What are the parameters of logistic regression?

 $L(\hat{y}, y) = -(ylog(\hat{y}) + (1 - y)log(1 - \hat{y}))$
- If y = 1: $L(\hat{y}, y) = -log(\hat{y}) \Rightarrow log(\hat{y}) \approx 0 \rightarrow \hat{y}  \approx 1$
- If y = 0: $L(\hat{y}, y) = -log(1 - \hat{y}) \Rightarrow log(1 - \hat{y}) \approx 0 \rightarrow \hat{y} \approx 0$

Logistic Regression Loss Function

To train the parameters W and B of the logistic regression model, you need to define a cost function.

$$J(w, b) = \frac{1}{m} \sum_{i=1}^{m} L(\hat{y}^{(i)}, y^{(i)})$$

$$=-\frac{1}{m} \sum_{i=1}^{m} [y^{(i)}log(\hat{y}^{(i)}) + (1 - y^{(i)})log(1 - \hat{y}^{(i)})]$$

This loss function is Convex.

Logistic Regression Cost Function

Neural networks are made up of nodes (each representing a Logistic regression) with layers. They are trained to map an input vector X to an output vector Y based on the training data. The hidden layers of a network contain neurons, connections, and activation functions that occur between the input and output. An activation function is applied at each layer to correctly connect the input and output. Different types of networks may use diverse types of activation functions to correctly identify the nature of how the input and output are related.
In the figure out of the 4 neurons, the Logistic regression representation of two of them are shown.

Artificial Neural Networks Formulation

Given X, we want $\hat{y}=P(y = 1|X)$   $0 \leq \hat{y} \leq 1$

Parameters: $w \in R^{n_{x}}, b \in R$

Output (Activation): $\hat{y}^{(i)} = a^{(i)} = \sigma (w^{T}x^{(i)} + b)$

Given $[(x^{(1)}, y^{(1)}), (x^{(2)}, y^{(2)}), ..., (x^{(m)}, y^{(m)})]$, we want $\hat{y}^{i} \approx y^{i}$

$\sigma(z^{(i)}) = \frac{1}{1 + e^{-z^{(i)}}}$

If z is large positive, $\sigma(z^{(i)}) \approx \frac{1}{1 + 0} = 1$

If z is large negative, $\sigma(z^{(i)}) \approx \frac{1}{1 + Big Num} \approx 0$

University of Michigan - Ann Arbor

Logistic regression is used to model discontinuous output choices - for example, when a variable is 'yes' or 'no' rather than a continuous value. In this case, logistic regression calculates the probability of a given categorical output, and decides on a threshold at which the model decides between categories to select for output. 

Logistic Regression

Different methods are required for fitting a model for classification. Two reasons:
1. If there are >2 classifications, simply assigning dummy numerical variables and making a prediction based on the nearest classifier would imply an ordering to the classifications; that is to say that classification 4 and 3 are more similar than 4 and 1. 
2. Even using a binary classification, 1 or 0, an OLS model can produce an output outside of the range [0, 1], which is difficult to interpret and again implies an ordering to the classifications

OLS fitting cannot be used for classification

LDA is more stable than logistic regression in instances where the classes are well separated or when "n is small and the distribution of the predictors X is approximately normal in each of the classes" (ISLR). 

Using LDA vs Logistic Regression

This video series breaks down logistic regression and its applications.

[Logistic Regression]. (2020, February 21). Logistic Regression [Video File]. Retrieved from [www.youtube.com/playlist?list=PLblh5JKOoLUKxzEP5HA2d-Li7IJkHfXSe](https://www.youtube.com/playlist?list=PLblh5JKOoLUKxzEP5HA2d-Li7IJkHfXSe) 

Logistic Regression Videos

There are three out of four binary classification metrics mentioned in the Book of Why; true positive, false positive, and false negative.  The fourth one that is not mentioned in the book is true negative.  The following are the two primary building blocks of such four metrics: 

1. True vs. False: These are about prediction's correctness whether a prediction outcome was made accurately/ correctly.  The correct prediction is True, and the incorrect prediction is False.
2. Positive vs. Negative: These are about a binary classification -(e.g.) In the mammogram example in the book, "positive" is positive test results (breast cancer), whereas "negative" is negative test results (not breast cancer).  

The combinations of the above concepts result in True positive, True negative, false positive, and false negative, each of which is explained in the subsequent nodes to follow.  


Binary Classification Metrics

The hypothesis we are using for the logistic regression is the sigmoid function:



Hypothesis

The hypothesis function for the logistic regression is the sigmoid function:


Hypothesis function

Logistic Regression Formulation

Like ridge and lasso regression, a regularization penalty on the model coefficients can also be applied with logistic regression, and is controlled with the parameter C. 

In fact, the same L2 regularization penalty used for ridge regression is turned on by default for logistic regression with a default value C = 1. With large values of C, logistic regression tries to fit the training data as well as possible. While with small values of C, the model tries harder to find model coefficients that are closer to 0, even if that model fits the training data a little bit worse. 

If the L1 and L2 norms are added to the logistic regression, the effect is: feature selection can be done, and overfitting can also be prevented to a certain extent. Reason: L1–LASSO can generate sparse solutions for feature selection; L2–Ridge constrains model parameters to prevent overfitting. In addition, L2 can get smooth weights.

Logistic Regression - Regularization

Linear regression is used for prediction and logical regression is used for classification. Linear regression is to fit the function, and logical regression is to predict the function. Linear regression uses least square method to calculate parameters, and logical regression uses maximum likelihood estimation to calculate parameters. Linear regression is more susceptible to outliers, while logical regression has better stability to outliers.


Linear Regression vs Logistic Regression

Softmax regression is the generalization of logistic regression to multiple classes. In other words, each data point belongs to one of multiple classes (rather than just two options, as is the case for logistic regression). Hence softmax regression is also called multi-class logistic regression or multinomial logistic regression.

Softmax Regression (Activation)

$$P(\hat{Y})=\frac{e^{\beta_{0}+\beta_{1}X}}{1+e^{\beta_{0}+\beta_{1}X}}$$ $$\Rightarrow P(\hat{Y})+P(\hat{Y})\times[e^{\beta_{0}+\beta_{1}X}]=e^{\beta_{0}+\beta_{1}X}$$ $$\Rightarrow P(\hat{Y})=[1-P(\hat{Y})]\times [e^{\beta_{0}+\beta_{1}X}]$$ $$\Rightarrow \frac{P(\hat{Y})}{1-P(\hat{Y})}=e^{\beta_{0}+\beta_{1}X}$$ The left hand side is called the *odds*. $$\Rightarrow\log{\left(\frac{P(\hat{Y})}{1-P(\hat{Y})}\right)}=\beta_{0}+\beta_{1}X$$ The left hand side is called the *log-odds* or *logit*. This models the log-odds as a linear function of $$X$$. The coefficient $$\beta_{1}$$ means: a one-unit increase in $$X$$ is associated with an increase in the log-odds by $$\beta_{1}$$ units.

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