Series of random numbers on a continuous function

Question

At one point, I read about a function used to generate random numbers that follow a continuous pattern. By this I mean random numbers that as a series is random, but in which items tend to be relatively close to their successors and predecessors. In other words, rather than generating a series like the following:

50, 1, 84, 23, 26, 56, 42, 92, 34, 11, 47

This function would generate random numbers like the following:

42, 45, 49, 48, 34, 31, 36, 42, 45, 53, 61

I can't seem to find this function anywhere, and without its name it isn't yielding to search engines. What is it?

Update - I Found the Function!

I was watching a series of videos on constructing graphics using almost random noise, when I realized that what the hosts were talking about was exactly what I was looking for back in April 2015: Perlin noise

Perlin noise is essentially a continuous "random" function (along any number of dimensions) in which the values at certain points correspond to the values at neighboring points, yielding a smooth, continuous selection of "random" values, associated by proximity. Thanks everyone.

Answer 1

Using Uniform Random Numbers to Simulate Various Distributions

Usually, by 'random number' or 'pseudorandom number', we mean an observation $U$ from a uniform distribution on the interval (0, 1). Other distributions are simulated by transforming $U$ to match some other distribution. A histogram of many observations $U$ would have several bars with bases in (0, 1) and heights nearly equal. Altogether, the histogram has a roughly rectangular shape, and sometimes uniform distributions are called 'rectangular'.

The simplest case might be to use $60U$ to simulate events randomly spread over an hour's time. This is another rectangular distribution but the base of the rectangle is the interval (0, 60).

If you wanted observations confined to the interval (60, 70), you could use the transformation $10U + 60$. Another rectangular distribution with base (60, 70).

Other transformations use nonlinear transformation. For example, many observations made with the square root transformation $\sqrt{U}$ would give a histogram that is roughly a right triangle with base (0,1) and the tallest end of the hypotenuse on the right side. Admittedly, it is not immediately obvious why taking the square root gives a triangular-shaped histogram, but there is a mathematical proof for that.

Your specification that you just want values crowded rather closely together mainly clustering around 45 is pretty vague. Most statistical software has pre-programmed transformations (some of them quite elaborate) to turn uniform $U$'s into various distributional shapes. So it is not necessary for the user to figure out the transformation needed to get a desired shape. If you can be more specific what you want, I might be able to tell you how to simulate the kind of data you want.

Below is an example of 50 observations from a normal population with population mean 45, population standard deviation 8, and rounded to integer values. This illustration was done with R statistical software, but many kinds of software would do the same job about as simply.

Please understand that this is a random procedure, so the next time I type 'round(rnorm(50, 45, 8))' into R, I will get entirely different numbers, but still integers in the 'general vicinity' of 45. Also, 50 observations is not nearly enough to yield a smooth histogram that closely matches the classical bell-shaped curve; 1000 would be better for that. (Numbers in brackets simply give the count out of 50 of the first number in each row.)

 > round(rnorm(50, 45, 8))
  [1] 46 50 51 51 50 51 29 50 48 55 35 41 31 43
 [15] 49 53 53 48 37 48 33 43 40 37 46 43 46 60
 [29] 35 64 57 48 53 48 39 61 44 43 41 33 46 39
 [43] 40 58 45 45 57 29 41 44