/Unix

Java

Oracle Data Miner

Yes

Win/Unix

Java

Orange

No

Win/Unix

C++/Python

PCP

Yes

Win/Unix

C/C++/Fortran

Pyml

No

Win/Unix

Python

R

Yes

Win/Unix

C/Fortran/R

RapidMiner

Yes

Win/Unix/OSX

Java/Groovy

Salford Systems

Yes

Win

C/C++/.NET?

SAS Enterprise Miner

Yes

Win/Unix

C

scikit-learn

Yes

Win/Unix/OSX

C/C++/Python/ Cython

Shogun

Yes

Win/Unix

C/C++/Python/ R/MATHLAB

Statistica

Yes

Win

.NET/R

support for additional programming languages as well. The most popular languages are Java [10], C/C++ [9] and Python [6], followed by .NET, FORTRAN, R etc.

CONCLUSION

Shortly after we started this survey, we have been overwhelmed by the large number of libraries, tools, projects addressing machine learning, showing huge interest in this topic among research teams in academia and industry, equally. Applying popular machine learning algorithms to large amounts of data raised new challenges for ML practitioners. Traditional ML libraries does not support well processing of huge data sets, so that new approaches are needed based on parallelization of time-consuming tasks using modern parallel computing frameworks, such as MPI, Map Reduce. A sequel survey will investigate machine learning solutions designed for distributed computing environments, such as grids or cloud computing.

Our future plan aims at building a smart platform for problem solving applied in the field of Machine Learning, which will be able to smartly support end- users in their activities by selecting the most appropriate method for a given data set, or tweaking algorithms parameters.

All the tools and libraries referred in table 3 are commercial, close-source products, while the others are licensed under various open-source licenses (GNU (L) GPL, Apache or MIT) with a strong preference towards GPL and LGPL. In terms of operating system (column OS), as most of them rely on virtual machines (Java, Python), they are running cross-platform (Windows, Unix, Mac OS X).[9][10]The few exceptions are large commercial applications developed for Windows operating system. The ability to handle large data sets (column HLD) is largely impacted by two factors: the programming language and environment used to develop the tool and the supported machine learning methods. [7] One can observe that most of the products originating in Python world, such as Mlpy, Pyml and YAPLF, have problems in handling large data sets, may be due to the lack of mature Python libraries for large data processing at the time tool development was started. [5] Machine learning methods also impact this criteria, some of them, such as neural networks, being not well suited candidates for large data sets handling. Programming language support and interfacing (column Language) is an important criterion when it comes to integrate a library in your own application.[8] All of the surveyed products are supporting at least one external interface, which usually are its native language / platform. Many of them offer

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