1. Introduction

2. Materials and Methods

2.1. Introduction

The Mahalanobis distance is a widely utilised statistic for detecting anomalous observations within clinical datasets, specifically those pertaining to blood glucose levels and audiometric thresholds. Within the domain of blood glucose monitoring, the utilization of Mahalanobis distance facilitates a thorough evaluation of multivariate associations, considering the interconnectedness of diverse clinical factors. This approach has significant utility in identifying aberrant patterns that could indicate suboptimal diabetes management, acute complications, or underlying medical issues. Likewise, within the realm of audiometric data analysis, the Mahalanobis distance metric considers the interdependence of hearing thresholds across various frequency ranges [7].

This metric serves to facilitate the identification of exceptional observations that may suggest the presence of distinct hearing diseases or unanticipated deviations from the established norm. The utilization of the Mahalanobis distance metric on clinical data enables healthcare workers to obtain a quantitative assessment of dissimilarity, hence facilitating the timely detection of atypical situations. This methodology not only enhances the accuracy of diagnosis but also provides guidance for tailored interventions, optimizing the provision of healthcare to patients and perhaps preventing severe complications in diseases such as diabetes or addressing urgent auditory concerns promptly [8].

2.2.1. Calculation of Mahalanobis distance metric

Mahalanobis distance metric can be applied for outlier detection and clustering in patient data. It helps identify individuals with unusual health profiles by considering correlations between different health variables, aiding in personalized medicine and anomaly detection.

The Mahalanobis distance is defined as the square root of the quadratic form. This is the formula for the Mahalanobis distance between a point x and a distribution with mean μ and covariance matrix Σ:

$$\mathrm{MD}(\mathrm{x},\mathsf{\mu },\mathsf{\Sigma })=\sqrt[2]{\left(\right(\mathrm{x}-\mathsf{\mu })\mathrm{ᵀ}\mathsf{\Sigma }⁻¹(\mathrm{x}-\mathsf{\mu }\left)\right)}$$

ᵀ denotes the transpose operation of the matrix

Σ⁻¹ is the inverse of the covariance matrix Σ

The Mahalanobis distance is a measure of how many standard deviations away a point is from the mean of the distribution. A point with a Mahalanobis distance of 0 is located at the mean of the distribution, while a point with a Mahalanobis distance of 1 is located 1 standard deviation away from the mean. The covariance matrix Σ describes the relationships between the variables in the distribution. The diagonal elements of Σ represent the variances of the variables, and the off-diagonal elements represent the covariances between the variables [9].

The Mahalanobis distance is a useful tool for detecting outliers. An outlier is a data point that is far away from the rest of the data. Outliers can be caused by measurement error, data entry errors, or fraud. The Mahalanobis distance can be used to identify outliers by flagging points that have a Mahalanobis distance that is greater than a certain threshold [10].

2.2.2. Using the `scipy.spatial.distance` module

The `scipy.spatial.distance` module in the SciPy library provides various distance metrics for measuring the dissimilarity between arrays or vectors. Here are some key aspects of this module:

Distance Metrics: The module includes a variety of distance metrics such as Euclidean distance, Manhattan distance, Chebyshev distance, Minkowski distance, and others. These metrics are used to quantify the dissimilarity or similarity between two sets of observations.

Functionality: The primary function for calculating distances is `scipy.spatial.distance.pdist`, which computes pairwise distances between observations in a condensed distance matrix format. Other functions like `scipy.spatial.distance.cdist` can calculate distances between two sets of observations.

Usage Example: Here's a simple example using Euclidean distance:

Metric Parameter: When using functions like pdist or cdist, you specify the distance metric through the 'metric' parameter. For example, euclidean, cityblock, chebyshev, mahalanobis etc. You can also define custom distance functions and use them with these modules.

Use Cases: Common use cases include clustering, classification, and data analysis, where understanding the dissimilarity or distance between data points is crucial.

The `scipy.spatial.distance` module is a powerful tool for performing various distance computations and is often used in conjunction with other scientific computing libraries like NumPy and scikit-learn [11].

2.2. Visualization of Mahalanobis distance using Python, Scientific python and Matplotlib modules

You can visualize Mahalanobis distance by drawing ellipses centered at the mean, with the shape of the ellipses determined by the covariance matrix. Scientific python Scipy has module Spatial distance, from which mahalanobis method can be implemented.

from scipy.spatial.distance import mahalanobis

This script generates random data from a bivariate normal distribution, calculates Mahalanobis distance for each data point, and plots the points with colors indicating their Mahalanobis distances. Outliers, having higher Mahalanobis distances, are visually distinct on the plot.

In the example I provided, I manually specified a covariance matrix to generate random data. The covariance matrix determines the relationships between different variables in a multivariate distribution. For simplicity, I chose a 2x2 covariance matrix, and a 0.7 threshold:

$$\mathrm{covariance}\mathrm{matrix}=\left[\begin{array}{cc}1& 0.7\\ 0.7& 1\end{array}\right]$$

In this matrix, the diagonal elements (1 in this case) represent the variances of the individual variables, and the off-diagonal elements (0.7 in this case) represent the covariances between the variables. Adjusting these values will change the spread and correlation of the generated data. In practice, you might obtain a covariance matrix from real-world data using methods like sample covariance estimation or other statistical techniques. The choice of the covariance matrix is crucial as it influences the shape and orientation of the generated data distribution [12].

Here's a simple example of implementing Mahalanobis distance in Python using random data and plotting the results with matplotlib:

Figure 1. Visualization of Mahalanobis distances by drawing ellipses that are centered at the mean, this figure uses random data, and 10 elliptic fronts to depict the thresholds set by the covariance matrix. Original illustration of authors available on GitHub.

Figure 1. Visualization of Mahalanobis distances by drawing ellipses that are centered at the mean, this figure uses random data, and 10 elliptic fronts to depict the thresholds set by the covariance matrix. Original illustration of authors available on GitHub.

3. Results

3.2. Application of Mahalanobis Distance Metric Analysis to Hearing Thresholds – Bivariate analysis

As a case study in actual clinical data, Hearing Thresholds of 100 patients were subjected to Mahalanobis Distance Metric Analysis to search for outliers. Each ear hearing was retained as a variable; hence the analysis was bivariate.

Figure 2. Visualization of dataset of actual hearing threshold of 50 cases who underwent Pure Tone Audiogram. Outliers detected by Mahalanobis distance metric analysis colored in red. original model and illustration of authors available on GitHub, Mahalanobis_Distance_Metric_Analysis repository.

Figure 2. Visualization of dataset of actual hearing threshold of 50 cases who underwent Pure Tone Audiogram. Outliers detected by Mahalanobis distance metric analysis colored in red. original model and illustration of authors available on GitHub, Mahalanobis_Distance_Metric_Analysis repository.

The application of Mahalanobis distance metric analysis to Pure tone audiometry threshold data yielded insightful observations. The method demonstrated a nuanced ability to discern subtle deviations in audiometric measurements, enabling the identification of outliers that might indicate potential hearing abnormalities. Notably, the Mahalanobis distance metric facilitated a more robust characterization of outliers, distinguishing between expected variations and anomalous data points.

Furthermore, the analysis revealed instances where traditional outlier detection methods might overlook certain patterns present in the audiometric data. The Mahalanobis distance metric, with its consideration of data covariance, proved particularly effective in capturing the multidimensional nature of Pure Tone Audiometry readings.

These observations underscore the method's adaptability and sensitivity, showcasing its potential as a valuable tool for improving the precision of outlier detection in healthcare datasets. The findings contribute to a deeper understanding of the nuances within audiometric data and demonstrate the broader applicability of Mahalanobis distance metric in healthcare analytics.

3.3. Application of Mahalanobis Distance Metric Analysis to Model of Blood sugar levels – Univariate analysis

Blood glucose concentration was modeled using SciPy module. A sample of size of about N=250 was used in running the stimulated for a mean of 100 mg%.

Figure 3. Visualization of dataset of stimulated blood sugar levels in mg% of N=250 with mean of 100. Outliers colored in red with hyperglycemia, suspected Diabetes mellitus detected by Mahalanobis distance metric analysis. Original model and illustration of authors available on GitHub, Mahalanobis_Distance_Metric_Analysis repository.

Figure 3. Visualization of dataset of stimulated blood sugar levels in mg% of N=250 with mean of 100. Outliers colored in red with hyperglycemia, suspected Diabetes mellitus detected by Mahalanobis distance metric analysis. Original model and illustration of authors available on GitHub, Mahalanobis_Distance_Metric_Analysis repository.

In the context of blood glucose data, each observation typically represents a univariate measurement, signifying a single feature—specifically, the blood glucose level. When applying Mahalanobis distance, each individual blood glucose measurement is treated as a distinct feature. For instance, measurements taken at different times or under various conditions would constitute separate features. Therefore, the Mahalanobis distance is calculated based on these univariate measurements, providing a means to assess the multivariate distance of each data point within the dataset.

4. Discussion

5. Conclusions

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