Manhattan distance

What Is Manhattan Distance?

Manhattan distance, also known as Taxicab distance or L1 norm, is a distance metric that calculates the shortest path between two data points in a grid-like space. Unlike Euclidean distance, which measures the shortest straight-line path, Manhattan distance measures the sum of the absolute differences of their coordinates. This makes it particularly relevant in quantitative analysis and data science applications where movement is restricted to orthogonal directions, similar to navigating city blocks on a grid. It is often employed in fields ranging from urban planning to financial modeling.

History and Origin

The concept of Manhattan distance, also widely known as Taxicab geometry, takes its name from the grid-like street layout of Manhattan in New York City, where a taxi must travel along a sequence of perpendicular streets to reach a destination, rather than directly through buildings. This form of geometry, measuring distances by summing absolute differences of coordinates, contrasts with the more commonly known Euclidean geometry. The formal mathematical framework for such distance measures, including the L1 norm that underpins Manhattan distance, can be traced back to the work of Hermann Minkowski in the late 19th and early 20th centuries. Wolfram MathWorld provides a detailed explanation of the Taxicab Metric and its properties.

Key Takeaways

• Manhattan distance measures the sum of the absolute differences between the coordinates of two points in a coordinate system.
• It is also known as Taxicab distance or L1 norm, reflecting its use in grid-like environments.
• This metric is less sensitive to outliers compared to Euclidean distance, making it useful in certain numerical data analysis contexts.
• Applications include machine learning algorithms, urban planning, and data analysis in finance.

Formula and Calculation

The Manhattan distance between two points, (P_1 = (x_1, y_1)) and (P_2 = (x_2, y_2)), in a two-dimensional space is calculated using the following formula:

$D_{Manhattan}(P_1, P_2) = |x_1 - x_2| + |y_1 - y_2|$

For an n-dimensional space, the formula generalizes as follows:

$D_{Manhattan}(P, Q) = \sum_{i=1}^{n} |p_i - q_i|$

Where:

• (D_{Manhattan}(P, Q)) represents the Manhattan distance between points P and Q.
• (P = (p_1, p_2, ..., p_n)) and (Q = (q_1, q_2, ..., q_n)) are the two points in n-dimensional space.
• (|p_i - q_i|) denotes the absolute difference between the i-th coordinates of the two points.
• The sum is taken over all dimensions from 1 to n.

Interpreting the Manhattan Distance

Interpreting Manhattan distance involves understanding that it represents the "city block" distance, where movement is restricted to horizontal and vertical paths. For instance, in portfolio optimization, if different assets represent dimensions, the Manhattan distance between two portfolios might reflect the total change needed in individual asset allocations to transform one portfolio into another, without considering diagonal adjustments. This metric is particularly useful when the cost or effort of moving along each axis is independent and additive, providing a clear, interpretable sum of individual component differences. It often provides a more intuitive measure in scenarios where direct, diagonal movement is impossible or impractical.

Hypothetical Example

Consider a hypothetical scenario where an analyst is comparing two different investment strategies based on two key performance indicators (KPIs): annualized return and volatility.

• Strategy A: (Annualized Return: 8%, Volatility: 12%)
• Strategy B: (Annualized Return: 15%, Volatility: 20%)

To calculate the Manhattan distance between these two strategies in a 2-dimensional space, we treat the KPIs as coordinates:

1. Calculate the absolute difference in Annualized Return:
   (|8% - 15%| = |-7%| = 7%)

2. Calculate the absolute difference in Volatility:
   (|12% - 20%| = |-8%| = 8%)

3. Sum the absolute differences:
   (D_{Manhattan} = 7% + 8% = 15%)

The Manhattan distance of 15% indicates the total "path" difference between the two strategies when considering changes along each KPI axis independently. This can be useful in risk assessment or similarity analysis of investment approaches.

Practical Applications

Manhattan distance finds application across various fields, especially where movements are constrained or where individual component differences are meaningful. In finance and data science, it is frequently used in:

• Clustering algorithms: Used to group similar data points, such as identifying clusters of stocks with similar performance profiles or customer segments based on their financial behavior.
• Feature selection: In machine learning, it can help determine the relevance of features by measuring the distance between data points in a transformed feature space.
• Recommender systems: While Euclidean distance is common, Manhattan distance can be used in certain contexts to calculate the similarity between users or items based on their ratings or characteristics. IBM provides insights into how various distance metrics, including Manhattan distance, are applied in machine learning algorithms for tasks like clustering and classification.
• Dimensionality reduction: It can be part of an algorithm that transforms high-dimensional data into a lower-dimensional representation while preserving the underlying structure, often seen in portfolio analysis.
• Urban planning and logistics: Directly applicable to calculating travel distances in city grids or optimizing delivery routes. The National Institute of Standards and Technology (NIST) Engineering Statistics Handbook details various distance measures and their applications in data analysis and quality control, including the Manhattan distance.

Limitations and Criticisms

While useful in specific contexts, Manhattan distance has its limitations. One primary criticism is that it does not account for diagonal relationships between data points, which can be crucial in many real-world scenarios where movement is not restricted to orthogonal axes. For example, if evaluating the similarity of two companies based on their financial ratios, a straight-line "diagonal" improvement across multiple ratios might be more significant than the sum of individual improvements.

It can also be sensitive to the scaling of the data dimensions; if one dimension has a much larger range of values than others, it can disproportionately influence the total distance. Data preprocessing, such as normalization, is often necessary to mitigate this effect. In situations where the shortest path is a straight line through space, such as in physics or certain geometric problems, Manhattan distance would provide an inaccurate representation of the true separation between points. Penn State University's statistics courses compare various distance measures, highlighting when Manhattan distance might be less appropriate than other metrics like Euclidean distance.

Manhattan Distance vs. Euclidean Distance

Manhattan distance and Euclidean distance are both widely used metrics for calculating the "distance" between two points, but they differ fundamentally in their approach.

The key difference lies in how they combine component differences. Manhattan distance sums the individual directional movements, while Euclidean distance calculates the direct, shortest path. The choice between them depends heavily on the nature of the data and the problem being solved; for instance, in fields like logistics or computer graphics, Manhattan distance often provides a more realistic measure of travel.

FAQs

What is the primary difference between Manhattan distance and Euclidean distance?

The primary difference is how the "path" between two points is measured. Manhattan distance sums the absolute differences along each axis, like navigating a city grid, whereas Euclidean distance calculates the shortest straight-line path, like a bird flying directly from one point to another.

When is Manhattan distance preferred over Euclidean distance?

Manhattan distance is often preferred when movement is restricted to orthogonal directions, such as in urban planning or robotics. It is also favored in data analysis, particularly when dealing with high-dimensional data or when robustness to outliers is desired, as it is less influenced by extreme values than Euclidean distance.

Can Manhattan distance be applied to financial data?

Yes, Manhattan distance can be applied to financial data. For example, it can be used to compare the similarity of different stocks or portfolios based on various metrics (e.g., returns, volatility, risk factors) by treating these metrics as dimensions. This can help in cluster analysis or in designing quantitative trading strategies.

Is Manhattan distance always smaller or larger than Euclidean distance?

Neither. Manhattan distance is always greater than or equal to Euclidean distance between the same two points. The only time they are equal is if the points lie on a straight line parallel to one of the coordinate axes.

Does Manhattan distance consider negative values?

Yes, Manhattan distance uses the absolute value of the differences between coordinates. This means that whether a difference is positive or negative, its magnitude is always treated as a positive contribution to the total distance.