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Before discussing the logistic regression problem (Logistic Regression) , let's discuss some real-life situations: determine whether an email is spam? Determine whether a transaction is a fraudulent transaction? To determine whether a document is a valid document? This kind of problem, we call it the classification problem (Classication Problem) **. In classification problems, we often try to predict whether the result belongs to a certain class (correct or false).
We start our discussion with a binary classification problem, i.e. whether the problem is true or false.
We call the two classes that the Dependent Variable may belong to as Negative Class and Positive Class, respectively , then the dependent variable :
Among them, 0 represents the negative class, 1 represents the positive class
We assume the **classification problem (Malignant or Benign) ** of predicting whether a tumor is a malignant tumor . We assume the relationship between the malignant or benign tumor and the size of the tumor, and a linear regression method can be used to find a straight line that fits the data:
According to the linear regression model, we can only predict continuous values, then for classification problems, we only need to output 0 or 1, we can predict:
For the data shown above, such a linear model seems to work well for the classification task. Suppose we observe another malignant tumor of a very large size and add it to our training set as a new instance, which will have some effect on our linear model and obtain a new line.
At this time, it is not so appropriate to use 0.5 as a threshold to predict whether the tumor is benign or malignant. It can be seen that the linear regression model, because its predicted value can go beyond the range of [0, 1], is not suitable for solving such a problem.
We introduce a new model, logistic regression, whose output variable ranges moderately between 0 and 1.
The assumptions of the logistic regression model are:
Among them, some flags are explained:
The image of the function is:
Combining the logistic function and the hypothesis function yields the hypothesis for the logistic regression model:
For the model, you can have the following understanding:
The function of hø(x) is, for a given input variable, to calculate the Estimated Probability ** of the output variable = 1 according to the selected parameters, namely:
For example, if for a given x, hø(x)=0.7 is calculated through the determined parameters, it means that there is a 70% probability that y is a positive class, and correspondingly, y is a negative class. The probability is thirty percent (1-0.7=0.3).
The above is the logistic regression model. Next time we discuss ** Decision Boundary and Cost Function **.