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The figure below plots the training data and the actual decision boundary.

Scatter Plot of Training Data

We split the training data set into 3 separate sets, training set with 600 examples, cross-validation set with 200 examples and test data set with 200 examples.


This training data was then fed into a logistic regression classifier to study the performance of the classifier. It is important to note that the objective of logistic regression classifier is maximizing the accuracy of labeling the data into two classes. Unlike linear regression, the decision boundary of the logistic regression classifier does not try to match the underlying true boundary which divides the data into two classes.

In addition to the two features that identify a training example, polynomial features up to a desired degree are generated. We start off the optimization with an initial parameter value of all 0. The optimization of the cost function is done using the Matlab's built-in fminunc function. A function costFunctionReg.m that outputs the regularized cost and regularized gradient, with the training data, parameter values and the regularization parameter as inputs, is given as input along with initial parameter values to this fminunc function.

Now we vary the maximum degree of the polynomial features to study the decision boundary of the classifier. Its important to note that the values of the parameters are obtained from the training data set, for a given value of maximum degree. The optimal values of maximum degree are determined by the performance on the cross-validation set. Finally, the decision boundary obtained by solving for the parameters with optimal values of maximum degree is used to evaluate the performance on a test data set, in order to see how well the classifier generalizes.

The decision boundary for a degree 1 polynomial is shown in [link] below. The accuracy on the cross-validation set was 84 . 50 .

Logistic Regression with 1st degree features

From [link] it is clear that 1 st degree features are not sufficient to capture both classes. So the maximum degree was increased to 2. [link] plots the decision boundary for degree 2. The accuracy of the classifier on cross-validation set in this case was 98 . 50 .

Logistic Regression with 2nd degree features

We now try the maximum degree of 3. [link] plots the decision boundary with maximum degree 3. The accuracy on cross-validation set is 98.00.

Logistic Regression with 3rd degree features

Due to its lower accuracy, the logistic regression classifier with maximum degree 2 is chosen from amongst the 3 classifiers. This classifier was then evaluated on test data set to study how well it generalizes. The accuracy of this classifier on test set was 98.00. Hence this logistic regression classifier generalizes very well.

To see the MATLAB code that generated these plots, download the following .zip file: MATLAB files for simulated data. !


As was stated, this collection is intended to be an introduction to regression analysis, but is sufficient in order to understand the application of logistic regression to an application. There are plenty of resources to learn more about more nuanced views of the key components of the theory, and more resources to see logistic regression in action!

For an application of Logistic Regression to a synthetic dataset and to a real-world problem in statistical physics, see Optimizing Logistic Regression for Particle Physics !

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Smarajit Reply
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Source:  OpenStax, Introductory survey and applications of machine learning methods. OpenStax CNX. Dec 22, 2011 Download for free at http://legacy.cnx.org/content/col11400/1.1
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