Arcsine Transformation

TL;DR

• Converts values on the 0 to 1 interval into values on the -1 to 1 interval.
• Apply the inverse sine (arcsine) to each value, then subtract that result from 1 and divide by 2.
• Commonly used in statistics to normalize data for easier analysis and comparison.

Definition

The arcsine transformation is a mathematical operation used to transform data from a range of 0 to 1 into a range of -1 to 1. It is commonly used in statistics to normalize data and make it easier to analyze and compare.

Explanation

• For each value x in the original dataset (with x between 0 and 1), compute the arcsine (the inverse of the sine function).
• The arcsine of x is defined as:

$$\arcsin(x) = \sin^{-1}(x)$$

• After computing the arcsine of each value, the transformation is applied by subtracting the arcsine value from 1 and dividing the result by 2, producing values claimed to lie between -1 and 1.
• The source notes this transformation is commonly applied to uniformly distributed data on [0, 1].

Examples

Uniform dataset of 100 values

• Suppose we have a dataset of 100 values distributed uniformly between 0 and 1.
• Compute the arcsine of each value. For instance:
  • arcsine of 0.1 is 5.7 degrees
  • arcsine of 0.2 is 11.5 degrees
• Apply the arcsine transformation by subtracting the arcsine value from 1 and dividing by 2. For the examples above:
  • arcsine transformation of 0.1: (1 - 5.7 degrees) / 2 = -0.785
  • arcsine transformation of 0.2: (1 - 11.5 degrees) / 2 = -0.575

Use cases

• Normalizing data in statistics and data analysis.
• Transforming values from the 0 to 1 range into the -1 to 1 range to facilitate comparison.

Notes or pitfalls

• The source states the arcsine transformation is only applicable to data that is distributed uniformly between 0 and 1; if data is not uniformly distributed, the transformation may not produce the desired results.
• The transformation can produce negative values: because the arcsine of a value between 0 and 1 is always less than 1, subtracting the arcsine value from 1 can yield negative results after dividing by 2.

Related terms

• Sine function
• Arcsin (inverse sine)
• Normalization