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Home > Data > Machine Learning > F# for Machine Learning Essentials
F# for Machine Learning Essentials
F# for Machine Learning Essentials

F# for Machine Learning Essentials: Get up and running with machine learning with F# in a fun and functional way

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F# for Machine Learning Essentials

Chapter 2. Linear Regression

"Honey! How much will gasoline cost next year?"

Linear regression is a technique to predict the value of a feature/attribute in a continuous range. It is similar to classification in a way that both types of algorithms solve a similar problem. But classification yields a discrete value as the tag while regression tries to predict a real value. In this chapter, you will learn how linear regression works and how it can be used in real-life settings.

Objective

After reading this chapter, you will be able to understand how several linear regression algorithms work and how to tune your linear regression model. You will also learn to use some parts of Math.NET and Accord.NET, which make implementing some of the linear regression algorithms simple. Along the way, you will also learn how to use FsPlot to plot various charts. All source code is made available at https://gist.github.com/sudipto80/3b99f6bbe9b21b76386d.

Different types of linear regression algorithms

Based on the approach used and the number of input parameters, there are several types of linear regression algorithms to determine the real value of the target variable. In this chapter, you will learn how to implement the following algorithms using F#:

  • Simple Least Square Linear Regression
  • Multiple Linear Regression
  • Weighted Linear regression
  • Ridge Regression
  • Multivariate Multiple Linear Regression

These algorithms will be implemented using a robust industry standard open source .NET mathematics API called Math.NET. Math.NET has an F# friendly wrapper.

APIs used

In this chapter, you will learn how to use the preceding APIs to solve problems using several linear regression methods and plot the result.

APIs used

FsPlot is a charting library for F# to generate charts using industry standard JavaScript charting APIs, such as HighCharts. FsPlot provides a nice interface to generate several combination charts, which is very useful when trying to understand the linear regression model. You can find more details about the API at its homepage at https://github.com/TahaHachana/FsPlot.

Math.NET Numerics for F# 3.7.0

Math.NET Numerics is the numerical foundation of the Math.NET project, aiming to provide methods and algorithms for numerical computations in science, in engineering, and in everyday use. It supports F# 3.0 on .Net 4.0, .Net 3.5, and Mono on Windows, Linux, and Mac; Silverlight 5 and Windows 8 with PCL portable profile 47; Android/iOS with Xamarin.

Math.NET Numerics for F# 3.7.0

You can get the API from the NuGet page at https://www.nuget.org/packages/MathNet.Numerics.FSharp/....

The basics of matrices and vectors (a short and sweet refresher)

Using Math.Net numerics, you can create matrices and vectors easily. The following section shows how. However, before you can create the vector or the matrix using Math.NET API, you have to reference the library properly. The examples in this chapter run using the F# script.

You have to write the following code at the beginning of the file and then run these in the F# interactive:

The basics of matrices and vectors (a short and sweet refresher)

Creating a vector

You can create a vector as follows:

Creating a vector

The vector values must always be float as per Math.NET. Once you run this in the F# interactive, it will create the following output:

Creating a vector

Creating a matrix

A matrix can be created in several ways using the Math.NET package. In the examples in this chapter, you will see the following ways most often:

  • Creating a matrix by hand: A matrix can be created manually using the Math.Net F# package, as follows:
    Creating a matrix

    Once you run this, you will get the following in the F# interactive:

    Creating a matrix
  • Creating a matrix from a list of rows...

QR decomposition of a matrix

The general linear regression model calculation requires us to find the inverse of the matrix, which can be computationally expensive for bigger matrices. A decomposition scheme, such as QR and SVD, helps in that regard.

QR decomposition breaks a given matrix into two different matrices—Q and R, such that when these two are multiplied, the original matrix is found.

QR decomposition of a matrix

In the preceding image, X is an n x p matrix with n rows and p columns, R is an upper diagonal matrix, and Q is an n x n matrix given by:

QR decomposition of a matrix

Here, Q1 is the first p columns of Q and Q2 is the last n – p columns of Q.

Using the Math.Net method QR you can find QR factorization:

QR decomposition of a matrix

Just to prove the fact that you will get the original matrix back, you can multiply Q and R to see if you get the original matrix back:

let myMatAgain = qr.Q * qr.R

SVD of a matrix

SVD stands for Single Value Decomposition. In this a matrix, X is represented by three matrices (the definition of SVD is taken from Wikipedia...

Objective

After reading this chapter, you will be able to understand how several linear regression algorithms work and how to tune your linear regression model. You will also learn to use some parts of Math.NET and Accord.NET, which make implementing some of the linear regression algorithms simple. Along the way, you will also learn how to use FsPlot to plot various charts. All source code is made available at https://gist.github.com/sudipto80/3b99f6bbe9b21b76386d.

Different types of linear regression algorithms

Based on the approach used and the number of input parameters, there are several types of linear regression algorithms to determine the real value of the target variable. In this chapter, you will learn how to implement the following algorithms using F#:

  • Simple Least Square Linear Regression

  • Multiple Linear Regression

  • Weighted Linear regression

  • Ridge Regression

  • Multivariate Multiple Linear Regression

These algorithms will be implemented using a robust industry standard open source .NET mathematics API called Math.NET. Math.NET has an F# friendly wrapper.

APIs used

In this chapter, you will learn how to use the preceding APIs to solve problems using several linear regression methods and plot the result.

FsPlot is a charting library for F# to generate charts using industry standard JavaScript charting APIs, such as HighCharts. FsPlot provides a nice interface to generate several combination charts, which is very useful when trying to understand the linear regression model. You can find more details about the API at its homepage at https://github.com/TahaHachana/FsPlot.

Math.NET Numerics for F# 3.7.0

Math.NET Numerics is the numerical foundation of the Math.NET project, aiming to provide methods and algorithms for numerical computations in science, in engineering, and in everyday use. It supports F# 3.0 on .Net 4.0, .Net 3.5, and Mono on Windows, Linux, and Mac; Silverlight 5 and Windows 8 with PCL portable profile 47; Android/iOS with Xamarin.

You can get the API from the NuGet page at https://www.nuget.org/packages/MathNet.Numerics.FSharp/. For...

The basics of matrices and vectors (a short and sweet refresher)

Using Math.Net numerics, you can create matrices and vectors easily. The following section shows how. However, before you can create the vector or the matrix using Math.NET API, you have to reference the library properly. The examples in this chapter run using the F# script.

You have to write the following code at the beginning of the file and then run these in the F# interactive:

Creating a vector

You can create a vector as follows:

The vector values must always be float as per Math.NET. Once you run this in the F# interactive, it will create the following output:

Creating a matrix

A matrix can be created in several ways using the Math.NET package. In the examples in this chapter, you will see the following ways most often:

  • Creating a matrix by hand: A matrix can be created manually using the Math.Net F# package, as follows:

    Once you run this, you will get the following in the F# interactive:

  • Creating a matrix from a list of rows...

QR decomposition of a matrix

The general linear regression model calculation requires us to find the inverse of the matrix, which can be computationally expensive for bigger matrices. A decomposition scheme, such as QR and SVD, helps in that regard.

QR decomposition breaks a given matrix into two different matrices—Q and R, such that when these two are multiplied, the original matrix is found.

In the preceding image, X is an n x p matrix with n rows and p columns, R is an upper diagonal matrix, and Q is an n x n matrix given by:

Here, Q1 is the first p columns of Q and Q2 is the last n – p columns of Q.

Using the Math.Net method QR you can find QR factorization:

Just to prove the fact that you will get the original matrix back, you can multiply Q and R to see if you get the original matrix back:

let myMatAgain = qr.Q * qr.R

SVD of a matrix

SVD stands for Single Value Decomposition. In this a matrix, X is represented by three matrices (the definition of SVD is taken from Wikipedia).

Suppose M...

Linear regression method of least square

Let's say you have a list of data point pairs such as the following:

You want to find out if there are any linear relationships between and .

In the simplest possible model of linear regression, there exists a simple linear relationship between the independent variable (also known as the predictor variable) and the dependent variable (also known as the predicted or the target variable). The independent variable is most often represented by the symbol and the target variable is represented by the symbol . In the simplest form of linear regression, with only one predictor variable, the predicted value of Y is calculated by the following formula:

is the predicted variable for . Error for a single data point is represented by:

and are the regression parameters that can be calculated with the following formula.

The best linear model minimizes the sum of squared errors. This is known as Sum of Squared Error (SSE).

For the best model, the regression...

Finding linear regression coefficients using F#

The following is an example problem that can be solved using linear regression.

For seven programs, the amount of disk I/O operations and processor times were measured and the results were captured in a list of tuples. Here is that list: (14,2), (16,5),(27,7) (42,9), (39, 10), (50,13), (83,20). The task for linear regression is to fit a model for these data points.

For this experiment, you will write the solution using F# from scratch, building each block one at a time.

  1. Create a new F# program script in LINQPad as shown and highlighted in the following image:

  2. Add the following variables to represent the data points:

  3. Add the following code to find the values needed to calculate b0 and b1:

  4. Once you do this, you will get the following output:

The following is the final output we receive:

Now in order to understand how good your linear regression model fits the data, we need to plot the actual data points as scatter plots and the regression line as a straight...

Finding the linear regression coefficients using Math.NET

In the previous example, we used F# to obtain b0 and b1 from the first principal. However, we can use Math.NET to find these values for us. The following code snippet does that for us in three lines of code:

This produces the following result in the F# interactive:

Take a moment to note that the values are exactly the same as we calculated earlier.

Putting it together with Math.NET and FsPlot

In this example, you will see how Math.NET and FsPlot can be used together to generate the linear regression coefficients and plot the result. For this example, we will use a known relation between Relative Humidity (RH) and Dew point temperature. The relationship between relative humidity and dew point temperature is given by the following two formulas:

Here, t and td are the temperatures in degrees Celsius.

td is dew point, which is a measure of atmospheric moisture. It is the temperature to which the air must be cooled in order to reach saturation (assuming the air pressure and the moisture content are constant).

Let's say the dew point is 10 degrees Celsius, then we can see how linear regression can be used to find a relationship between the temperature and RH.

The following code snippet generates a list of 50 random temperatures and then uses the formula to find the RH. It then feeds this data into a linear regression system to find the best...

Multiple linear regression

Sometimes it makes more sense to include more predictors (that is, the independent variables) to find the value of the dependent variable (that is, the predicted variable). For example, predicting the price of a house based only on the total area is probably not a good idea. Maybe the price also depends on the number of bathrooms and the distance from several required facilities, such as schools, grocery stores, and so on.

So we might have a dataset as shown next, and the problem we pose for our linear regression model is to predict the price of a new house given all the other parameters:

In this case, the model can be represented as a linear combination of all the predictors, as follows:

Here, the theta values represent the parameters we must select to fit the model. In vectorized form, this can be written as:

Theta can be calculated by the following formula:

So using the MathNet.Fsharp package, this can be calculated as follows:

Previously, in Chapter 1, Introduction...

Multiple linear regression and variations using Math.NET

As mentioned earlier, a regression model can be found using matrix decomposition such as QR and SVD.

The following code finds the theta from QR decomposition for the same data:

let qrlTheta = MultipleRegression.QR( created2 ,Y_MPG)

When run, this shows the following result in the F# interactive:

Now using these theta values, the predicted miles per gallon for the unknown car can be found by the following code snippet:

let mpgPredicted = qrlTheta * vector[8.;360.;215.;4615.;14.]

Similarly, SVD can be used to find the linear regression coefficients as done for QR:

let svdTheta = MultipleRegression.SVD( created2 ,Y_MPG)

Weighted linear regression

Sometimes each sample, or in other words, each row of the predictor variable matrix, is treated with different weightage. Normally, weights are given by a diagonal matrix where each element on the diagonal represent the weight for the row. If the weight is represented by W, then theta (or the linear regression coefficient) is given by the following formula:

Math.NET has a special class called WeightedRegression to find theta. If all the elements of the weight matrix are 1 then we have the same linear regression model as before.

The weight matrix is normally determined by taking a look at the new value for which the target variable has to be evaluated. If the new value is depicted as x, then the weights are normally calculated using the following formula:

The numerator of the weight matrix can be calculated as the Euclidean distance between two vectors. The first vector is from the training data and the other is the new vector depicting the new entry for which the...

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Key benefits

  • Design algorithms in F# to tackle complex computing problems
  • Be a proficient F# data scientist using this simple-to-follow guide
  • Solve real-world, data-related problems with robust statistical models, built for a range of datasets

Description

The F# functional programming language enables developers to write simple code to solve complex problems. With F#, developers create consistent and predictable programs that are easier to test and reuse, simpler to parallelize, and are less prone to bugs. If you want to learn how to use F# to build machine learning systems, then this is the book you want. Starting with an introduction to the several categories on machine learning, you will quickly learn to implement time-tested, supervised learning algorithms. You will gradually move on to solving problems on predicting housing pricing using Regression Analysis. You will then learn to use Accord.NET to implement SVM techniques and clustering. You will also learn to build a recommender system for your e-commerce site from scratch. Finally, you will dive into advanced topics such as implementing neural network algorithms while performing sentiment analysis on your data.

Who is this book for?

If you are a C# or an F# developer who now wants to explore the area of machine learning, then this book is for you. Familiarity with theoretical concepts and notation of mathematics and statistics would be an added advantage.

What you will learn

  • Use F# to find patterns through raw data
  • Build a set of classification systems using Accord.NET, Weka, and F#
  • Run machine learning jobs on the Cloud with MBrace
  • Perform mathematical operations on matrices and vectors using Math.NET
  • Use a recommender system for your own problem domain
  • Identify tourist spots across the globe using inputs from the user with decision tree algorithms
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ISBN-13 : 9781783989348
Category :
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Languages :
F#
Concepts :
Machine Learning
Tools :
Visual Studio

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Table of Contents

8 Chapters
1. Introduction to Machine Learning Chevron down icon Chevron up icon
1. Introduction to Machine Learning
Objective
Why use F#?
Unsupervised learning
Machine learning frameworks
Machine learning for fun and profit
Recognizing handwritten digits – your "Hello World" ML program
Summary
2. Linear Regression Chevron down icon Chevron up icon
2. Linear Regression
Objective
Different types of linear regression algorithms
APIs used
The basics of matrices and vectors (a short and sweet refresher)
QR decomposition of a matrix
Linear regression method of least square
Finding linear regression coefficients using F#
Finding the linear regression coefficients using Math.NET
Putting it together with Math.NET and FsPlot
Multiple linear regression
Multiple linear regression and variations using Math.NET
Weighted linear regression
Plotting the result of multiple linear regression
Ridge regression
Multivariate multiple linear regression
Feature scaling
Summary
3. Classification Techniques Chevron down icon Chevron up icon
3. Classification Techniques
Objective
Different classification algorithms you will learn
Some interesting things you can do
Understanding logistic regression
Multiclass classification using logistic regression
Multiclass classification using decision trees
Predicting a traffic jam using a decision tree: a case study
Challenge yourself!
Summary
4. Information Retrieval Chevron down icon Chevron up icon
4. Information Retrieval
Objective
Different IR algorithms you will learn
What interesting things can you do?
Information retrieval using tf-idf
5. Collaborative Filtering Chevron down icon Chevron up icon
5. Collaborative Filtering
Objective
Different classification algorithms you will learn
Vocabulary of collaborative filtering
Baseline predictors
Item-item collaborative filtering
Top-N recommendations
Evaluating recommendations
Ranking accuracy metrics
Working with real movie review data (Movie Lens)
Summary
6. Sentiment Analysis Chevron down icon Chevron up icon
6. Sentiment Analysis
Objective
What you will learn
A baseline algorithm for SA using SentiWordNet lexicons
Handling negations
Identifying praise or criticism with sentiment orientation
Pointwise Mutual Information
Using SO-PMI to find sentiment analysis
Summary
7. Anomaly Detection Chevron down icon Chevron up icon
7. Anomaly Detection
Objective
Detecting point anomalies using IQR (Interquartile Range)
Detecting point anomalies using Grubb's test
Grubb's test for multivariate data using Mahalanobis distance
Chi-squared statistic to determine anomalies
Detecting anomalies using density estimation
Strategy to convert a collective anomaly to a point anomaly problem
Dealing with categorical data in collective anomalies
Summary
Index Chevron down icon Chevron up icon
Index

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Oct 2022 698 pages
Full star icon 4.6 (45)
eBook
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  • eBook eBook €8.99
  • Paperback Paperback €37.99
€8.99 €29.99
€37.99
Machine Learning Engineering with Python
Machine Learning Engineering with Python
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Aug 2023 462 pages
Full star icon 4.6 (36)
eBook
eBook
  • eBook eBook €8.99
  • Paperback Paperback €37.99
€8.99 €29.99
€37.99
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About the author

Profile icon Mukherjee
Mukherjee
Sudipta Mukherjee was born in Kolkata and migrated to Bangalore. He is an electronics engineer by education and a computer engineer/scientist by profession and passion. He graduated in 2004 with a degree in electronics and communication engineering. He has a keen interest in data structure, algorithms, text processing, natural language processing tools development, programming languages, and machine learning at large. His first book on Data Structure using C has been received quite well. Parts of the book can be read on Google Books. The book was also translated into simplified Chinese, available from Amazon.cn. This is Sudipta's second book with Packt Publishing. His first book, .NET 4.0 Generics , was also received very well. During the last few years, he has been hooked to the functional programming style. His book on functional programming, Thinking in LINQ, was released in 2014. He lives in Bangalore with his wife and son. Sudipta can be reached via e-mail at sudipto80@yahoo.com and via Twitter at @samthecoder.
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Shipping Details

USA:

'

Economy: Delivery to most addresses in the US within 10-15 business days

Premium: Trackable Delivery to most addresses in the US within 3-8 business days

UK:

Economy: Delivery to most addresses in the U.K. within 7-9 business days.
Shipments are not trackable

Premium: Trackable delivery to most addresses in the U.K. within 3-4 business days!
Add one extra business day for deliveries to Northern Ireland and Scottish Highlands and islands

EU:

Premium: Trackable delivery to most EU destinations within 4-9 business days.

Australia:

Economy: Can deliver to P. O. Boxes and private residences.
Trackable service with delivery to addresses in Australia only.
Delivery time ranges from 7-9 business days for VIC and 8-10 business days for Interstate metro
Delivery time is up to 15 business days for remote areas of WA, NT & QLD.

Premium: Delivery to addresses in Australia only
Trackable delivery to most P. O. Boxes and private residences in Australia within 4-5 days based on the distance to a destination following dispatch.

India:

Premium: Delivery to most Indian addresses within 5-6 business days

Rest of the World:

Premium: Countries in the American continent: Trackable delivery to most countries within 4-7 business days

Asia:

Premium: Delivery to most Asian addresses within 5-9 business days

Disclaimer:
All orders received before 5 PM U.K time would start printing from the next business day. So the estimated delivery times start from the next day as well. Orders received after 5 PM U.K time (in our internal systems) on a business day or anytime on the weekend will begin printing the second to next business day. For example, an order placed at 11 AM today will begin printing tomorrow, whereas an order placed at 9 PM tonight will begin printing the day after tomorrow.


Unfortunately, due to several restrictions, we are unable to ship to the following countries:

  1. Afghanistan
  2. American Samoa
  3. Belarus
  4. Brunei Darussalam
  5. Central African Republic
  6. The Democratic Republic of Congo
  7. Eritrea
  8. Guinea-bissau
  9. Iran
  10. Lebanon
  11. Libiya Arab Jamahriya
  12. Somalia
  13. Sudan
  14. Russian Federation
  15. Syrian Arab Republic
  16. Ukraine
  17. Venezuela