What is the difference between linear regression and logistic regression

Question

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

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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

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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

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Linear regression output as probabilities  It s tempting to use the linear regression output as probabilities but it s a mistake because the output can be negative  and greater than 1 whereas probability can not  As regression might actually produce probabilities that could be less than 0  or even bigger than 1  logistic regression was introduced    Source  http   gerardnico com wiki data mining simple logistic regression   Outcome  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   The dependent variable  Logistic regression is used when the response variable is categorical in nature  For instance  yes no  true false  red green blue  1st 2nd 3rd 4th  etc     Linear regression is used when your response variable is continuous  For instance  weight  height  number of hours   etc  Equation  Linear regression gives an equation which is of the form Y   mX   C  means equation with degree 1    However  logistic regression gives an equation which is of the form  Y   eX   e-X Coefficient interpretation  In linear regression  the coefficient interpretation of independent variables are quite straightforward  i e  holding all other variables constant  with a unit increase in this variable  the dependent variable is expected to increase decrease by xxx     However  in logistic regression  depends on the family  binomial  Poisson  etc   and link  log  logit  inverse-log  etc   you use  the interpretation is different   Error minimization technique  Linear regression uses ordinary least squares method to minimise the errors and arrive at a best possible fit  while logistic regression uses maximum likelihood method to arrive at the solution   Linear regression is usually solved by minimizing the least squares error of the model to the data  therefore large errors are penalized quadratically    Logistic regression is just the opposite  Using the logistic loss function causes large errors to be penalized to an asymptotically constant   Consider linear regression on categorical  0  1  outcomes to see why this is a problem  If your model predicts the outcome is 38  when the truth is 1  you ve lost nothing  Linear regression would try to reduce that 38  logistic wouldn t  as much 2

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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

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Just to add on the previous answers  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   Normal equation  direct way to solve the problem  Gradient descent  Iterative approach   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

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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

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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

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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

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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

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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

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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

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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

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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

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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  lt  0 So we use Logistic regression were 0 lt  h x  lt  1

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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

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

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