Name: ロジスティック回帰 (Logistic regression)
Uploaded: 2021-01-14T05:25:22.000Z
Duration: 51 min 46 s
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In statistics, logistic regression, or logit regression, is a type of probabilistic statistical

classification model. It is also used to predict a binary response from a binary predictor,

used for predicting the outcome of a categorical dependent variable based on one or more predictor

variables. That is, it is used in estimating the parameters of a qualitative response model.

The probabilities describing the possible outcomes of a single trial are modeled, as

a function of the explanatory variables, using a logistic function. Frequently "logistic

regression" is used to refer specifically to the problem in which the dependent variable

is binary—that is, the number of available categories is two—while problems with more

than two categories are referred to as multinomial logistic regression or, if the multiple categories

are ordered, as ordered logistic regression. Logistic regression measures the relationship

between a categorical dependent variable and one or more independent variables, which are

usually continuous, by using probability scores as the predicted values of the dependent variable.

As such it treats the same set of problems as does probit regression using similar techniques.

Fields and examples of applications Logistic regression was put forth in the 1940s

as an alternative to Fisher's 1936 classification method, linear discriminant analysis. It is

used extensively in numerous disciplines, including the medical and social science fields.

For example, the Trauma and Injury Severity Score, which is widely used to predict mortality

in injured patients, was originally developed by Boyd et al. using logistic regression.

Logistic regression might be used to predict whether a patient has a given disease, based

on observed characteristics of the patient. Another example might be to predict whether

an American voter will vote Democratic or Republican, based on age, income, gender,

race, state of residence, votes in previous elections, etc. The technique can also be

used in engineering, especially for predicting the probability of failure of a given process,

system or product. It is also used in marketing applications such as prediction of a customer's

propensity to purchase a product or cease a subscription, etc. In economics it can be

used to predict the likelihood of a person's choosing to be in the labor force, and a business

application would be to predict the likehood of a homeowner defaulting on a mortgage. Conditional

random fields, an extension of logistic regression to sequential data, are used in natural language

Logistic regression can be binomial or multinomial. Binomial or binary logistic regression deals

with situations in which the observed outcome for a dependent variable can have only two

possible types. Multinomial logistic regression deals with situations where the outcome can

have three or more possible types. In binary logistic regression, the outcome is usually

coded as "0" or "1", as this leads to the most straightforward interpretation. If a

particular observed outcome for the dependent variable is the noteworthy possible outcome

it is usually coded as "1" and the contrary outcome as "0". Logistic regression is used

to predict the odds of being a case based on the values of the independent variables.

The odds are defined as the probability that a particular outcome is a case divided by

the probability that it is a noncase. Like other forms of regression analysis, logistic

regression makes use of one or more predictor variables that may be either continuous or

categorical data. Unlike ordinary linear regression, however, logistic regression is used for predicting

binary outcomes of the dependent variable rather than continuous outcomes. Given this

difference, it is necessary that logistic regression take the natural logarithm of the

odds of the dependent variable being a case to create a continuous criterion as a transformed

version of the dependent variable. Thus the logit transformation is referred to as the

link function in logistic regression—although the dependent variable in logistic regression

is binomial, the logit is the continuous criterion upon which linear regression is conducted.

The logit of success is then fit to the predictors using linear regression analysis. The predicted

value of the logit is converted back into predicted odds via the inverse of the natural

logarithm, namely the exponential function. Therefore, although the observed dependent

variable in logistic regression is a zero-or-one variable, the logistic regression estimates

the odds, as a continuous variable, that the dependent variable is a success. In some applications

the odds are all that is needed. In others, a specific yes-or-no prediction is needed

for whether the dependent variable is or is not a case; this categorical prediction can

be based on the computed odds of a success, with predicted odds above some chosen cut-off

value being translated into a prediction of a success.

An explanation of logistic regression begins with an explanation of the logistic function,

which always takes on values between zero and one:

and viewing as a linear function of an explanatory variable , the logistic function can be written

This will be interpreted as the probability of the dependent variable equalling a "success"

or "case" rather than a failure or non-case. We also define the inverse of the logistic

A graph of the logistic function is shown in Figure 1. The input is the value of and

the output is . The logistic function is useful because it can take an input with any value

from negative infinity to positive infinity, whereas the output is confined to values between

0 and 1 and hence is interpretable as a probability. In the above equations, refers to the logit

function of some given linear combination of the predictors, denotes the natural logarithm,

is the probability that the dependent variable equals a case, is the intercept from the linear

regression equation, is the regression coefficient multiplied by some value of the predictor,

and base denotes the exponential function. The formula for illustrates that the probability

of the dependent variable equaling a case is equal to the value of the logistic function

of the linear regression expression. This is important in that it shows that the value

of the linear regression expression can vary from negative to positive infinity and yet,

after transformation, the resulting expression for the probability ranges between 0 and 1.

The equation for illustrates that the logit is equivalent to the linear regression expression.

Likewise, the next equation illustrates that the odds of the dependent variable equaling

a case is equivalent to the exponential function of the linear regression expression. This

illustrates how the logit serves as a link function between the probability and the linear

regression expression. Given that the logit ranges between negative infinity and positive

infinity, it provides an adequate criterion upon which to conduct linear regression and

the logit is easily converted back into the odds.

Multiple explanatory variables If there are multiple explanatory variables,

then the above expression can be revised to Then when this is used in the equation relating

the logged odds of a success to the values of the predictors, the linear regression will

be a multiple regression with m explanators; the parameters for all j = 0, 1, 2, ..., m

The regression coefficients are usually estimated using maximum likelihood estimation. Unlike

linear regression with normally distributed residuals, it is not possible to find a closed-form

expression for the coefficient values that maximizes the likelihood function, so an iterative

process must be used instead, for example Newton's method. This process begins with

a tentative solution, revises it slightly to see if it can be improved, and repeats

this revision until improvement is minute, at which point the process is said to have

converged. In some instances the model may not reach

convergence. When a model does not converge this indicates that the coefficients are not

meaningful because the iterative process was unable to find appropriate solutions. A failure

to converge may occur for a number of reasons: having a large proportion of predictors to

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ロジスティック回帰 (Logistic regression)

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