Hi all,

I decided to start a new series of posts now focusing on general machine learning with several snippets for anyone to use with real problems or real datasets. Since I am studying machine learning again with a great course online offered this semester by Stanford University, one of the best ways to review the content learned is to write some notes about what I learned. The best part is that it will include examples with Python, Numpy and Scipy. I expect you enjoy all those posts!

**Linear Regression**

In this post I will implement the linear regression and get to see it work on data. Linear Regression is the oldest and most widely used predictive model in the field of machine learning. The goal is to minimize the sum of the squared errros to fit a straight line to a set of data points. (You can find further information at Wikipedia).

The linear regression model fits a linear function to a set of data points. The form of the function is:

*Y*=

*β*

_{0}+

*β*

_{1}*

*X*

_{1}+

*β*

_{2}*

*X*

_{2}+ … +

*β*

_{n}*

*X*

_{n}

_{}

Where

*Y*is the target variable, and*X*_{1},*X*_{2}, ...*X*_{n }are the predictor variables and*β*_{1},*β*_{2}, …*β*_{n }are the coefficients that multiply the predictor variables.*β*_{0 }is constant.
For example, suppose you are the CEO of a big company of shoes franchise and are considering different cities for opening a new store. The chain already has stores in various cities and you have data for profits and populations from the cities. You would like to use this data to help you select which city to expand next. You could use linear regression for evaluating the parameters of a function that predicts profits for the new store.

The final function would be:

Y = -3.63029144 + 1.16636235 *

*X*_{1}
There are two main approaches for linear regression: with one variable and with multiple variables. Let's see both!

**Linear regression with one variable**

Considering our last example, we have a file that contains the dataset of our linear regression problem. The first column is the population of the city and the second column is the profit of having a store in that city. A negative value for profit indicates a loss.

Before starting, it is useful to understand the data by visualizing it. We will use the scatter plot to visualize the data, since it has only two properties to plot (profit and population). Many other problems in real life are multi-dimensional and can't be plotted on 2-d plot.

If you run this code above (you must have the Matplotlib package installed in order to present the plots), you will see the scatter plot of the data as shown at Figure 1.

Now you must fit the linear regression parameters to our dataset using gradient descent. The objective of linear regression is to minimize the cost function:

where the hypothesis H0 is given by the linear model:

The parameters of your model are the θ values. These are the values you will adjust to minimize cost J(θ). One way to do it is to use the batch gradient descent algorithm. In batch gradient, each iteration performs the update:

With each step of gradient descent, your parameters θ, come close to the optimal values that will achieve the lowest cost J(θ).

For our initial inputs we start with our initial fitting parameters θ, our data and add another dimmension to our data to accommodate the θo intercept term. As also our learning rate alpha to 0.01.

As you perform gradient descent to learn minimize the cost function J(θ), it is helpful to monitor the convergence by computing the cost. The function cost is show below:

A good way to verify that gradient descent is working correctly is to look at the value of J(θ) and check that it is decreasing with each step. It should converge to a steady valeu by the end of the algorithm.

Your final values for θ will be used to make predictions on profits in areas of 35.000 and 70.000 people. For that we will use some matrix algebra functions with the packages Scipy and Numpy, powerful Python packages for scientific computing.

Our final values as shown below:

Y = -3.63029144 + 1.16636235 *

*X*_{1}
Now you can use this function to predict your profits! If you use this function with our data we will come with plot:

Another interesting plot is the contour plots, it will give you how J(θ) varies with changes in θo and θ1. The cost function J(θ) is bowl-shaped and has a global mininum as you can see in the figure below.

This minimum is the optimal point for θo and θi, and each step of gradient descent moves closer to this point.

**Linear regression with multiple variables**

Ok, but when you have multiple variables ? How do we work with them using linear regression ? That comes the linear regression with multiple variables. Let's see an example:

Suppose you are selling your house and you want to know what a good market price would be. One way to do this is to first collect information on recent houses sold and make a model of housing prices.

Our training set of housing prices in Recife, Pernambuco, Brazil are formed by three columns (three variables). The first column is the size of the house (in square feet), the second column is the number of bedrooms, and the third column is the price of the house.

But before going directly to the linear regression it is important to analyze our data. By looking at the values, note that house sizes are about 1000 times the number of bedrooms. When features differ by orders of magnitude, it is important to perfom a feature scaling that can make gradient descent converge much more quickly.

The basic steps are:

- Subtract the mean value of each feature from the dataset.
- After subtracting the mean, additionally scale (divide) the feature values by their respective “standard deviations.”

The standard deviation is a way of measuring how much variation there is in the range of values of a particular feature (most data points will lie within ±2 standard deviations of the mean); this is an alternative to taking the range of values (max-min).

Now that you have your data scaled, you can implement the gradient descent and the cost function.

Previously, you implemented gradient descent on a univariate regression problem. The only difference now is that there is one more feature in the matrix X. The hypothesis function and the batch gradient descent update rule remain unchanged.

In the multivariate case, the cost function can also be written in the following vectorized form:

J(θ)=12m(Xθ−y)T(Xθ−y)

After running our code, it will come with following function:

215810.61679138, 61446.18781361, 20070.13313796

The gradient descent will run until convergence to find the final values of θ. Next, we will this value of θ to predict the price of a house with 1650 square feet and 3 bedrooms.

θ:=θ−α1mxT(xθT−y)

θ:=θ−α1mxT(xθT−y)

**Predicted price of a 1650 sq-ft, 3 br house:**183865.197988

If you plot the convergence plot of the gradient descent you may see that convergence will decrease as the number of iterations grows.

**Extra Notes**

The Scipy package comes with several tools for helping you in this task, even with a module that has a linear regression implemented for you to use!

The module is scipy.stats.linregress and implements several other techniques for updating the theta parameters. Check more about it here.

**Conclusions**

The goal of regression is to determine the values of the ß parameters that minimize the sum of the squared residual values (difference betwen predicted and the observed) for the set of observations. Since linear regression is restricted to fiting linear (straight line/plane) functions to data, it's not adequate to real-world data as more general techniques such as neural networks which can model non-linear functions. But linear regression has some interesting advantages:

- Linear regression is the most widely used method, and it is well understood.
- Training a linear regression model is usually much faster than methods such as neural networks.
- Linear regression models are simple and require minimum memory to implement, so they work well on embedded controllers that have limited memory space.
- By examining the magnitude and sign of the regression coefficients (
*β*) you can infer how predictor variables affect the target outcome. - It's is one of the simplest algorithms and available in several packages, even Microsoft Excel!

I hope you enjoyed this simple post, and in the next one I will explore another field of machine learning with Python! You can download the code at this link.

Marcel Caraciolo

Muito bom, continue assim.

ReplyDeleteAbraço.

Excelente Marcel, good job though!

ReplyDeleteCongratulations on Machine Learning in Python! http://www.scn.org/~mentifex/AiMind.html achieves Machine Learning in JavaScript by asking questions for the human user to answer.

ReplyDeleteEnjoy reading your post. Great article, thank you very much! Really nice and impressive blog i found today... Thx for sharing this

ReplyDeleteThis comes in handy for me! Thank you

ReplyDeletethanks so much man - I am really struggling with the stanford course and have been wishing it was in python! this is awesome...

ReplyDeleteis there a video tuorial on this?

ReplyDeleteYour code is a bit confusing in that you do not use at all your compute_cost function!

ReplyDeleteThe result of the cost function is stored in the variable J_history which is never used again.

Furthermore your gradient descent method never checks on a minimal error but just iterates a fixed number of times towards a minimum. This can also lead to bad results.

great tutorial. your code helps my comprehension of the process. do you have any examples of this executed using pandas?

ReplyDeletehi,

ReplyDeleteit[:,1] = X does not seem to work ...

A new column was not added to the array.

Nice work. Thanks!

ReplyDeleteExcellent material can be found in www.KautilyaClasses.com

DeleteI'm pretty sure gradient descent isn't actually linear regression, its a more general solver thats actually more advanced and used with non-linear data. Linear regression will fit only the simplest models but its FAST. Gradient descent is far slower.

ReplyDeleteyour are clear the error for python programming language.your site is very useful clear the error for programming.Thank you for sharing your paragraph.Best Python training institute in Chennai

ReplyDeleteI have read you article very useful information for python training.Thank you for sharing you article.Best Python Training in Chennai

ReplyDeleteWhy would you not include the datatypes of the inputs in your comments instead of some useless phrase lilke "Comput cost for linear regression"...

ReplyDeleteHow can I use that code in mongodb ? Sry I am quite new, I have a MongoDB Database with a collection of "documents". How can I run this code against my collection. The Objects are only filled with 2 attributes and the attributes are numeric. I want to run a linear regression over these :) Thanks

ReplyDeleteWelcome to Wiztech Automation - Embedded System Training in Chennai. We have knowledgeable Team for Embedded Courses handling and we also are after Job Placements offer provide once your Successful Completion of Course. We are Providing on Microcontrollers such as 8051, PIC, AVR, ARM7, ARM9, ARM11 and RTOS. Free Accommodation, Individual Focus, Best Lab facilities, 100% Practical Training and Job opportunities.

ReplyDelete✔ Embedded System Training in chennai

✔ Embedded System Training Institute in chennai

✔ Embedded Training in chennai

✔ Embedded Course in chennai

✔ Best Embedded System Training in chennai

✔ Best Embedded System Training Institute in chennai

✔ Best Embedded System Training Institutes in chennai

✔ Embedded Training Institute in chennai

✔ Embedded System Course in chennai

✔ Best Embedded System Training in chennai

This is really good share,

ReplyDelete"blueapplecourses"

Embedded system training: Wiztech Automation Provides Excellent training in embedded system training in Chennai - IEEE Projects - Mechanical projects in Chennai. Wiztech provide 100% practical training, Individual focus, Free Accommodation, Placement for top companies. The study also includes standard microcontrollers such as Intel 8051, PIC, AVR, ARM, ARMCotex, Arduino, etc.

ReplyDeleteEmbedded system training in chennai

Embedded Course training in chennai

Matlab training in chennai

Android training in chennai

LabVIEW training in chennai

Robotics training in chennai

Oracle training in chennai

Final year projects in chennai

Mechanical projects in chennai

ece projects in chennai

WIZTECH Automation, Anna Nagar, Chennai, has earned reputation offering the best automation training in Chennai in the field of industrial automation. Flexible timings, hands-on-experience, 100% practical. The candidates are given enhanced job oriented practical training in all major brands of PLCs (AB, Keyence, ABB, GE-FANUC, OMRON, DELTA, SIEMENS, MITSUBISHI, SCHNEIDER, and MESSUNG)

ReplyDeletePLC training in chennai

Automation training in chennai

Best plc training in chennai

PLC SCADA training in chennai

Process automation training in chennai

Final year eee projects in chennai

VLSI training in chennai

This comment has been removed by the author.

ReplyDeleteWiztech Automation is the Leading Best IEEE Final year project Centre in Chennai and the final year students are provided complete guidance and support in their final year projects. The IEEE projects in Chennai that Wiztech Automation offers guidance and support for include complete range of system domains – such as PLC projects, embedded projects, VLSI projects, software projects, IT projects, Civil projects. Students looking for specific projects pertaining to departments of ECE, EEE, E&I, Mechanical, Mechatronics, bio-medical, IT, Computer, Civil projects in B.E, M.E, B.Tech, M.Tech, B.SC., and M.Sc Electronics, could also get turnkey solutions at Wiztech Automation Solutions to turn out successful project outcomes and models. Since the students at Wiztech Automation gain thorough theoretical and practical knowledge and skills as they pursue their final year projects and develop 2015 and 2016 Latest IEEE Projects portraying them well.

ReplyDeleteFinal year projects in chennai

Mechanical projects in chennai

ece projects in chennai

Final year eee projects in chennai

VLSI project center in chennai

Industrial projects in chennai

Fianl year CSE projects in chennai

This can be a single case in which your Uk teacher had been right. But if your article is actually riddled together with punctuation blunders in addition to grammatical mishaps, you'll almost instantly get rid of reliability.nurse personal statement

ReplyDeleteHi admin thanks for sharing informative article on hadoop technology. In coming years, hadoop and big data handling is going to be future of computing world. This field offer huge career prospects for talented professionals. Thus, taking Hadoop & Spark Training in Hyderabad will help you to enter big data hadoop & spark technology.

ReplyDeleteCould you share your ex1data1.txt data ? Thanks

ReplyDeletegot the file @ Coursera Data science tutorial by andrew ng ..excecise week 2 assignment

DeleteExcellent material can be found in www.KautilyaClasses.com

ReplyDeleteHadoop Training Institutes in Noida

ReplyDeleteHadoop Training Institutes in Noida

ReplyDeleteParis airport transfer - Parisairportransfer is very common in Paris that provides facilities to both the businessmen and the tourists. We provide airport transfers from London to any airport in London and also cruise transfer services at very affordable price to our valuable clients.

ReplyDeleteParis taxi

Paris airport shuttle

paris hotel transfer

paris airport transfer

paris shuttle

paris car service

paris airport service

disneyland paris transfer

paris airport transportation

beauvais airport transfer

taxi beauvais airport

taxi cdg airport

taxi orly airport

ReplyDeleteMIDSUMMER SEASON WEDDING WEAR SHOES

Handbag & Clutches For Hot Girls

Front Open Double Shirt

Fashion Gallery Lehenga Choli

Stylo Best Mehndi Designs

HANDBAGS FOR WOMEN FASHION

Latest Sherwani Designs

Bridal Jewellery Set

Zara Shahjahan Eid Dresses

Mehndi Patterns for EID

SUMMER SEASON LADIES DRESSES FASHION

Sophia Tolli Collection

Earrings In Gold Collection

Actress Maya Ali – Fashion Collection

Bridal Gowns Collection

LADIES BLAZER STYLES OUTFITS

MEHNDI DRESS DESIGNS

BRIDAL SHOES

SHIRTS GRAY MAXI SKIRT SKIRTS

BRIDAL DRESSES WESTERN STYLE

LAWN AND CHIFFON OUTFITS

classic lawn suits

mix eid dresses

midsummer kurta

anarkali suits

REVLON NAIL POLISH COLORS

ReplyDeletelehnga choli dresses

bridal makeup

ball garments

babydoll night wear dresses

MEN WEAR WEDDING SHERWANI

Jewelry Women Wear

Saheli Couture By Preity Zinta Dresses

Parties Hairstyle

Zainab Chottani Pretty Suits

STYLISH SUNGLASSES DESIGNS

FROCKS DESIGNS FASHION

LONG GOWNS OUTFITS FASHION

HUMAN SALMAN KHAN STYLISH DRESSES

UTSAV FASHION NET INDIAN SAREES

Fancy Lawn Clothes

Nail Designs For UK Girls

Girls Footwear Selection

Pakistani Lehenga Clothes

Saheli Couture By Preity Zinta Dresses

ReplyDeleteParties Hairstyle

Outfits Fashion For Ladies By Zainab Chottani

Elegant Nail Designs Fashion

Churidar Clothes Fashion Style

Indian Lehenga Choli Bridal Dress

lehnga choli dresses

bridal makeup

STYLISH SUNGLASSES DESIGNS

FROCKS DESIGNS FASHION

Girls Footwear Selection

Pakistani Lehenga Clothes

Fashion Gallery Lehenga Choli

Stylo Best Mehndi Designs

Zara Shahjahan Eid Dresses

Mehndi Patterns for EID

Sophia Tolli Collection

Earrings In Gold Collection

SHIRTS GRAY MAXI SKIRT SKIRTS

midsummer kurta

anarkali suits

WALKS THE RAMP FOR LALA TEXTILES

ReplyDeleteElegant Nail Designs Fashion

Churidar Clothes Fashion Style

Indian Lehenga Choli Bridal Dress

STYLISH SUNGLASSES DESIGNS

FROCKS DESIGNS FASHION

Keep on posting these types of articles. I like your blog design as well. Cheers!!!MATLAB training in noida

ReplyDeleteUseful Information

ReplyDeleteone and only affiliate agency in south INDIA, earn money online from affiliate network in india

ReplyDeletenice post and site, good work! Data Scientist online

Thanku for sharing this excellent posts..

ReplyDeleteSAP GRC training in hyderabad

Thanku for sharing..

ReplyDeletesap fiori online training

Java Training Institute in Noida - Croma Campus imparts the most effective JAVA Training in Noida which is based on the principle write once and run anywhere which means that the code which runs on one platform does not need to be complied again to run on the other.

ReplyDelete

ReplyDeleteI actually enjoyed reading through this posting.Many thanks.

Function Point Estimation Training

This comment has been removed by the author.

ReplyDeleteInformatica training institutes in noida - Croma campus offers best Informatica Training in noida with most experienced professionals. Our Instructors are working in Informatica and joint technologies for more years in MNC’s. We aware of industry needs and we are offering Informatica Training in noida.

ReplyDeleteInformatica training institutes in noida - Croma campus offers best Informatica Training in noida with most experienced professionals. Our Instructors are working in Informatica and joint technologies for more years in MNC’s. We aware of industry needs and we are offering Informatica Training in noida.

ReplyDeleteThis comment has been removed by the author.

ReplyDeleteCroma campus has been NO.1 & Best Android training institute in noida offering 100% Guaranteed JOB Placements, Cost-Effective, Quality & Real time Training courses Croma campus provide all IT course like JAVA, DOT NET, ANDROID APPS, PHP, PLC SCADA, ROBOTICS and more IT training then joining us Croma campus and your best futures.

ReplyDeleteCroma campus has been NO.1 & Best Android training institute in noida offering 100% Guaranteed JOB Placements, Cost-Effective, Quality & Real time Training courses Croma campus provide all IT course like JAVA, DOT NET, ANDROID APPS, PHP, PLC SCADA, ROBOTICS and more IT training then joining us Croma campus and your best futures.

ReplyDeleteNice Information:

ReplyDeleteTelugu Cinema Contains Telugu Cinema News, Latest Movie Reviews, Actor, Actress, Movie Galleries And Many More Telugu Cinema News

Nice Information

ReplyDeleteone and only affiliate agency in south INDIA, earn money online from affiliate network in india

Useful Information……

ReplyDeleteRecruitment voice contains Daily GK Updates, Bank Recruitment, Government jobs, Bank jobs, Interview Tips, Banking News, GK Updates Bank Recruitment

This is one of the valuable information share by you about embedded linux course. Thanks for sharing it with us. Keep it on...

ReplyDeleteTraining on MATLAB|Training on VLSIvery useful information..

ReplyDeletebe projects in chennai

ieee projects in chennai

very useful information..

ReplyDeletebe projects in chennai

ieee projects in chennai

As a full-fledged Industrial automation training in Hyderabad company SOS India offers complete training on PLC & SCADA with advanced hardware facilities.Unlimited Practices-Industrial Tours-Excellent Placements.

ReplyDeleteLearn Big Data from Basics ... Hadoop Training in Hyderabad

ReplyDeleteLearn Big Data from Basics ... Hadoop Training in Hyderabad

ReplyDelete