Multiplication Rule, Factorials and Combinations

In your statistics class, you probably came across multiplication rule, factorials, and combinations, while solving probability questions. It's understandable that it can be confusing when to use which of these operations. Here's a brief overview along with examples to clarify their usage.

Multiplication Rule

Use the multiplication rule when you need to determine the total number of outcomes for a series of independent choices.

Example: Choosing 3 students from a classroom, with each student being either male (M) or female (F).

• Each choice has 2 possible outcomes (M or F).
• Total sequences = 2×2×2=8.

Example: Choosing a type of drink (coffee, tea, juice) and a type of pastry (croissant, muffin). Total combinations = 3×2 = 6.

Factorials

Use factorials when you need to count the number of ways to arrange a set of items where the order matters.

Example: Arranging 5 books on a shelf.

• The number of ways to arrange 5 books = 5! = 5×4×3×2×1=120.
  
  
• To understand this, imagine there are five slots on the shelf for the books. You have five choices for the first slot since any of the books can go there. After placing a book in the first slot, you have four remaining choices for the second slot, and so on.

Example: How many ways can 4 runners finish a race? Total arrangements = 4! = 24.

Combinations

Use combinations when you need to count the number of ways to choose items from a set without regard to the order.

Example: Choosing 3 students from a class of 10 to form a committee.

• The number of ways to choose 3 students from 10 = C(10, 3) = 10! / (3! x 7!) = 120

Example: Choosing 3 toppings from 5 available toppings for a pizza.

• The number of ways to choose 3 toppings from 5 = C(5, 3) = 5! / (3! x 2!) = 10

Summary

• Use the multiplication rule when you have independent choices.
• Use factorials when arranging items in a specific order.
• Use combinations when choosing items without regard to order.

 

Scenario: Planning a School Science Fair

Let's create a comprehensive scenario that involves a combination of the multiplication rule, factorials, and combinations to solve different aspects of the problem.

Imagine you are organizing a school science fair. You have to decide on several aspects:

1. Booth Assignment: Assign booths to different science projects. There are 3 science projects and 5 available booths. Each project can be assigned to any booth, but each project must have its own booth.
   
   
2. Volunteer Scheduling: Arrange volunteers to manage the event. There are 4 volunteers to manage different roles during the fair. Each volunteer will be assigned a specific role.
   
   
3. Selecting Judges: Choose a subset of teachers to judge the projects. There are 10 teachers, and you need to select 3 to be judges. The order of selection does not matter.

 The solution to this problem is provided below but you may want to first try yourself.

Solution:

1. Booth Assignment (Multiplication Rule):

For each project, you can choose any of the 5 booths. Since each project gets its own booth, we have:

• Project 1: 5 choices
• Project 2: 4 remaining choices (after assigning Project 1)
• Project 3: 3 remaining choices (after assigning Projects 1 and 2)

Total number of ways to assign the booths (using the multiplication rule): 5×4×3 = 60

2. Volunteer Scheduling (Factorials):

There are 4 volunteers and 4 different roles. There are 4 volunteers available for the first role. You can choose any one of them, so you have 4 choices. After assigning the first role, 3 volunteers remain. You can choose any one of them for the second role, giving you 3 choices. For the next two roles you will have 2 and 1 choices respectively. Therefore the number of ways to assign these roles is: 4! = 4×3×2×1 = 24

3. Selecting Judges (Combinations):

Out of 10 teachers, you need to select 3 to be judges. The order in which they are selected does not matter. The number of ways to choose the judges is: C (10, 3) = 120

This scenario demonstrates how different counting principles can be applied to various aspects of a single problem.

Intuition behind using Combination:

You might be rightfully wondering why we did not just use multiplication rule for finding the number of ways for the selection of judges. Let's explain the logic by considering how many choices you have at each step and why we divide by certain numbers.

1. Choosing the First Judge: You have 10 teachers to start with. So, there are 10 possible choices for the first judge.
   
   
2. Choosing the Second Judge: After selecting the first judge, 9 teachers remain. So, there are 9 possible choices for the second judge.
   
   
3. Choosing the Third Judge: After selecting the second judge, 8 teachers remain. So, there are 8 possible choices for the third judge.

Total Number of Selections (Considering Order):

If we consider the order in which we select the judges (which means we are considering different orders as different selections), we would multiply the number of choices at each step. This gives us the number of ways to select 3 judges considering different orders: 10×9×8 = 720

Adjusting for Order (Why We Divide):

However, in this scenario, the order in which we select the judges does not matter. For example, selecting Teachers A, B, and C is the same as selecting B, A, and C or C, B, and A. Each group of 3 judges can be arranged in 3×2×1 = 6 different ways (since there are 3 judges, the first can be any of the 3, the second any of the remaining 2, and the last is fixed).

To account for this and avoid counting the same group multiple times, we divide the total number of ordered selections by the number of ways to arrange 3 judges: 10×9×8 / 3×2×1= 120. So, there are 120 unique ways to choose 3 judges from 10 teachers when the order does not matter.

This division to handle non-unique groups is provided by the combination operator.

  

Ensemble Classifiers

Ensemble Techniques in Machine Learning

Ensemble techniques are powerful methods in machine learning that combine multiple models to produce a single, superior predictive model. By leveraging the strengths of different models, ensembles can achieve higher accuracy, better generalization, and increased robustness compared to individual models. These methods are particularly useful in dealing with complex datasets and improving the performance of machine learning models. Ensemble techniques work by training multiple models and then combining their predictions in various ways. This approach helps to reduce the risk of overfitting and increases the stability and reliability of the predictions. Below are some common ensemble techniques along with brief descriptions of how they work.

Bagging: Uses bootstrapped datasets to train multiple models and averages their predictions. Random Forest is a well-known example.

AdaBoost: Builds a sequence of models, each correcting the errors of the previous ones. The final model is a weighted sum of these models.

Gradient Boosting: Iteratively builds models that correct the errors of the previous models by optimizing a loss function.

XGBoost: An efficient implementation of Gradient Boosting with regularization to prevent overfitting and handle missing data.

Stacking: Combines multiple models via a meta-model trained on the outputs of the base models to improve performance.

Voting: Aggregates predictions from multiple models by averaging or majority voting to produce the final prediction.

Blending: Uses a holdout dataset to train a meta-model on the predictions of base models, similar to stacking.

In the remainder of this article, we will delve deeper into these techniques, explaining their underlying methods. Additionally, we will demonstrate their effectiveness in a binary classification problem using on a well-known Breast Cancer dataset from the UCI Machine Learning Repository.

Bagging in Machine Learning

Bagging, which stands for Bootstrap Aggregating, is an ensemble learning method designed to improve the accuracy and robustness of machine learning models. This technique involves training multiple models on different subsets of the training data, which are generated by random sampling with replacement. The predictions from these models are then combined, typically by averaging (for regression) or voting (for classification), to produce a final prediction.

Key Benefits of Bagging:

1. Reduces Variance: By averaging multiple models, bagging helps to smooth out the predictions, reducing the impact of individual model errors and thereby decreasing variance.

2. Increases Stability: It makes the model less sensitive to the noise in the training data, leading to more reliable predictions.

3. Improves Accuracy: The combination of multiple models often results in better overall performance compared to any single model.

Example: Applying Bagging on the Breast Cancer Dataset

To illustrate the power of bagging, let's use the Breast Cancer dataset, which is a well-known dataset for classification tasks. We'll compare the performance of a single Decision Tree model with a Bagging Classifier that uses Decision Trees as its base models.

Here is the Python code to perform this comparison:

import pandas as pdfrom sklearn.datasets import load_breast_cancerfrom sklearn.model_selection import train_test_split from sklearn.tree import DecisionTreeClassifier from sklearn.ensemble import BaggingClassifier from sklearn.metrics import accuracy_score   # Load the Breast Cancer dataset breast_cancer = load_breast_cancer() X = breast_cancer.data y = breast_cancer.target # Split the dataset into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # Train a single Decision Tree model dt = DecisionTreeClassifier(random_state=42) dt.fit(X_train, y_train) y_pred_dt = dt.predict(X_test) accuracy_dt = accuracy_score(y_test, y_pred_dt) # Train a Bagging Classifier with Decision Trees bagging = BaggingClassifier(base_estimator=DecisionTreeClassifier(), n_estimators=50, random_state=42) bagging.fit(X_train, y_train) y_pred_bagging = bagging.predict(X_test) accuracy_bagging = accuracy_score(y_test, y_pred_bagging) accuracy_dt, accuracy_bagging

Results:

- Single Decision Tree Model Accuracy: 94.15%

- Bagging Classifier Accuracy: 95.91%

As the results indicate, the Bagging Classifier achieved a higher accuracy compared to the single Decision Tree model. This demonstrates the effectiveness of bagging in enhancing the performance of machine learning models by combining the strengths of multiple models and mitigating their individual weaknesses.

Bagging is a powerful technique that can be applied to various machine learning tasks to achieve better and more stable results. By leveraging the diversity and collective wisdom of multiple models, bagging ensures that the final predictions are more accurate and reliable.

Random Forest is an ensemble learning method that falls under the category of Bagging (Bootstrap Aggregating) ensembles. It constructs multiple decision trees during training and combines their predictions to improve accuracy and control overfitting. Each tree in the forest is trained on a random subset of the data with replacement (bootstrap sampling) and considers a random subset of features when splitting nodes. The final prediction is made by aggregating the predictions of all the trees, typically using majority voting for classification tasks and averaging for regression tasks. This method leverages the strengths of bagging to create a robust and reliable predictive model.

 

Boosting in Machine Learning

Boosting is an ensemble technique that combines multiple weak learners to form a strong learner. The primary idea is to train models sequentially, each trying to correct the errors of its predecessor. This iterative process focuses on the difficult cases that previous models failed to predict correctly.

Key Boosting Techniques:

1. AdaBoost (Adaptive Boosting):

- Mechanism: AdaBoost assigns weights to each instance in the dataset. Initially, all instances have equal weights. In each iteration, the model focuses more on the instances that were misclassified by the previous model, adjusting the weights accordingly.

- Prediction: The final prediction is a weighted vote of the predictions from all the models.

2. Gradient Boosting Machines (GBM):

- Mechanism: GBM builds models sequentially, where each new model tries to minimize the residual errors made by the previous models.

- Optimization: It uses gradient descent, a method to find the minimum of a function by iteratively moving towards the steepest descent, to optimize the loss function, which measures how well the model's predictions match the actual outcomes.

3. XGBoost (Extreme Gradient Boosting):

- Mechanism: XGBoost is an optimized implementation of gradient boosting designed for speed and performance.

- Regularization: It includes regularization techniques, which add penalties to the model complexity to prevent overfitting, where the model performs well on training data but poorly on unseen data, making it more robust and scalable (able to handle larger datasets and more complex models efficiently).

Base Classifier

In these boosting techniques, the base classifier is typically a decision tree. Specifically, shallow decision trees, also known as decision stumps (trees with a depth of one) or trees with limited depth, are used. These weak learners are essential for the boosting process to be effective. While decision trees are the most commonly used base classifiers, other classifiers such as support vector machines, linear models, and neural networks can also be employed, but their use is less common and often more complex to implement.

Comparison of Base Classifier and Boosting Techniques

1. Unrestricted Decision Tree Example

from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.tree import DecisionTreeClassifier from sklearn.metrics import accuracy_score # Load the Breast Cancer dataset cancer = load_breast_cancer() X = cancer.data y = cancer.target # Split the dataset into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # Train a Decision Tree model without depth limit dt = DecisionTreeClassifier(random_state=42) dt.fit(X_train, y_train) y_pred_dt = dt.predict(X_test) accuracy_dt = accuracy_score(y_test, y_pred_dt) accuracy_dt

2. Decision Stump Example (Base Classifier for Boosting)

# Train a Decision Tree model with max_depth=1 dt_stump = DecisionTreeClassifier(max_depth=1, random_state=42) dt_stump.fit(X_train, y_train) y_pred_dt_stump = dt_stump.predict(X_test) accuracy_dt_stump = accuracy_score(y_test, y_pred_dt_stump) accuracy_dt_stump

3. AdaBoost Example

from sklearn.ensemble import AdaBoostClassifier # Train an AdaBoost Classifier with Decision Stumps ada = AdaBoostClassifier(base_estimator=DecisionTreeClassifier(max_depth=1), n_estimators=50, random_state=42) ada.fit(X_train, y_train) y_pred_ada = ada.predict(X_test) accuracy_ada = accuracy_score(y_test, y_pred_ada) accuracy_ada

4. Gradient Boosting Machines (GBM) Example

from sklearn.ensemble import GradientBoostingClassifier # Train a Gradient Boosting Classifier with shallow Decision Trees gbm = GradientBoostingClassifier(n_estimators=50, max_depth=3, random_state=42) gbm.fit(X_train, y_train) y_pred_gbm = gbm.predict(X_test) accuracy_gbm = accuracy_score(y_test, y_pred_gbm) accuracy_gbm

5. XGBoost Example

import xgboost as xgb # Train an XGBoost Classifier with shallow Decision Trees xgb_model = xgb.XGBClassifier(n_estimators=50, max_depth=3, random_state=42) xgb_model.fit(X_train, y_train) y_pred_xgb = xgb_model.predict(X_test) accuracy_xgb = accuracy_score(y_test, y_pred_xgb) accuracy_xgb

 

Results:

- Unrestricted Decision Tree Accuracy: 94.15%
- Base Classifier (Decision Stump) Accuracy:  61.11%
- AdaBoost Classifier Accuracy: 97.66%
- Gradient Boosting Classifier Accuracy: 95.91%
- XGBoost Classifier Accuracy: 95.91%

These results demonstrate that the boosting techniques (AdaBoost, Gradient Boosting, and XGBoost) can indeed achieve higher accuracy compared to a single unrestricted Decision Tree on a more challenging dataset. AdaBoost, in particular, outperformed the unrestricted Decision Tree, showing its effectiveness in enhancing model performance. This highlights the power of boosting methods in handling complex datasets and improving predictive accuracy.

 

Stacking (Stacked Generalization) in Machine Learning

Stacking, also known as stacked generalization, is an ensemble technique that combines multiple machine learning models to create a stronger predictive model. It works by training several base models (also called level-0 models) and then combining their predictions using a meta-model (also called level-1 model). The meta-model learns how to best combine the base models' predictions to improve overall performance.

Key Points:

1. Base Models (Level-0): These are the initial models that make predictions on the dataset. They can be any machine learning algorithms, such as decision trees, logistic regression, or neural networks. Multiple base models are used to capture different patterns in the data.

2. Meta-Model (Level-1): This model takes the predictions of the base models as input and learns how to combine them to produce the final prediction. The meta-model is usually a simple model like linear regression or logistic regression, but more complex models can also be used.

Implementation on the Breast Cancer Dataset

Dataset Information

The Breast Cancer dataset is a binary classification problem with features that are more complex and less likely to be perfectly fit by a single decision tree, making it suitable for demonstrating the power of stacking.

1. Base Models and Unrestricted Decision Tree

from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.tree import DecisionTreeClassifier from sklearn.svm import SVC from sklearn.neighbors import KNeighborsClassifier from sklearn.naive_bayes import GaussianNB from sklearn.metrics import accuracy_score # Load the Breast Cancer dataset cancer = load_breast_cancer() X = cancer.data y = cancer.target # Split the dataset into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # Train a Decision Tree model without depth limit dt = DecisionTreeClassifier(random_state=42) dt.fit(X_train, y_train) y_pred_dt = dt.predict(X_test) accuracy_dt = accuracy_score(y_test, y_pred_dt) # Train a Support Vector Machine model svm = SVC(probability=True, random_state=42) svm.fit(X_train, y_train) y_pred_svm = svm.predict(X_test) accuracy_svm = accuracy_score(y_test, y_pred_svm) # Train a K-Neighbors Classifier model knn = KNeighborsClassifier() knn.fit(X_train, y_train) y_pred_knn = knn.predict(X_test) accuracy_knn = accuracy_score(y_test, y_pred_knn) # Train a Gaussian Naive Bayes model nb = GaussianNB() nb.fit(X_train, y_train) y_pred_nb = nb.predict(X_test) accuracy_nb = accuracy_score(y_test, y_pred_nb) accuracy_dt, accuracy_svm, accuracy_knn, accuracy_nb

2. Stacking Example

from sklearn.ensemble import StackingClassifier from sklearn.linear_model import LogisticRegression # Define base models base_models = [     ('dt', DecisionTreeClassifier(random_state=42)),     ('svm', SVC(probability=True, random_state=42)),     ('knn', KNeighborsClassifier()),     ('nb', GaussianNB()) ] # Define meta-model meta_model = LogisticRegression() # Train a Stacking Classifier stacking = StackingClassifier(estimators=base_models, final_estimator=meta_model, cv=5) stacking.fit(X_train, y_train) y_pred_stacking = stacking.predict(X_test) accuracy_stacking = accuracy_score(y_test, y_pred_stacking) accuracy_stacking

 

Results:

- Decision Tree Accuracy: 94.15%
- Support Vector Machine Accuracy: 93.57%
- K-Neighbors Classifier Accuracy: 95.91%
- Gaussian Naive Bayes Accuracy: 94.15%
- Stacking Model Accuracy: 97.08%

These results demonstrate that the stacking model achieved a higher accuracy compared to each individual base model on the Breast Cancer dataset. This highlights the effectiveness of stacking in combining the strengths of multiple base models to improve predictive performance.

 

 

Voting Ensembles in Machine Learning

Voting ensembles combine the predictions of multiple models and make a final prediction based on a majority vote (for classification) or average (for regression). There are two main types of voting:

1. Hard Voting (Majority Voting): Each model in the ensemble makes a prediction (vote), and the final prediction is the one that gets the majority of the votes.

2. Soft Voting (Weighted Voting): Each model in the ensemble outputs a probability for each class, and the final prediction is made by averaging these probabilities (optionally weighted by model performance).

We provide an example of using both hard and soft voting strategies with the following classifiers as base models: Decision Tree, Support Vector Machine, K-Nearest Neighbors, and Gaussian Naive Bayes.

1. Base Models and Unrestricted Decision Tree

from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.tree import DecisionTreeClassifier from sklearn.svm import SVC from sklearn.neighbors import KNeighborsClassifier from sklearn.naive_bayes import GaussianNB from sklearn.metrics import accuracy_score # Load the Breast Cancer dataset cancer = load_breast_cancer() X = cancer.data y = cancer.target # Split the dataset into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # Train a Decision Tree model without depth limit dt = DecisionTreeClassifier(random_state=42) dt.fit(X_train, y_train) y_pred_dt = dt.predict(X_test) accuracy_dt = accuracy_score(y_test, y_pred_dt) # Train a Support Vector Machine model svm = SVC(probability=True, random_state=42) svm.fit(X_train, y_train) y_pred_svm = svm.predict(X_test) accuracy_svm = accuracy_score(y_test, y_pred_svm) # Train a K-Neighbors Classifier model knn = KNeighborsClassifier() knn.fit(X_train, y_train) y_pred_knn = knn.predict(X_test) accuracy_knn = accuracy_score(y_test, y_pred_knn) # Train a Gaussian Naive Bayes model nb = GaussianNB() nb.fit(X_train, y_train) y_pred_nb = nb.predict(X_test) accuracy_nb = accuracy_score(y_test, y_pred_nb) accuracy_dt, accuracy_svm, accuracy_knn, accuracy_nb

 

2. Voting Ensemble Example

from sklearn.ensemble import VotingClassifier # Define base models base_models = [     ('dt', DecisionTreeClassifier(random_state=42)),     ('svm', SVC(probability=True, random_state=42)),     ('knn', KNeighborsClassifier()),     ('nb', GaussianNB()) ] # Train a Voting Classifier (Hard Voting) voting_hard = VotingClassifier(estimators=base_models, voting='hard') voting_hard.fit(X_train, y_train) y_pred_voting_hard = voting_hard.predict(X_test) accuracy_voting_hard = accuracy_score(y_test, y_pred_voting_hard) # Train a Voting Classifier (Soft Voting) voting_soft = VotingClassifier(estimators=base_models, voting='soft') voting_soft.fit(X_train, y_train) y_pred_voting_soft = voting_soft.predict(X_test) accuracy_voting_soft = accuracy_score(y_test, y_pred_voting_soft) accuracy_voting_hard, accuracy_voting_soft

 

Results:

- Decision Tree Accuracy: 94.15%
- Support Vector Machine Accuracy: 93.57%
- K-Neighbors Classifier Accuracy: 95.91%
- Gaussian Naive Bayes Accuracy: 94.15%
- Voting Classifier (Hard Voting) Accuracy: 98.25%
- Voting Classifier (Soft Voting) Accuracy: 98.25%

These results demonstrate that both the hard voting and soft voting classifiers achieved higher accuracy compared to each individual base model on the Breast Cancer dataset. This highlights the effectiveness of voting ensembles in combining the strengths of multiple models to improve predictive performance.

 

Blending in Machine Learning

Blending is an ensemble technique that combines the predictions of multiple base models. The base models are trained on a training dataset, and their predictions are used as features to train a meta-model. The main difference between blending and stacking is that in blending, the meta-model is trained on a separate holdout set, not on the entire training set through cross-validation.

Key Points:

1. Base Models: Multiple base models are trained on the training dataset.

2. Holdout Set: A portion of the training data is set aside as a holdout set.

3. Meta-Model: The meta-model is trained on the predictions of the base models on the holdout set.

 

1. Base Models and Unrestricted Decision Tree

from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.tree import DecisionTreeClassifier from sklearn.svm import SVC from sklearn.neighbors import KNeighborsClassifier from sklearn.naive_bayes import GaussianNB from sklearn.metrics import accuracy_score # Load the Breast Cancer dataset cancer = load_breast_cancer() X = cancer.data y = cancer.target # Split the dataset into training, validation, and testing sets X_train, X_temp, y_train, y_temp = train_test_split(X, y, test_size=0.3, random_state=42) X_val, X_test, y_val, y_test = train_test_split(X_temp, y_temp, test_size=0.5, random_state=42) # Train a Decision Tree model without depth limit dt = DecisionTreeClassifier(random_state=42) dt.fit(X_train, y_train) y_pred_dt = dt.predict(X_test) accuracy_dt = accuracy_score(y_test, y_pred_dt) # Train a Support Vector Machine model svm = SVC(probability=True, random_state=42) svm.fit(X_train, y_train) y_pred_svm = svm.predict(X_test) accuracy_svm = accuracy_score(y_test, y_pred_svm) # Train a K-Neighbors Classifier model knn = KNeighborsClassifier() knn.fit(X_train, y_train) y_pred_knn = knn.predict(X_test) accuracy_knn = accuracy_score(y_test, y_pred_knn) # Train a Gaussian Naive Bayes model nb = GaussianNB() nb.fit(X_train, y_train) y_pred_nb = nb.predict(X_test) accuracy_nb = accuracy_score(y_test, y_pred_nb) accuracy_dt, accuracy_svm, accuracy_knn, accuracy_nb

 

2. Blending Example with Logistic Regression Meta-Model

import numpy as np from sklearn.linear_model import LogisticRegression # Generate predictions on the validation set using the base models val_preds = np.zeros((X_val.shape[0], 4)) base_models = [     ('dt', DecisionTreeClassifier(random_state=42)),     ('svm', SVC(probability=True, random_state=42)),     ('knn', KNeighborsClassifier()),     ('nb', GaussianNB()) ] for i, (name, model) in enumerate(base_models):     model.fit(X_train, y_train)     val_preds[:, i] = model.predict(X_val) # Train a Logistic Regression meta-model on the predictions of the base models on the validation set meta_model = LogisticRegression() meta_model.fit(val_preds, y_val) # Generate predictions on the testing set using the base models test_preds = np.zeros((X_test.shape[0], 4)) for i, (name, model) in enumerate(base_models):     test_preds[:, i] = model.predict(X_test) # Evaluate the blended model on the testing set y_pred_blend = meta_model.predict(test_preds) accuracy_blend_adjusted = accuracy_score(y_test, y_pred_blend) accuracy_blend_adjusted

 

Results:

- Decision Tree Accuracy: 94.15%
- Support Vector Machine Accuracy: 93.57%
- K-Neighbors Classifier Accuracy: 95.91%
- Gaussian Naive Bayes Accuracy: 94.15%
- Blended Model Accuracy: 97.67%

These results demonstrate that the blended model, using a larger holdout set and a Logistic Regression classifier as the meta-model, achieved a higher accuracy compared to the individual base models. This highlights the effectiveness of blending in combining the strengths of multiple models to improve predictive performance.

  

Comparison of Ensemble Techniques

The table below is based on general observations and experiences with these ensemble methods. It provides a qualitative comparison rather than quantitative data derived from the specific implementation on a specific dataset.

 

Method	Accuracy	Robustness	Computational Complexity	Ease of Implementation
Random Forest	High	High	Moderate	Easy
AdaBoost	Moderate	Moderate	Moderate	Easy
Gradient Boosting	High	High	High	Moderate
XGBoost	High	High	High	Moderate
Stacking	High	High	High	Moderate
Voting	Moderate	High	Low	Easy
Blending	High	High	High	Moderate

 

Accuracy: Gradient Boosting and XGBoost typically achieve the highest accuracy.

Robustness: Random Forest, Stacking, and Blending are generally robust to overfitting.

Computational Complexity: XGBoost and Gradient Boosting are computationally intensive. Random Forests and Voting are less so.

Ease of Implementation: Voting and Bagging are easiest to implement. Stacking and Blending are more complex due to the need for meta-models.

This comparative analysis helps identify the most suitable ensemble method based on the specific requirements of your project.

To provide a more detailed comparison, we present the results using the Breast Cancer dataset. Initially, due to the high cleanliness and accuracy of this data, all models performed exceptionally well, with Random Forest achieving the highest accuracy. However, when we introduced some noise into the dataset to increase the difficulty, AdaBoost emerged as the top performer. This highlights an important point: no single classifier is universally superior in all scenarios. Therefore, it is prudent to experiment with various techniques before selecting the final classifier for a specific application. We have also shown typical execution times for different algorithms executed on Google Colab. These times can vary depending on factors such as the software and hardware platform and the use of specific libraries.

 

Method	Accuracy (Original Data)	Accuracy (Noisy Data)	Execution Time (Seconds)
Random Forest	97.67%	95.32%	0.172581
AdaBoost	95.61%	96.49%	0.240852
Gradient Boosting	96.49%	94.74%	0.347579
XGBoost	96.78%	95.91%	0.137494
Stacking	97.08%	95.32%	4.486909
Voting	96.78%	95.32%	8.325362
Blending	97.67%	95.32%	14.68894