How is Euclidean distance calculated between two points in a multi-dimensional space?
The Euclidean distance is a fundamental concept in mathematics and plays a important role in various fields, including artificial intelligence and machine learning. It is a measure of the straight-line distance between two points in a multi-dimensional space. In the context of machine learning, the Euclidean distance is often used as a similarity measure to quantify the dissimilarity between data points.
To calculate the Euclidean distance between two points in a multi-dimensional space, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In a two-dimensional space, the Euclidean distance between two points (x1, y1) and (x2, y2) can be calculated using the following formula:
distance = sqrt((x2 – x1)^2 + (y2 – y1)^2)
Here, sqrt denotes the square root function, and the "^" symbol represents exponentiation. By taking the square root of the sum of the squared differences in the x and y coordinates, we obtain the Euclidean distance.
This concept can be extended to higher-dimensional spaces as well. In a d-dimensional space, the Euclidean distance between two points (x1, y1, …, zd1) and (x2, y2, …, zd2) can be calculated using the following formula:
distance = sqrt((x2 – x1)^2 + (y2 – y1)^2 + … + (zd2 – zd1)^2)
The formula remains the same, with the addition of the squared differences in the remaining coordinates.
Let's consider an example to illustrate the calculation of Euclidean distance in a three-dimensional space. Suppose we have two points A(1, 2, 3) and B(4, 5, 6). We can calculate the Euclidean distance between these points as follows:
distance = sqrt((4 – 1)^2 + (5 – 2)^2 + (6 – 3)^2)
= sqrt(3^2 + 3^2 + 3^2)
= sqrt(9 + 9 + 9)
= sqrt(27)
≈ 5.196
Therefore, the Euclidean distance between points A and B in this example is approximately 5.196.
The Euclidean distance is a versatile metric and finds applications in various machine learning algorithms. For instance, it can be used in clustering algorithms such as k-means, where it helps determine the similarity between data points. Additionally, it is often employed in dimensionality reduction techniques like principal component analysis (PCA) to measure the dissimilarity between high-dimensional data points.
The Euclidean distance is a fundamental concept in mathematics and machine learning. It provides a measure of the straight-line distance between two points in a multi-dimensional space. By using the Pythagorean theorem, we can calculate the Euclidean distance by taking the square root of the sum of squared differences in the coordinates of the points. This distance metric is widely used in various machine learning algorithms for tasks such as clustering and dimensionality reduction.
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