The RandomForestClassifier is a machine learning algorithm that is highly effective and known for its robustness in various tasks. It belongs to the ensemble learning family, which combines the strength of multiple models to enhance predictive performance. During training, this classifier builds many decision trees and merges their predictions by voting for classification tasks or averaging for regression tasks. One significant feature of the RandomForestClassifier is the introduction of randomness in its training process. It achieves this by selecting random subsets of features for each tree and training each tree on a bootstrapped sample of the data, a technique known as bagging.

This randomness helps prevent overfitting and increases the diversity among the individual trees, which leads to the overall model’s generalization capability. The hyperparameters of the RandomForestClassifier provide flexibility in tailoring the model to specific needs. Parameters like the number of trees (`n_estimators`), the depth of each tree (`max_depth`), and the number of features considered for each split (`max_features`) allow users to fine-tune the model for optimal performance on their datasets.

In practice, RandomForestClassifier is widely used for classification tasks because of its ability to handle complex relationships in data, resist overfitting, and provide robust predictions. Its versatility, ease of use, and effectiveness make it a popular choice for many machine-learning applications.

Exploring the dataset, the urban evolution of Boston with the Approved Building Permits dataset. This comprehensive collection of information provides details on construction activities across the city, ranging from minor additions like awnings to significant new constructions. The dataset includes permit numbers, applicant names, project valuations, and expiration dates, providing a vivid narrative of the construction landscape in Boston’s neighborhoods.

The data is useful for urban planners, real estate enthusiasts, and the public, fostering transparency and awareness about the ongoing transformations shaping Boston’s skyline. Each entry in the dataset represents more than just a construction permit; it tells the story of Boston’s neighborhoods. The latitude and longitude details add a spatial dimension, allowing users to map out the geographical distribution of these projects.

Whether it’s deciphering temporal trends, understanding the financial aspects of construction projects, or simply staying informed about the ebb and flow of development, this dataset provides a wealth of insights. From the bustling streets of Downtown to the serene corners of West Roxbury, each entry unveils a chapter in Boston’s ongoing narrative of growth and change. In essence, the Approved Building Permits dataset is a living document that encapsulates the dynamic rhythm of construction activities, providing both a historical record and a guide to the city’s future landscape.

Information Gain is a widely used concept in machine learning and decision trees. It helps measure how effective a feature is in classifying or predicting data and is commonly associated with the ID3 algorithm for constructing decision trees. The basic idea behind Information Gain is to determine how well a particular feature separates the data into different classes. This helps decide which feature should be used to split the data at a given node in a decision tree. Therefore, the feature with the highest Information Gain is chosen as the splitting criterion. Here’s a step-by-step explanation of how Information Gain is calculated:

1. Entropy (H): Entropy measures the impurity or disorder in a set of data. In the context of decision trees, it represents the uncertainty associated with classifying an instance in a given dataset.

2. Information Gain (IG): Information Gain is the reduction in entropy, or the amount of uncertainty removed from the dataset when a dataset is split by a specific feature.

3. Selection of Feature: The feature with the highest Information Gain is chosen as the splitting criterion at each node of the decision tree.

Pruning a decision tree is a technique used to make it simpler and prevent it from becoming too complex. This can happen when the tree is tailored to the training data, making it perform poorly on new, unseen data. The goal of pruning is to simplify the tree structure by removing unnecessary branches while keeping its predictive power. There are two types of pruning: pre-pruning and post-pruning.

Pre-pruning, also known as early stopping, involves setting limits on the tree-building process. For example, you can limit the maximum depth of the tree, the minimum number of required samples to split a node or the minimum number of samples allowed in a leaf node. These limits prevent the tree from growing too deep or becoming too specific to the training data.

Post-pruning, also known as cost-complexity pruning, involves building the full tree and then removing branches that do not significantly improve predictive performance. The decision tree is grown without limits first, and then nodes are pruned based on a cost-complexity measure. This measure considers the accuracy of the tree and its size. Nodes that do not contribute sufficiently to accuracy are pruned to simplify the model.

Decision tree is a powerful machine-learning algorithm that is widely used for both classification and regression tasks. It is a supervised learning method that helps to predict the outcome of a new data point based on the patterns learned from the training data. In the context of classification, a decision tree is a graphical representation of a set of rules that helps to classify the data into different categories. It is a tree-like structure where each internal node represents a feature or attribute, and each leaf node represents the outcome or class label.

The branches of the tree represent the decision rules that are used to split the data into subsets based on the values of the features. The primary goal of a decision tree is to create a model that can accurately predict the class label of a new data point. To achieve this, the algorithm follows a series of steps: selecting the best feature to split the data, creating a tree structure, and assigning class labels to the leaf nodes. The process starts at the root node, where the algorithm selects the feature that best splits the data into subsets. The feature selection is based on various criteria, such as Gini impurity and information gain.

Once the feature is selected, the data is split into subsets based on certain conditions, and each branch represents a possible outcome of the decision rule associated with the selected feature. The process is then applied recursively to each subset of data until a stopping condition is met, such as reaching a maximum depth or a minimum number of samples in a leaf node. Once the tree is constructed, each leaf node is associated with a class label, and when new data is presented to the tree, it traverses the tree based on the feature values of the data. The final prediction is the class label associated with the leaf node reached.

Random Forests is a popular ensemble learning technique used in machine learning for both classification and regression tasks. It belongs to the broader class of ensemble methods, which combine the predictions of multiple individual models to improve overall performance and robustness. Here are the key concepts associated with Random Forests:

Ensemble Learning: Ensemble learning combines the predictions of multiple models to produce a more accurate and robust prediction than any individual model. The idea is that by aggregating the predictions of multiple models, the weaknesses of individual models can be mitigated, leading to better overall performance.

Decision Trees: Random Forests are built on top of decision trees, simple models that make decisions based on a set of rules. Individual decision trees are considered weak learners, as they may overfit the training data.

Random Forests Construction: Random Forests use a technique called bagging, where multiple decision trees are trained on different random subsets of the training data. In addition to using different subsets of data, Random Forests also introduce randomness by considering only a random subset of features at each split in the decision tree.

Voting Mechanism: The final prediction is often determined by a majority vote among the individual decision trees for classification tasks. For regression tasks, the final prediction may be the average of the predictions from individual trees. Random Forests tend to be more robust against overfitting compared to individual decision trees. They provide a measure of feature importance based on the contribution of each feature to the overall model performance.

Hyperparameters: The number of decision trees in the forest is a crucial hyperparameter. The maximum depth of each decision tree is another important parameter. The number of features considered at each split influences the level of feature randomization.

Applications: Random Forests are widely used in various applications, including classification, regression, and feature selection. They are robust and work well in practice for a diverse range of datasets.

Limitations: Despite their advantages, Random Forests may not always outperform other algorithms, and their performance can be affected by noisy data or irrelevant features Random Forests are a powerful and versatile tool in machine learning, and their effectiveness often makes them a go-to choice for many practical applications.

Time series analysis involves examining patterns and dependencies within a sequence of data points collected over time. The process starts with collecting and visualizing time-stamped data to identify trends and outliers. An initial understanding of the data is gained through descriptive statistics such as mean and standard deviation.

Decomposition techniques break down the time series into components like trend, seasonality, and residual error. Ensuring stationarity, often through differencing, is crucial for many time series models. To understand the temporal dependencies in the data, autocorrelation and partial autocorrelation functions are used. Model selection involves choosing appropriate models such as ARIMA or SARIMA based on the characteristics of the time series. For more complex patterns, machine learning models like Random Forests or LSTM can be employed.

Evaluation metrics such as Mean Squared Error or Mean Absolute Error are used to assess the accuracy of the model on a test set. Once the model is trained, it can be used for forecasting future values. Continuous monitoring of model performance is essential, and periodic updates with new data ensure the model remains relevant. The process is dynamic, and the choice of techniques depends on the specific nature and goals of the time series analysis. Various tools and libraries such as pandas and statsmodels in Python or their counterparts in R facilitate the implementation of these techniques.

A decision tree is a popular algorithm in machine learning used for classification and regression tasks. The algorithm works by partitioning the input space recursively into regions and assigning a label or predicting a value for each region. The decision tree structure takes the form of a tree where each internal node represents a decision based on a specific feature, each branch shows the outcome of that decision, and each leaf node represents the final prediction or classification.

There are several key concepts related to decision trees. The root node is the topmost node in the tree, representing the best feature to split the data. Internal nodes are nodes that represent decisions based on features. They lead to branches corresponding to different outcomes. Branches are the edges connecting nodes that show the possible outcomes of a decision. Leaf nodes are terminal nodes that represent the final prediction or classification.

Splitting is the process of dividing a node into two or more child nodes. Entropy is a measure of impurity or disorder in a set of data. Decision trees aim to minimize entropy. Information gain is a measure of the effectiveness of a feature in reducing entropy. Features with higher information gain are preferred for splitting. Gini impurity is another measure of impurity used in decision trees. It measures the probability of misclassifying an element.

Pruning is the process of removing branches that do not provide significant predictive power. It helps prevent overfitting. The process of building a decision tree involves selecting the best feature to split the data at each node. This is done based on criteria like information gain or Gini impurity. The tree is constructed recursively until a stopping condition is met, such as reaching a maximum depth or having nodes with a minimum number of data points.

Decision trees have several advantages, including simplicity, interpretability, and the ability to handle both numerical and categorical data. However, they can be prone to overfitting, especially when the tree is deep. Techniques like pruning and setting a maximum depth can help mitigate this issue.

Today, I spent time reviewing the new dataset, this dataset is for Boston in 2013 and presents a comprehensive overview of key economic indicators. Tourism highlights the passenger traffic and international flight activity at Logan Airport, offering insights into the city’s connectivity and appeal to visitors. This information is crucial for understanding the dynamics of the local tourism industry. Shifting the focus to the hotel market and labor sector, the dataset provides a detailed examination of hotel occupancy rates, average daily rates, total jobs, and unemployment rates. These metrics offer a nuanced understanding of the city’s hospitality and labor landscapes, shedding light on factors influencing employment and economic stability.

Furthermore, the dataset digs into the real estate domain and explores approved development projects, foreclosure rates, housing sales, and construction permits. This section paints a different picture of the city’s real estate dynamics, capturing trends in housing demand, affordability, and development activities. Overall, the dataset proves to be a valuable resource for anyone seeking to grasp the multifaceted facets of Boston’s economy in 2013.

PCA is a powerful technique that enables data analysts and machine learning experts to reduce the complexity of high-dimensional data while retaining critical information. This dimensionality reduction method is achieved by transforming the data to a lower-dimensional space, making it easier to analyze. The process of PCA involves seven steps, starting with the standardization of data to ensure that all features contribute equally to the analysis. Next, the covariance matrix is calculated to determine how different features vary in relation to each other. Afterward, the eigenvectors and eigenvalues are computed, where the former represents the directions of maximum variance in the data, while the latter indicates the magnitude of variance in those directions. Sorting the eigenvectors by eigenvalues is a crucial step because it allows analysts to identify the most important directions of variance. The top k eigenvectors, where k is the desired number of dimensions for the reduced data, are then selected to form the principal components.

The selected eigenvectors are used to create a projection matrix, which serves as the tool to transform the original data into a new, lower-dimensional space. Finally, the original data is multiplied by the projection matrix to obtain the lower-dimensional representation of the data, which is often easier to analyze and interpret. PCA is a widely used technique for data visualization, noise reduction, and feature extraction, and it has practical applications in various fields, including image processing, facial recognition, and bioinformatics. Its most significant advantage is its ability to reduce the complexity of high-dimensional data, which can be challenging to analyze and interpret.

k-Nearest Neighbors (k-NN) is a method that finds nearby data points based on distance. In some cases, you might want to find all the points within a certain distance from a specific point, and that’s where ε (epsilon) comes in.

Epsilon (ε) is like a boundary or a distance limit. You use ε to say, Find all the points within ε distance of this point. It helps you define a neighborhood around your point.

If you set ε small, you’ll only find points very close to your point. If you set ε big, you’ll find farther away points. It’s a way to adjust how far you want to look for neighbors.

So, k-NN with ε helps you find points that are not just the k closest ones, but all the points that fall within a specific distance from your chosen point. It’s useful for tasks where you care about a certain range of proximity in your data.

• An epsilon (ε) value of 13 means that are considering points within a distance of 13 units from each other as part of the same neighborhood. This value defines how densely packed your clusters are in terms of proximity.
• A minPts value of 40 sets a minimum number of data points required within the ε-distance to form a cluster. In other words, to be considered a cluster, a group of points must have at least 40 neighbors within the 13-unit distance.

These parameter values indicate that are looking for relatively large and dense clusters in data. When running DBSCAN with these values, will identify clusters that meet these criteria.

DBSCAN is a clever way to group similar data points together, especially when we don’t know how many groups there are. It works by looking at how close points are to each other. If a point has many nearby points, it’s a core point. DBSCAN starts with one point and adds all its friends to a group. Then it moves to another point, and the process repeats until all points are assigned to groups or marked as loners. This helps find groups of different shapes and sizes in our data, even when there are some lonely, ungrouped points.

DBSCAN is great at handling messy data and doesn’t require us to guess the number of groups in advance. It’s like finding clusters of stars in the night sky, where some stars are closer to others, forming groups, while some are all by themselves.

1. Core Points: In DBSCAN, a “core point” is a data point that has at least a specified number of other data points (minPts) within a certain distance (epsilon, ε) from it. Core points are typically located within dense regions of a cluster.
2. Border Points: A “border point” is a data point within ε distance of a core point but does not have enough neighboring data points to be considered a core point. Border points are part of a cluster but are located on its periphery.
3. Noise Points: Data points that are neither core nor border points are classified as “noise points” or outliers. They do not belong to any cluster.

Parameters: DBSCAN has two primary parameters:

• ε (epsilon): The radius or maximum distance that defines the neighborhood around each data point. It determines which points are considered neighbors.
• minPts: The minimum number of data points required to form a cluster. A core point must have at least minPts neighbors to define a cluster.

This bar chart represents the number of shootings in different U.S. states. The height of each bar on the chart corresponds to the number of shootings in a particular state. California (CA) has the highest number of shootings, while Rhode Island (RI) has the lowest number of shootings. The code uses the value_counts method to count the occurrences of each state in the dataset and then plots this information in the form of a bar chart for visualization.

The pie chart: “With Signs of Mental Illness” and “Without Signs of Mental Illness.” The pie chart visually represents the distribution of shootings in the dataset, making it clear that a minority of shootings have signs of mental illness (20.9%) while the majority do not (79.1%). This visualization provides a quick and easy way to understand the prevalence of mental illness signs in the context of these shootings.

Elbow method is a technique used to find the best number of clusters in a K-means clustering algorithm. K-means is a method that groups data points into clusters. The elbow method helps you find the suitable value of k by examining how the within-cluster sum of squares changes as k increases.

To apply the elbow method in K-means clustering, follow these steps:

1. Select a range of possible values for the number of clusters you want to find, such as k values from 1 to a certain maximum number.

2. Run the K-means algorithm for each value of k in the chosen range.

3. Calculate the WCSS for each k, which is the sum of squared distances between data points and their assigned cluster centroids using the formula: WCSS(k) = Σ(Σ(||x – μ||^2)).

4. Create a plot with k on the x-axis and the corresponding WCSS on the y-axis.

5. Look for a point on the plot where the rate of decrease in WCSS starts to slow down. This point is called the “elbow” point.

6. Based on the elbow point in the plot, choose the best value of k for your K-means clustering. The elbow point is where the WCSS starts to level off, indicating that it’s a good compromise between having too few or too many clusters. Remember, the optimal k value is subjective and may require some domain knowledge or interpretation. The elbow method provides a useful heuristic, but it’s not always clear-cut, especially if the data doesn’t have a clear elbow in the WCSS plot. In those cases, we may need to use other evaluation metrics or techniques, such as silhouette analysis, to find the best value of k for your specific problem.

Silhouette analysis is a method of examining how well a clustering algorithm, like K-means, groups data points. It measures the distance between each data point and its own cluster, as well as the distance to neighboring clusters. A higher score means that the clusters are well-defined and separate from each other, while a lower or negative score indicates that the clustering may not be optimal. Silhouette analysis can be used to evaluate the quality of clustering without relying on a specific example. It is a useful tool to assess whether the clusters created are meaningful and distinct.

Simple histogram plot to show the distribution of ages in the dataset. The plot indicates that most data points are clustered around a central age value, forming a distribution somewhat similar to a normal distribution but with a slight right skew. This suggests that most individuals fall within a specific age range, with fewer individuals being older.

Simple bar chart that shows the relationship between the presence or absence of “Signs of Mental Illness” (1 for true, 0 for false) and the “Manner of Death.” It visualizes the count of incidents in each category.

• When “Signs of Mental Illness” is 0 (false), the most common “Manner of Death” is “shot” or “shot and Tasered,” with a higher count.
• When “Signs of Mental Illness” is 1 (true), the count of incidents for “shot” or “shot and Tasered” is lower compared to when there are no signs of mental illness.

When individuals do not show signs of mental illness, the most common manners of death are “shot” and “shot and Tasered.” This means that incidents where individuals are shot or shot and Tasered are more frequently observed when there are no signs of mental illness.

Hyperparameter tuning is a crucial aspect of the K-means clustering algorithm to ensure optimal performance on specific datasets. K-means has several hyperparameters that can be adjusted to optimize its performance. Here is a guide on how to tune hyperparameters for K-means:

Number of Clusters (K): The number of clusters we want to divide the data into is the most critical hyperparameter in K-means. Selecting the appropriate value for K is often the most challenging part. Techniques such as the Elbow Method or Silhouette Score can be used to determine an appropriate value for K. The goal is to find a point where increasing the number of clusters does not significantly reduce the cost function or improve the clustering quality.

Initialization Method: K-means is sensitive to the initial placement of cluster centers. Different initialization methods can lead to different results. Common initialization techniques are “random” and “k-means++.” The k-means++ method usually produces better results as it initializes cluster centers in a way that is more likely to lead to a good final solution.

Max Iterations (n_init): K-means uses an iterative process to converge to a solution. The algorithm stops when the cluster assignments no longer change significantly or after a certain number of iterations. This can adjust the n_init parameter, which determines how many times the algorithm runs with different initializations and chooses the best result. Increasing this number can help to  find a more stable solution.

Tolerance (tol): This parameter specifies when the algorithm should stop iterating. It is often set to a small value like 1e-4, meaning that if the change in cluster centers between iterations is smaller than this value, the algorithm stops.

Distance Metric: K-means uses Euclidean distance by default to measure the dissimilarity between data points and cluster centers. Depending on the data, we may consider using a different distance metric, such as Manhattan distance or cosine similarity.

Preprocessing: Scaling or normalizing data can impact the performance of K-means. It is often a good idea to preprocess data to have features with similar scales, especially when working with distance-based algorithms like K-means.

Parallelization: Some implementations of K-means offer the ability to parallelize the computation, which can significantly speed up the algorithm, especially when dealing with large datasets. That can adjust the number of CPU cores or threads used for parallelization.

Mini-batch K-means: If are dealing with large datasets, consider using mini-batch K-means. This variant of K-means can be faster but might require tuning additional parameters like batch size and learning rate.

K-medoids are the most typical representatives in a group of data. They are selected so that the distance from each data point to the chosen medoid is as small as possible. To understand medoids, let’s compare them to central points in K-Means. The relationship between central points and medoids is similar to that of averages and middle values in a list. The key difference is that medoids and middle values are always actual data points, whereas central points and averages may not be. The main difference between K-Means and K-Medoids is how they group data. K-Means cluster the data based on the distances between data points and central points, while K-Medoids cluster it based on the distances to medoids. Unusual data points can influence k-means, but K-medoids are more resilient and don’t rely on central points, making them better at handling outliers.

DBSCAN is a clustering algorithm that identifies groups of data points that are close to each other, even if they do not have a circular or square shape. It can also detect data points that do not belong to any group. The algorithm works by measuring the distance between data points. If the distance is less than or equal to a predetermined value ε, the data points may be considered part of the same group. Additionally, a minimum number of data points, MinPts, must be within the distance ε for a group to be formed. Based on these criteria, DBSCAN classifies data points as Core Points, Border Points, or Outliers. Core Points have enough data points within the distance ε to form a cluster. Border Points can be reached from a Core Point but have fewer data points within the distance ε. Outliers are data points that do not belong to any group and cannot be reached from any Core Point.

Clustering is like sorting or grouping similar things together. Clustering in the context of computers and data means using special techniques to do this automatically, without having to decide in advance how many piles or groups there should be.

It’s a way to find patterns or groups in data, which can be helpful for many things like organizing information, understanding customer behavior, or even organizing photos on your computer.

K-means Clustering:

• K-means is like sorting things into groups based on their similarity.
• We can decide how many groups we want, and the computer puts things in those groups.
• It keeps moving things around until it finds the best groups.

Hierarchical Clustering:

• Hierarchical clustering is like building a family tree for data.
• It starts with every item in its own family, and then it joins them together step by step, like merging families in a family tree.
• This is how things are related at different levels, from big groups to smaller ones.

Logistic regression is a math method used for saying “yes” or “no” things. Like, it can help tell if an email is spam or not, or if a person will buy something or not. In this, we use special numbers called “coefficients” to make predictions.

Here’s what these coefficients do:

1. Coefficient Values: For every piece of information we have, there’s a special number (let’s call it “β”) that goes with it. These numbers help us guess how likely the event is. They show how things change when we change the info slightly, keeping everything else the same.
2. Intercept (Bias): Besides the info numbers, we also have one more number called the “intercept” or “bias.” It’s like the starting point for our guessing. When all the info is zero, this number gives us the basic chance of the event happening.
3. Log-Odds: To make a guess, we find the log-odds. It’s a way to see how likely the event is. We add up the info numbers times the actual info we have, and then we add the intercept.Log-Odds = β₀ + β₁ * x₁ + β₂ * x₂ + … + βn * xn
4. Odds Ratio: We can use the odds ratio to compare how different info affects our guess. If the odds ratio is more than 1, it means the info makes the event more likely. If it’s less than 1, the event is less likely.Odds Ratio = exp(β)
5. Probability: Finally, to get a proper chance of the event happening, we use a special function (sigmoid function) to change the log-odds into a number between 0 and 1.Probability (p) = 1 / (1 + e^(-Log-Odds))In all of this:
   • β₀ is the starting point.
   • β₁, β₂, and so on are the info numbers.
   • x₁, x₂, and so on are the actual info we have.
   • “e” is a special number (around 2.71828) used in math.

Making a bar plot showing age and race involves using bars to display the average age for different racial groups. Each bar represents a group, and its height shows the average age for that group. This helps compare the average ages between different races, making it easier to see their age differences. white and unknown races are mostly shot at the same age.

Clustering algorithms are used in machine learning to group similar data points based on their features. There are different types of clustering algorithms, each with its own way of identifying clusters. K-means divides data into a set number of clusters by optimizing mean distances. Hierarchical clustering creates a hierarchy of clusters by merging or splitting them. DBSCAN identifies clusters by density, while Gaussian Mixture Models use Gaussian distributions to model data. Mean Shift finds dense regions without predefining the number of clusters. Spectral clustering uses eigenvectors to cluster data, and OPTICS ranks points according to density to discover clusters of various shapes. The algorithm choice depends on the data’s characteristics and the desired clustering outcomes.

Logistic regression is a versatile statistical technique utilized for analyzing datasets where the dependent variable involves two distinct outcomes, typically denoted as 0 and 1. The application of logistic regression is widespread, finding utility in fields ranging from healthcare to finance and beyond. The essence lies in its ability to model the probability of a binary event occurring based on one or more independent variables. This makes it invaluable for scenarios like predicting patient survival, customer purchase behavior, or whether a credit card transaction is fraudulent.

At its core, logistic regression transforms a linear combination of independent variables through the logistic function. The logistic function maps the sum of the products of the independent variables and their corresponding coefficients to a probability value between 0 and 1. This probability represents the likelihood of the event belonging to the positive class. The logistic function’s mathematical expression is essential for converting the linear prediction into a probability, enabling a clear interpretation of the prediction. During the training phase, logistic regression estimates the parameters that maximize the likelihood of the observed data given the logistic function. This process involves iteratively adjusting the coefficients to minimize the difference between the predicted probabilities and the actual binary outcomes in the training dataset. The resultant model encapsulates the relationships between the independent variables and the log odds of the event, providing a sound basis for making predictions.

Today, I spent time reviewing the new dataset, This data is about the Washington Post started tracking police-involved killings in the US by gathering information from various sources such as news reports, law enforcement websites, social media, and databases like Fatal Encounters and Killed by Police. They made sure to keep track of key details such as the race of the person killed, the circumstances surrounding the shooting, whether the person was armed, and whether they were experiencing a mental health crisis. In 2022, they have improved its database by adding the names of the police departments involved in each shooting, aiming to assess accountability at the department level. The death of Michael Brown in 2014 in Ferguson, Missouri, sparked the Black Lives Matter movement and increased attention to police accountability. The Post’s database specifically focuses on cases in which a police officer, while on duty, shoots and kills a civilian. The FBI and the Centers for Disease Control and Prevention also track fatal police shootings, but their data is incomplete. Since 2015, The Post has documented more than double the amount of fatal police shootings annually, compared to federal records. This disparity has grown, with the FBI tracking only a third of departments’ fatal shootings in 2021.

The post aims to create comprehensive records and update its database regularly with information on fatal shootings and individual cases.

Principal Component Analysis (PCA) is a method used in data analysis and machine learning to simplify complex datasets while retaining essential information. It starts by standardizing the data and then calculating a covariance matrix, revealing how features in the dataset are related. The eigenvectors and eigenvalues of this matrix point out the directions of maximum variance, called principal components. These components are arranged by the amount of variance they represent, enabling the selection of a smaller set of key components that effectively summarize the data. PCA is beneficial for tasks like visualization, noise reduction, and feature extraction, achieved by projecting the original data into a lower-dimensional space defined by these principal components.

On the other hand, the t-test is a statistical tool used to determine if there’s a significant difference between the means of two groups. There are two main types: the one-sample t-test compares a single sample mean to a known or hypothesized population mean, while the two-sample t-test examines the difference between the means of two independent samples. The t-test computes a t-statistic, which is then compared to a critical t-value from a t-distribution based on the desired level of significance and degrees of freedom. If the t-statistic surpasses the critical t-value, it suggests a significant difference between the means of the groups. This makes the t-test a valuable tool for hypothesis testing in various scientific and analytical domains.

Box-Cox transformation is a widely used mathematical transformation used in statistics and data analysis. It’s primarily used to stabilize variance and make the data more normally distributed, both of which are essential assumptions for many statistical techniques.

The purpose of the Box-Cox transformation is to stabilize variance and make the data closer to a normal distribution. This can be particularly useful for data analysis techniques that assume normally distributed data, such as linear regression. To determine the optimal value of  $\lambda$,  typically search for the value that maximizes the log-likelihood function, which measures how well the transformed data fit a normal distribution. This is often done using numerical optimization techniques.

Yeo-Johnson transformation is a mathematical formula used to transform data in statistics and data analysis. It’s an extension of the Box-Cox transformation and is designed to handle a broader range of data, including both positive and negative values, as well as zero.

The Yeo-Johnson transformation can handle a wider range of data than the original Box-Cox transformation, which is limited to positive data or data with positive shifts. The Yeo-Johnson transformation can be applied to both positive and negative data, making it more versatile in practical applications . Choosing the appropriate value of $\lambda$ is critical to obtaining a meaningful transformation. Typically, this is done through a search for the optimal $\lambda$ that maximizes the log-likelihood function, or by using other criteria such as minimizing the mean squared error. 

Outlier detection involves identifying unusual or abnormal data points that significantly deviate from the majority of the dataset. Various approaches like statistical methods, machine learning algorithms, or domain-specific rules can be used to pinpoint these outliers. Statistical techniques such as Z-score or interquartile range are commonly applied, whereas machine learning models like Isolation Forest or One-Class SVM utilize the data’s underlying patterns to detect outliers. Once identified, outliers can be either removed from the dataset or handled through imputation or transformation, depending on the context and purpose of the analysis.

In the process of outlier removal, careful consideration is essential to maintain data integrity and ensure that the removal or treatment aligns with the overall goals of the analysis. Removing outliers can help enhance the accuracy and reliability of statistical analysis or machine learning models by reducing the influence of erroneous or extreme data points. However, it’s crucial to strike a balance and use domain knowledge to determine the appropriate course of action, as excessive outlier removal may lead to loss of valuable information or distortion of the true data distribution.

Isolation Forest is an efficient and effective algorithm for outlier detection. It operates by isolating anomalies within a dataset by constructing a random forest of decision trees. The key idea is that anomalies are likely to require fewer random splits to be isolated, making them stand out in the tree structure. By measuring the average path length to isolate each data point, Isolation Forest identifies outliers as those with shorter average path lengths, representing their higher degree of separability from the majority of the data. This method is particularly adept at handling high-dimensional data and offers quick and accurate outlier detection, making it widely used in various domains including fraud detection, network security, and anomaly detection.

Variable interaction and nonlinearity are fundamental concepts in statistical analysis and machine learning that help model complex relationships between variables.

Variable interaction occurs when the effect of one variable on the outcome is influenced by another variable. Understanding and accounting for interactions are essential for building accurate predictive models, as neglecting them can lead to biased estimates and erroneous conclusions about the relationships between variables.

Nonlinearity refers to a situation where the relationship between predictor variables and the response is not linear. In linear relationships, changes in predictors lead to proportional changes in the response. However, in nonlinear relationships, this proportionality does not hold. Detecting and modeling nonlinear relationships are critical for creating accurate models.

Dealing with Nonlinearity

1. Utilize nonlinear models such as decision trees, random forests, neural networks, etc., that can capture complex relationships.
2. Apply feature engineering techniques, like adding polynomial features or transformations, to better represent nonlinearities.
3. Leverage kernel methods to implicitly map data into higher-dimensional spaces where linear models can capture nonlinear patterns.
4. Use ensemble methods that combine various models to capture different aspects of the nonlinear relationship, resulting in more accurate predictions.

Cross-validation and bootstrap are two commonly used techniques in statistics and machine learning for assessing the performance of models and estimating parameters. 

Cross-validation is a resampling technique used to evaluate machine learning models on a limited data sample. It involves splitting the dataset into training and validation sets, where the model is trained on the training set and then evaluated on the validation set. This process is repeated multiple times, with different subsets of the data used for training and validation. Common types of cross-validation include k-fold cross-validation, leave-one-out cross-validation, and stratified k-fold cross-validation.

• K-fold cross-validation: The data is divided into k subsets, and the model is trained and validated k times, with each subset used as the validation data once.
• Leave-one-out cross-validation (LOOCV): Each observation is used as the validation set once while the rest of the data forms the training set.
• Stratified k-fold cross-validation: Data is divided into k folds, ensuring that each fold has a similar distribution of the target variable.

Bootstrap is a resampling technique that involves random sampling with replacement from the original dataset to create new samples of the same size. This technique is often used for estimating the sampling distribution of a statistic like mean or variance, or for constructing confidence intervals.

The key steps in bootstrap resampling are:

• Sample with replacement: Randomly select observations from the original dataset with replacement to create a bootstrap sample of the same size as the original dataset.
• Calculate statistic: Calculate the statistic of interest (e.g., mean, median, standard deviation) on the bootstrap sample.
• Repeat: Repeat the above steps a large number of times to create a bootstrap distribution of the statistic.

Bootstrap is particularly useful when the underlying distribution of the data is unknown or when you want to estimate the sampling distribution of a statistic without making strong assumptions about the data.

Cross-validation is a technique used in machine learning and statistics to evaluate the performance of a predictive model. It involves partitioning a dataset into subsets, training the model on some of these subsets, and evaluating its performance on the remaining subset.

K-fold cross-validation is a specific approach to cross-validation where the original dataset is divided into K equal-sized folds. The model is trained on K-1 of these folds and validated on the remaining fold. This process is repeated K times, with each fold used as the validation set exactly once. 

• Split into K equal parts.
• For training and validation: Train on K-1 parts, and validate on the first part.
• Repeat this K times with a different validation part each time.
• To evaluate performance: Measure accuracy, mean squared error, etc.
• To get the overall performance: Average the performance metrics from all K iterations.

In our project, we used the Kruskal-Wallis H test which is a non-parametric statistical test used to determine whether there are statistically significant differences between the medians of three or more independent variables.

• The statistic value is a measure of the overall difference in ranks among the groups being compared. The calculated H statistic is approximately 896.813.
• The p-value is a measure of the probability of observing the data, assuming the null hypothesis is true. It tells us how likely it is to observe such an extreme H statistic by chance alone if there were no actual differences between the groups.
• The p-value in this output is approximately 1.817×(10)^−195, an extremely small value close to zero.
• The small p-value suggests strong evidence against the null hypothesis. With such a small p-value, it’s safe to reject the null hypothesis and conclude that there are significant differences in medians among the groups involved in the Kruskal-Wallis H test.

We combined information from three different sets: one for inactive individuals, another for people with diabetes, and a third for those dealing with obesity.

Histogram Plot for Inactive and Diabetics Data: When we looked at the data for inactive individuals and those with diabetes, the histogram plot showed a “normal distribution.” This means that the data forms a bell-shaped curve, and the mean and standard deviation can fully describe the characteristics of this distribution. It’s like how most people’s heights cluster around an average height, with some variability.

Histogram for Obesity Data: However, when we examined the data for obesity, the histogram looked different. It was a “left-skewed histogram,” meaning it tilted more towards the left side. The mean was typically less than the median. The longer tail on the left side of the graph indicated that there were some unusually low values in the data, which is common in obesity data.

By understanding these different distributions, we can analyze data more accurately, especially when the data follows a normal distribution.

A t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups. It is commonly used when you have a small sample size and want to infer if the difference between the groups is likely due to chance.
There are two types:

1. Independent Samples T-Test: Compares means of two separate groups.
2. Paired Samples T-Test: Compares means of paired data.

• Null Hypothesis (H0​): No significant difference.
• Alternative Hypothesis (H1​): Significant difference.

The test calculates a statistic (t) based on sample means and standard deviations. If the p-value is less than a chosen significance level, we reject the null hypothesis, implying a significant difference. Check assumptions like normality and equal variances.

The Crab Molt model looks at measurements of crabs before and after molting. Our main goal was to predict the crab size before molting based on their post-molt measurements. Using a simple model, we got an impressive R-squared value of 0.98, indicating the model predicts well based on the data. And also analyzed the pre-molt and post-molt data. They were similar in distribution, with a small mean difference of about 14.7 units.

Multiple linear regression is a statistical method that is an extension of simple linear regression in which more than one independent variable (X) is used to predict a single dependent variable (Y). The predicted value of Y is a linear transformation of the X variables such that the sum of squared deviations of the observed and predicted Y is a minimum. The computations are more complex, however, because the interrelationships among all the variables must be considered in the weights assigned to the variables. The interpretation of the results of a multiple regression analysis is also more complex for the same reason. With two independent variables the prediction of Y is expressed by the following equation:

Y’_i = b₀ + b₁X_1i + b₂X_2i

This transformation is similar to the linear transformation of two variables discussed in the previous chapter except that the w’s have been replaced with b’s and the X’_i has been replaced with a Y’_i.

The “b” values are called regression weights and are computed in a way that minimizes the sum of squared deviations

Multiple linear regression for Obesity, Inactivity, and diabetics. The relationship between two independent variables, “inactive” and “obesity”, and a dependent variable “diabetics”. R-squared, is a statistical measure used in regression analysis to evaluate the goodness of fit of a regression model. It provides an indication of the proportion of variance in the dependent variable that can be explained by the independent variables in the model. An R-squared value of 0.34 in multiple linear regression indicates that the independent variables included in the model collectively explain about 34% of the variability in the dependent variable. 

Residuals
In statistical analysis, residuals are the differences between observed values and the corresponding values predicted by a statistical model. These differences are fundamental for understanding how well a statistical model fits the observed data and for diagnosing the appropriateness of the model assumptions.

Breusch-Pagan test for heteroskedasticity
The Breusch-Pagan test is a statistical test used to determine whether the variance of the errors in a regression model is constant or varies with respect to the predictor variables. In regression analysis, heteroscedasticity refers to the unequal scatter of residuals. Specifically, it refers to the case where there is a systematic change in the spread of the residuals over the range of measured values.

Heteroscedasticity is a problem because ordinary least squares (OLS) regression assumes that the residuals come from a population that has homoscedasticity, which means constant variance. When heteroscedasticity is present in a regression analysis, the results of the analysis become hard to trust. One way to determine if heteroscedasticity is present in a regression analysis is to use a Breusch -Pagan Test.

Multi Linear Regression for Obesity, Inactivity, and Diabetics is a generalization of simple linear regression, in the sense that this approach makes it possible to evaluate the linear relationships between a response variable and several explanatory variables.

P-Value:

A p-value, a “probability value,” is a statistical measure used in hypothesis testing to determine the strength of evidence against a null hypothesis. The null hypothesis is a statement that there is no significant effect or relationship in a given set of data.

• For example, the Null hypothesis (H0) would be that the coin is fair, meaning it has an equal chance of landing on heads or tails (probability of each = 0.5).  An alternative hypothesis (Ha) would be that the coin is biased towards tails, meaning it’s more likely to land on tails than heads.

A linear regression plot on two variables, Obesity, and Diabetic, helps you visualize the relationship between these two variables and understand how they are related linearly. The formula y = c + b*x, where y = estimated dependent variable score, c = constant, b = coefficient, and x = score on the independent variable. Regression analysis draws a line through these points that minimizes their overall distance from the line. More specifically, least squares regression minimizes the sum of the squared differences between the data points and the line. Following the practice in statistics, the Y-axis displays the dependent variable, % DIABETIC. The X-axis shows the independent variable, which is the % OBESE. The Pearson correlation coefficient is used to measure the strength of a linear association between Obesity and Diabetes. The values have a moderate positive correlation.

array([[1.        , 0.38532577],
       [0.38532577, 1.        ]])

The Breusch-Pagan test is a statistical test used to detect heteroskedasticity in a regression model. Heteroskedasticity occurs when the variance of the residuals in the regression model is not constant across all levels of the independent variables, violating one of the assumptions of linear regression.

The histogram for obesity is a left-skewed histogram is known negatively skewed histogram. It’s a graphical representation of data in which the tail of the distribution extends to the left. The mean is typically less than the median. The longer tail on the left side indicates some lower extreme values in the data.

The histogram for diabetes is a normal distribution. The mean and standard deviation of a normal distribution fully describe characteristics. When data is approximately normal, these tests tend to be more accurate.

QQ plot for Diabetes and obesity compares the quantiles from the dataset to the quantiles expected from a theoretical distribution. These pairs of quantiles are then plotted against each other on a scatterplot, with the theoretical quantiles on the x-axis and the observed quantiles on the y-axis. QQ plot for diabetes is S-shaped which indicates skewness.QQ plots for obesity points are consistently below the line, which suggests that obesity data has lighter tails.