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!

The series:

In this post I will cover the Logistic Regression and Regularization.

**Logistic Regression**

Logistic Regression is a type of regression that predicts the probability of ocurrence of an event by fitting data to a logit function (logistic function). Like many forms of regression analysis, it makes use of several predictor variables that may be either numerical or categorical. For instance, the probability that a person has a heart attack within a specified time period might be predicted from knowledege of the person's age, sex and body mass index. This regression is quite used in several scenarios such as prediction of customer's propensity to purchase a product or cease a subscription in marketing applications and many others.

**Visualizing the Data**

Let's explain the logistic regression by example. Consider you are the administrator of a university department and you want to determine each applicant's chance of admission based on their results on two exams. You have the historical data from previous applicants that you can use as a trainning set for logistic regression. For each training example, you have the applicant's scores on two exams and the admissions decision. We will use logistic regression to build this model that estimates the probability of admission based the scores from those two exams.

Let's first visualize our data on a 2-dimensional plot as show below. As you can see the axes are the two exam scores, and the positive and negative examples are shown with different markers.

Sample training visualization |

The code

**Costing Function and Gradient**

The logistic regression hypothesis is defined as:

where the function g is the sigmoid function. It is defined as:

The sigmoid function has special properties that can result values in the range [0,1]. So you have large positive values of X, the sigmoid should be close to 1, while for large negative values, the sigmoid should be close to 0.

The cost function and gradient for logistic regression is given as below:

and the gradient of the cost is a vector theta where the j element is defined as follows:

You may note that the gradient is quite similar to the linear regression gradient, the difference is actually because linear and logistic regression have different definitions of h(x).

Let's see the code:

Now to find the minimum of this cost function, we will use a scipy built-in function called fmin_bfgs. It will find the best parameters theta for the logistic regression cost function given a fixed dataset (of X and Y values).

The parameters are:

- The initial values of the parameters you are trying to optimize;
- A function that, when given the training set and a particular theta, computes the logistic regression cost and gradient with respect to theta for the dataset (X,y).

The final theta value will then be used to plot the decision boundary on the training data, resulting in a figure similar to the figure below.

**Evaluating logistic regression**

Now that you learned the parameters of the model, you can use the model to predict whether a particular student will be admited. For a student with an Exam1 score of 45 and an Exam 2 score of 85, you should see an admission probability of 0.776.

But you can go further, and evaluate the quality of the parameters that we have found and see how well the learned model predicts on our training set. If we consider the threshold of 0.5 using our sigmoid logistic function, we can consider that:

Where 1 represents admited and -1 not admited.

Going to the code and calculate the training accuracy of our classifier we can evaluate the percentage of examples it got correct. Source code.

89% , not bad hun?!

**Regularized logistic regression**

But when your data can not be separated into positive and negative examples by a straight-line trought the plot ? Since our logistic regression will be only be able to find a linear decision boundary, we will have to fit the data in a better way. Let's go through an example.

Suppose you are the product manager of the factory and you have the test results for some microships of two different tests. From these two tests you would like to determine whether the microships should be accepted or rejected. We have a dataset of test results on past microships, from which we can build a logistic regression model.

**Visualizing the data**

Let's visualize our data. As you can see in the figure below, the axes are the two test scores, and the positive (y = 1, accepted) and negative (y = 0, rejected) examples are shown with different markers.

You may see that the model built for this task may predict perfectly all training data and sometimes it migh cause some troubling cases. Just because ithe model can perfectly reconstruct the training set does not mean that it had everything figured out. This is known as overfitting. You can imagine that if you were relying on this model to make important decisions, it would be desirable to have at least of regularization in there. Regularization is a powerful strategy to combat the overfitting problem. We will see it in action at the next sections.

**Feature mapping**

One way to fit the data better is to create more features from each data point. We will map the features into all polynomial terms of x1 tand x2 up to the sixth power.

As a result of this mapping, our vector of two features (the scores on two QA tests) has been transformed into a 28-dimmensional vector. A logistic regression classifier trained on this higher dimension feature vector will have a more complex decision boundary and will appear nonlinear when drawn in our 2D plot.

Although the feature mapping allows us to buid a more expressive classifier, it also me susceptible to overfitting. That comes the regularized logistic regression to fit the data and avoid the overfitting problem.

Source code.

Source code.

**Cost function and gradient**

The regularized cost function in logistic regression is :

Note that you should not regularize the parameter theta, so the final summation is for j = 1 to n, not j= 0 to n. The gradient of the cost function is a vector where the jn element is defined as follows:

Now let's learn the optimal parameters theta. Considering now those new functions and our last numpy optimization function we will be able to learn the parameters theta.

The all code now provided (code)

**Plotting the decision boundary**

Let's visualize the model learned by the classifier. The plot will display the non-linear decision boundary that separates the positive and negative examples.

**Scikit-learn**

Scikit-learn is an amazing tool for machine learning providing several modules for working with classification, regression and clustering problems. It uses python, numpy and scipy and it is open-source!

If you want to use logistic regression and linear regression you should take consider the scikit-learn. It has several examples and several types of regularization strategies to work with. Take a look at this link and see by yourself! I recommend!

**Conclusions**

Logistic regression has several advantages over linear regression, one specially it is more robust and does not assume linear relationship since it may handle nonlinear effects. However it requires much more data to achieve stable, meaningful results. There are another machine learning techniques to handle with non-linear problems and we will see in the next posts. I hope you enjoyed this article!

Regards,

Marcel Caraciolo

Amazing numerics.........

ReplyDeletewow..i am also doing the stanford exercises in python..i was stuck with the optimization part..thank you very much for the article..

ReplyDeletethe source code posted is giving errors..

ReplyDeleteWarning: divide by zero encountered in log

Warning: overflow encountered in power

Warning: overflow encountered in power

please check out! i am trying to debug it

Hi Anonymous:

ReplyDeleteDid you try feature scaling (mean normalization)?

Here some quick code for that (n.B. I'm using other data, so the axes may have to change):

def normalize(X):

mu = numpy.mean(X, axis=0)

Smin = numpy.amin(X, axis=0)

Smax = numpy.amax(X, axis=0)

x = (X - mu) / (Smax - Smin)

return x

Hi Marcel

ReplyDeleteGreat work posting this.

I am getting similar warnings as anonymous using the above code (primarily when I expand the number of thetas to be estimated). The code works just fine for about 10 parameters or so.

The following warning also appears:

Warning: overflow encountered in double_scalars

More concerning I suppose, the value returned maximum likelihood from fmin_bfgs (fmin_l_bfgs_b in my case) is nan and the following error occurs ABNORMAL_TERMINATION_IN_LNSRCH.

Also, my features have already all be scaled as well.

Any thoughts on what could possibly be occurring?

fantastic presentation of Logistic Regression..

ReplyDeleteCan you stop linking to this image http://www.mblondel.org/tlml/_images/math/9dd37d56c18555e80d91e8f57a1ceeb83fc72a5a.png? (the bandwidth on my server is not for free)

ReplyDeleteThanks.

Hey, nice site you have here! Keep up the excellent work!

ReplyDeleteFunction Point Estimation Training

This is a great tutorial, but I am confused with the first example. Why is the theta vector of length 3? Shouldn't it be of length 2? The theta vector you are trying to optimize is the slope and y-intercept, correct?

ReplyDeleteThanks for the help.

How would I use fmin bfgs if I'm training for a Neural Network? The cost function over there has more than 1 theta. How would I provide a list of thetas to "decorated cost" function. I tried doing it but, I get errors in scipy optimize (the thetas don't change; program crashes after a couple of iterations.

ReplyDeleteHi. Nice post. I am wondering if it is possible to tweak a little bit of LogisticRegression in scikit-learn to get a "Regressor" rather that a "Classifier" like LogisticRegression? I went through all the codes. It seems that one of the main base class BaseLibLinear can only train different set of coefficients for different y. I really appreciate if you happy to get an answer. thanks.

ReplyDeleteThis comment has been removed by the author.

ReplyDeleteLike several of the commenters, I get:

ReplyDeleteWarning: divide by zero encountered in log

because the elements of theta very quickly get big enough that sigmoid returns 1.0.

Has anyone gotten the basic logistic regression code to actually work (without the regularization)?

I figured it out. Line 26 of compute_cost():

ReplyDeletereturn - 1 * J.sum()

This negates the entire cost function, which makes it difficult for LBFGS to minimize it. (:

This explains why the thetas go through the roof.

I seem to be having an issue with the code. Downloaded from GitHub and run it. I would assume that in log_reg.py that the output from decorated_cost() function would be the theta values defining our boundary. In fact, the code hard codes those theta values rather than using the model output. If you use what is returned by decorated_cost(), it is not accurate. How did you generate the hard coded values? Am I missing something?

ReplyDeleteThis is an informative post review. I am so pleased to get this post article and nice information. I was looking forward to get such a post which is very helpful to us. A big thank for posting this article in this website. Keep it up.

ReplyDeletemind control

Thanks for sharing such kind of nice and wonderful collection......Nice post Dude keep it up.

ReplyDeleteI have appreciate with getting lot of good and reliable and legislative information with your post......

scripts, NLP, vance, advertisement

I like totally and agree. And I think that in order to be comfortable with your style is to wear it more often. So wear your style to the lab on days that you don't have to do anything bloody, muddy or otherwise gross!

ReplyDeletesubliminal advertising

Heya¡my very first comment on your site. ,I have been reading your blog for a while and thought I would completely pop in and drop a friendly note. .

ReplyDeleteFunction Point Estimation Training

Hi all I solved the issue related to logistic regression, for a simple misunderstood I replaced the cost_function with wrong J , since the f_min receives only a single value and also the negative value which was wrong from the problem (minimization).

ReplyDeleteHello Marcel, I can not make either one work.

ReplyDeletethe log_reg.py shows the "RuntimeWarning: overflow encountered in exp"

for the log_reg_regular.py, I changed the maxfun to maxiter

but it still shows thetaR = theta[1:, 0]

IndexError: too many indices"

anybody got it work? Can I have the code please?

Very Good information on property dealing. This site has very useful inputs related to Real Estate. Well Done & Keep it up to the team of Property Bytes….

ReplyDeleteFunction Point Estimation Training in Chennai

Found your article and is very intersting after some effort to understand logistic regression. I notice that if the h[it] in predict function is changed from 0.5 to 0.2 or 0.3, the test accuracy result is sky rocketing to 0.92! Can you explain why ? How can we understand if that is a correct result or not ?

ReplyDeleteThanks for any feedback.

In first example, what did you use to draw decision boundary?

ReplyDeleteIn theory example 1 should yield better accuracy if we added more features the same way it's done in example 2. After adding additional features for some reason minimizing function doesn't want to converge and stays at 60% any ideas why?

ReplyDeleteThank You For The Information

ReplyDeleteI agree with your post, the Introduction of automation testing product shortens the development life cycle. It helps the software developers and programmers to validate software application performance and behavior before deployment. You can choose testing product based on your testing requirements and functionality. QTP Training Chennai

ReplyDeleteNice site....Please refer this site also nice

ReplyDeleteDot Net Training in Chennai,

dotnettrainingchennai

Its new for me, i will try learn this and we introduce my web design training institute

ReplyDeleteHi, While running

ReplyDeletescipy.optimize.fmin_bfgs(costFunction(theta, X, y), initial_theta,maxiter = 10),

it throws error - 'tuple' object is not callable.

My cost function is:

def costFunction(theta, X, y):

J = 0

grad = zeros((size(theta)))

z = sigmoid(np.dot(X,theta))

cost = -(y*log(z)) - (1-y)*log(1 - z)

J = sum(cost)/m

grad = np.dot(X.T, (z - y))/m

return grad,J

Can anyone help me with this?

hi,

ReplyDeleteI am trying to do event recommendation by tags of events and i want to use logistic regresion as an algorith of the system. But logistic regression using vectors (x,y), but i could not transform tags to vectors. Does anyone can help me ?

thnx

Thanks for sharing this informative blog. Recently I have completed Digital Marketing courses at a leading digital marketing company. It's really useful for me to make a bright career. If anyone wants to get Digital Marketing Course in Chennai visit infiniX located at Chennai. Rated as No.1 digital marketing company in Chennai.

ReplyDeleteYour blog is really useful for me. Thanks for sharing this informative blog. If anyone wants to get real time Oracle Training in Chennai reach FITA located at Chennai. They give professional and job oriented training for all students.

Thanks for sharing this informative blog. If anyone wants to get Unix Training in Chennai, Please visit Fita Academy located at Chennai, Velachery.

ReplyDeleteThanks for sharing this valuable information..If anyone wants to get SAP Training in Chennai, please visit FITA Academy located at Chennai..

ReplyDeleteYour posts is really helpful for me.Thanks for your wonderful post.It is really very helpful for us and I have gathered some important information from this blog.If anyone wants to get Dot Net Training in Chennai reach FITA, rated as No.1 Dot Net Training Institutes in Chennai.

ReplyDeleteJava is one of the popular technologies with improved job opportunity for hopeful professionals. Java Training in Chennai helps you to study this technology in details.

ReplyDeleteThanks for sharing this informative blog. If anyone wants to get Android Course in Chennai reach FITA Academy located at Chennai, Velachery.

ReplyDeleteThanks for sharing this informative blog.. If anyone want to get HTML Training in Chennai please visit FITA academy located at Chennai, Velachery. Rated as No.1 training and placement academy in Chennai.

ReplyDeleteThanks for sharing this informative blog..If anyone want to get Cloud Computing Training in Chennai reach FITA academy located at Chennai, Velachery.

ReplyDeleteThanks for sharing this informative blog..If anyone want to get Salesforce Training in Chennai reach FITA academy.

ReplyDeleteSEO is one of the digital marketing techniques which is used to increase website traffic and organic search results. If anyone wants to get SEO Training in Chennai visit FITA Academy located at Chennai. Rated as No.1 Training institutes in Chennai.

ReplyDeleteThanks for your informative article on digital marketing trends. I hardly stick with SEO techniques in boosting my online presence as its cost efficient and deliver long term results. SEO Course in Chennai

ReplyDeleteIts really awesome blog..If anyone wants to get Software Testing Training in Chennai visit FITA IT academy located at Chennai.

ReplyDeleteI stick with Social Media Marketing. This promotional strategy is ideal for start-up and small organizations to enjoy maximum leads with minimal investment amount. However, you need to run effective marketing campaign to be successful. SEO Training Center in Chennai

ReplyDeleteYour posts is really helpful for me.Thanks for your wonderful post.It is really very helpful for us and I have gathered some important information from this blog.If anyone wants to get Dot Net Training in Chennai reach FITA, rated as No.1 Dot Net Training Institutes in Chennai.

ReplyDeleteyour pots is really useful for me....i hope to really understand easy..hadoop training in chennai

ReplyDeletethis sites information excellent..oracle training in chennai

ReplyDeleteSelenium Training in Chennai,

ReplyDeleteSelenium is an open source web automation tool developed by Thoughtworks. Since it is based on JavaScript so it can be operated from any of the platforms like Windows, Linux, Mac, Android (Mobile OS developed by Google) , iOS (OS for iPhone and iPad) along with the supported web browsers such as Firefox, Internet Explorer, Chrome, Safari, Opera etc. Visit Us, Selenium Training in Chennai

QTP Training in Chennai,

ReplyDeleteQTP is widely used test automation tool mainly for functional testing. QTP has many more advanced options and HP recommends that all existing and new users should begin with Quick Test Professional(QTP) instead of Win Runner.

Visit Us, QTP Training in Chennai

QTP Training in Chennai,

ReplyDeleteAutomated software testing is a process in which software tools execute pre-scripted tests on a software application before it is released into production Visit Us, QTP Training in Chennai

Learn how to use Selenium from beginner level to advanced techniques which is taught by experienced working professionals. Best Training in Chennai

ReplyDeleteHi, I wish to be a regular contributor of your blog. I have read your blog. Your information is really useful for beginner. I did QTP Training Chennai at Fita training and placement academy which offer best Selenium Training Chennai with years of experienced professionals. This is really useful for me to make a bright career.

ReplyDeleteThanks for your wonderful post.It is really very helpful for us and I have gathered some important information from this blog.If anyone wants to get Dot Net Course in Chennai reach FITA, rated as No.1 Dot Net Training Institute in Chennai.

ReplyDeleteThanks for sharing this informative blog. Recently I did Digital Marketing Courses in Chennai at a leading digital marketing company. It's really useful for me to make a bright career. To know more details about this course please visit FITA.

ReplyDeleteThanks for sharing this information. SEO is one of the digital marketing techniques which is used to increase website traffic and organic search results. . If anyone wants to get SEO Training in Chennai visit FITA Academy located at Chennai. Rated as No.1 SEO Training Institute in Chennai.

ReplyDeleteThanks for your informative article. With the world is totally dependent on internet, the future of digital marketing is on positive note. It also assures lucrative career opportunity for professionals looking for job in digital marketing. Digital Marketing Training in Chennai | Digital Marketing Course in Chennai

ReplyDelete

ReplyDeleteI have read all the articles in your blog; was really impressed after reading it. FITA is glad

To inform you that; we provide Salesforcecrm practical training with MNC exports. We Assure you that through our training the students will gain all the sufficient knowledge to have a voyage in IT industry.

Salesforce training in Chennai | Salesforce courses in Chennai | Salesforce training

This comment has been removed by the author.

ReplyDeleteI have read your article very useful python programming language.you are clearing the errors for python programming.Thank you for sharing your article.Python training center in Chennai

ReplyDeleteI have read your blog and i got a very useful and knowledgeable information from your blog.its really a very nice article. I did Loadrunner Course in Chennai. This is really useful for me. Suppose if anyone interested to learn Manual Testing Course in Chennai reach FITA academy located at Chennai Velachery.

ReplyDeleteRegards.....

Testing Training in Chennai

Digital Marketing Chennai

ReplyDeleteThanks for sharing this informative blog. Recently I did Digital Marketing courses at a leading Digital Marketing Training Institutes in Chennai. It's really useful for me to make a bright career. If anyone wants to get Digital Marketing Training in Chennai visit infiniX.

Regards...

Digital Marketing Courses in Chennai

I have read your article very useful information for python training.python training is very useful course for job.Thank you for sharing your article.

ReplyDeletePython Training in Chennai

The SAS system is a powerful software program designed to give researchers a wide variety of both data management and data analysis capabilities. SAS Training in Chennai Although SAS has millions of users worldwide.That process involves designing studies, collecting good data, describing the data with numbers and graphs, analyzing the data, and then making Decisions / conclusions. SAS Training in Chennai All these can be accomplished by using SAS software.

ReplyDeleteHTML5 Training in Chennai

ReplyDeleteYour blog is really awesome. Thank you for your sharing this informative blog. Recently I did PHP course at a leading academy. If you are looking for best PHP Training Institute in Chennai visit FITA IT training academy which offer real time PHP Training in Chennai.

PHP Course in Chennai

Wiztech Automation Solutions is the best Training institute in Chennai, started in the year 2006 and it extended its circle through providing the best education as per the global quality standards. Hence our PLC training Center in Chennai was recognized by IAO and ISO for its inspiring Education quality standards. Wiztech Automation Solution, the PLC SCADA Training Academy in Chennai offers both PLC, SCADA, DCS, VFD, Drives, Control Panels, HMI, Pneumatics, Embedded systems, VLSI Training courses in chennai with latest various brands. As we know that how PLC and SCADA technologies plays a vital role in the Automation Industry

ReplyDeletewww.automationtraining.info

ReplyDeleteThanks for sharing this valuable information to our vision. You have posted a trust worthy blog keep sharing.Nice article i was really impressed by seeing this article, it was very interesting and it is very useful for Learners.. CCNA training in chennai | CCNA training chennai | CCNA course in chennai | CCNA course chennai

I thought Python is old. So there is no use of learning that language. But now only I realized. Thanks for sharing nice article to us.

ReplyDeleteAMC Square Learning Chennai Reviews|AMC Square Learning Chennai|AMC Square LearningVery nice and informative article. Hope all your readers have got something useful from it.

ReplyDeleteschool websites design

Thanks for your informative article on software testing. Your post helped me to understand the future and career prospects in software testing. Keep on updating your blog with such awesome article. Best Python Training in chennai

ReplyDeleteThanks for sharing this valuable information to our vision. You have posted a trust worthy blog keep sharing. PHP Training in chennai | PHP Training chennai | PHP course in chennai | PHP course chennai

ReplyDeleteThanks for sharing such a great information..Its really nice and informative.

ReplyDeletePython Training in Chennai | python and django training in chennai

Thank you provide valuable informations and iam seacrching same informations,and saved my time

ReplyDeleteSharepoint Coaching in Chennai

Thanks for giving important information to training seekers,Keep posting useful information

ReplyDeletePython Training In Chennai | python and django training in chennai

Thanks for your informative article on digital marketing trends. I hardly stick with SEO techniques in boosting my online presence as its cost efficient and deliver long term results. SEO Training Institutes in Chennai

ReplyDeleteThanks for sharing such a great information..Its really nice and informative.Best Python Training in Chennai

ReplyDeleteThanks for your informative article on software testing. Your post helped me to understand the future and career prospects in software testing. Keep on updating your blog with such awesome article.

ReplyDeletePython Training Chennai | Python Training Institute in Chennai

Excellent post!!! Digital marketing training provides students with meaning skills to promote a business in digital age. This training covers search engine optimization, SEM, social media marketing, online reputation management, etc. Digital Marketing Training

ReplyDeleteThanks for the long and well formatted article on Python.... ;-)

ReplyDeleteWeb Designing Course in Chennai

Thanks for sharing this valuable information to our vision. You have posted a trust worthy blog keep sharing.

ReplyDeleteRegards,

Informatica courses in Chennai|Informatica institutes in Chennai|Salesforce training institute in Chennai

Thanks for sharing; Salesforce crm cloud application provides special cloud computing tools for your client management problems. It’s a fresh technology in IT industries for the business management.

ReplyDeleteRegards,

Salesforce training in Chennai|Salesforce training |Salesforce training institute in Chennai

It’s too informative blog and I am getting conglomerations of info’s. Thanks for sharing; I would like to see your updates regularly so keep blogging. If anyone looking PHP just get here

ReplyDeleteRegards,

PHP Training in Chennai|PHP Course in Chennai|PHP Training Institute in Chennai

I feel satisfied to read your blog, you have been delivering a useful & unique information to our vision even you have explained the concept as deep clean without having any uncertainty, keep blogging.

ReplyDeleteWeb designing course in chennai|Web design training in chennai|Web design course in Chennai

I always get satisfied reading this blog, the content in this site is always satisfying and truly helpful, I also wanted to share few links related to Python training

ReplyDeleteCheck this siteTeKslate for indepth Python training.Go hereif you’re looking for information on Python training.This data is magnificent. I am impressed with your writing style and how properly you define this topic. After studying your post, my understanding has improved substantially. Thanks for taking the time to discuss this, I feel strongly about it and love learning more on this topic.

ReplyDeleteRegards,

ccna course in Chennai|ccna training in Chennai

Nice blog Very useful information is providing by ur blog. find Great beginning html tutorials Very clear and helpful for beginners.

ReplyDeleteThanks for the wonderful tutorial. I learned quite a few things from this.

ReplyDeletebest web designing course in chennai

Thanks for sharing . I have learned something from this blog . Thanks once again . I just want to introduce my site Software Testing Courses Chennai

ReplyDeleteawesome information Thanks for sharing us http://jobslatest14.blogspot.in/2015/07/top-16-sites-to-prepare-resume-in-online.html

ReplyDelete