Logistic Regression

Logistic regression is used to model the probability of a certain class or classes based on modeling data.

The term logistic refers to the logistic curve used as the basis for logistic regression analysis.

Prediction results are assigned a probability between 0 (lowest score) and 1 (highest score).

Logistic (s curve) functions are used, which have the shape:

Mathematical Model

Key aspects of the model include:

Odds of Something Occurring

As example is the odds that a sports team will win a give game. See a detailed discussion of odds here.

Log-odds (logit) of Odds

This is the logarithm to a given base of odds. See a detailed discussion of odds here.

Logarithms

This is the inverse of exponentiation. See a detailed discussion of odds here.

Probabilities

This is the likelihood that an event will occur expressed in a range from 0 to 1. See a detailed discussion of odds here.

Model Equations

See a detailed discussion of odds here.

$\\ d = \beta _{0}+\beta_{1}x_{1}+...\,\,\beta_{n}x_{n} \\ \\ o = \frac{p}{1-p}=b^{d} \\ \\ l=log_{b}\left (o\right )=d \\ \\ p=\frac{d}{d+1} \\ \\ \\ \textup{where:} \\ \\ d\,\,\,\,\,\textup{dot product of the vectors }\beta ,x \\ o \,\,\,\,\,\,\textup{odds, which range from 0 to infinity} \\ l \,\,\,\,\,\,\,\textup{log-odds (logarithm of the odds) } \\ b \,\,\,\textup{\,\,\,base - a positive real number not equal to 1, typically 2, 10, or} \,e \\ p \,\,\,\textup{\,\,\,probability, which ranges from 0 to 1, that the predicted variable equals 1} \\ \beta _{i} \,\textup{\,\,\,parameters of the model} \\ x _{i} \,\textup{\,\,\,predictors of the model}$

Python Example

To download the code below, click here.

"""
logistic_regression_with_scikit_learn.py
trains and uses a model to predict one of three classes for each input
"""

# Import needed libraries.
import random
from sklearn.datasets import load_iris
from sklearn.linear_model import LogisticRegression

# Set parameters.
number_of_prediction_inputs = 100

# Load test data.
X, y = load_iris(return_X_y=True)
print("X - Data Features:")
print(X)
print("y - Data Classes:")
print(y)

# Instantiate a model.
model = LogisticRegression(random_state=0)

# Train the model.
estimator = model.fit(X, y)

# Get the training score (accuracy).
score = estimator.score(X, y)
print("Score:")
print(score)

# Create shuffled prediction input data.
shuffled_input_data = X
random.shuffle(shuffled_input_data)
print("Shuffled Input Data:")
print(shuffled_input_data)

# Get prediction input data from the shuffled training data.
prediction_input = shuffled_input_data[:number_of_prediction_inputs, :]
print("Prediction Input:")
print(prediction_input)

# Make predictions.
predicted_classes = estimator.predict(prediction_input)
print("Predicted Classes: ")
print(predicted_classes)

# Get prediction probabilities for each class.
probabilities = estimator.predict_proba(prediction_input)
print("Probabilities: ")
print(probabilities)

Output is below:

X - Data Features:
[[5.1 3.5 1.4 0.2]
 [4.9 3.  1.4 0.2]
 [4.7 3.2 1.3 0.2]
 [4.6 3.1 1.5 0.2]
 [5.  3.6 1.4 0.2]
 [5.4 3.9 1.7 0.4]
 [4.6 3.4 1.4 0.3]
 [5.  3.4 1.5 0.2]
 [4.4 2.9 1.4 0.2]
 [4.9 3.1 1.5 0.1]
 [5.4 3.7 1.5 0.2]
 [4.8 3.4 1.6 0.2]
 [4.8 3.  1.4 0.1]
 [4.3 3.  1.1 0.1]
 [5.8 4.  1.2 0.2]
 [5.7 4.4 1.5 0.4]
 [5.4 3.9 1.3 0.4]
 [5.1 3.5 1.4 0.3]
 [5.7 3.8 1.7 0.3]
 [5.1 3.8 1.5 0.3]
 [5.4 3.4 1.7 0.2]
 [5.1 3.7 1.5 0.4]
 [4.6 3.6 1.  0.2]
 [5.1 3.3 1.7 0.5]
 [4.8 3.4 1.9 0.2]
 [5.  3.  1.6 0.2]
 [5.  3.4 1.6 0.4]
 [5.2 3.5 1.5 0.2]
 [5.2 3.4 1.4 0.2]
 [4.7 3.2 1.6 0.2]
 [4.8 3.1 1.6 0.2]
 [5.4 3.4 1.5 0.4]
 [5.2 4.1 1.5 0.1]
 [5.5 4.2 1.4 0.2]
 [4.9 3.1 1.5 0.1]
 [5.  3.2 1.2 0.2]
 [5.5 3.5 1.3 0.2]
 [4.9 3.1 1.5 0.1]
 [4.4 3.  1.3 0.2]
 [5.1 3.4 1.5 0.2]
 [5.  3.5 1.3 0.3]
 [4.5 2.3 1.3 0.3]
 [4.4 3.2 1.3 0.2]
 [5.  3.5 1.6 0.6]
 [5.1 3.8 1.9 0.4]
 [4.8 3.  1.4 0.3]
 [5.1 3.8 1.6 0.2]
 [4.6 3.2 1.4 0.2]
 [5.3 3.7 1.5 0.2]
 [5.  3.3 1.4 0.2]
 [7.  3.2 4.7 1.4]
 [6.4 3.2 4.5 1.5]
 [6.9 3.1 4.9 1.5]
 [5.5 2.3 4.  1.3]
 [6.5 2.8 4.6 1.5]
 [5.7 2.8 4.5 1.3]
 [6.3 3.3 4.7 1.6]
 [4.9 2.4 3.3 1. ]
 [6.6 2.9 4.6 1.3]
 [5.2 2.7 3.9 1.4]
 [5.  2.  3.5 1. ]
 [5.9 3.  4.2 1.5]
 [6.  2.2 4.  1. ]
 [6.1 2.9 4.7 1.4]
 [5.6 2.9 3.6 1.3]
 [6.7 3.1 4.4 1.4]
 [5.6 3.  4.5 1.5]
 [5.8 2.7 4.1 1. ]
 [6.2 2.2 4.5 1.5]
 [5.6 2.5 3.9 1.1]
 [5.9 3.2 4.8 1.8]
 [6.1 2.8 4.  1.3]
 [6.3 2.5 4.9 1.5]
 [6.1 2.8 4.7 1.2]
 [6.4 2.9 4.3 1.3]
 [6.6 3.  4.4 1.4]
 [6.8 2.8 4.8 1.4]
 [6.7 3.  5.  1.7]
 [6.  2.9 4.5 1.5]
 [5.7 2.6 3.5 1. ]
 [5.5 2.4 3.8 1.1]
 [5.5 2.4 3.7 1. ]
 [5.8 2.7 3.9 1.2]
 [6.  2.7 5.1 1.6]
 [5.4 3.  4.5 1.5]
 [6.  3.4 4.5 1.6]
 [6.7 3.1 4.7 1.5]
 [6.3 2.3 4.4 1.3]
 [5.6 3.  4.1 1.3]
 [5.5 2.5 4.  1.3]
 [5.5 2.6 4.4 1.2]
 [6.1 3.  4.6 1.4]
 [5.8 2.6 4.  1.2]
 [5.  2.3 3.3 1. ]
 [5.6 2.7 4.2 1.3]
 [5.7 3.  4.2 1.2]
 [5.7 2.9 4.2 1.3]
 [6.2 2.9 4.3 1.3]
 [5.1 2.5 3.  1.1]
 [5.7 2.8 4.1 1.3]
 [6.3 3.3 6.  2.5]
 [5.8 2.7 5.1 1.9]
 [7.1 3.  5.9 2.1]
 [6.3 2.9 5.6 1.8]
 [6.5 3.  5.8 2.2]
 [7.6 3.  6.6 2.1]
 [4.9 2.5 4.5 1.7]
 [7.3 2.9 6.3 1.8]
 [6.7 2.5 5.8 1.8]
 [7.2 3.6 6.1 2.5]
 [6.5 3.2 5.1 2. ]
 [6.4 2.7 5.3 1.9]
 [6.8 3.  5.5 2.1]
 [5.7 2.5 5.  2. ]
 [5.8 2.8 5.1 2.4]
 [6.4 3.2 5.3 2.3]
 [6.5 3.  5.5 1.8]
 [7.7 3.8 6.7 2.2]
 [7.7 2.6 6.9 2.3]
 [6.  2.2 5.  1.5]
 [6.9 3.2 5.7 2.3]
 [5.6 2.8 4.9 2. ]
 [7.7 2.8 6.7 2. ]
 [6.3 2.7 4.9 1.8]
 [6.7 3.3 5.7 2.1]
 [7.2 3.2 6.  1.8]
 [6.2 2.8 4.8 1.8]
 [6.1 3.  4.9 1.8]
 [6.4 2.8 5.6 2.1]
 [7.2 3.  5.8 1.6]
 [7.4 2.8 6.1 1.9]
 [7.9 3.8 6.4 2. ]
 [6.4 2.8 5.6 2.2]
 [6.3 2.8 5.1 1.5]
 [6.1 2.6 5.6 1.4]
 [7.7 3.  6.1 2.3]
 [6.3 3.4 5.6 2.4]
 [6.4 3.1 5.5 1.8]
 [6.  3.  4.8 1.8]
 [6.9 3.1 5.4 2.1]
 [6.7 3.1 5.6 2.4]
 [6.9 3.1 5.1 2.3]
 [5.8 2.7 5.1 1.9]
 [6.8 3.2 5.9 2.3]
 [6.7 3.3 5.7 2.5]
 [6.7 3.  5.2 2.3]
 [6.3 2.5 5.  1.9]
 [6.5 3.  5.2 2. ]
 [6.2 3.4 5.4 2.3]
 [5.9 3.  5.1 1.8]]
y - Data Classes:
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2
 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 2 2]
Score:
0.96
Shuffled Input Data:
[[5.1 3.5 1.4 0.2]
 [5.1 3.5 1.4 0.2]
 [4.7 3.2 1.3 0.2]
 [4.7 3.2 1.3 0.2]
 [4.7 3.2 1.3 0.2]
 [5.1 3.5 1.4 0.2]
 [4.7 3.2 1.3 0.2]
 [5.  3.6 1.4 0.2]
 [4.9 3.  1.4 0.2]
 [4.6 3.1 1.5 0.2]
 [5.  3.4 1.5 0.2]
 [5.4 3.9 1.7 0.4]
 [4.6 3.4 1.4 0.3]
 [4.8 3.  1.4 0.1]
 [4.8 3.  1.4 0.1]
 [5.  3.4 1.5 0.2]
 [5.4 3.9 1.3 0.4]
 [4.9 3.  1.4 0.2]
 [4.6 3.1 1.5 0.2]
 [4.6 3.4 1.4 0.3]
 [5.4 3.9 1.3 0.4]
 [5.1 3.7 1.5 0.4]
 [4.8 3.  1.4 0.1]
 [5.4 3.7 1.5 0.2]
 [5.7 4.4 1.5 0.4]
 [5.8 4.  1.2 0.2]
 [5.4 3.9 1.7 0.4]
 [5.4 3.9 1.7 0.4]
 [5.2 3.4 1.4 0.2]
 [5.8 4.  1.2 0.2]
 [4.8 3.  1.4 0.1]
 [5.4 3.7 1.5 0.2]
 [4.7 3.2 1.3 0.2]
 [4.6 3.6 1.  0.2]
 [5.8 4.  1.2 0.2]
 [4.8 3.4 1.9 0.2]
 [5.1 3.8 1.5 0.3]
 [5.  3.6 1.4 0.2]
 [5.1 3.7 1.5 0.4]
 [4.9 3.  1.4 0.2]
 [4.7 3.2 1.6 0.2]
 [5.1 3.5 1.4 0.3]
 [4.4 2.9 1.4 0.2]
 [4.9 3.1 1.5 0.1]
 [5.  3.5 1.3 0.3]
 [5.4 3.4 1.7 0.2]
 [4.8 3.  1.4 0.3]
 [5.  3.2 1.2 0.2]
 [4.9 3.1 1.5 0.1]
 [5.  3.4 1.6 0.4]
 [5.4 3.9 1.3 0.4]
 [4.7 3.2 1.6 0.2]
 [4.6 3.1 1.5 0.2]
 [4.8 3.  1.4 0.1]
 [5.  3.5 1.3 0.3]
 [4.5 2.3 1.3 0.3]
 [5.1 3.5 1.4 0.3]
 [5.  3.2 1.2 0.2]
 [4.9 2.4 3.3 1. ]
 [5.  3.4 1.5 0.2]
 [5.1 3.4 1.5 0.2]
 [5.7 2.8 4.5 1.3]
 [5.1 3.4 1.5 0.2]
 [5.1 3.5 1.4 0.2]
 [5.1 3.8 1.6 0.2]
 [6.7 3.1 4.4 1.4]
 [5.1 3.5 1.4 0.2]
 [4.8 3.  1.4 0.3]
 [4.5 2.3 1.3 0.3]
 [4.9 3.  1.4 0.2]
 [6.2 2.2 4.5 1.5]
 [5.7 4.4 1.5 0.4]
 [5.1 3.5 1.4 0.2]
 [5.2 2.7 3.9 1.4]
 [4.9 3.  1.4 0.2]
 [5.  3.4 1.6 0.4]
 [6.2 2.2 4.5 1.5]
 [5.1 3.5 1.4 0.2]
 [5.2 4.1 1.5 0.1]
 [5.1 3.8 1.6 0.2]
 [5.4 3.4 1.7 0.2]
 [4.4 3.  1.3 0.2]
 [6.1 2.8 4.  1.3]
 [5.  3.4 1.6 0.4]
 [5.2 3.4 1.4 0.2]
 [6.1 2.9 4.7 1.4]
 [5.8 4.  1.2 0.2]
 [6.  3.4 4.5 1.6]
 [6.4 2.9 4.3 1.3]
 [5.1 3.8 1.6 0.2]
 [5.8 4.  1.2 0.2]
 [5.7 2.8 4.5 1.3]
 [4.6 3.1 1.5 0.2]
 [5.1 3.3 1.7 0.5]
 [5.4 3.4 1.7 0.2]
 [5.1 3.5 1.4 0.3]
 [5.5 4.2 1.4 0.2]
 [5.4 3.  4.5 1.5]
 [6.2 2.2 4.5 1.5]
 [5.4 3.4 1.5 0.4]
 [4.6 3.4 1.4 0.3]
 [6.7 3.1 4.4 1.4]
 [5.2 4.1 1.5 0.1]
 [6.9 3.1 4.9 1.5]
 [5.4 3.4 1.7 0.2]
 [5.1 3.8 1.6 0.2]
 [5.  3.4 1.6 0.4]
 [5.8 2.6 4.  1.2]
 [4.3 3.  1.1 0.1]
 [5.1 3.5 1.4 0.3]
 [5.1 3.8 1.6 0.2]
 [5.8 2.7 5.1 1.9]
 [5.9 3.  4.2 1.5]
 [5.8 2.7 4.1 1. ]
 [6.3 3.3 4.7 1.6]
 [5.4 3.  4.5 1.5]
 [5.4 3.4 1.5 0.4]
 [6.3 3.3 4.7 1.6]
 [5.2 3.4 1.4 0.2]
 [5.7 2.6 3.5 1. ]
 [6.2 2.2 4.5 1.5]
 [6.3 3.3 6.  2.5]
 [5.6 2.7 4.2 1.3]
 [6.5 3.  5.5 1.8]
 [5.2 3.5 1.5 0.2]
 [5.1 3.8 1.6 0.2]
 [5.1 3.5 1.4 0.2]
 [7.7 3.8 6.7 2.2]
 [5.4 3.9 1.3 0.4]
 [7.7 3.8 6.7 2.2]
 [6.8 2.8 4.8 1.4]
 [6.3 2.7 4.9 1.8]
 [5.9 3.  4.2 1.5]
 [5.8 2.6 4.  1.2]
 [5.2 3.4 1.4 0.2]
 [5.8 2.7 5.1 1.9]
 [5.1 3.8 1.9 0.4]
 [6.3 3.3 4.7 1.6]
 [6.8 3.  5.5 2.1]
 [6.1 2.8 4.7 1.2]
 [4.9 2.5 4.5 1.7]
 [5.7 2.5 5.  2. ]
 [6.9 3.1 5.4 2.1]
 [5.5 3.5 1.3 0.2]
 [5.  2.3 3.3 1. ]
 [5.8 2.8 5.1 2.4]
 [5.5 4.2 1.4 0.2]
 [5.7 2.9 4.2 1.3]
 [4.9 3.  1.4 0.2]
 [5.  3.5 1.6 0.6]]
Prediction Input:
[[5.1 3.5 1.4 0.2]
 [5.1 3.5 1.4 0.2]
 [4.7 3.2 1.3 0.2]
 [4.7 3.2 1.3 0.2]
 [4.7 3.2 1.3 0.2]
 [5.1 3.5 1.4 0.2]
 [4.7 3.2 1.3 0.2]
 [5.  3.6 1.4 0.2]
 [4.9 3.  1.4 0.2]
 [4.6 3.1 1.5 0.2]
 [5.  3.4 1.5 0.2]
 [5.4 3.9 1.7 0.4]
 [4.6 3.4 1.4 0.3]
 [4.8 3.  1.4 0.1]
 [4.8 3.  1.4 0.1]
 [5.  3.4 1.5 0.2]
 [5.4 3.9 1.3 0.4]
 [4.9 3.  1.4 0.2]
 [4.6 3.1 1.5 0.2]
 [4.6 3.4 1.4 0.3]
 [5.4 3.9 1.3 0.4]
 [5.1 3.7 1.5 0.4]
 [4.8 3.  1.4 0.1]
 [5.4 3.7 1.5 0.2]
 [5.7 4.4 1.5 0.4]
 [5.8 4.  1.2 0.2]
 [5.4 3.9 1.7 0.4]
 [5.4 3.9 1.7 0.4]
 [5.2 3.4 1.4 0.2]
 [5.8 4.  1.2 0.2]
 [4.8 3.  1.4 0.1]
 [5.4 3.7 1.5 0.2]
 [4.7 3.2 1.3 0.2]
 [4.6 3.6 1.  0.2]
 [5.8 4.  1.2 0.2]
 [4.8 3.4 1.9 0.2]
 [5.1 3.8 1.5 0.3]
 [5.  3.6 1.4 0.2]
 [5.1 3.7 1.5 0.4]
 [4.9 3.  1.4 0.2]
 [4.7 3.2 1.6 0.2]
 [5.1 3.5 1.4 0.3]
 [4.4 2.9 1.4 0.2]
 [4.9 3.1 1.5 0.1]
 [5.  3.5 1.3 0.3]
 [5.4 3.4 1.7 0.2]
 [4.8 3.  1.4 0.3]
 [5.  3.2 1.2 0.2]
 [4.9 3.1 1.5 0.1]
 [5.  3.4 1.6 0.4]
 [5.4 3.9 1.3 0.4]
 [4.7 3.2 1.6 0.2]
 [4.6 3.1 1.5 0.2]
 [4.8 3.  1.4 0.1]
 [5.  3.5 1.3 0.3]
 [4.5 2.3 1.3 0.3]
 [5.1 3.5 1.4 0.3]
 [5.  3.2 1.2 0.2]
 [4.9 2.4 3.3 1. ]
 [5.  3.4 1.5 0.2]
 [5.1 3.4 1.5 0.2]
 [5.7 2.8 4.5 1.3]
 [5.1 3.4 1.5 0.2]
 [5.1 3.5 1.4 0.2]
 [5.1 3.8 1.6 0.2]
 [6.7 3.1 4.4 1.4]
 [5.1 3.5 1.4 0.2]
 [4.8 3.  1.4 0.3]
 [4.5 2.3 1.3 0.3]
 [4.9 3.  1.4 0.2]
 [6.2 2.2 4.5 1.5]
 [5.7 4.4 1.5 0.4]
 [5.1 3.5 1.4 0.2]
 [5.2 2.7 3.9 1.4]
 [4.9 3.  1.4 0.2]
 [5.  3.4 1.6 0.4]
 [6.2 2.2 4.5 1.5]
 [5.1 3.5 1.4 0.2]
 [5.2 4.1 1.5 0.1]
 [5.1 3.8 1.6 0.2]
 [5.4 3.4 1.7 0.2]
 [4.4 3.  1.3 0.2]
 [6.1 2.8 4.  1.3]
 [5.  3.4 1.6 0.4]
 [5.2 3.4 1.4 0.2]
 [6.1 2.9 4.7 1.4]
 [5.8 4.  1.2 0.2]
 [6.  3.4 4.5 1.6]
 [6.4 2.9 4.3 1.3]
 [5.1 3.8 1.6 0.2]
 [5.8 4.  1.2 0.2]
 [5.7 2.8 4.5 1.3]
 [4.6 3.1 1.5 0.2]
 [5.1 3.3 1.7 0.5]
 [5.4 3.4 1.7 0.2]
 [5.1 3.5 1.4 0.3]
 [5.5 4.2 1.4 0.2]
 [5.4 3.  4.5 1.5]
 [6.2 2.2 4.5 1.5]
 [5.4 3.4 1.5 0.4]]
Predicted Classes:
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1
 0 0 1 0 0 0 0 0 1 0 0 1 0 2 1 0 0 1 0 0 0 0 0 2 1 0]
Probabilities:
[[8.79681649e-01 1.20307538e-01 1.08131372e-05]
 [8.79681649e-01 1.20307538e-01 1.08131372e-05]
 [8.53796795e-01 1.46177302e-01 2.59031285e-05]
 [8.53796795e-01 1.46177302e-01 2.59031285e-05]
 [8.53796795e-01 1.46177302e-01 2.59031285e-05]
 [8.79681649e-01 1.20307538e-01 1.08131372e-05]
 [8.53796795e-01 1.46177302e-01 2.59031285e-05]
 [8.97323628e-01 1.02665167e-01 1.12050036e-05]
 [7.99706325e-01 2.00263292e-01 3.03825365e-05]
 [8.25383127e-01 1.74558937e-01 5.79356669e-05]
 [8.61839691e-01 1.38141399e-01 1.89095833e-05]
 [9.26986574e-01 7.30004562e-02 1.29693872e-05]
 [8.95064974e-01 1.04895775e-01 3.92506205e-05]
 [7.88177618e-01 2.11794929e-01 2.74526810e-05]
 [7.88177618e-01 2.11794929e-01 2.74526810e-05]
 [8.61839691e-01 1.38141399e-01 1.89095833e-05]
 [9.40906153e-01 5.90890027e-02 4.84421830e-06]
 [7.99706325e-01 2.00263292e-01 3.03825365e-05]
 [8.25383127e-01 1.74558937e-01 5.79356669e-05]
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