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

scatter
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.

Classification

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 .

regress
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 .

regress2
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.

regress3
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. !

Conclusion

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

a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
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y=10×
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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J, combine like terms 7x-4y
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f(n)= 2n + 1
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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