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Logistic Regression

Or how I learned to love the sigmoid "squishification" function for categorical data classification James Andrew Godwin April 13, 20215 min read Logistic Regression | Photo by Denise Bossarte Logistic Regression | Photo by Denise Bossarte This article is a brief continuation of my regression series. So far the regression examples I have been illustrating have been numeric, of numbers: predicting a continuous variable. With the Galton family height dataset, we were predicting children's height - a continuously varying parameter. Yet, note how a regression line fails to fit a binary outcome, a classification example: Line of best fit for binary outcomes Line of best fit for binary outcomes In the above example, there appears to be a threshold affecting the classification outcomes. It does not make sense to have a continuous regression line for this data. Why does it not make sense? Look at the two dependent (y) variables: they take on one of two values: 0 and 1. There is no value for 0.25 or 0.5, just those two extrema. This means that for this classification problem we want a function that takes as an input, or a series of inputs, and gives us a probability of it being one of the two classes: namely 0 and 1. We can give any arbitrary label to these classes: yes or no, cat or not cat, send a promotional email or do not send a promotional email, sales lead or unlikely sales lead. Logistic regression is a powerful tool for data scientists and machine learning engineers alike. I would say, for sanity checks, that it should be the baseline model for classification purposes - one that is compared against more complex models (like PCR, neural networks, etc).

History

Logistic regression

To get to logistic regression, we first begin with our classic regression example: Classic regression Classic regression To develop a model for classification we will borrow the sigmoid function from math, commonly expressed below as: Sigmoid formula Sigmoid formula Rearranging and substituting into the regression: Formula for logistic regression Formula for logistic regression What this family of functions does is give a probability of assigning the observation to the target class. Once the logistic regression function assigns a probability, this requires the model to be fit and trained on the dataset, an example further down, the data point is projected into a class based on the probability value. For probability values greater than 0.5 the value is projected onto 1, for us the target class, and for probability values less than 0.5, they are projected onto the 0 class, or not the target class. A standard sigmoid function looks as follows: Standard logistic function | Photo source Wikipedia. Standard logistic function | Photo source Wikipedia.) This transforms our original dataset regression line into: The inflection point is at 0.5 The inflection point is at 0.5

What just happened?

This "s" shape is a powerful tool for classification purposes. I like to think that given probability value output for a given observation, say 0.36, that the logistic regression is "squishing" the probability down to binary value. Interpreting this, the model is saying that there is only a 36% chance that the given observation belongs to the target class the model was trained on - let us say that is 1. Meaning that there is a 1 - .36 or 64% chance that this observation does not belong to the target class, so it is labeled as 0 ->the observation is projected down to the dependent variable outcome label of 0. The logistic regression line is similar to our previous regression examples, but we interpret the output as a probability of assigning the data point to the target class. The most common flavor of logistic regression is binary logistic regression, but multinomial logit models do exist for multiclass regression analysis. N.B: It is technically possible to achieve a value of 0.5, but vanishingly small. Regardless, the algorithm you are using will have some predefined instruction on how to classify this edge case.

Python example: Logistic regression

For illustrative purposes, I will be using the famous iris dataset included with sklean to demonstrate the classification power in the following Jupyter notebook cells: text

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import accuracy_score

text

%matplotlib inline

Create a df and transform all the raw data into a format familiar to sklearn: text

from sklearn.datasets import load_iris
iris = load_iris()
df = pd.DataFrame(data= np.c_[iris['data'], iris['target']],
 columns= ["sepel_length_cm", "sepal_width_cm",
 "petal_length_cm", "petal_width_cm", "target"])

text

df.head()

Data is well imported Data is well imported text

df.describe()

Basic summary statistics Basic summary statistics The data set contains the true labels, and we can see how nice and separable the three classes are: text

sns.pairplot(df, height=2.5, hue='target')
plt.show()

The classifications for the three flower types based on their measurables The classifications for the three flower types based on their measurables text

df['target'].value_counts()

Note how we have three classes of flowers, each with 50 observations. This dataset is very well balanced. text

X, y = load_iris(return_X_y=True)

text

X[:5]

text

y[:5]

The above five flowers are all of the 0 class. text

from sklearn.linear_model import LogisticRegression

text

# instantiate the model and fit
log = LogisticRegression(multi_class='warn', solver='warn')
log_reg = log.fit(X, y)

text

log_reg

text

# score the model
log.score(X, y)

As we can see, this is a very good score for the model! Next, I will talk about Time Series Analysis! Find me on Linkedin Physicist cum Data Scientist- Available for new opportunity | SaaS | Sports | Start-ups | Scale-ups

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