Logistic Regression


title: Logistic Regression

Logistic Regression

Logistic Function

Logistic regression is very similar to linear regression. As in both algorithms, models are trained to find a regression line to define function for futher prediction. Therefore, it’s a form of supervised learning, which tries to predict the responses of unlabeled, unseen data by first training with labeled data, on a set of observations which consists of both independent (X) and dependent (Y) variables. But while Linear Regression assumes that the response variable (Y) is quantitative or continuous, Logistic Regression is used specifically when the response variable is qualititative, or discrete.

Linear vs Logistic

How it Works

Logistic regression models the probability that Y, the response variable, belongs to a certain category. In many cases, the response variable will be a binary one, so logistic regression will want to model a function y = f(x) that outputs a normalized value that ranges from, say, 0 to 1 for all values of X, corresponding to the two possible values of Y. It does this by using the logistic function.
Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary), but it has another form such as mutivalued logistic regression which is used to classify for more than two classes. Like all regression analyses, the logistic regression is a predictive analysis. Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.

Cost Function


Logistic regression is used to solve classification problems, where the output is of the form y∈{0,1}. Here, 0 is a negative class and 1 is a positive class. Let’s say we have a hypothesis hθ(x), where x is our dataset(a matrix) of length m. θ is the parameteric matrix. We have : 0 < hθ(x) < 1

In Logistic regression, hθ(x) is a sigmoid function, thus hθ(x) = g(θ’x).
A Sigmoid function is an appropriate function developed by statician to classify which always result between 0 or 1 depending on its
input attributes and weights(which is theta θ here)
g(θ’x) = 1/ 1 + e^(-θ’x)

Note: θ’ is θ transpose.
Transpose is used to multiply here it with attribute array It will be easily understoodable once you get in a bit depth with Linear algebra.

Cost function

Cost Function is a measure of How much our, Hypothetic function is farther to the observed one .
The cost function used for Logistic regression is:

J(θ)=(1/m)∑Cost(hθ(x(i)),y(i)) , where summation is from i=1 to m.

Where hθ(x) is = hypothetic value calculated in accordance with attributes and weights which are calculated and balanced via algorithm such as gradient descent.
y = is the corresponding value from observation data set

Here cost function is not a proper sigmoid function in use but in place, two log functions which performs with greater efficieny without penalizing the learning algorithms are used.

  • Cost(hθ(x),y)=−log(hθ(x)) if y = 1
  • Cost(hθ(x),y)=−log(1−hθ(x)) if y = 0

Which we can simplify as:

Cost(hθ(x),y) = −log(hθ(x)) − log(1 − hθ(x))

Refer to this article for clearing your basics https://www.analyticsvidhya.com/blog/2017/06/a-comprehensive-guide-for-linear-ridge-and-lasso-regression/

Predictions using logistic regression:

Logistic regression models the probability of the default class (i.e. the first class).
You can classify results given by:

y = e^(b0 + b1X) / (1 + e^(b0 + b1X))

into two classes.Like for sigmoid function 0.5 is set as the decision boundary all x for which y≥0.5 are classified as class A and for which y<0.5 are classified as class B.

Multi class logistic regression:

Although you will see logistic regression usually being used in case of binary classification, you can also use it in case of classification into multiple classes by:

one vs one method:

Here a classifier for each class is created separately and the classifier with the highest score is considered as output.

one vs all method:

Here multiple(N*N(N-1)/2 where N=no. of classes) binary classifiers are made and then by comparing their scores the output is obtained.

Applications of logistic regression:

1) To classify mail as spam or not spam.

2) To determine presence or absence of certain disease, like cancer, based on symptoms and other medical data
Like- benign or malignant.

3) Classify images based on pixel data.

Logistic Regression Assumptions

Binary logistic regression requires the dependent variable to be binary.
For a binary regression, the factor level 1 of the dependent variable should represent the desired outcome.
Only the meaningful variables should be included.
The independent variables should be independent of each other. That is, the model should have little or no multicollinearity.
The independent variables are linearly related to the log odds.
Logistic regression requires quite large sample sizes.

More Information:

For further reading to build logistic regression step by step:

  • Click here for an article about building a Logistic Regression in Python.
  • Click here for another article on building a Logical Regression.
  • Click here for another article on mathematics and intuition behind Logical Regression.

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