Lecture Note
University
Stanford UniversityCourse
CS229 | Machine LearningPages
2
Academic year
2023
anon
Views
48
Understanding the Decision Boundary for Logistic Regression For binary classiﬁcation issues, logistic regression is a popular approach in machinelearning. The input features are mapped to a probability of the positive class using a logisticfunction. We shall examine the computation of predictions via logistic regression and therepresentation of the decision boundary in more detail in this article. Logistic Regression Review A value termed z, which is equal to the dot product of the input features and weights plus abias component, is calculated as the ﬁrst step in logistic regression. The anticipatedprobability of the positive class is then obtained by passing the value z through the Sigmoidfunction, g(z). The Sigmoid function's formula is as follows: 𝑔(𝑧) = 1 1+𝑒 −𝑧 Then, using a threshold, the ﬁnal forecast, y hat, is generated. The most common thresholdis 0.5, with y hat being adjusted to 1 if and 0 otherwise. 𝑔(𝑧) ≥ 0. 5 Making Sense of the Decision Boundary Logistic regression establishes a decision boundary to distinguish between the positive andnegative samples in a two-feature classiﬁcation problem. Depending on the quantity ofcharacteristics, the equation of a line or a curve determines the decision boundary. Set g(z) =0.5 and solve for one feature in terms of the other to obtain the equation of the decisionboundary. The decision boundary, where and are the weights for the two features and b is the 𝑤 1 𝑤 2 bias term, occurs when . 𝑤 1 · 𝑥 1 + 𝑤 2 𝑥 2 + 𝑏 = 0 We can plot the aﬃrmative and negative examples and depict the boundary between themby drawing a line or curve. The positive class is represented by the side of the line or curvewhere , and the negative class is represented by the side where . 𝑔(𝑧) =≥ 0. 5 𝑔(𝑧) < 0. 5 Using the Decision Boundary to InterpretThe decision border is a handy tool for deciphering the predictions made by the logisticregression model. We may learn more about the relative weights of each characteristic inestablishing the class labels by examining the coeﬃcients of the weights and the bias term. For instance, if a feature's weight is high and positive, it suggests that a feature's value iscorrelated with a higher likelihood of belonging to the positive class. A larger value of thatattribute is connected to a reduced probability of the positive class, however, if the weight islarge and negative.
The decision border is moved up or down by the bias term, but its slope is unaffected. Thedecision boundary is shifted upward by a signiﬁcant positive bias term, which makes itsimpler for samples to fall into the positive category. A signiﬁcant negative bias term lowersthe decision limit, making it more diﬃcult for samples to fall inside the positive category.
Understanding the Decision Boundary for Logistic Regression
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