This chapter gives you a quick overview of one of the most popular data analysis libraries in Scala, how to get them, and their most frequently used functions and data structures.

We will be focusing on Breeze in this first chapter, which is one of the most popular and powerful linear algebra libraries. Spark MLlib, which we will be seeing in the subsequent chapters, builds on top of Breeze and Spark, and provides a powerful framework for scalable machine learning.

In simple terms, Breeze (http://www.scalanlp.org) is a Scala library that extends the Scala collection library to provide support for vectors and matrices in addition to providing a whole bunch of functions that support their manipulation. We could safely compare Breeze to NumPy (http://www.numpy.org/) in Python terms. Breeze forms the foundation of MLlibâ€”the Machine Learning library in Spark, which we will explore in later chapters.

In this first recipe, we will see how to pull the Breeze libraries into our project using **Scala Build Tool** (**SBT**). We will also see a brief history of Breeze to better appreciate why it could be considered as the "go to" linear algebra library in Scala.

### Note

For all our recipes, we will be using Scala 2.10.4 along with Java 1.7. I wrote the examples using the Scala IDE, but please feel free to use your favorite IDE.

Let's add the Breeze dependencies into our `build.sbt`

so that we can start playing with them in the subsequent recipes. The Breeze dependencies are just twoâ€”the `breeze`

(core) and the `breeze-native`

dependencies.

Under a brand new folder (which will be our project root), create a new file called

`build.sbt`

.Next, add the

`breeze`

libraries to the project dependencies:organization := "com.packt" name := "chapter1-breeze" scalaVersion := "2.10.4" libraryDependencies ++= Seq( "org.scalanlp" %% "breeze" % "0.11.2", //Optional - the 'why' is explained in the How it works section "org.scalanlp" %% "breeze-natives" % "0.11.2" )

From that folder, issue a

`sbt compile`

command in order to fetch all your dependencies.### Note

You could import the project into your Eclipse using

`sbt eclipse`

after installing the`sbteclipse`

plugin https://github.com/typesafehub/sbteclipse/. For IntelliJ IDEA, you just need to import the project by pointing to the root folder where your`build.sbt`

file is.

Let's look into the details of what the `breeze`

and `breeze-native`

library dependencies we added bring to us.

Breeze has a long history in that it isn't written from scratch in Scala. Without the native dependency, Breeze leverages the power of `netlib-java`

that has a Java-compiled version of the FORTRAN Reference implementation of BLAS/LAPACK. The `netlib-java`

also provides gentle wrappers over the Java compiled library. What this means is that we could still work without the native dependency but the performance won't be great considering the best performance that we could leverage out of this FORTRAN-translated library is the performance of the FORTRAN reference implementation itself. However, for serious number crunching with the best performance, we should add the `breeze-natives`

dependency too.

With its native additive, Breeze looks for the machine-specific implementations of the BLAS/LAPACK libraries. The good news is that there are open source and (vendor provided) commercial implementations for most popular processors and GPUs. The most popular open source implementations include ATLAS (http://math-atlas.sourceforge.net) and OpenBLAS (http://www.openblas.net/).

If you are running a Mac, you are in luckâ€”Native BLAS libraries come out of the box on Macs. Installing NativeBLAS on Ubuntu / Debian involves just running the following commands:

sudo apt-get install libatlas3-base libopenblas-basesudo update-alternatives --config libblas.so.3sudo update-alternatives --config liblapack.so.3

### Tip

**Downloading the example code**

You can download the example code files from your account at http://www.packtpub.com for all the Packt Publishing books you have purchased. If you purchased this book elsewhere, you can visit http://www.packtpub.com/support and register to have the files e-mailed directly to you.

For Windows, please refer to the installation instructions on https://github.com/xianyi/OpenBLAS/wiki/Installation-Guide.

There are subtle yet powerful differences between Breeze vectors and Scala's own `scala.collection.Vector`

. As we'll see in this recipe, Breeze vectors have a lot of functions that are linear algebra specific, and the more important thing to note here is that Breeze's vector is a Scala wrapper over `netlib-java`

and most calls to the vector's API delegates the call to it.

Vectors are one of the core components in Breeze. They are containers of homogenous data. In this recipe, we'll first see how to create vectors and then move on to various data manipulation functions to modify those vectors.

In this recipe, we will look at various operations on vectors. This recipe has been organized in the form of the following sub-recipes:

Creating vectors:

Creating a vector from values

Creating a zero vector

Creating a vector out of a function

Creating a vector of linearly spaced values

Creating a vector with values in a specific range

Creating an entire vector with a single value

Slicing a sub-vector from a bigger vector

Creating a Breeze vector from a Scala vector

Vector arithmetic:

Scalar operations

Calculating the dot product of a vector

Creating a new vector by adding two vectors together

Appending vectors and converting a vector of one type to another:

Concatenating two vectors

Converting a vector of int to a vector of double

Computing basic statistics:

Mean and variance

Standard deviation

Find the largest value

Finding the sum, square root and log of all the values in the vector

In order to run the code, you could either use the Scala or use the Worksheet feature available in the Eclipse Scala plugin (or Scala IDE) or in IntelliJ IDEA. The reason these options are suggested is due to their quick turnaround time.

Let's look at each of the above sub-recipes in detail. For easier reference, the output of the respective command is shown as well. All the classes that are being used in this recipe are from the `breeze.linalg`

package. So, an `"import breeze.linalg._"`

statement at the top of your file would be perfect.

Let's look at the various ways we could construct vectors. Most of these construction mechanisms are through the `apply`

method of the vector. There are two different flavors of vectorâ€”`breeze.linalg.DenseVector`

and `breeze.linalg.SparseVector`

â€”the choice of the vector depends on the use case. The general rule of thumb is that if you have data that is at least 20 percent zeroes, you are better off choosing `SparseVector`

but then the 20 percent is a variant too.

**Creating a dense vector from values**: Creating a`DenseVector`

from values is just a matter of passing the values to the`apply`

method:**val dense=DenseVector(1,2,3,4,5)****println (dense) //DenseVector(1, 2, 3, 4, 5)****Creating a sparse vector from values**: Creating a`SparseVector`

from values is also through passing the values to the`apply`

method:**val sparse=SparseVector(0.0, 1.0, 0.0, 2.0, 0.0)****println (sparse) //SparseVector((0,0.0), (1,1.0), (2,0.0), (3,2.0), (4,0.0))**

Notice how the `SparseVector`

stores values against the index.

Obviously, there are simpler ways to create a vector instead of just throwing all the data into its `apply`

method.

Calling the vector's `zeros`

function would create a zero vector. While the numeric types would return a `0`

, the object types would return `null`

and the Boolean types would return `false`

:

val denseZeros=DenseVector.zeros[Double](5) //DenseVector(0.0, 0.0, 0.0, 0.0, 0.0) val sparseZeros=SparseVector.zeros[Double](5) //SparseVector()

Not surprisingly, the `SparseVector`

does not allocate any memory for the contents of the vector. However, the creation of the `SparseVector`

object itself is accounted for in the memory.

The `tabulate`

function in vector is an interesting and useful function. It accepts a size argument just like the `zeros`

function but it also accepts a function that we could use to populate the values for the vector. The function could be anything ranging from a random number generator to a naÃ¯ve index based generator, which we have implemented here. Notice how the return value of the function (`Int`

) could be converted into a vector of `Double`

by using the `type`

parameter:

val denseTabulate=DenseVector.tabulate[Double](5)(index=>index*index) //DenseVector(0.0, 1.0, 4.0, 9.0, 16.0)

The `linspace`

function in `breeze.linalg`

creates a new `Vector[Double]`

of linearly spaced values between two arbitrary numbers. Not surprisingly, it accepts three argumentsâ€”the start, end, and the total number of values that we would like to generate. Please note that the start and the end values are inclusive while being generated:

val spaceVector=breeze.linalg.linspace(2, 10, 5) //DenseVector(2.0, 4.0, 6.0, 8.0, 10.0)

The `range`

function in a vector has two variants. The plain vanilla function accepts a start and end value (start inclusive):

val allNosTill10=DenseVector.range(0, 10) //DenseVector(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

The other variant is an overloaded function that accepts a "step" value:

val evenNosTill20=DenseVector.range(0, 20, 2) // DenseVector(0, 2, 4, 6, 8, 10, 12, 14, 16, 18)

Just like the `range`

function, which has all the arguments as integers, there is also a `rangeD`

function that takes the start, stop, and the step parameters as `Double`

:

val rangeD=DenseVector.rangeD(0.5, 20, 2.5) // DenseVector(0.5, 3.0, 5.5, 8.0, 10.5, 13.0, 15.5)

Filling an entire vector with the same value is child's play. We just say HOW BIG is this vector going to be and then WHAT value. That's it.

val denseJust2s=DenseVector.fill(10, 2) // DenseVector(2, 2, 2, 2, 2, 2 , 2, 2, 2, 2)

Choosing a part of the vector from a previous vector is just a matter of calling the slice method on the bigger vector. The parameters to be passed are the start index, end index, and an optional "step" parameter. The step parameter adds the step value for every iteration until it reaches the end index. Note that the end index is excluded in the sub-vector:

val allNosTill10=DenseVector.range(0, 10) //DenseVector(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) val fourThroughSevenIndexVector= allNosTill10.slice(4, 7) //DenseVector(4, 5, 6) val twoThroughNineSkip2IndexVector= allNosTill10.slice(2, 9, 2) //DenseVector(2, 4, 6)

A Breeze vector object's `apply`

method could even accept a Scala Vector as a parameter and construct a vector out of it:

val vectFromArray=DenseVector(collection.immutable.Vector(1,2,3,4)) // DenseVector(Vector(1, 2, 3, 4))

Operations with scalars work just as we would expect, propagating the value to each element in the vector.

Adding a scalar to each element of the vector is done using the `+`

function (surprise!):

val inPlaceValueAddition=evenNosTill20 +2 //DenseVector(2, 4, 6, 8, 10, 12, 14, 16, 18, 20)

Similarly the other basic arithmetic operationsâ€”subtraction, multiplication, and division involves calling the respective functions named after the universally accepted symbols (`-`

, `*`

, and `/`

):

//Scalar subtraction val inPlaceValueSubtraction=evenNosTill20 -2 //DenseVector(-2, 0, 2, 4, 6, 8, 10, 12, 14, 16) //Scalar multiplication val inPlaceValueMultiplication=evenNosTill20 *2 //DenseVector(0, 4, 8, 12, 16, 20, 24, 28, 32, 36) //Scalar division val inPlaceValueDivision=evenNosTill20 /2 //DenseVector(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

Each vector object has a function called `dot`

, which accepts another vector of the same length as a parameter.

Let's fill in just `2s`

to a new vector of length `5`

:

val justFive2s=DenseVector.fill(5, 2) //DenseVector(2, 2, 2, 2, 2)

We'll create another vector from `0`

to `5`

with a step value of `1`

(a fancy way of saying `0`

through `4`

):

val zeroThrough4=DenseVector.range(0, 5, 1) //DenseVector(0, 1, 2, 3, 4)

Here's the `dot`

function:

val dotVector=zeroThrough4.dot(justFive2s) //Int = 20

It is to be expected of the function to complain if we pass in a vector of a different length as a parameter to the dot product - Breeze throws an `IllegalArgumentException`

if we do that. The full exception message is:

Java.lang.IllegalArgumentException: Vectors must be the same length!

The `+`

function is overloaded to accept a vector other than the scalar we saw previously. The operation does a corresponding element-by-element addition and creates a new vector:

val evenNosTill20=DenseVector.range(0, 20, 2) //DenseVector(0, 2, 4, 6, 8, 10, 12, 14, 16, 18) val denseJust2s=DenseVector.fill(10, 2) //DenseVector(2, 2, 2, 2, 2, 2, 2, 2, 2, 2) val additionVector=evenNosTill20 + denseJust2s // DenseVector(2, 4, 6, 8, 10, 12, 14, 16, 18, 20)

There's an interesting behavior encapsulated in the addition though. Assuming you try to add two vectors of different lengths, if the first vector is smaller and the second vector larger, the resulting vector would be the size of the first vector and the rest of the elements in the second vector would be ignored!

val fiveLength=DenseVector(1,2,3,4,5) //DenseVector(1, 2, 3, 4, 5) val tenLength=DenseVector.fill(10, 20) //DenseVector(20, 20, 20, 20, 20, 20, 20, 20, 20, 20) fiveLength+tenLength //DenseVector(21, 22, 23, 24, 25)

On the other hand, if the first vector is larger and the second vector smaller, it would result in an `ArrayIndexOutOfBoundsException`

:

tenLength+fiveLength // java.lang.ArrayIndexOutOfBoundsException: 5

There are two variants of concatenation. There is a `vertcat`

function that just vertically concatenates an arbitrary number of vectorsâ€”the size of the vector just increases to the sum of the sizes of all the vectors combined:

val justFive2s=DenseVector.fill(5, 2) //DenseVector(2, 2, 2, 2, 2) val zeroThrough4=DenseVector.range(0, 5, 1) //DenseVector(0, 1, 2, 3, 4) val concatVector=DenseVector.vertcat(zeroThrough4, justFive2s) //DenseVector(0, 1, 2, 3, 4, 2, 2, 2, 2, 2)

No surprise here. There is also the `horzcat`

method that places the second vector horizontally next to the first vector, thus forming a matrix.

val concatVector1=DenseVector.horzcat(zeroThrough4, justFive2s)//breeze.linalg.DenseMatrix[Int]0 21 22 23 24 2

### Note

While dealing with vectors of different length, the `vertcat`

function happily arranges the second vector at the bottom of the first vector. Not surprisingly, the `horzcat`

function throws an exception:

`java.lang.IllegalArgumentException`

, meaning all vectors must be of the same size!

The conversion of one type of vector into another is not automatic in Breeze. However, there is a simple way to achieve this:

val evenNosTill20Double=breeze.linalg.convert(evenNosTill20, Double)

Other than the creation and the arithmetic operations that we saw previously, there are some interesting summary statistics operations that are available in the library. Let's look at them now:

### Note

Needs import of `breeze.linalg._`

and `breeze.numerics._`

. The operations in the Other operations section aim to simulate the NumPy's `UFunc`

or universal functions.

Now, let's briefly look at how to calculate some basic summary statistics for a vector.

Calling the
`stddev`

on a `Double`

vector could give the standard deviation:

stddev(evenNosTill20Double) //Double = 6.0553007081949835

The `max`

universal function inside the `breeze.linalg`

package would help us find the maximum value in a vector:

val intMaxOfVectorVals=max (evenNosTill20) //18

The same as with
`max`

, the `sum`

universal function inside the `breeze.linalg`

package calculates the `sum`

of the vector:

val intSumOfVectorVals=sum (evenNosTill20) //90

The functions `sqrt`

, `log`

, and various other universal functions in the `breeze.numerics`

package calculate the square root and log values of all the individual elements inside the vector:

val sqrtOfVectorVals= sqrt (evenNosTill20) // DenseVector(0.0, 1. 4142135623730951, 2.0, 2.449489742783178, 2.8284271247461903, 3.16227766016 83795, 3.4641016151377544, 3.7416573867739413, 4.0, 4.242640687119285)

As we discussed in the *Working with vectors* recipe, you could use the Eclipse or IntelliJ IDEA Scala worksheets for a faster turnaround time.

There are a variety of functions that we have in a matrix. In this recipe, we will look at some details around:

Creating matrices:

Creating a matrix from values

Creating a zero matrix

Creating a matrix out of a function

Creating an identity matrix

Creating a matrix from random numbers

Creating from a Scala collection

Matrix arithmetic:

Addition

Multiplication (also element-wise)

Appending and conversion:

Concatenating a matrix vertically

Concatenating a matrix horizontally

Converting a matrix of Int to a matrix of Double

Getting column vectors

Getting row vectors

Getting values inside the matrix

Getting the inverse and transpose of a matrix

Computing basic statistics:

Mean and variance

Standard deviation

Finding the largest value

Finding the sum, square root and log of all the values in the matrix

Calculating the eigenvectors and eigenvalues of a matrix

Let's first see how to create a matrix.

The simplest way to create a matrix is to pass in the values in a row-wise fashion into the `apply`

function of the matrix object:

val simpleMatrix=DenseMatrix((1,2,3),(11,12,13),(21,22,23))//Returns a DenseMatrix[Int]1 2 311 12 1321 22 23

There's also a Sparse version of the matrix tooâ€”the **Compressed Sparse Column Matrix** (**CSCMatrix**):

val sparseMatrix=CSCMatrix((1,0,0),(11,0,0),(0,0,23))//Returns a SparseMatrix[Int](0,0) 1(1,0) 11(2,2) 23

Creating a zero matrix is just a matter of calling the matrix's `zeros`

function. The first integer parameter indicates the rows and the second parameter indicates the columns:

val denseZeros=DenseMatrix.zeros[Double](5,4)//Returns a DenseMatrix[Double]0.0 0.0 0.0 0.00.0 0.0 0.0 0.00.0 0.0 0.0 0.00.0 0.0 0.0 0.00.0 0.0 0.0 0.0val compressedSparseMatrix=CSCMatrix.zeros[Double](5,4)//Returns a CSCMatrix[Double] = 5 x 4 CSCMatrix

The `tabulate`

function in a matrix is very similar to the vector's version. It accepts a row and column size as a tuple (in the example `(5,4)`

). It also accepts a function that we could use to populate the values for the matrix. In our example, we generated the values of the matrix by just multiplying the row and column index:

val denseTabulate=DenseMatrix.tabulate[Double](5,4)((firstIdx,secondIdx)=>firstIdx*secondIdx)Returns a DenseMatrix[Double] =0.0 0.0 0.0 0.00.0 1.0 2.0 3.00.0 2.0 4.0 6.00.0 3.0 6.0 9.00.0 4.0 8.0 12.0

The `type`

parameter is needed only if you would like to convert the type of the matrix from an `Int`

to a `Double`

. So, the following call without the parameter would just return an `Int`

matrix:

val denseTabulate=DenseMatrix.tabulate(5,4)((firstIdx,secondIdx)=>firstIdx*secondIdx)0 1 2 30 2 4 60 3 6 90 4 8 12

The `eye`

function of the matrix would generate an identity square matrix with the given dimension (in the example's case, `3`

):

val identityMatrix=DenseMatrix.eye[Int](3)Returns a DenseMatrix[Int]1 0 00 1 00 0 1

The `rand`

function in the matrix would generate a matrix of a given dimension (4 rows * 4 columns in our case) with random values between `0`

and `1`

. We'll have an in-depth look into random number generated vectors and matrices in a subsequent recipe.

val randomMatrix=DenseMatrix.rand(4, 4)Returns DenseMatrix[Double]0.09762565779429777 0.01089176285376725 0.2660579009292807 0.194281939619856740.9662568115400412 0.718377391997945 0.8230367668470933 0.39575408543931690.9080090988364429 0.7697780247035393 0.49887760321635066 0.267220191056544153.326843165250004E-4 0.447925644082819 0.8195838733418965 0.7682752255172411

We could create a matrix out of a Scala array too. The constructor of the matrix accepts three argumentsâ€”the rows, the columns, and an array with values for the dimensions. Note that the data from the array is picked up to construct the matrix in the column first order:

val vectFromArray=new DenseMatrix(2,2,Array(2,3,4,5))Returns DenseMatrix[Int]2 43 5

If there are more values than the number of values required by the dimensions of the matrix, the rest of the values are ignored. Note how `(6,7)`

is ignored in the array:

val vectFromArray=new DenseMatrix(2,2,Array(2,3,4,5,6,7))DenseMatrix[Int]2 43 5

However, if fewer values are present in the array than what is required by the dimensions of the matrix, then the constructor call would throw an `ArrayIndexOutOfBoundsException`

:

val vectFromArrayIobe=new DenseMatrix(2,2,Array(2,3,4))//throws java.lang.ArrayIndexOutOfBoundsException: 3

Now let's look at the basic arithmetic that we could do using matrices.

Let's consider a simple 3*3 `simpleMatrix`

and a corresponding identity matrix:

val simpleMatrix=DenseMatrix((1,2,3),(11,12,13),(21,22,23))//DenseMatrix[Int]1 2 311 12 1321 22 23val identityMatrix=DenseMatrix.eye[Int](3)//DenseMatrix[Int]1 0 00 1 00 0 1

Adding two matrices will result in a matrix whose corresponding elements are summed up.

val additionMatrix=identityMatrix + simpleMatrix// Returns DenseMatrix[Int]2 2 311 13 1321 22 24

Now, as you would expect, multiplying a matrix with its identity should give you the matrix itself:

val simpleTimesIdentity=simpleMatrix * identityMatrix//Returns DenseMatrix[Int]1 2 311 12 1321 22 23

Breeze also has an alternative element-by-element operation that has the format of prefixing the operator with a colon, for example, `:+`

,`:-`

, `:*`

, and so on. Check out what happens when we do an element-wise multiplication of the identity matrix and the simple matrix:

val elementWiseMulti=identityMatrix :* simpleMatrix//DenseMatrix[Int]1 0 00 12 00 0 23

Let's briefly see how to append two matrices and convert matrices of one numeric type to another.

Similar to vectors, matrix has a `vertcat`

function, which vertically concatenates an arbitrary number of matricesâ€”the row size of the matrix just increases to the sum of the row sizes of all matrices combined:

val vertConcatMatrix=DenseMatrix.vertcat(identityMatrix, simpleMatrix)//DenseMatrix[Int]1 0 00 1 00 0 11 2 311 12 1321 22 23

Attempting to concatenate a matrix of different columns would, as expected, throw an `IllegalArgumentException`

:

java.lang.IllegalArgumentException: requirement failed: Not all matrices have the same number of columns

Not surprisingly, the `horzcat`

function concatenates the matrix horizontallyâ€”the column size of the matrix increases to the sum of the column sizes of all the matrices:

val horzConcatMatrix=DenseMatrix.horzcat(identityMatrix, simpleMatrix)// DenseMatrix[Int]1 0 0 1 2 30 1 0 11 12 130 0 1 21 22 23

Similar to the vertical concatenation, attempting to concatenate a matrix of a different row size would throw an `IllegalArgumentException`

:

java.lang.IllegalArgumentException: requirement failed: Not all matrices have the same number of rows

Let's create a simple 2*2 matrix that will be used for the rest of this section:

val simpleMatrix=DenseMatrix((4.0,7.0),(3.0,-5.0))//DenseMatrix[Double] =4.0 7.03.0 -5.0

The first column vector could be retrieved by passing in the column parameter as `0`

and using `::`

in order to say that we are interested in all the rows.

val firstVector=simpleMatrix(::,0) //DenseVector(4.0, 3.0)

Getting the second column vector and so on is achieved by passing the correct zero-indexed column number:

val secondVector=simpleMatrix(::,1) //DenseVector(7.0, -5.0)

Alternatively, you could explicitly pass in the columns to be extracted:

val firstVectorByCols=simpleMatrix(0 to 1,0) //DenseVector(4.0, 3.0)

While explicitly stating the range (as in `0`

to `1`

), we have to be careful not to exceed the matrix size. For example, the following attempt to select 3 columns (`0`

through `2`

) on a 2 * 2 matrix would throw an `ArrayIndexOutOfBoundsException`

:

val errorTryingToSelect3ColumnsOn2By2Matrix=simpleMatrix(0,0 to 2) //java.lang.ArrayIndexOutOfBoundsException

If we would like to get the row vector, all we need to do is play with the row and column parameters again. As expected, it would give a transpose of the column vector, which is simply a row vector.

Like the column vector, we could either explicitly state our columns or pass in a wildcard (`::`

) to cover the entire range of columns:

val firstRowStatingCols=simpleMatrix(0,0 to 1) //Transpose(DenseVector(4.0, 7.0)) val firstRowAllCols=simpleMatrix(0,::) //Transpose(DenseVector(4.0, 7.0))

Getting the second row vector is achieved by passing the second row (1) and all the columns (`::`

) in that vector:

val secondRow=simpleMatrix(1,::) //Transpose(DenseVector(3.0, -5.0))

Assuming we are just interested in the values within the matrix, pass in the exact row and the column number of the matrix. In order to get the first row and first column of the matrix, just pass in the row and the column number:

val firstRowFirstCol=simpleMatrix(0,0) //Double = 4.0

Getting the inverse and the transpose of a matrix is a little counter-intuitive in Breeze. Let's consider the same matrix that we dealt with earlier:

val simpleMatrix=DenseMatrix((4.0,7.0),(3.0,-5.0))

On the one hand, `transpose`

is a function on the matrix object itself, like so:

val transpose=simpleMatrix.t4.0 3.07.0 -5.0

`inverse`

, on the other hand is a universal function under the `breeze.linalg`

package:

val inverse=inv(simpleMatrix)0.12195121951219512 0.170731707317073180.07317073170731708 -0.0975609756097561

Let's do a matrix product to its inverse and confirm whether it is an identity matrix:

simpleMatrix * inverse1.0 0.0-5.551115123125783E-17 1.0

As expected, the result is indeed an identity matrix with rounding errors when doing floating point arithmetic.

Now, just like vectors, let's briefly look at how to calculate some basic summary statistics for a matrix.

### Tip

This needs import of `breeze.linalg._`

, `breeze.numerics._`

and, `breeze.stats._`

. The operations in the "Other operations" section aims to simulate the NumPy's `UFunc`

or universal functions.

Calculating the mean and variance of a matrix could be achieved by calling the `meanAndVariance`

universal function in the `breeze.stats`

package. Note that this needs a matrix of `Double`

:

meanAndVariance(simpleMatrixAsDouble) // MeanAndVariance(12.0,75.75,9)

Alternatively, converting an `Int`

matrix to a `Double`

matrix and calculating the mean and variance for that Matrix could be merged into a one-liner:

meanAndVariance(convert(simpleMatrix, Double))

Calling the `stddev`

on a `Double`

vector could give the standard deviation:

stddev(simpleMatrixAsDouble) //Double = 8.703447592764606

Next up, let's look at some basic aggregation operations:

val simpleMatrix=DenseMatrix((1,2,3),(11,12,13),(21,22,23))

The (`apply`

method of the) `max`

object (a universal function) inside the `breeze.linalg`

package will help us do that:

val intMaxOfMatrixVals=max (simpleMatrix) //23

The same as with `max`

, the
`sum`

object inside the `breeze.linalg`

package calculates the sum of all the matrix elements:

val intSumOfMatrixVals=sum (simpleMatrix) //108

The functions `sqrt`

, `log`

, and various other objects (universal functions) in the `breeze.numerics`

package calculate the square root and log values of all the individual values inside the matrix.

val sqrtOfMatrixVals= sqrt (simpleMatrix)//DenseMatrix[Double] =1.0 1.4142135623730951 1.73205080756887723.3166247903554 3.4641016151377544 3.6055512754639894.58257569495584 4.69041575982343 4.795831523312719

val log2MatrixVals=log(simpleMatrix)//DenseMatrix[Double]0.0 0.6931471805599453 1.09861228866810982.3978952727983707 2.4849066497880004 2.56494935746153673.044522437723423 3.091042453358316 3.1354942159291497

Calculating eigenvectors is straightforward in Breeze. Let's consider our `simpleMatrix`

from the previous section:

val simpleMatrix=DenseMatrix((4.0,7.0),(3.0,-5.0))

Calling the `breeze.linalg.eig`

universal function on a matrix returns a `breeze.linalg.eig.DenseEig`

object that encapsulate eigenvectors and eigenvalues:

val denseEig=eig(simpleMatrix)

This line of code returns the following:

Eig(DenseVector(5.922616289332565, -6.922616289332565),DenseVector(0.0, 0.0),0.9642892971721949 -0.5395744865143975 0.26485118719604456 0.8419378679586305)

We could extract the eigenvectors and eigenvalues by calling the corresponding functions on the returned `Eig`

reference:

val eigenVectors=denseEig.eigenvectors//DenseMatrix[Double] =0.9642892971721949 -0.53957448651439750.26485118719604456 0.8419378679586305

The two `eigenValues`

corresponding to the two `eigenvectors`

could be captured using the `eigenvalues`

function on the `Eig`

object:

val eigenValues=denseEig.eigenvalues//DenseVector[Double] = DenseVector(5.922616289332565, -6.922616289332565)

Let's validate the eigenvalues and the vectors:

Let's multiply the matrix with the first eigenvector:

val matrixToEigVector=simpleMatrix*denseEig.eigenvectors (::,0) //DenseVector(5.7111154990610915, 1.568611955536362)

Then let's multiply the first eigenvalue with the first eigenvector. The resulting vector will be the same with a marginal error when doing floating point arithmetic:

val vectorToEigValue=denseEig.eigenvectors(::,0) * denseEig.eigenvalues (0) //DenseVector(5.7111154990610915, 1.5686119555363618)

The same as with vectors, the initialization of the Breeze matrices are achieved by way of the `apply`

method or one of the various methods in the matrix's `Object`

class. Various other operations are provided by way of polymorphic functions available in the `breeze.numeric`

, `breeze.linalg`

and `breeze.stats`

packages.

The `breeze.stats.distributions`

package supplements the random number generator that is built into Scala. Scala's default generator just provides the ability to get the random values one by one using the "next" methods. Random number generators in Breeze provide the ability to build vectors and matrices out of these generators. In this recipe, we'll briefly see three of the most common distributions of random numbers.

In this recipe, we will cover at the following sub-recipes:

*Creating vectors with uniformly distributed random values**Creating vectors with normally distributed random values**Creating vectors with random values that have a Poisson distribution**Creating a matrix with uniformly random values**Creating a matrix with normally distributed random values**Creating a matrix with random values that has a Poisson distribution*

Before we delve into how to create the vectors and matrices out of random numbers, let's create instances of the most common random number distribution. All these generators are under the `breeze.stats.distributions`

package:

//Uniform distribution with low being 0 and high being 10 val uniformDist=Uniform(0,10) //Gaussian distribution with mean being 5 and Standard deviation being 1 val gaussianDist=Gaussian(5,1) //Poission distribution with mean being 5 val poissonDist=Poisson(5)

We could actually directly sample from these generators. Given any distribution we created previously, we could sample either a single value or a sequence of values:

//Samples a single value println (uniformDist.sample()) //eg. 9.151191360491392 //Returns a sample vector of size that is passed in as parameter println (uniformDist.sample(2)) //eg. Vector(6.001980062275654, 6.210874664967401)

With no generator parameter, the `DenseVector.rand`

method accepts a parameter for the length of the vector to be returned. The result is a vector (of length `10`

) with uniformly distributed values between `0`

and `1`

:

val uniformWithoutSize=DenseVector.rand(10)println ("uniformWithoutSize \n"+ uniformWithoutSize)//DenseVector(0.1235038023750481, 0.3120595941786264, 0.3575638744660876, 0.5640844223813524, 0.5336149399548831, 0.1338053814330793, 0.9099684427908603, 0.38690724148973166, 0.22561993631651522, 0.45120359622713657)

The `DenseVector.rand`

method optionally accepts a distribution object and generates random values using that input distribution. The following line generates a vector of `10`

uniformly distributed random values that are within the range `0`

and `10`

:

val uniformDist=Uniform(0,10)val uniformVectInRange=DenseVector.rand(10, uniformDist)println ("uniformVectInRange \n"+uniformVectInRange)//DenseVector(1.5545833905907314, 6.172564377264846, 8.45578509265587, 7.683763574965107, 8.018688137742062, 4.5876187984930406, 3.274758584944064, 2.3873947264259954, 2.139988841403757, 8.314112884416943)

In the place of the `uniformDist`

generator, we could also pass the previously created Gaussian generator, which is configured to yield a distribution that has a mean of `5`

and standard deviation of `1`

:

val gaussianVector=DenseVector.rand(10, gaussianDist)println ("gaussianVector \n"+gaussianVector)//DenseVector(4.235655596913547, 5.535011377545014, 6.201428236839494, 6.046289604188366, 4.319709374229152,4.2379652913447154, 2.957868021601233, 3.96371080427211, 4.351274306757224, 5.445022658876723)

Similarly, by passing the previously created Poisson random number generator, a vector of values that has a mean of `5`

could be generated:

val poissonVector=DenseVector.rand(10, poissonDist)println ("poissonVector \n"+poissonVector)//DenseVector(5, 5, 7, 11, 7, 6, 6, 6, 6, 6)

We saw how easy it is to create a vector of random values. Now, let's proceed to create a matrix of random values. Similar to `DenseVector.rand`

to generate vectors with random values, we'll use the `DenseMatrix.rand`

function to generate a matrix of random values.

The `DenseMatrix.rand`

defaults to the uniform distribution and generates a matrix of random values given the row and the column parameter. However, if we would like to have a distribution within a range, then as in vectors, we could use the optional parameter:.

//Uniform distribution, Creates a 3 * 3 Matrix with random values from 0 to 1val uniformMat=DenseMatrix.rand(3, 3)println ("uniformMat \n"+uniformMat)0.4492155777289115 0.9098840386699856 0.82030222529882920.0888975848853315 0.009677790736892788 0.60588859059342370.6201415814136939 0.7017492438727635 0.08404147915159443//Creates a 3 * 3 Matrix with uniformly distributed random values with low being 0 and high being 10val uniformMatrixInRange=DenseMatrix.rand(3,3, uniformDist)println ("uniformMatrixInRange \n"+uniformMatrixInRange)7.592014659345548 8.164652560340933 6.9664452944644018.35949395084735 3.442654641743763 3.67616402409384429.42626645215854 0.23658921372298636 7.327120138868571

Just as in vectors, in place of the `uniformDist`

generator, we could also pass the previously created Gaussian generator to the `rand`

function to generate a matrix of random values that has a mean of `5`

and standard deviation of `1`

:

//Creates a 3 * 3 Matrix with normally distributed random values with mean being 5 and Standard deviation being 1val gaussianMatrix=DenseMatrix.rand(3, 3,gaussianDist)println ("gaussianMatrix \n"+gaussianMatrix)5.724540885605018 5.647051873430568 5.3379061351070986.2228893721489875 4.799561665187845 5.124697794898335.136960834730864 5.176410360757703 5.262707072950913

Similarly, by passing the previously created Poisson random number generator, a matrix of random values that has a mean of `5`

could be generated:

//Creates a 3 * 3 Matrix with Poisson distribution with mean being 5val poissonMatrix=DenseMatrix.rand(3, 3,poissonDist)println ("poissonMatrix \n"+poissonMatrix)4 11 36 6 56 4 2

Reading and writing a CSV file in Breeze is really a breeze. We just have two functions in `breeze.linalg`

package to play with. They are very intuitively named `csvread`

and `csvwrite`

.

In this recipe, you'll see how to:

Read a CSV file into a matrix

Save selected columns of a matrix into a new matrix

Write the newly created matrix into a CSV file

Extract a vector out of the matrix

Write the vector into a CSV

There are just two functions that we need to remember in order to read and write data from and to CSV files. The signatures of the functions are pretty straightforward too:

csvread(file, separator, quote, escape, skipLines) csvwrite(file, mat, separator, quote, escape, skipLines)

Let's look at the parameters by order of importance:

`file`

:`java.io.File`

: Represents the file location.`separator`

: Defaults to a comma so as to represent a CSV. Could be overridden when needed.`skipLines`

: This is the number of lines to be skipped while reading the file. Generally, if there is a header, we pass a`skipLines=1`

.`mat`

: While writing, this is the matrix object that is being written.`quote`

: This defaults to double quotes. It is a character that implies that the value inside is one single value.`escape`

: This defaults to a backspace. It is a character used to escape special characters.

Let's see these in action. For the sake of clarity, I have skipped the quote and the escape parameter while calling the `csvread`

and `csvwrite`

functions. For this recipe, we will do three things:

Read a CSV file as a matrix

Extract a sub-matrix out of the read matrix

Write the matrix

Read the CSV as a matrix:

Let's use the

`csvread`

function to read a CSV file into a 100*3 matrix. We'll also skip the header while reading and print 5 rows as a sample:**val usageMatrix=csvread(file=new File("WWWusage.csv"), separator=',', skipLines=1)****//print first five rows****println ("Usage matrix \n"+ usageMatrix(0 to 5,::))****Output :****1.0 1.0 88.0****2.0 2.0 84.0****3.0 3.0 85.0****4.0 4.0 85.0****5.0 5.0 84.0****6.0 6.0 85.0**Extract a sub-matrix out of the read matrix:

For the sake of generating a submatrix let's skip the first column and save the second and the third column into a new matrix. Let's call it

`firstColumnSkipped`

:**val firstColumnSkipped= usageMatrix(::, 1 to usageMatrix.cols-1)****//Sample some data so as to ensure we are fine****println ("First Column skipped \n"+ firstColumnSkipped(0 to 5, ::))****Output :****1.0 88.0****2.0 84.0****3.0 85.0****4.0 85.0****5.0 84.0****6.0 85.0**Write the matrix:

As a final step, let's write the

`firstColumnSkipped`

matrix to a new CSV file named`firstColumnSkipped.csv`

://Write this modified matrix to a file csvwrite(file=new File ("firstColumnSkipped.csv"), mat=firstColumnSkipped, separator=',')