Mahalanobis Distance

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What Is Mahalanobis Distance?

Mahalanobis distance is a statistical measure used to determine the similarity between two data points in a multidimensional space. It is instrumental in data analysis, pattern recognition, and classification tasks. This distance metric takes into account the covariance structure of the data, which makes it suitable for situations where the variables are correlated.

It can help investors construct portfolios that are well-diversified across different assets. By considering the covariance between asset returns, it can identify which assets are most similar or dissimilar to each other. This information helps in selecting assets that provide the best risk-return trade-off for a given level of portfolio risk.

What Is Mahalanobis Distance?

Mahalanobis distance is a multivariate measure that quantifies the dissimilarity between two data points in a multidimensional space, considering the covariance structure of the data.
It takes into account the correlations between variables, making it suitable for datasets where variables are interrelated.
It is scale-invariant, meaning it is not affected by the scaling of variables, making it versatile for different units of measurement.
Thresholds for identifying outliers or anomalies can be customized based on the specific application, allowing flexibility in analysis.

Mahalanobis Distance Explained

Mahalanobis distance is a mathematical measure that quantifies the dissimilarity between two data points in a multivariate dataset. named after the Indian statistician Prasanta Chandra Mahalanobis. It's a versatile tool for data analysis and pattern recognition, originating in the field of statistics during the early 20th century.

Prasanta Chandra Mahalanobis, an influential Indian scientist, introduced this concept in the 1930s. He played a pivotal role in establishing the Indian Statistical Institute (ISI) and contributed significantly to the development of statistical methods in India. Mahalanobis recognized the limitations of using Euclidean distance for multivariate data analysis, especially when dealing with correlated variables. To address this, he proposed a distance metric that incorporates the covariance structure of the data. His work aimed to develop statistical tools to aid in diverse fields, including agriculture, economics, and social sciences. The Mahalanobis distance became one of his most enduring contributions to statistics.

The Mahalanobis distance formula considers the mean vector and the covariance matrix of the dataset to calculate the distance between data points. It standardizes the data, transforming it into a space where variables are uncorrelated and have unit variances.

Formula

The Mahalanobis distance formula measures the number of standard deviations that are one data point away from the mean of the dataset in a multidimensional space. The formula is as follows:

Mahalanobis Distance (D) = √((X - μ)' Σ^(-1) (X - μ))

Where:

D is the Mahalanobis distance between the two data points.
X represents the vector of values for the data point one wants to measure the distance.
μ (mu) is the mean vector of the multivariate dataset, containing the mean values of each variable.
Σ (Sigma) is the covariance matrix of the dataset, which captures the relationships and variances between variables.
Σ^(-1) is the inverse of the covariance matrix.

Here's a step-by-step breakdown of the formula:

Subtract the Mean: (X - μ) calculates the difference between the values of the data point one is interested in (X) and the mean vector (μ). This step standardizes the data by centering it around the mean.
Covariance Matrix Inverse: Σ^(-1) is the inverse of the covariance matrix. It accounts for the correlations between variables and their variances. Inverting the covariance matrix allows us to give more importance to variables that have higher variances or are more relevant to the analysis.
Matrix Multiplication: (X - μ)' Σ^(-1) (X - μ) performs matrix multiplication between the transposed (X - μ) vector and Σ^(-1), and then the result is again multiplied by (X - μ). This step computes the weighted squared differences between the data point and the mean, with the weights determined by the covariance matrix.
Square Root: Finally, taking the square root of the result gives the Mahalanobis distance, which represents how far the data point X is from the mean, considering the correlations and variances of the variables in the dataset.

Examples

Let us understand it through the following examples.

Example #1

Let's consider an imaginary scenario where a bank is using Mahalanobis distance for fraud detection. The bank has a dataset of customer transactions, including information such as transaction amount, location, time of day, and customer history.

The bank calculates the Mahalanobis distance for each transaction from the mean transaction profile of legitimate customer behavior. If a transaction's Mahalanobis distance is significantly higher than the average, it may be a potentially fraudulent transaction. This approach helps the bank identify unusual patterns of behavior that might indicate fraud, even if the transaction amount is not extraordinarily high.

Example #2

In a report from CNBC dated February 5, 2020, a study conducted by researchers from MIT and State Street suggests a concerning economic outlook, highlighting the application of statistical tools like Mahalanobis distance in financial analysis. The study indicates a 70% chance of a recession occurring within the next six months. This finding raises alarm bells as the global economy faces uncertainties and potential headwinds.

The research takes into account various economic indicators and financial market data, including sophisticated analytical methods like Mahalanobis distance, to arrive at this prediction. Mahalanobis distance, a statistical measure, factors in the covariance structure of economic variables, offering insights into data similarity and dissimilarity in multidimensional space.

Factors such as trade tensions, geopolitical instability, and slowing global economic growth have contributed to the heightened recession risk, as highlighted by Mahalanobis's distance-based analysis.

How To Interpret?

Here's how to interpret Mahalanobis distance:

Magnitude of Distance: The Mahalanobis distance is a positive value that quantifies the dissimilarity between a data point and the mean of the data set. A smaller distance indicates that the data point is closer to the mean and is more similar to the dataset as a whole. In contrast, a more considerable distance signifies more significant dissimilarity.
Standard Deviations: One can think of the Mahalanobis distance in terms of standard deviations. A Mahalanobis distance of 1 corresponds to a distance of 1 standard deviation away from the mean in each dimension. More considerable distances represent deviations that are multiple standard deviations away.
Thresholds: To interpret the Mahalanobis distance effectively, one needs to establish a threshold. The choice of threshold depends on the specific application and the desired level of sensitivity to outliers.
Multivariate Analysis: Mahalanobis distance is instrumental in multivariate analysis because it accounts for correlations between variables. If a data point has a considerable Mahalanobis distance from the mean, it suggests that it deviates significantly from the expected behavior, considering the relationships between variables.
Context Matters: Interpretation should always consider the context of the analysis. For example, in fraud detection, a high Mahalanobis distance may indicate a suspicious transaction, while in medical diagnosis, it could signal a patient's health anomaly.
Decision Making: In practical applications, decisions are based on the Mahalanobis distance. For example, if the distance of a financial portfolio from the average risk profile is too high, it might warrant a review or adjustment of the portfolio composition.

Applications

Some of its known applications are:

Outlier Detection: It is helpful in anomaly detection. In finance, for instance, Mahalanobis distance can identify unusual market behaviors or fraudulent transactions by flagging data points with distances significantly more significant than the norm.
Portfolio Optimization: In finance, it aids in constructing well-diversified portfolios by quantifying the distance of individual assets or investments from the portfolio's mean risk-return profile. Investors use it to allocate assets effectively.
Credit Scoring: Lenders use Mahalanobis distance to assess the creditworthiness of applicants. It helps in comparing an applicant's financial attributes to historical data, identifying deviations that may signify credit risk.
Quality Control: In manufacturing, Mahalanobis distance monitors product quality. It can flag products with measurements that deviate significantly from the production process mean, indicating potential defects.
Image Recognition: In computer vision, it classifies and recognizes objects based on their features. Mahalanobis distance helps measure the similarity between feature vectors extracted from images.
Healthcare: Medical professionals employ it for patient diagnosis. For example, it can help identify patients whose health characteristics deviate significantly from the norm, aiding in early disease detection.
Market Research: Researchers use Mahalanobis distance in clustering and classification tasks to group similar market segments or customer profiles based on various attributes.

Advantages And Disadvantages

Following is a comparison of the advantages and disadvantages of using Mahalanobis distance:

Advantages	Disadvantages
1. Accounts for Covariance: Mahalanobis distance considers the covariance structure of data, making it suitable for correlated variables.	1. Sensitive to Outliers: It can be sensitive to extreme values or outliers in the data, which might skew the results.
2. Multivariate Analysis: Useful for multivariate data analysis, allowing for the assessment of data points in multidimensional space.	2. Data Dimensionality: Performance decreases with high-dimensional data due to increased computational complexity and data sparsity.
3. Customizable Thresholds: One can set custom thresholds to identify outliers or anomalies, offering flexibility in applications.	3. Requires Sufficient Data: It's most effective with a sufficiently large dataset to estimate the mean and covariance matrix accurately.
4. Robust to Scaling: Mahalanobis distance is scale-invariant, meaning it is not affected by the scaling of variables.	4. Computationally Intensive: Calculating the covariance matrix inverse can be computationally expensive, especially for large datasets.
5. Widely Applicable: It finds applications in various fields, including finance, healthcare, quality control, and image recognition.	5. Sensitivity to Data Distribution: Performance can be influenced by the distribution of data; it assumes multivariate normality.
6. Identifies Relationships: Helps identify relationships and similarities between data points, making it useful in clustering and classification tasks.	6. Subject to Assumptions: It assumes that data follows a multivariate normal distribution, which may not hold for all datasets.

Mahalanobis Distance vs Euclidean Distance

Below is a comparison between Mahalanobis distance and Euclidean distance:

Aspect	Mahalanobis Distance	Euclidean Distance
Definition and Formula	Measures dissimilarity while considering the covariance structure of the data. It is calculated using the mean vector, covariance matrix, and data point vector.	Measures the straight-line distance between two data points in a multidimensional space. It is calculated as the square root of the sum of squared differences along each dimension.
Sensitivity to Data Distribution	Assumes that the data follows a multivariate normal distribution.	Assumes no specific data distribution; it is applicable to a wide range of data types and distributions.
Robustness to Scaling	Scale-invariant; it is not affected by the scaling of variables.	Sensitive to outliers, extreme values can significantly affect distance calculations.
Handling Correlated Variables	Suitable for datasets with correlated variables; considers variable correlations in the covariance matrix.	Treats variables independently; does not account for correlations between variables.
Dimensionality	Becomes less effective with high-dimensional data due to increased computational complexity and potential data sparsity.	Generally applicable to high-dimensional data, although interpretation can become challenging as dimensions increase.
Outlier Sensitivity	May be less sensitive to outliers due to covariance structure consideration.	It may be less sensitive to outliers due to covariance structure consideration.
Customization of Thresholds	Customizable thresholds can be set to identify outliers or anomalies, providing flexibility.	Thresholds are typically not customized, and outliers are identified based on distance magnitude alone.
Applications	Widely used in various fields, including finance, healthcare, quality control, and image recognition, where correlations between variables are important.	Commonly applied in geometric and spatial analysis, machine learning, and data clustering tasks when correlations between variables are less critical.

Frequently Asked Questions (FAQs)

1. What does a high Mahalanobis distance indicate?

A high Mahalanobis distance suggests that a data point is significantly dissimilar from the mean of the dataset, considering variable correlations. This could indicate an outlier or an unusual data point.

2. Can Mahalanobis distance be used with non-normal data?

While Mahalanobis distance assumes multivariate normality, it is also applicable to non-normal data. However, the results may be less reliable in such cases, and alternative distance metrics may be taken into consideration.

3. How do I set a threshold for Mahalanobis distance for outlier detection?

Thresholds for Mahalanobis distance depend on domain knowledge, simulation, or statistical methods. One can choose a threshold that balances sensitivity and specificity, depending on the specific application and tolerance for false positives/negatives.