Probability distributions

A probability distribution gives a relative weight to each available option in the space of possible outcomes; this is the probability density function (pdf). For a discrete set of possible outcomes, there is an explicit probability mass for each; for a continuous set (e.g., any number from the range 0.0 to 1.0) a mass can be associated with any interval. The sum of the probability density is normalized to 1. 

From a probability density function we can create a cumulative distribution function (cdf) which allows us to  calculate which output we get on a particular occasion (particular roll of the dice). 

//Helper functions for investigating pdfs and cdfs

(

//normalize total sum of an array to 1.0

~normalize = {|array|   array/(array.sum)}; //note, not safe if array is empty or otherwise sums to zero

//(could also use normalizeSum, just showing this explicitly)

//create array of cumulative values, assuming input is normalized (sums to 1.0)

~cumulative = {|array| 

var num = array.size; 

var accumulate = 0.0; 

var cumul; 

cumul = array.collect{|val, i|  var old= accumulate; accumulate = accumulate+val; old};

cumul

}; 

//use cumulative distribution to find an output value for an input

//assumes array is a cumulative distribution function, and array size is at least 1

~draw= {|input, array|

var nextindex;

nextindex= array.indexOfGreaterThan(input); //also see indexInBetween if want linearly interpolated index

//get nil if input greater than anything in array

if (nextindex.isNil,{nextindex= array.size;});

nextindex= nextindex-1; //get index before; we 'went past' and have to go one back to find the slot our input falls in

//nextindex should never be less than 0

nextindex/(array.size); //get position proportional within array length

}

)

Plotting approximations to different probability distributions:

//1. Uniform distribution, equal chance for any equal subsection of the range:

p= ~normalize.value(Array.fill(1000,1.0));

p.plot2(minval:0.0, maxval:2.0/1000)  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)}).sort.plot2 //choose values; driven by uniform random number, could also just provide increasing uniform numbers from 0.0 to 1.0 (and then wouldn't need the sort) 

//we do this a thousand times to simulate 'rolling the dice' many times; the distribution only really shows itself over many trials (and can do it more times for better approximations)

Array.fill(1000,{ 1.0.rand }).sort.plot2 //create values directly

//2. Linear distribution, probability density drops linearly, so more likely to get lower values:

p= ~normalize.value(Array.fill(1000,{|i| 1000-i}));

p.plot2  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)}).sort.plot2 //choose values 

Array.fill(1000,{ 1.0.linrand }).sort.plot2 //create values directly

//3. Negative exponential distribution, probability density drops exponentially, so much more likely to get lower values:

//there is a parameter here for the rate of fall off of the distribution 

~alpha= 10.0; 

p= ~normalize.value(Array.fill(1000,{|i| exp((i.neg/1000)*~alpha) })); 

p.plot2  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)}).sort.plot2 //choose values 

//go back and try ~alpha= 100.0; note the quantisation caused by only working with arrays of limited size as we create a discrete approximation to the ideal continuous distribution

//4. Normal distribution (Gaussian)

//two parameters, mean ('centre') and standard deviation ('spread'); here we take sensible values to plot the distribution easily

~mu= 0.5; //mean

~sigma=  0.17; //standard deviation; most of probability mass within 3 standard deviations, so this makes the Gaussian fit the 0.0 to 1.0 range easily for our plotting; try changing these parameters later to see the effect. 

//normalization constant calculated automatically, though there is a mathematical expression for it

p= ~normalize.value(Array.fill(1000,{|i|  exp((((i/1000)-~mu)/~sigma).squared.neg)})); 

p.plot2  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)}).sort.plot2 //choose values 

Array.fill(1000,{  0.5.gauss(0.17).max(0.0).min(1.0)  }).sort.plot2 //create values directly, clamping within +-3 standard deviations

//5. Arbitrary distribution

//let's make up our own function

p= ~normalize.value(Array.fill(1000,{|i|  var prop= (i/1000.0); if(prop<0.2,{(0.2-prop)**(0.3)},{(prop-0.2)**2}) })); 

p.plot2  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)}).sort.plot2 //choose values 

//example in use; 20 notes drawn using the custom distribution (0.0 to 1.0 range output rescaled to MIDI notes 48 to 72)

(

{

20.do{

Synth(\acsound,[\freq, 48+(24*~draw.value(1.0.rand, c))]); 

//could quantise the notes to discrete pitches, e.g., with .round(1.0)

0.25.wait;

}

}.fork;

)

There are many other interesting distributions you might investigate, see for example: 

Charles Ames. A catalog of statistical distributions: Techniques for transforming random, determinate and chaotic sequences. Leonardo Music Journal, 1(1):55–70, 1991. 

Denis Lorrain. A panoply of stochastic 'cannons'. Computer Music Journal, 41(1):53–81, 1980. 

In practice, we often work with a relatively small discrete set of options. It is easy to create a custom probability distribution over a set: 

[0,1,2,3,4,5].wchoose([0.2,0.3,0.1,0.1,0.05,0.25]) //make sure weights add to 1.0, or use normalizeSum

//examination using our machinery for pdf, cdf, and draws:

p= ~normalize.value([0.2,0.3,0.1,0.1,0.05,0.25]); 

p.plot2  //pdf

c= ~cumulative.value(p); 

c.plot2 //cdf

Array.fill(1000,{ ~draw.value(1.0.rand, c)*6}).sort.plot2 //choose values; multiply by 6 to get back integer indices