[ANN] Multidimensional Array - MDArray 0.5.5


R

Rodrigo Botafogo

Announcement
============

MDArray version 0.5.5 has Just been released. MDArray is a multi dimensional array implemented
for JRuby inspired by NumPy (www.numpy.org) and Masahiro Tanaka´s Narray (narray.rubyforge.org).
MDArray stands on the shoulders of Java-NetCDF and Parallel Colt. At this point MDArray has
libraries for linear algebra, mathematical, trigonometric and descriptive statistics methods.

NetCDF-Java Library is a Java interface to NetCDF files, as well as to manyother types of
scientific data formats. It is developed and distributed by Unidata (http://www.unidata.ucar.edu).

Parallel Colt (https://sites.google.com/site/piotrwendykier/software/parallelcolt is a
multithreaded version of Colt (http://acs.lbl.gov/software/colt/). Colt provides a set of
Open Source Libraries for High Performance Scientific and Technical Computing in Java.
Scientific and technical computing is characterized by demanding problem sizes and a need for
high performance at reasonably small memory footprint.


What´s new:
===========

Class MDMatrix and Linear Algebra Methods
-----------------------------------------

This version of MDArray introduces class MDMatrix. MDMatrix is a matrix class wrapping many
linear algebra methods from Parallel Colt (see below). MDMatrix support only the following
types: i) int; ii) long; iii) float and iv) double.

Differently from other libraries, in which matrix is a subclass of array, MDMatrix is a
twin class of MDArray. MDMatrix is a twin class of MDArray as every time an MDMatrix is
instantiated, an MDArray class is also instantiated. In reality, there is only one backing
store that can be viewed by either MDMatrix or MDArray.

Creation of MDMatrix follows the same API as MDArray API. For instance, creating a double
square matrix of size 5 x 5 can be done by:

matrix = MDMatrix.double([5, 5], [2.0, 0.0, 8.0, 6.0, 0.0,\
1.0, 6.0, 0.0, 1.0, 7.0,\
5.0, 0.0, 7.0, 4.0, 0.0,\
7.0, 0.0, 8.0, 5.0, 0.0,\
0.0, 10.0, 0.0, 0.0, 7.0])

Creating an int matrix filled with zero can be done by:

matrix = MDMatrix.int([4, 3])

MDMatrix also supports creation based on methods such as fromfunction, linspace, init_with,
arange, typed_arange and ones:


array = MDArray.typed_arange("double", 0, 15)
array = MDMatrix.fromfunction("double", [4, 4]) { |x, y| x + y }

An MDMatrix can also be created from an MDArray as follows:

d2 = MDArray.typed_arange("double", 0, 15)
double_matrix = MDMatrix.from_mdarray(d2)

An MDMatrix can only be created from MDArrays of one, two or three dimensions. However,
one can take a view from an MDArray to create an MDMatrix, as long as the view is at most
three dimensional:

# Instantiate an MDArray and shape it with 4 dimensions
d1 = MDArray.typed_arange("double", 0, 420)
d1.reshape!([5, 4, 3, 7])
# slice the array, getting a three dimensional array and from that, make a matrix
matrix = MDMatrix.from_mdarray(d1.slice(0, 0))
# get a region from the array, with the first two dimensions of size 0,reduce it to a
# size two array and then build a two dimensional matrix
matrix = MDMatrix.from_mdarray(d1.region:)spec => "0:0, 0:0, 0:2,0:6").reduce)

printing the above two dimensional matrix gives us:
matrix.print
3 x 7 matrix
0,00000 1,00000 2,00000 3,00000 4,00000 5,00000 6,00000
7,00000 8,00000 9,00000 10,0000 11,0000 12,0000 13,0000
14,0000 15,0000 16,0000 17,0000 18,0000 19,0000 20,0000

Every MDMatrix instance has a twin MDArray instance that uses the same backing store. This
allows the user to treat the data as either a matrix or an array and use methods either from
matrix or array. The above matrix can be printed as an array:

matrix.mdarray.print
[[0.00 1.00 2.00 3.00 4.00 5.00 6.00]
[7.00 8.00 9.00 10.00 11.00 12.00 13.00]
[14.00 15.00 16.00 17.00 18.00 19.00 20.00]]

With an MDMatrix, many linear algebra methods can be easily calculated:

# basic operations on matrix can be done, such as, ‘+’,‘-‘, ´*’, ‘/’
# make a 4 x 4 matrix and fill it with ´double´ 2.5
a = MDMatrix.double([4, 4])
a.fill(2.5)
make a 4 x 4 matrix ´b´ from a given function (block)
b = MDMatrix.fromfunction("double", [4, 4]) { |x, y| x + y } # add both matrices
c = a + b # multiply by scalar
c = a * 2
# divide two matrices. Matrix ´b´ has to be non-singular, otherwise an exception is
# raised.
# generate a non-singular matrix from a given matrix
b.generate_non_singular!
c = a / b

Linear algebra methods:

# create a matrix with the given data
pos = MDArray.double([3, 3], [4, 12, -16, 12, 37, -43, -16, -43, 98])
matrix = MDMatrix.from_mdarray(pos)
# Cholesky decomposition from wikipedia example
chol = matrix.chol
assert_equal(2, chol[0, 0])
assert_equal(6, chol[1, 0])
assert_equal(-8, chol[2, 0])
assert_equal(5, chol[2, 1])
assert_equal(3, chol[2, 2])

All other linear algebra methods are called the same way.


MDArray and SciRuby:
====================

MDArray subscribes fully to the SciRuby Manifesto (http://sciruby.com/).

“Ruby has for some time had no equivalent to the beautifully constructed NumPy, SciPy, and
matplotlib libraries for Python.

We believe that the time for a Ruby science and visualization package has come. Sometimes
when a solution of sugar and water becomes super-saturated, from it precipitates a pure,
delicious, and diabetes-inducing crystal of sweetness, induced by no more than the tap of a
finger. So is occurring now, we believe, with numeric and visualization libraries for Ruby.â€

MDArray main properties are:
============================

+ Homogeneous multidimensional array, a table of elements (usually numbers), all of the
same type, indexed by a tuple of positive integers;
+ Support for many linear algebra methods (see bellow);
+ Easy calculation for large numerical multi dimensional arrays;
+ Basic types are: boolean, byte, short, int, long, float, double, string,structure;
+ Based on JRuby, which allows importing Java libraries;
+ Operator: +,-,*,/,%,**, >, >=, etc.;
+ Functions: abs, ceil, floor, truncate, is_zero, square, cube, fourth;
+ Binary Operators: &, |, ^, ~ (binary_ones_complement), <<, >>;
+ Ruby Math functions: acos, acosh, asin, asinh, atan, atan2, atanh, cbrt,cos, erf, exp,
gamma, hypot, ldexp, log, log10, log2, sin, sinh, sqrt, tan, tanh, neg;
+ Boolean operations on boolean arrays: and, or, not;
+ Fast descriptive statistics from Parallel Colt (complete list found bellow);
+ Easy manipulation of arrays: reshape, reduce dimension, permute, section, slice, etc.;
+ Support for reading and writing NetCDF-3 files;
+ Reading of two dimensional arrays from CSV files (mainly for debugging and simple testing
purposes);
+ StatList: a list that can grow/shrink and that can compute Parallel Coltdescriptive
statistics;
+ Experimental lazy evaluation (still slower than eager evaluation).

Supported linear algebra methods:
=================================

+ backwardSolve: Solves the upper triangular system U*x=b;
+ chol: Constructs and returns the cholesky-decomposition of the given matrix.
+ cond: Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
+ det: Returns the determinant of matrix A.
+ eig: Constructs and returns the Eigenvalue-decomposition of the given matrix.
+ forwardSolve: Solves the lower triangular system L*x=b;
+ inverse: Returns the inverse or pseudo-inverse of matrix A.
+ kron: Computes the Kronecker product of two real matrices.
+ lu: Constructs and returns the LU-decomposition of the given matrix.
+ mult: Inner product of two vectors; Sum(x * y).
+ mult: Linear algebraic matrix-vector multiplication; z = A * y.
+ mult: Linear algebraic matrix-matrix multiplication; C = A x B.
+ multOuter: Outer product of two vectors; Sets A[i,j] = x * y[j].
+ norm1: Returns the one-norm of vector x, which is Sum(abs(x)).
+ norm1: Returns the one-norm of matrix A, which is the maximum absolute column sum.
+ norm2: Returns the two-norm (aka euclidean norm) of vector x; equivalent to Sqrt(mult(x,x)).
+ norm2: Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
+ normF: Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A2)).
+ normF: Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
+ normInfinity: Returns the infinity norm of vector x, which is Max(abs(x)).
+ normInfinity: Returns the infinity norm of matrix A, which is the maximum absolute row sum.
+ pow: Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
+ qr: Constructs and returns the QR-decomposition of the given matrix.
+ rank: Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
+ solve: Solves A*x = b.
+ solve: Solves A*X = B.
+ solveTranspose: Solves X*A = B, which is also A'*X' = B'.
+ svd: Constructs and returns the SingularValue-decomposition of the given matrix.
+ trace: Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
+ trapezoidalLower: Modifies the matrix to be a lower trapezoidal matrix.
+ vectorNorm2: Returns the two-norm (aka euclidean norm) of vector X.vectorize();
+ xmultOuter: Outer product of two vectors; Returns a matrix with A[i,j] = x * y[j].
+ xpowSlow: Linear algebraic matrix power; B = Ak <==> B = A*A*....*A.

Properties´ methods tested on matrices:
=======================================

+ density: Returns the matrix's fraction of non-zero cells; A.cardinality() / A.size().
+ generate_non_singular!: Modifies the given square matrix A such that itis diagonally dominant by row and column, hence non-singular, hence invertible.
+ diagonal?: A matrix A is diagonal if A[i,j] == 0 whenever i != j.
+ diagonally_dominant_by_column?: A matrix A is diagonally dominant by column if the absolute value of each diagonal element is larger than the sum of the absolute values of the off-diagonal elements in the corresponding column.
+ diagonally_dominant_by_row?: A matrix A is diagonally dominant by row if the absolute value of each diagonal element is larger than the sum of theabsolute values of the off-diagonal elements in the corresponding row.
+ identity?: A matrix A is an identity matrix if A[i,i] == 1 and all other cells are zero.
+ lower_bidiagonal?: A matrix A is lower bidiagonal if A[i,j]==0 unless i==j || i==j+1.
+ lower_triangular?: A matrix A is lower triangular if A[i,j]==0 whenever i < j.
+ nonnegative?: A matrix A is non-negative if A[i,j] >= 0 holds for allcells.
+ orthogonal?: A square matrix A is orthogonal if A*transpose(A) = I.
+ positive?: A matrix A is positive if A[i,j] > 0 holds for all cells.
+ singular?: A matrix A is singular if it has no inverse, that is, iff det(A)==0.
+ skew_symmetric?: A square matrix A is skew-symmetric if A = -transpose(A), that is A[i,j] == -A[j,i].
+ square?: A matrix A is square if it has the same number of rows and columns.
+ strictly_lower_triangular?: A matrix A is strictly lower triangular if A[i,j]==0 whenever i <= j.
+ strictly_triangular?: A matrix A is strictly triangular if it is triangular and its diagonal elements all equal 0.
+ strictly_upper_triangular?: A matrix A is strictly upper triangular if A[i,j]==0 whenever i >= j.
+ symmetric?: A matrix A is symmetric if A = tranpose(A), that is A[i,j] == A[j,i].
+ triangular?: A matrix A is triangular iff it is either upper or lower triangular.
+ tridiagonal?: A matrix A is tridiagonal if A[i,j]==0 whenever Math.abs(i-j) > 1.
+ unit_triangular?: A matrix A is unit triangular if it is triangular andits diagonal elements all equal 1.
+ upper_bidiagonal?: A matrix A is upper bidiagonal if A[i,j]==0 unless i==j || i==j-1.
+ upper_triangular?: A matrix A is upper triangular if A[i,j]==0 whenever i > j.
+ zero?: A matrix A is zero if all its cells are zero.
+ lower_bandwidth: The lower bandwidth of a square matrix A is the maximum i-j for which A[i,j] is nonzero and i > j.
+ semi_bandwidth: Returns the semi-bandwidth of the given square matrix A..
+ upper_bandwidth: The upper bandwidth of a square matrix A is the maximum j-i for which A[i,j] is nonzero and j > i.

Descriptive statistics methods imported from Parallel Colt:
===========================================================

+ auto_correlation, correlation, covariance, durbin_watson, frequencies, geometric_mean,
+ harmonic_mean, kurtosis, lag1, max, mean, mean_deviation, median, min, moment, moment3,
+ moment4, pooled_mean, pooled_variance, product, quantile, quantile_inverse,
+ rank_interpolated, rms, sample_covariance, sample_kurtosis, sample_kurtosis_standard_error,
+ sample_skew, sample_skew_standard_error, sample_standard_deviation, sample_variance,
+ sample_weighted_variance, skew, split, standard_deviation, standard_error, sum,
+ sum_of_inversions, sum_of_logarithms, sum_of_powers, sum_of_power_deviations,
+ sum_of_squares, sum_of_squared_deviations, trimmed_mean, variance, weighted_mean,
+ weighted_rms, weighted_sums, winsorized_mean.

Double and Float methods from Parallel Colt:
============================================

+ acos, asin, atan, atan2, ceil, cos, exp, floor, greater, IEEEremainder,inv, less, lg,
+ log, log2, rint, sin, sqrt, tan.

Double, Float, Long and Int methods from Parallel Colt:
=======================================================

+ abs, compare, div, divNeg, equals, isEqual (is_equal), isGreater (is_greater),
+ isles (is_less), max, min, minus, mod, mult, multNeg (mult_neg), multSquare (mult_square),
+ neg, plus (add), plusAbs (plus_abs), pow (power), sign, square.

Long and Int methods from Parallel Colt
=======================================

+ and, dec, factorial, inc, not, or, shiftLeft (shift_left), shiftRightSigned
(shift_right_signed), shiftRightUnsigned (shift_right_unsigned), xor.

MDArray installation and download:
==================================

+ Install Jruby
+ jruby –S gem install mdarray

MDArray Homepages:
==================

+ http://rubygems.org/gems/mdarray
+ https://github.com/rbotafogo/mdarray/wiki

Contributors:
=============
Contributors are welcome.

MDArray History:
================

+ 14/11/2013: Version 0.5.5 - Support for linear algebra methods
+ 07/08/2013: Version 0.5.4 - Support for reading and writing NetCDF-3 files
+ 24/06/2013: Version 0.5.3 - Over 90% Performance improvements for methods imported
from Parallel Colt and over 40% performance improvements for all other methods
(implemented in Ruby);
+ 16/05/2013: Version 0.5.0 - All loops transferred to Java with over 50%performance
improvements. Descriptive statistics from Parallel Colt;
+ 19/04/2013: Version 0.4.3 - Fixes a simple, but fatal bug in 0.4.2. Nonew features;
+ 17/04/2013: Version 0.4.2 - Adds simple statistics and boolean operators;
+ 05/04/2013: Version 0.4.0 - Initial release.
 
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