Machine Learning and Neural Networks - page 69
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5.3 Object Oriented Programming & Python Classes (L05: Machine Learning with Scikit-Learn)
5.3 Object Oriented Programming & Python Classes (L05: Machine Learning with Scikit-Learn)
Before we delve into the topic of machine learning with scikit-learn in the next lecture, let's take a moment to discuss object-oriented programming, specifically Python classes. Understanding classes will be highly relevant as scikit-learn heavily relies on object-oriented programming concepts. Towards the end of this video, I will demonstrate the implementation of K-nearest neighbors using the scikit-learn API, which is the approach scikit-learn uses for implementing estimators like classifiers.
So, let's begin by discussing Python classes. To better comprehend the scikit-learn API, it's important to understand the basics of classes. In simple terms, a class can be thought of as a blueprint for creating objects. Objects are instances of a class and can be visualized as different variations of the same cookie cutter shape used to make cookies. The class itself acts as the cookie cutter template, while the cookies represent the objects created from the class.
In Python, we define a class using the class keyword, followed by the class name. Inside the class, we define different class methods. Class methods are similar to functions, but they have a mandatory first argument called self, which refers to the object itself. This self argument allows us to access the attributes and methods of the object. In other words, it enables us to interact with the object's data and behavior.
In the context of the vehicle example, let's consider a simple naive vehicle class. This class represents different types of vehicles such as cars, motorcycles, or trucks. The class has various methods to define its behavior. The first method is the __init__ method, also known as the constructor. This method is automatically executed when a new object is created from the class. It accepts the self argument and any additional arguments required to initialize the object.
In the __init__ method, we define an attribute called horsepower, which is assigned the value provided as an argument. This attribute represents the horsepower of the vehicle. When a new object is created, it will have a horsepower attribute that can be accessed to retrieve the horsepower value.
In addition to the __init__ method, we can define other methods that modify the attributes of the object. For example, the tune_motor method doubles the horsepower attribute of the vehicle, simulating a motor tune-up. By calling this method on the vehicle object, its horsepower attribute will be modified accordingly.
Furthermore, we can define methods that return values based on the object's attributes. In the example, the horsepower_to_torque method calculates the torque value based on the object's horsepower and a provided RPM value. This method demonstrates how the object's attributes can be utilized to perform calculations and return useful results.
It's worth noting that in Python, there are conventions for indicating the visibility of methods. Methods with a single underscore prefix, such as _private_method, are considered private and are not intended for direct use by users of the class. However, users can still access and call these methods, although it's generally discouraged. Methods with a double underscore prefix, such as __very_private_method, are even more restricted and require a specific syntax to access them.
Additionally, Python supports class inheritance, allowing us to create child classes that inherit properties and methods from a parent class. This concept enables us to create specialized classes with additional attributes and behaviors while leveraging the existing functionality defined in the parent class. For example, we could create a specialized Car class that inherits from the Vehicle class and adds a number_of_wheels attribute specifically for cars.
To illustrate the concepts discussed, an example of a K-nearest neighbors classifier is provided. This implementation follows the scikit-learn API conventions and demonstrates the use of an estimator class in scikit-learn. Here's a simplified implementation:
In this example, KNNClassifier is a class representing a K-nearest neighbors classifier. The constructor takes a parameter k, which specifies the number of nearest neighbors to consider.
The fit method is used to train the classifier. It takes two arguments: X_train (the training data) and y_train (the corresponding labels). The method simply stores the training data and labels in the object's attributes for later use.
The predict method is used to make predictions on new data. It takes X_test (the test data) as an argument and returns the predicted labels for the test data. For each data point in X_test, the method calculates the distances to all data points in the training set using the _calculate_distance method. It then selects the k nearest neighbors and determines the majority class using the _majority_vote method. The predicted label is appended to the predictions list.
The _calculate_distance method is a private method (denoted by the leading underscore) that calculates the distance between two data points. This could be the Euclidean distance or any other distance metric suitable for the problem.
The _majority_vote method is another private method that determines the majority class among a set of neighbors. This can be done by counting the occurrences of each class label and selecting the label with the highest count.
This example demonstrates the basic structure of an estimator class in scikit-learn. Of course, scikit-learn provides a more sophisticated and optimized implementation of K-nearest neighbors in the KNeighborsClassifier class, but this simplified version illustrates the underlying principles.
5.4 Intro to Scikit-learn (L05: Machine Learning with Scikit-Learn)
5.4 Intro to Scikit-learn (L05: Machine Learning with Scikit-Learn)
In this relatively short video, the goal is to introduce machine learning with scikit-learn. Scikit-learn is a widely used machine learning library in Python that provides a comprehensive set of tools and algorithms for various machine learning tasks. While you may have seen scikit-learn before in the context of k-nearest neighbor (KNN) lectures, this video aims to take a step back and properly introduce the library.
After this short video, there will be a longer video that dives deeper into preparing the training dataset using scikit-learn. This will cover techniques and tools that make data preparation more convenient and efficient compared to traditional approaches.
In the subsequent video, we will explore some of the cool concepts in scikit-learn, such as combining pre-processing techniques, machine learning classifier fitting, and training, using scikit-learn pipelines. This allows for a more streamlined and efficient workflow.
Now, let's discuss machine learning with scikit-learn in more detail. Scikit-learn is widely considered the main machine learning library for Python due to its established reputation, large user base, and user-friendly nature. It is a well-designed library that offers a consistent and intuitive API. Scikit-learn is also actively maintained and regularly updated, with numerous contributors making it a robust and reliable choice for machine learning tasks.
It's important to note that scikit-learn is primarily focused on traditional machine learning techniques and is not meant for deep learning. Deep learning is a separate field with its own specialized libraries. For deep learning, other libraries like TensorFlow or PyTorch are typically used. However, for traditional machine learning tasks, scikit-learn is often the go-to library.
Scikit-learn has been around for a while, with its initial release dating back to 2007. Despite its age, it remains a popular and actively maintained library. It started as a Google Summer of Code project by David Cournapeau and gained contributions from many other developers over time. With more than 1,875 contributors on GitHub and nearly 150,000 users, it's evident that scikit-learn is a highly regarded library with substantial community support.
You can find the official website of scikit-learn, which includes documentation, tutorials, and other helpful resources. If you use scikit-learn in your research projects, it's good practice to cite the library as a reference, acknowledging the efforts put into its development.
To understand scikit-learn's Estimator API, let's delve into its main components. The Estimator API is used for supervised learning tasks and includes regressors for regression analysis and classifiers for classification tasks. When using scikit-learn, you typically initialize an estimator with specific hyperparameters, which are set in the constructor of the class.
The fitting process is crucial for an estimator. After initialization, you need to call the fit method, providing the training data (features) and their corresponding labels. The fit method trains the estimator on the given data, enabling it to make predictions later on. During the fitting process, certain attributes are assigned to the estimator, denoted by a trailing underscore, indicating that they are created during model fitting.
Once the model is fitted, you can use the predict method to make predictions on new data. The predict method takes the test data (with the same features as the training data) as input and returns the predicted labels.
Additionally, scikit-learn provides a score method that computes the performance of the model. For classifiers, it often represents the accuracy, while for regressors, it typically calculates the coefficient of determination (R^2 score). This method serves as a convenient way to evaluate the model's performance.
In addition to these core components, scikit-learn also offers a wide range of preprocessing techniques and utilities to enhance your machine learning workflow.
One important aspect of machine learning is data preprocessing, which involves transforming raw data into a suitable format for training a model. Scikit-learn provides various preprocessing modules that can handle tasks such as feature scaling, handling missing values, encoding categorical variables, and more.
For example, the StandardScaler class can be used to standardize features by subtracting the mean and scaling to unit variance. This is important when working with features that have different scales, as it helps algorithms to converge faster and produce more accurate results.
Another useful preprocessing technique is handling missing values. The SimpleImputer class provides strategies to replace missing values with suitable alternatives, such as using the mean, median, or most frequent values of the corresponding features.
When dealing with categorical variables, scikit-learn offers the OneHotEncoder and LabelEncoder classes. The LabelEncoder converts categorical labels into numerical values, while the OneHotEncoder transforms categorical features into a binary vector representation, enabling algorithms to work with categorical data effectively.
To streamline your machine learning workflow, scikit-learn provides a powerful tool called pipelines. A pipeline combines multiple preprocessing steps and a machine learning model into a single object, making it easier to manage and apply the entire workflow consistently.
Pipelines ensure that the same preprocessing steps are applied consistently to both the training and test datasets, avoiding data leakage and potential errors. They also simplify the deployment process, as you can save the entire pipeline object and reuse it on new data without worrying about individual preprocessing steps.
By using scikit-learn's pipeline functionality, you can chain together multiple preprocessing techniques, such as scaling, imputing missing values, and encoding categorical variables, with your desired machine learning model. This results in a more streamlined and efficient workflow, allowing you to focus on the core aspects of your machine learning project.
Scikit-learn supports a wide range of machine learning algorithms, including linear regression, logistic regression, support vector machines, decision trees, random forests, gradient boosting, and many more. Each algorithm is implemented as an estimator class with consistent methods such as fit, predict, and score.
To select the appropriate algorithm for your task, scikit-learn provides various tools for model selection and evaluation. These include techniques for cross-validation, hyperparameter tuning, and model evaluation metrics. Cross-validation helps assess the performance of your model by splitting the data into multiple train-test splits, training and evaluating the model on different subsets of the data.
Hyperparameter tuning involves finding the optimal values for the hyperparameters of a model, which are parameters that are not learned from the data but are set before training. Scikit-learn provides methods such as grid search and random search to automate the process of searching for the best hyperparameter values.
Model evaluation metrics, such as accuracy, precision, recall, F1 score, and area under the ROC curve, are crucial for assessing the performance of your model on different tasks. Scikit-learn offers a wide range of such metrics that can be easily computed and compared.
Scikit-learn is a powerful and popular machine learning library in Python that provides a wide range of tools and algorithms for various machine learning tasks. Its user-friendly API, extensive documentation, and active community make it an excellent choice for both beginners and experienced practitioners. Whether you need to preprocess your data, build pipelines, select models, or evaluate performance, scikit-learn has the tools you need to get the job done efficiently and effectively.
5.5 Scikit-learn Transformer API (L05: Machine Learning with Scikit-Learn)
5.5 Scikit-learn Transformer API (L05: Machine Learning with Scikit-Learn)
In the video, the presenter delves into the process of preparing a training data set using utilities from scikit-learn and introduces the transformer API, which is closely related to the estimator API discussed in a previous video. The estimator API is primarily used for supervised learning models such as classifiers and regression analysis models, while the transformer API is designed for data transformations.
The presenter begins by addressing the issues associated with random sub-sampling, which involves dividing a data set into two subsets, typically a training set and a test set. They explain that randomly dividing the data set can lead to changes in the distribution of class labels, resulting in a misrepresentation of classes in both the training and test sets. To overcome this challenge, the presenter suggests using stratified splitting, which ensures that the class distribution or proportions are maintained in the subsets. They proceed to demonstrate how to achieve stratified splitting using the train_test_split function from scikit-learn's model_selection sub-module.
Moving on, the presenter delves into the concept of data normalization, with a specific focus on two techniques: min-max scaling and standardization. Min-max scaling involves scaling a feature so that all its values fall within the range of 0 to 1. The presenter provides the formula for min-max scaling, which entails subtracting each value in the feature column by the minimum feature value and then dividing it by the difference between the maximum and minimum feature values.
In contrast, standardization involves transforming a feature to have a mean of zero and a standard deviation of one. The presenter explains the formula for standardization, which consists of subtracting each value in the feature column by the mean of the feature and then dividing it by the standard deviation of the feature. They mention that standardization is more commonly used in machine learning and is particularly useful for certain optimization algorithms.
To illustrate the practical application of min-max scaling and standardization, the presenter provides examples using toy feature columns. They emphasize that the choice between using the sample or population standard deviation does not significantly impact machine learning outcomes, as long as the features are centered and have approximately unit variance. Additionally, they highlight the importance of using the parameters (mean and standard deviation) computed from the training set to scale the validation and test sets since the validation and test sets represent unseen data.
The video proceeds to explore various techniques for data transformation and handling in machine learning. It acknowledges that for simple procedures like standardization, it is feasible to perform the transformations manually without relying on separate libraries or advanced techniques. However, for more complex transformations like feature selection, feature dimensionality reduction, and feature extraction, using tools such as the Transformer API can offer greater convenience and efficiency.
Next, the presenter focuses on dealing with categorical data. They introduce a toy dataset comprising three feature columns and one label column, emphasizing that categorical variables can be classified into two types: ordinal and nominal. Ordinal variables possess a specific order or hierarchy, while nominal variables do not. As an illustration, they highlight the "size" column as an ordinal variable, where t-shirt sizes like M, L, and XXL exhibit a clear order. To handle ordinal variables, the video recommends employing a mapping dictionary to transform the values into numeric representations.
On the other hand, the video presents class labels as an example of nominal categorical data. Since there is no inherent order among class labels, the presenter suggests using the label encoder to assign unique integer values to each label. The label encoder is fitted on the class label column to create a mapping dictionary, which is then used to transform the string labels into integer labels.
For nominal feature columns like "color," where no order is implied, utilizing the label encoder may introduce misleading information. In such cases, the video introduces one-hot encoding as a suitable alternative. This technique involves creating a new feature column for each distinct value in the nominal feature column and assigning 0s and 1s to indicate the presence or absence of a particular value. It is mentioned that dropping one of the resulting feature columns can eliminate redundancy without losing essential information.
The video briefly touches upon missing data and proposes some basic approaches to handle it. One strategy involves dropping rows or columns containing missing values if they occur randomly and are not indicative of a systematic issue. This can be accomplished using the dropna() method in the pandas library. Another approach is to impute missing data by filling in the gaps with statistical measures such as the mean or median, employing tools like the SimpleImputer transformer. However, the video cautions that imputation should be carefully considered, as it may introduce unintended biases.
Additionally, the video mentions the possibility of predicting missing values by treating the problem as a supervised learning task. In this scenario, the missing feature column can be considered the target variable, and the rows without missing data can be utilized as training data to fit a regression model.
The video provides a comprehensive overview of data transformation techniques, including the importance of preparing a training data set, the application of stratified splitting, and the normalization of data using min-max scaling and standardization. It further covers handling categorical data, distinguishing between ordinal and nominal variables, and introduces techniques such as mapping dictionaries, label encoders, and one-hot encoding. Additionally, the video briefly addresses missing data and outlines approaches such as dropping or imputing missing values, while also mentioning the possibility of predicting missing values through supervised learning.
5.6 Scikit-learn Pipelines (L05: Machine Learning with Scikit-Learn)
5.6 Scikit-learn Pipelines (L05: Machine Learning with Scikit-Learn)
Alright, everyone, we have finally reached the last part of Lecture 5, which is, in my opinion, the most interesting one: sacred learn pipelines. These pipelines are incredibly useful objects or classes that allow us to combine data processing and prediction steps seamlessly. To give you an overview, I'll present a flowchart of how pipelines work, although I won't delve into all the details just yet.
Think of a pipeline as an estimator with a transformer API. Similar to an estimator, a pipeline has a fit method that we can use on the training set. Internally, the pipeline goes through multiple fit and transform steps, and eventually arrives at the final fit step. Depending on what we define in our pipeline, it can perform various tasks such as data scaling (e.g., standardization or min-max scaling), dimensionality reduction, training a learning algorithm, and returning a predictive model that we can use on the test set.
When we call the predict method on the test set, similar to the estimator API, the pipeline internally uses the transform step to apply the same scaling that was performed on the training set. This ensures consistency in the data processing. Finally, the pipeline uses the fitted model to make predictions, such as returning class labels in the case of a classifier.
Let me illustrate this with a simple code example of a pipeline. Here, I'm creating a pipeline using the make_pipeline function from the sacred learn's sklearn.pipeline submodule. This is a convenient way to create pipelines. In this example, I'm building a basic pipeline consisting of a standard scaler and a K-nearest neighbor classifier. The standard scaler, as we discussed in the previous video, standardizes the data by making it have zero mean and unit variance. We initialize the standard scaler and the K-nearest neighbor classifier as components of the pipeline.
Once the pipeline is created, we can use the fit method to train the pipeline on the training data. Then, we can use the predict method to make predictions on the test set. When we call fit, the pipeline first applies the standard scaler to learn the mean and standard deviation from the training data. It then uses this information to scale the data, which is passed to the next step, the K-nearest neighbor classifier. The classifier receives the standardized training set and performs its learning algorithm.
During prediction, the same process happens, except there is no need for a fit step. The pipeline reuses the statistics learned during training to scale the test data and applies the learned classifier to make predictions.
To summarize, a pipeline allows us to chain together multiple objects, such as transformers and estimators, to create a cohesive workflow. It provides a convenient and efficient way to handle data processing and prediction steps in machine learning tasks.
Now, let's take a look at a pipeline in action by applying the simple holdout method for model selection. Model selection involves tuning hyperparameters to select the best model. In the holdout method, we split the data into a training set and a test set. Within the training set, we further split it into a subset for learning and another subset for validation, where we evaluate different models with various hyperparameter settings.
In scikit-learn, we can perform the holdout method and hyperparameter tuning using a method called grid search. Grid search involves creating a parameter grid that defines the hyperparameter combinations we want to evaluate. For example, in the case of k-nearest neighbors, we may consider different values for the number of neighbors (k) and the distance metric (p). Grid search iterates over all possible combinations, fitting the models on the training set and evaluating them on the validation set.
Although grid search is typically used with K-fold cross-validation, in this example, we'll focus on the holdout method. To apply the holdout method and grid search using a pipeline, we can utilize the GridSearchCV class from scikit-learn. This class allows us to define the parameter grid and the pipeline, and it handles the process of fitting the models and evaluating them.
Here's an example code snippet to demonstrate how to use GridSearchCV with a pipeline:
Next, we define the parameter grid param_grid that specifies the different hyperparameter combinations we want to evaluate. In this case, we vary the number of neighbors (n_neighbors) and the weight function (weights) for the K-nearest neighbor classifier. Note that the parameter names in the grid are prefixed with the name of the pipeline component followed by a double underscore (__).
We create a GridSearchCV object, passing the pipeline, parameter grid, and the desired number of cross-validation folds (cv). The GridSearchCV class automatically performs the grid search by fitting the pipeline on the training data and evaluating the models on the validation set.
After the grid search is complete, we can access the best parameters and best score using the best_params_ and best_score_ attributes of the GridSearchCV object. We print these values to see which hyperparameter combination yielded the best performance.
Finally, we evaluate the best model on the test set by accessing it from the best_estimator_ attribute of the GridSearchCV object and calculating the accuracy using the score method.
This example demonstrates how we can leverage pipelines and grid search to efficiently explore different hyperparameter settings and select the best model using the holdout method. By combining data processing steps and model training within a pipeline, we can streamline the machine learning workflow and easily experiment with different components and hyperparameters.
6.1 Intro to Decision Trees (L06: Decision Trees)
6.1 Intro to Decision Trees (L06: Decision Trees)
Finally, we are going to cover a new topic - decision trees. Decision tree algorithms can be considered as iterative top-down construction methods for classifiers and regression models. To keep the videos manageable and not too long, I have split the lecture into seven parts.
In the first part, we will provide a brief overview of decision trees, their conceptual representation, and introduce core terminology. Next, we will discuss recursive algorithms in the context of decision trees and briefly explain what recursive algorithms are. This understanding will help us analyze the runtime complexity (Big O) of decision trees.
Moving on, we will explore different types of decision trees. Currently, the second learner implements the CART (Classification and Regression Trees) decision tree algorithm developed by Leo Breiman. However, there are other algorithms like ID3 and C4.5, each with their own advantages and disadvantages. We will touch upon these different methods and might assign a homework exercise, possibly implementing the ID3 decision tree.
Within the lecture, we will also explore various splitting criteria used for decision trees. Currently, the second learner uses CART, which is typically based on Gini impurity. However, it allows for mixing and matching different splitting criteria. These splitting criteria refer to the functions or measures used to determine a good feature for splitting a decision tree. We will cover the Gini impurity and entropy criteria and discuss why they are preferred over the misclassification error when evaluating tree growth and split quality.
Once we have covered these details, we will delve into the improvements that can make decision trees more efficient in terms of computational runtime and classification accuracy. We will also address the challenge of overfitting and explore techniques to mitigate it.
Lastly, we will provide a code example that demonstrates working with decision trees in scikit-learn, a popular machine learning library. This practical example will help solidify your understanding of decision trees and their implementation.
Now, let's dive into the basic concept of a decision tree. While not strictly a machine learning problem, decision trees relate to everyday life scenarios involving decision-making. For instance, we can consider the decision of what to do at a given moment. If we have work to do, we may choose to stay inside and complete the work. Conversely, if we don't have any pending work, we can consider going outside, depending on the weather outlook.
Let's say, if it's sunny and not too cold, we may opt to go to the beach. However, if it's overcast, we might choose to go for a run instead. In the case of rain or snow, we could ask our friends about their availability. If our friends are busy, we might decide to stay inside and engage in activities like reading a book. On the other hand, if our friends are free, we can explore options like going to the movie theater, playing video games online, or having an online chat.
This basic outline represents a decision tree, where each node corresponds to a question or decision, and the branches lead to different options. The root node represents the first question or decision in the tree, while the leaf nodes are the final outputs. The internal nodes are the intermediary decisions that guide the flow of the tree.
Furthermore, the nodes in a decision tree can be categorized into binary or multi-category nodes. Binary nodes offer only two options, while multi-category nodes support more than two options. However, multi-category splits can be converted into binary splits, as discussed earlier.
Decision trees can also be interpreted as sets of rules, similar to if-else statements in programming. In fact, decision trees can be converted into rule sets, although the reverse conversion is not always possible. This characteristic of decision trees makes them highly interpretable and explainable, which is crucial in domains where interpretability and explainability are important, such as in legal and medical domains.
To construct a decision tree, we need a dataset that contains examples of inputs and corresponding outputs. Each example consists of a set of features (also known as attributes or predictors) and a target variable (the output we want to predict). The decision tree algorithm analyzes the features and their relationships to determine the best way to split the data and make predictions.
The process of building a decision tree involves selecting the best feature to split the data at each node. The goal is to find the feature that provides the most information gain or the best split in terms of classification accuracy. This process is typically performed recursively, starting from the root node and continuing down to the leaf nodes.
At each step, the algorithm evaluates different splitting criteria, such as Gini impurity or entropy. Gini impurity measures the probability of misclassifying a randomly chosen element from the set, while entropy measures the average amount of information required to identify the class label of a randomly chosen element from the set. These criteria help determine the purity of the resulting subsets after a split.
Once a feature is selected for splitting, the data is divided into subsets based on the possible feature values. This process is repeated recursively for each subset until a stopping criterion is met. The stopping criterion could be reaching a maximum tree depth, reaching a minimum number of samples in a leaf node, or achieving a certain level of purity in the leaf nodes.
To predict the target variable for a new instance using a trained decision tree, we start at the root node and traverse down the tree by following the appropriate branches based on the feature values of the instance. Eventually, we reach a leaf node, which provides the predicted output.
Decision trees have several advantages. They are easy to understand and interpret, making them a popular choice for visualizing and explaining the decision-making process. They can handle both numerical and categorical features, and they can be used for both classification and regression tasks. Decision trees are also robust to missing values and outliers, as they can handle them effectively during the splitting process.
However, decision trees also have some limitations. They can easily overfit the training data, resulting in poor generalization to unseen examples. This can be mitigated by using pruning techniques or employing ensemble methods like random forests or gradient boosting. Decision trees are also sensitive to small changes in the training data, and different splits can be generated for similar datasets. Additionally, decision trees may struggle to capture complex relationships between features.
In conclusion, decision trees are powerful and interpretable models that can be used for classification and regression tasks. They provide a clear decision-making framework by representing the data as a tree structure with nodes and branches. The choice of splitting criteria and stopping criteria, along with techniques to handle overfitting, play important roles in building accurate and robust decision trees.
6.2 Recursive algorithms & Big-O (L06: Decision Trees)
6.2 Recursive algorithms & Big-O (L06: Decision Trees)
In this video, our discussion revolves around recursive algorithms, which are closely connected to the concept of divide and conquer. Divide and conquer involves breaking down a problem into smaller subproblems, solving them individually, and then combining the solutions. Decision tree training and prediction, as well as various divide and conquer algorithms, are linked to this concept. Recursion is a common technique used to implement divide and conquer algorithms, although it's not the only approach.
To grasp the idea of recursion, let's examine an example of a recursive algorithm implemented in Python. For the sake of discussion, I have intentionally hidden the actual name of the function to encourage you to analyze its purpose before we delve into its details. I encourage you to take a moment to think about what this function might do. You can pause the video or even experiment with it in a Jupyter notebook to understand its behavior.
Assuming you've taken the time to analyze it, let's explore the function together. This particular function operates on Python lists. For instance, consider a list like [1, 2, 3]. The purpose of this function is to determine the length of an array or list. Let's examine how it works. The function takes an input, denoted as 'X' here, and checks two conditions. First, it checks if the list is empty. If it is, the function returns 0 because an empty list has a length of zero, which acts as the stopping condition. Otherwise, if the list is not empty, the function returns 1 and calls itself with a smaller input.
If this seems abstract, let's break it down step by step. Suppose we start with a complete list, such as [1, 2, 3]. Initially, the function checks if the list is empty, which it isn't. Consequently, it proceeds to the 'else' statement, where it returns 1 and recursively calls itself with a smaller input. In this case, the input becomes [2, 3] since we remove the first element from the original list. We repeat the process: the function returns 1 again, calls itself with the new input [3], and ultimately calls itself with an empty list.
Upon reaching the empty list, the function once again checks the 'if' condition, which is now true. As a result, it returns 0. When we evaluate the entire expression, we obtain the value 3. Therefore, this function computes the length of an array using recursion, where the function calls itself within its own definition. It's worth noting that while recursion is a conceptually elegant solution in computer science theory, it may not always be the most practical approach for implementation. In Python, recursion has limitations on the number of self-calls, and excessively large lists can cause a Stack Overflow error.
Moving on, let's explore another example that employs recursion. Here, we tackle a divide and conquer problem—sorting a list or array—using the quicksort algorithm. Similar to the previous function, quicksort uses recursion as a means of implementation. The algorithm employs a stopping condition, where the function returns the array as is if it contains fewer than two elements. Otherwise, the algorithm executes the main section of the code.
How does quicksort work? First, we select a pivot, typically the first element of the array. Then, we create two new lists: one to hold the elements smaller than the pivot and another for the elements larger than the pivot. We iterate through the array, excluding the pivot, and distribute each element to either the smaller or larger list based on its comparison with the pivot. Next, we recursively call quicksort on both the smaller and larger lists, using the pivot as the central element. Eventually, the
recursive calls will reach the stopping condition when the lists have fewer than two elements. At that point, the function simply returns the sorted lists.
Let's walk through an example to understand the process. Suppose we have an unsorted list [7, 2, 5, 1, 9, 3]. The quicksort algorithm will proceed as follows:
By recursively dividing the list into smaller sublists and sorting them, the quicksort algorithm efficiently sorts the entire list.
In conclusion, recursive algorithms play a crucial role in divide and conquer approaches. They break down problems into smaller subproblems and solve them individually, eventually combining the solutions to solve the original problem. Recursive functions call themselves within their own definition, repeatedly working on smaller inputs until reaching a stopping condition. However, it's important to consider the termination condition to avoid infinite recursion and ensure the algorithm converges to a solution.
6.3 Types of decision trees (L06: Decision Trees)
6.3 Types of decision trees (L06: Decision Trees)
In the previous videos, we focused on introducing decision trees and recursive algorithms. Now, let's delve into different types of decision trees. We'll explore how altering certain design choices can lead to different implementations of decision tree algorithms.
First, let's recap the generic decision tree algorithm in pseudocode, which we discussed in the previous video. We dealt with a binary classification problem, where the class labels were only 1 and 0. Our tree followed a binary structure, involving binary splitting. This means that each node was divided into exactly two child nodes. Additionally, we only considered binary features, where the feature values could be either 0 or 1.
However, as we demonstrated earlier using the decision tree visualization in scikit-learn, we can also utilize continuous features and convert them into binary splits. For example, we can select a feature, let's call it xj, and split it into two parts using a threshold, denoted as 't'. This splitting criterion can be defined as xj smaller than t or xj greater or equal to t, which can be represented as true or false. This allows us to perform a binary split even with continuous features, as we can adjust the decision threshold during the splitting process. You will have an opportunity to work on such a split in the homework.
Now, let's focus on the decision tree algorithm implementation. At each node in the decision tree, we consider a single feature, denoted as xj, where 'j' ranges from 1 to m, representing up to m features. When we split a parent node into two child nodes, let's say child zero and child one, we need to determine which feature to choose for splitting. For continuous features, we also need to consider the decision threshold, where we compare if xj is greater than or equal to a specific value 't'. Selecting the appropriate feature and threshold is crucial, and we require a measure of goodness to evaluate the quality of a split.
To summarize the generic tree growing algorithm, we select the feature that results in the largest information gain when the parent node is split. Information gain is a measure of goodness for a split. The higher the information gain, the better the split and the feature chosen, including its splitting threshold for continuous features. In the next video, we will discuss two commonly used measures to assess the goodness of a split: entropy and Gini impurity.
The algorithm follows certain stopping conditions. It stops if the child nodes are pure, meaning all the data points within a node have the same class label. Alternatively, it halts if the information gain is less than or equal to zero, indicating no improvement. Once we reach a pure node or fail to make progress, we stop growing the tree further.
After growing the decision tree, we can use it to make predictions. Suppose we have a tree with multiple levels, comprising parent and leaf nodes. To predict the class label of a new data point, we traverse the tree based on the feature values of the data point. At each node, we follow the corresponding branch based on the feature conditions until we reach a leaf node. For leaf nodes, we use the majority vote approach to determine the class label. This means we predict the class label that appears most frequently within that leaf node.
It's important to note that decision trees can handle both binary and multi-class classification problems. The pseudocode we discussed in the previous slides focused on binary classification, but decision trees can handle an arbitrary number of class labels. The majority vote approach is applicable regardless of the number of classes.
When developing decision tree algorithms, we encounter various design choices. One crucial question is how to split the nodes. We need to define the criterion for splitting and determine how to compare and evaluate different splits. The two commonly used measures for assessing the quality of a split are entropy and Gini impurity.
Entropy is a measure of the impurity or disorder within a node. It quantifies the uncertainty associated with the class labels of the data points in that node. The entropy of a node is calculated using the following formula:
Entropy(node) = - sum(p(i) * log2(p(i))), for all classes i
where p(i) represents the proportion of data points in the node that belong to class i. The entropy value ranges from 0 to 1, where 0 indicates a pure node (all data points belong to the same class) and 1 indicates maximum impurity (equal distribution of data points across all classes).
To evaluate the quality of a split, we compute the weighted average of the entropy values for the resulting child nodes. This is known as the information gain. The information gain is calculated as follows:
Information Gain = Entropy(parent) - sum((|Sv| / |S|) * Entropy(Sv)), for all child nodes v
where Entropy(parent) is the entropy of the parent node, |Sv| represents the number of data points in child node v, and |S| is the total number of data points in the parent node. The information gain measures the reduction in entropy achieved by splitting the node based on a particular feature.
Gini impurity is another measure used for assessing the quality of a split. It quantifies the probability of misclassifying a randomly chosen data point in a node if we assign a class label based on the distribution of class labels in that node. The Gini impurity of a node is calculated using the following formula:
Gini(node) = 1 - sum(p(i)^2), for all classes i
where p(i) represents the proportion of data points in the node that belong to class i. Similar to entropy, the Gini impurity value ranges from 0 to 1, where 0 indicates a pure node and 1 indicates maximum impurity.
To evaluate the quality of a split, we calculate the weighted average of the Gini impurity values for the resulting child nodes. This is known as the Gini impurity index. The Gini impurity index is calculated as follows:
Gini Index = sum((|Sv| / |S|) * Gini(Sv)), for all child nodes v
where |Sv| represents the number of data points in child node v, and |S| is the total number of data points in the parent node. The Gini index measures the reduction in Gini impurity achieved by splitting the node based on a particular feature.
Both entropy and Gini impurity are commonly used in decision tree algorithms, and the choice between them depends on the specific problem and data characteristics. In scikit-learn, you can select the criterion parameter to specify either 'entropy' or 'gini' when building decision tree models.
In the next video, we will delve deeper into these measures and discuss how to use them to determine the best splits in a decision tree algorithm.
6.4 Splitting criteria (L06: Decision Trees)
6.4 Splitting criteria (L06: Decision Trees)
In the video, the speaker delves into the intricacies of decision trees and introduces the concept of splitting criteria within this framework. Splitting criteria are essentially the criteria or measures employed to determine the most suitable feature for splitting a parent node into its child nodes. Generally, a dataset encompasses multiple features, denoted as x1, x2, x3, ..., xm, where j represents a value ranging from 1 to m.
The speaker emphasizes that at each node in the decision tree, a critical decision must be made regarding the feature to utilize for the splitting process. To identify the optimal feature, certain criteria or measures are defined to compare and evaluate the available features. The objective is to select a feature that yields a better split, thereby enhancing the predictive accuracy of the decision tree.
To illustrate the workings of decision trees, the speaker presents a toy dataset that consists of three features: x1, x2, x3, and a column denoting the class label, y. The class labels in this dataset are binary, taking on the values of either ones or zeros. The speaker notes that by employing only two features, it is feasible to achieve a training accuracy of 100% for this particular dataset.
Challenging the audience, the speaker prompts them to find two rules based on the three features that can lead to a training accuracy of 100%. They suggest pausing the video to contemplate the solution. Subsequently, the speaker reveals the solution, explaining that only x1 and x2 are the relevant and useful features, while x3 is seemingly random and does not contribute to the accuracy.
Moving forward, the speaker visually represents the dataset by plotting the values of x1 and x2 on a graph. The red dots on the graph represent data points belonging to class label one, whereas the blue squares represent data points labeled as zero. Based on the pattern observed in the data points, the speaker proceeds to create a split, resulting in the formation of a decision tree.
The initial split is based on x1 being greater than 5.5. This division partitions the data into two regions, one labeled blue and the other red. The speaker notes that while this split correctly classifies some data points, it also misclassifies others. The subsequent split is based on x2 being greater than 10.5. This further refines the classification process and ultimately leads to a decision tree that attains 100% training accuracy.
To enhance clarity, the speaker provides a clearer visual representation of the decision tree, elucidating the information associated with each node. Each node symbolizes a parent node that undergoes splitting, resulting in the creation of child nodes. For each node, the splitting value (represented by the feature value and threshold), entropy (a measure of information content), number of training examples (samples), class label distribution (venue), and majority class are displayed.
The decision tree is depicted in a hierarchical structure, with parent nodes giving rise to child nodes. The speaker underscores the significance of each node, highlighting that decision trees employ these nodes to make predictions based on the input features.
Lastly, the speaker mentions an alternative approach to achieve 100% training accuracy by solely utilizing feature two. They demonstrate a decision tree based on this alternative approach, showcasing how feature two can be split at values 7.5 and 10 to accurately segregate the data points into the desired classes.
In an ideal scenario where we have continuous features and correctly set thresholds, the aforementioned concepts would apply. Instead of using the variable "V," we would employ a variable xj, representing a continuous feature that is smaller or equal to a threshold. The second child node would symbolize values greater than the threshold. In this case, if we possess continuous features, a similar formula as before can be employed, but now we need to incorporate a threshold value into the function. We can compare the values to check if they are greater than or equal to the threshold.
Hence, if we have a continuous feature and a binary tree similar to CART (Classification and Regression Tree), we only need to sum over two child nodes instead of multiple ones. One child node corresponds to values greater than the threshold, while the other represents values smaller or equal to the threshold. This simplification is logical as we concentrate on the two possible outcomes based on the threshold value. Nevertheless, it is crucial to acknowledge that this explanation might appear dense, and a significant aspect that is still missing is the concept of entropy, which will be explored in the subsequent slides.
In this context, entropy pertains to the Shannon entropy, introduced by Claude Shannon in the realm of information theory. It diverges from the entropy employed in biophysics or thermodynamics. Shannon entropy serves as a metric for quantifying the impurity or disorder of child nodes within decision trees. It quantifies the amount of information conveyed by a discrete random variable that possesses two outcomes, akin to a Bernoulli distribution. In the Bernoulli distribution, a probability parameter denoted as p represents the likelihood of an event occurring.
Shannon defined information as the number of bits required to encode the value 1/p. In simpler terms, it gauges the level of certainty or uncertainty associated with an event. The number of bits needed can be calculated as the logarithm base 2 of 1/p. As the probability p increases, the number of bits required diminishes, indicating a higher degree of certainty. Conversely, as the probability approaches zero, the number of bits required escalates, implying a higher level of uncertainty.
To exemplify this concept, let's consider a few examples. If we have a probability of 0.5, the number of bits required would amount to 1. If the probability is 0, the number of bits required would be infinity, signifying absolute certainty. Conversely, if the probability is 1, the number of bits required would be 0, indicating complete uncertainty. Hence, the range of values for the term -log2(p) spans from minus infinity to 0.
Shannon entropy is computed as the average information across all possible events. It represents the weighted average of the information associated with each event, with the weights being the respective probabilities of the events. In the case of decision trees, the concept of entropy can be applied to gauge the impurity of a child node. If a node exhibits a well-balanced distribution of class labels, it will possess higher entropy, signifying greater impurity. Conversely, if a node displays a skewed distribution where one class dominates, it will possess lower entropy, indicating lower impurity. This notion aligns intuitively, as a node with higher impurity provides less information for classification purposes.
Entropy offers a means of measuring the impurity or disorder within child nodes of decision trees. It enables the evaluation of a node's purity based on the distribution of class labels. A node with higher entropy suggests a more diverse distribution, while a node with lower entropy indicates a more homogeneous distribution. By considering the entropy of child nodes, more informed decisions can be made when constructing decision trees.
6.5 Gini & Entropy versus misclassification error (L06: Decision Trees)
6.5 Gini & Entropy versus misclassification error (L06: Decision Trees)
In the previous video, we discussed the various splitting criteria that can be used to grow a decision tree. Now, let's delve into why two of the splitting criteria, namely Gini impurity and entropy, are preferred over the third criterion, misclassification error.
To recap, we have three impurity measures: entropy, scaled entropy (scaled by 0.5 for comparison with Gini impurity), and misclassification error. The shape of these impurity measures differs. Entropy appears as a concave function, while misclassification error has a sharp peak at 0.5 with linear slopes.
The question arises: why do we use entropy and Gini impurity instead of misclassification error for growing decision trees? This question applies not only to entropy but also to Gini impurity. In this video, we will focus on entropy, but the concept equally applies to Gini impurity.
Let's consider the information gain equation. We have an impurity function for the parent node, which represents the dataset D at the parent node. When we split this dataset based on feature values, we generate different child nodes. This concept applies to both categorical and continuous features. For continuous features, we can split by creating bins based on a threshold.
The impurity measure is used for both the parent and child nodes, and we sum them up while considering the size of the original dataset at the parent node and the current child node. Generally, if we choose one impurity measure for the parent, we keep it consistent for the child nodes as well.
In practice, we tend to avoid using misclassification error as it has a disadvantage, which we will discuss in this video. Let's briefly revisit how entropy and misclassification error are calculated. Entropy is calculated by summing up the product of the proportion of labels for each class and the logarithm of the proportion. On the other hand, misclassification error is based on the 0/1 loss, which measures the proportion of labels incorrectly classified.
Now, focusing on entropy versus misclassification error, one key difference is their shapes. Entropy is concave, whereas misclassification error is not. This difference influences why entropy is favored over misclassification error for growing decision trees.
To illustrate this, let's consider a simple toy dataset. At the parent node, we have 40 examples of class one and 80 examples of class zero. If we split the dataset based on feature values, we end up with child nodes that have different class distributions. We need to evaluate whether this split improves the purity of the nodes compared to the parent.
If we calculate the entropy for the parent and child nodes, we find that the entropy for child node two is lower than the parent, indicating improved purity. However, child node one is worse than the parent. On average, we need to determine if the split is beneficial.
To measure the quality of the split, we use the information gain, which considers the entropy values for the parent and child nodes. If the information gain is positive, it indicates a good split. In our example, the information gain is positive, indicating a favorable split based on entropy.
Now, let's examine the same scenario using misclassification error. The error for the parent, child node one, and child node two is calculated based on the proportion of misclassified labels. If we plug in these error values into the information gain formula, we find that the information gain is zero. A zero information gain implies that the split is not beneficial. Consequently, these nodes will not exist, and we would need to consider other features.
In conclusion, the main reason why entropy is preferred over misclassification error for growing decision trees is that entropy captures the uncertainty and disorder in the data more effectively. The concave shape of entropy allows for better differentiation between different class distributions and provides a more nuanced measure of impurity. On the other hand, misclassification error is a more simplistic measure that only considers the proportion of misclassified labels and does not capture the uncertainty and distribution of the classes.
The concave shape of entropy allows it to penalize both small and large class imbalances. It is sensitive to changes in class proportions, giving higher weights to more evenly distributed classes. This property makes entropy particularly useful when dealing with datasets that have imbalanced class distributions.
In contrast, misclassification error has a linear shape, with a sharp peak at 0.5. It treats all misclassifications equally and does not distinguish between different degrees of misclassification. This makes misclassification error more sensitive to class imbalances and less effective in scenarios with imbalanced datasets.
Furthermore, the difference in shapes between entropy and misclassification error impacts the decision tree learning process. Decision trees aim to find splits that maximize information gain or decrease impurity. Since entropy provides a more fine-grained measure of impurity, it allows decision trees to make more informed and accurate splits.
By using entropy as the impurity measure, decision trees are able to capture complex relationships between features and classes. They can handle both continuous and categorical features and can discover intricate patterns in the data.
In summary, entropy is preferred over misclassification error for growing decision trees because it captures the uncertainty and disorder in the data more effectively. Its concave shape allows for better differentiation between different class distributions, and it is more robust to imbalanced datasets. By using entropy, decision trees can make more informed splits and capture complex relationships between features and classes.
6.6 Improvements & dealing with overfitting (L06: Decision Trees)
6.6 Improvements & dealing with overfitting (L06: Decision Trees)
In the previous video, we discussed the different splitting criteria that can be used to grow a decision tree. Now, we will delve into why two of the splitting criteria, namely Gini impurity and entropy, are preferred over the third criterion, misclassification error.
To refresh our memory, let's recall the three splitting criteria: entropy, scaled entropy (for comparison with Gini impurity), and misclassification error. The shape of these impurity measures can be visualized as follows: the entropy is a concave function, represented by a tall black line; the scaled entropy is also concave and is obtained by multiplying the entropy by 0.5; and the misclassification error exhibits a sharp peak at 0.5 and linear slopes.
Now, the question arises: why do we prefer using entropy and Gini impurity instead of the misclassification error when growing decision trees? This question applies to both entropy and Gini impurity, but for simplicity, we will focus on entropy in this discussion.
Let's recap the information gain equation. We start with a parent node, denoted as D, and split this dataset based on feature values, creating different child nodes. The impurity function is used for both the parent and child nodes, and we sum up the impurity values while taking into account the size of the datasets. If we choose entropy for the parent node, we also use entropy for the child nodes. The same principle applies to Gini impurity.
Practically, we would prefer not to use the misclassification error because it has a disadvantage, which we will explore in this video. To understand this further, let's briefly review the formulas for entropy, Gini impurity, and misclassification error.
Entropy is computed by plugging in the proportions of class labels and summing them up. It uses logarithms and is represented by the formula where we sum over the classes and multiply the proportion of each class by the logarithm of that proportion.
Gini impurity, on the other hand, squares the class label proportion and subtracts it from one. It avoids using logarithms and is denoted as 1 minus the sum of squared class label proportions.
The misclassification error is based on the 0-1 loss, which measures the proportion of misclassified labels. For example, if we have a node with labels 001111, the majority vote would predict 0 because it is the majority class. However, considering the distribution, we would only be correct four out of six times, resulting in an accuracy of 4/6 or 66.6%. The error would be 2/6 or 33.3%.
To compare entropy and misclassification error, we observe that entropy is concave, while the misclassification error exhibits a sharp peak at 0.5 and linear slopes. This difference is crucial in understanding why entropy is preferred over misclassification error for growing decision trees.
To illustrate this, let's consider a simple toy dataset with a parent node containing 40 examples from class one and 80 examples from class zero. Assuming the best feature to split is x1, we divide the dataset based on whether the feature values are one or zero. We obtain two child nodes: child node one and child node two. Upon analyzing the class distribution, we find that child node two is more pure than the parent, while child node one is worse.
The key question is whether it is worth splitting on this feature or not. To determine this, we calculate the information gain using entropy. We compute the entropy values for the parent, child node one, and child node two. Comparing these values, we observe that child node two is better, while child node one is worse than the parent. By applying the information gain formula, we find that the information gain for this split is positive, indicating that the split improves the overall purity of the dataset.
Now, let's consider the misclassification error as the impurity measure instead of entropy. We calculate the misclassification error for the parent, child node one, and child node two. Comparing these values, we find that child node two has a lower misclassification error than the parent, while child node one has a higher misclassification error.
However, when we compute the information gain using the misclassification error, we encounter a problem. The information gain formula involves subtracting the weighted misclassification errors of the child nodes from the misclassification error of the parent. Since the misclassification error is a linear function, the information gain can be negative if the misclassification error of the child nodes is higher than that of the parent.
In our example, even though child node two has a lower misclassification error than the parent, the misclassification error of child node one is higher, resulting in a negative information gain. This means that using the misclassification error as the impurity measure would discourage the split, even though it improves the purity of one of the child nodes.
On the other hand, when we use entropy or Gini impurity as the impurity measures, the information gain will always be non-negative. Both entropy and Gini impurity are concave functions, which means that the impurity values of the child nodes are always smaller than or equal to the impurity value of the parent node. This ensures that the information gain will be positive whenever the split improves the purity of at least one child node.
By using entropy or Gini impurity as the impurity measures, decision trees can make splits based on the information gain, which provides a principled approach to growing the tree and improving its predictive power. The misclassification error, on the other hand, may lead to suboptimal splits and less accurate decision trees.
In summary, the preference for entropy and Gini impurity over misclassification error in decision tree algorithms is rooted in their mathematical properties. The concave nature of entropy and Gini impurity ensures that the information gain will be positive for splits that improve the purity of the child nodes, while the linear nature of the misclassification error can result in negative information gains and suboptimal splits.