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.
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]
[8.95064974e-01 1.04895775e-01 3.92506205e-05]
[9.40906153e-01 5.90890027e-02 4.84421830e-06]
[9.21914602e-01 7.80675598e-02 1.78384021e-05]
[7.88177618e-01 2.11794929e-01 2.74526810e-05]
[8.92083069e-01 1.07910759e-01 6.17176870e-06]
[9.64535656e-01 3.54620850e-02 2.25877936e-06]
[9.28349898e-01 7.16491356e-02 9.66254924e-07]
[9.26986574e-01 7.30004562e-02 1.29693872e-05]
[9.26986574e-01 7.30004562e-02 1.29693872e-05]
[8.60034106e-01 1.39955486e-01 1.04082979e-05]
[9.28349898e-01 7.16491356e-02 9.66254924e-07]
[7.88177618e-01 2.11794929e-01 2.74526810e-05]
[8.92083069e-01 1.07910759e-01 6.17176870e-06]
[8.53796795e-01 1.46177302e-01 2.59031285e-05]
[9.26584671e-01 7.34068679e-02 8.46162713e-06]
[9.28349898e-01 7.16491356e-02 9.66254924e-07]
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