Transforming To A Uniform Distribution
If we observe a set of n values X1, ..., Xn with no ties (i.e. there are n distinct values), we can replace Xi with the transformed value Yi = k, where k is defined such that Xi is the kth largest among all the X values. This is called the rank transform, and creates data with a perfect fit to a uniform distribution. This approach has a population analogue. If X is any random variable, and F is the cumulative distribution function of X, then as long as F is invertible, the random variable U = F(X) follows a uniform distribution on the unit interval .
From a uniform distribution, we can transform to any distribution with an invertible cumulative distribution function. If G is an invertible cumulative distribution function, and U is a uniformly distributed random variable, then the random variable G−1(U) has G as its cumulative distribution function.
Read more about this topic: Data Transformation (statistics)
Famous quotes containing the words transforming, uniform and/or distribution:
“America is the civilization of people engaged in transforming themselves. In the past, the stars of the performance were the pioneer and the immigrant. Today, it is youth and the Black.”
—Harold Rosenberg (1906–1978)
“Odors from decaying food wafting through the air when the door is opened, colorful mold growing between a wet gym uniform and the damp carpet underneath, and the complete supply of bath towels scattered throughout the bedroom can become wonderful opportunities to help your teenager learn once again that the art of living in a community requires compromise, negotiation, and consensus.”
—Barbara Coloroso (20th century)
“In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
—Ralph Waldo Emerson (1803–1882)