# [machine-learning] What is the difference between linear regression and logistic regression?

When we have to predict the value of a categorical (or discrete) outcome we use logistic regression. I believe we use linear regression to also predict the value of an outcome given the input values.

Then, what is the difference between the two methodologies?

This question is related to `machine-learning` `data-mining` `linear-regression`

In linear regression the outcome is continuous whereas in logistic regression, the outcome has only a limited number of possible values(discrete).

example: In a scenario,the given value of x is size of a plot in square feet then predicting y ie rate of the plot comes under linear regression.

If, instead, you wanted to predict, based on size, whether the plot would sell for more than 300000 Rs, you would use logistic regression. The possible outputs are either Yes, the plot will sell for more than 300000 Rs, or No.

Logistic Regression is used in predicting categorical outputs like Yes/No, Low/Medium/High etc. You have basically 2 types of logistic regression Binary Logistic Regression (Yes/No, Approved/Disapproved) or Multi-class Logistic regression (Low/Medium/High, digits from 0-9 etc)

On the other hand, linear regression is if your dependent variable (y) is continuous. y = mx + c is a simple linear regression equation (m = slope and c is the y-intercept). Multilinear regression has more than 1 independent variable (x1,x2,x3 ... etc)

Cannot agree more with the above comments. Above that, there are some more differences like

In Linear Regression, residuals are assumed to be normally distributed. In Logistic Regression, residuals need to be independent but not normally distributed.

Linear Regression assumes that a constant change in the value of the explanatory variable results in constant change in the response variable. This assumption does not hold if the value of the response variable represents a probability (in Logistic Regression)

GLM(Generalized linear models) does not assume a linear relationship between dependent and independent variables. However, it assumes a linear relationship between link function and independent variables in logit model.

Simply put, linear regression is a regression algorithm, which outpus a possible continous and infinite value; logistic regression is considered as a binary classifier algorithm, which outputs the 'probability' of the input belonging to a label (0 or 1).

They are both quite similar in solving for the solution, but as others have said, one (Logistic Regression) is for predicting a category "fit" (Y/N or 1/0), and the other (Linear Regression) is for predicting a value.

So if you want to predict if you have cancer Y/N (or a probability) - use logistic. If you want to know how many years you will live to - use Linear Regression !

The basic difference between Linear Regression and Logistic Regression is : Linear Regression is used to predict a continuous or numerical value but when we are looking for predicting a value that is categorical Logistic Regression come into picture.

Logistic Regression is used for binary classification.

Regression means continuous variable, Linear means there is linear relation between y and x. Ex= You are trying to predict salary from no of years of experience. So here salary is independent variable(y) and yrs of experience is dependent variable(x). y=b0+ b1*x1 We are trying to find optimum value of constant b0 and b1 which will give us best fitting line for your observation data. It is a equation of line which gives continuous value from x=0 to very large value. This line is called Linear regression model.

Logistic regression is type of classification technique. Dnt be misled by term regression. Here we predict whether y=0 or 1.

Here we first need to find p(y=1) (wprobability of y=1) given x from formuale below.

Probaibility p is related to y by below formuale

Ex=we can make classification of tumour having more than 50% chance of having cancer as 1 and tumour having less than 50% chance of having cancer as 0.

Here red point will be predicted as 0 whereas green point will be predicted as 1.

The basic difference :

Linear regression is basically a regression model which means its will give a non discreet/continuous output of a function. So this approach gives the value. For example : given x what is f(x)

For example given a training set of different factors and the price of a property after training we can provide the required factors to determine what will be the property price.

Logistic regression is basically a binary classification algorithm which means that here there will be discreet valued output for the function . For example : for a given x if f(x)>threshold classify it to be 1 else classify it to be 0.

For example given a set of brain tumour size as training data we can use the size as input to determine whether its a benine or malignant tumour. Therefore here the output is discreet either 0 or 1.

*here the function is basically the hypothesis function

To put it simply, if in linear regression model more test cases arrive which are far away from the threshold(say =0.5)for a prediction of y=1 and y=0. Then in that case the hypothesis will change and become worse.Therefore linear regression model is not used for classification problem.

Another Problem is that if the classification is y=0 and y=1, h(x) can be > 1 or < 0.So we use Logistic regression were 0<=h(x)<=1.

In linear regression, the outcome (dependent variable) is continuous. It can have any one of an infinite number of possible values. In logistic regression, the outcome (dependent variable) has only a limited number of possible values.

For instance, if X contains the area in square feet of houses, and Y contains the corresponding sale price of those houses, you could use linear regression to predict selling price as a function of house size. While the possible selling price may not actually be any, there are so many possible values that a linear regression model would be chosen.

If, instead, you wanted to predict, based on size, whether a house would sell for more than \$200K, you would use logistic regression. The possible outputs are either Yes, the house will sell for more than \$200K, or No, the house will not.

In case of Linear Regression the outcome is continuous while in case of Logistic Regression outcome is discrete (not continuous)

To perform Linear regression we require a linear relationship between the dependent and independent variables. But to perform Logistic regression we do not require a linear relationship between the dependent and independent variables.

Linear Regression is all about fitting a straight line in the data while Logistic Regression is about fitting a curve to the data.

Linear Regression is a regression algorithm for Machine Learning while Logistic Regression is a classification Algorithm for machine learning.

Linear regression assumes gaussian (or normal) distribution of dependent variable. Logistic regression assumes binomial distribution of dependent variable.

In short: Linear Regression gives continuous output. i.e. any value between a range of values. Logistic Regression gives discrete output. i.e. Yes/No, 0/1 kind of outputs.

``````| Basis                                                           | Linear                                                                         | Logistic                                                                                                            |
|-----------------------------------------------------------------|--------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------|
| Basic                                                           | The data is modelled using a straight line.                                    | The probability of some obtained event is represented as a linear function of a combination of predictor variables. |
| Linear relationship between dependent and independent variables | Is required                                                                    | Not required                                                                                                        |
| The independent variable                                        | Could be correlated with each other. (Specially in multiple linear regression) | Should not be correlated with each other (no multicollinearity exist).                                              |
``````

Linear regression

Is meant to resolve the problem of predicting/estimating the output value for a given element X (say f(x)). The result of the prediction is a continuous function where the values may be positive or negative. In this case you normally have an input dataset with lots of examples and the output value for each one of them. The goal is to be able to fit a model to this data set so you are able to predict that output for new different/never seen elements. Following is the classical example of fitting a line to set of points, but in general linear regression could be used to fit more complex models (using higher polynomial degrees):

Resolving the problem

Linear regression can be solved in two different ways:

1. Normal equation (direct way to solve the problem)

Logistic regression

Is meant to resolve classification problems where given an element you have to classify the same in N categories. Typical examples are, for example, given a mail to classify it as spam or not, or given a vehicle find to which category it belongs (car, truck, van, etc ..). That's basically the output is a finite set of discrete values.

Resolving the problem

Logistic regression problems could be resolved only by using Gradient descent. The formulation in general is very similar to linear regression the only difference is the usage of different hypothesis function. In linear regression the hypothesis has the form:

``````h(x) = theta_0 + theta_1*x_1 + theta_2*x_2 ..
``````

where theta is the model we are trying to fit and [1, x_1, x_2, ..] is the input vector. In logistic regression the hypothesis function is different:

``````g(x) = 1 / (1 + e^-x)
``````

This function has a nice property, basically it maps any value to the range [0,1] which is appropiate to handle propababilities during the classificatin. For example in case of a binary classification g(X) could be interpreted as the probability to belong to the positive class. In this case normally you have different classes that are separated with a decision boundary which basically a curve that decides the separation between the different classes. Following is an example of dataset separated in two classes.