RANDPR

Generate or learn various random processes.

In its generation mode, RANDPR will generate a first-order, K-state Markov process given its first-order, KxK transition probabilites. RANDPR also computes equilibrium probabilities (if they exist), mean recurrence times, time to first absorption, absorption probabilites and the statistical auto-covariance function. RANDPR can create ergodic, regular, absorptive processes, Weiner (stationary in first increments), and wide-sense stationary processes.

In its learning mode, RANDPR produces supervised and unsupervised estimates: MLEs of the underlying transition probabilities, number of coherence intervals (and related SNR), and the underlying intensity profile. (MLE for the Weiner process, e.g. first time to exit, have not yet been implemented).

Both discrete- and continious-time models are supported in either forward or reverse mode.

RANDPR can be customized with onStep(), onBatch(), onEnd(), and filter() methods to sample metrics in both forward and reverse modes. Both modes can make use of piped streams to minimize memory usage.

RANDPR supports the following processes.

Bayes

K-state process governed by a prescribed conditional independency network:

	eqP: [pr, ...] the K equilibrium probs 
	net: [ {var: probs, ...}, ... ] the conditional dependencies 
	
or expressed as a DAG:

	dag: { ... }

Gillespie

Inhomogenious K-state process with specified transition probabilties:

	states: number of states
	
where its K^2 transition probs are synthesized using the gillespie model.

Markov

K-state process with specified transition probabilities:

	TxPrs: [ [...], ....] the K^2 (K^2-K independent) transition probs 
	
or:

	states: K
	TxPrs: { from: {to: pr, ... } , ... "from, ..." : "to, ..." }	
	
where from-to transition probs must be specified to conserve prob, i.e. sum_k TxPrs[n][k] = 1.

Gauss

Correlated, stateless random process whose parameters are typically derived (see man) 
for a process with known correlation intervals M = T/Tc or SNR = sqrt{ M / ( 1 + deltaC / M) }.

	values: [ ... ] pc eigen values  [unitless]
	vectors: [ [... ], ...] pc eigen values	[sqrt Hz]
	ref: reference eigenvalue 
	dim: max pc dimension (M = T/Tc )
	mean: mean count in observation interval T

Wiener

Stateless process with moving 2nd moment (but stationary in 1st increments) where:

	walks: number of walks at each time step (0 disables)
	
This still needs a quick debug.  May need to define	false Markov parms to init it.

Ornstein

Stateless Ornstein-Ulenbeck process with:

	theta: 0-pi
	a: sigma/sqrt(2 theta)
	

Mixing

Gauss mixing process with specified mu,sigma (mean, covar), or specified snr, cone, mixes, oncov, offcov

Refs

• www.statslab.cam.ac.uk/~rrw1
• www.stat.yale.edu/~pollard
• people.math.gatech.edu/~randall
• www.stat.berkeley.edu/~pitman
• www.math.dartmouth.edu/~pw

Manage

npm install @totemstan/randpr	# install
npm run start [ ? | $ | ...]	# Unit test
npm run verminor				# Roll minor version
npm run vermajor				# Roll major version
npm run redoc					# Regen documentation

Usage

Require the module:

var RAN = require("randpr");

then create a new instance:

var ran = new RAN({
	key: value, 						// set key
	"key.key": value, 					// indexed set
	"key.key.": value					// indexed append
}, function (err) {
	console.log( err ? "something evil is lurking" : "look mom - Im running!");
});

where configuration keys follow ENUMS deep copy conventions

The instance can then be piped:

ran.pipe()

to genenerate a process (if configured in the forward/generate mode), or learn process parameters (if configured in the reverse/learning mode).

Program Reference

RANDPR

Generates various random processes.

Requires: module:man, module:stream
Example

R2.1 - config methods:

	var ran = new RAN({
		p: [.4],
		//markov: [[0.1, 0.9], [0.1, 0.9]]
		//markov: { states: 3, 0: {1: 0.8, 2: 0.1}, 1: {0: 0.1} }
	});

Example

R2.3 - config methods:

	var ran = new RAN({
		emP: {
			dims: [3,3],
			weights: [1,1]
		},
		markov: { states: 9, 0: {1: 0.8, 2: 0.1}, 1: {0: 0.1}, "0,1": { "1,0": .4} }
	});
		

Example

R2.4 - config methods:

	var ran = new RAN({
		emP: {
			dims: [2,2,2],
			weights: [1,1,1]
		},
		markov: "random"
	});
	

Example

R3 sync pipe with various textbook examples, custom filtering with supervised learning validation:
	
	var ran = new RAN({
		// these have same eqprs [.5, .5] (symmetry -> detailed balance --> eqP[k] = 1/K  eqpr)
		//markov: [[.6, .4],[.4, .6]],

		//markov: [[0.83177, 0.16822], [0.17152, 0.82848]],

		//markov: [[.5, .5], [.5, .5]],
		//markov: [[0.1, 0.9], [0.9, 0.1]],

		// textbook exs
		markov: [[0.1, 0.9], [0.1, 0.9]],  // pg142 ex3

		//markov: [[1/2, 1/3, 1/6], [3/4, 0, 1/4], [0,1,0]],  // pg142 ex2  eqpr [.5, .333, .1666]
		//markov: [[1,0,0], [1/4, 1/2, 1/4], [0,0,1]],  // pg143 ex8  no eqprs

		// these have different eqprs
		//markov: [[0.9,0.1],[0.1,0.9]],
		//markov: [[0.1, 0.9], [0.1, 0.9]],  // bernoulli scheme has identical rows
		//markov: [[0.1, 0.9], [0.3, 0.7]],
		//markov: [[0.1, 0.9], [0.4, 0.6]],

		// textbook exs 
		//markov: [[0,1],[1,0]],  // pg433 ex16  regular (all states reachable) absorbing/non on even/odd steps non-regular non-absorbing but ergodic so --> eqpr [.5, .5]
		//markov: [[0.5,0.25,0.25],[0.5,0,0.5],[0.25,0.25,0.5]],  // pg406 ex1  regular (after 2 steps) thus ergodic so eqpr [.4, .2, .4]
		//markov: [[0,1,0,0,0], [0.25,0,0.75,0,0], [0,0.5,0,0.5,0], [0,0,0.75,0,0.25], [0,0,0,1,0]],  // pg433 ex17  non-absorbing non-regular but ergodic so eqpr [.0625, .25, .375]
		//markov: [[1,0,0,0,0],[0.5,0,0.5,0,0],[0,0.5,0,0.5,0],[0,0,0.5,0,0.5],[0,0,0,0,1]],    // 2 absorbing states; non-ergodic so 3 eqpr = [.75 ... .25], [.5 ... .5], [.25 ...  .75]

		//markov: [[1-.2, .1, .1], [0, 1-.1, .1], [.1, .1, 1-.2]],
		//markov: [[1-.2, .1, .1], [0.4, 1-.5, .1], [.1, .1, 1-.2]],
		//markov: [[1-.6, .2, .2,.2], [.1, 1-.3, .1,.1], [.1, .1, 1-.4,.2],[.1,.1,1-.8,.6]],  // non-ergodic

		batch: 50,  // supervised learning every 50 steps

		filter: function (str, ev) {  
			switch (ev.at) {
				case "config":
					//Trace(ev);
					str.push(ev);
					break;

				case "batch":
					//Trace(ev.s,ev.rel_txpr_error);
					Trace(ev);
					break;

				case "end":
					Trace(ev);
					var
						A = ev.stats.mle_tr_probs,
						B = ev.stats.mle_em_probs,
						H = ev.stats.mle_holding_times;

					Trace("MLEs", {
						holdTimes: JSON.stringify(H),
						emProbs: JSON.stringify(B),
						trProbs: JSON.stringify(A)
					});

					str.push(ev);
					break;
				}
			},

			N: 500,
			steps: 500
		});

		ran.pipe( function (store) { 
			Trace(store);
		});
		

Example

R3.1 - gen process for R3.2 with async pipe to stdout:
	
	var ran = new RAN({

		markov: [[0.1, 0.9], [0.1, 0.9]],  // pg142 ex3

		batch: 800,  // supervised learning every 50 steps

		N: 1000, 
		filter: function (str,ev) {
			switch (ev.at) {
				case "batch":
				case "config":
				case "end":
					Trace(JSON.stringify(ev));
			}
		},
		steps: 800  
	});

	ran.pipe(process.stdout);
		

Example

R3.2 - gen process for R3.3 using async pipe to stdout:

	var ran = new RAN({

		markov: [[0.1, 0.9], [0.1, 0.9]],  // pg142 ex3

		//batch: 50,  // supervised learning every 50 steps

		N: 10,
		//keys: {state:"u", index: "n"},
		filter: function (str,ev) {
			switch (ev.at) {
				case "jump":
					Trace(ev);
					break;
				default:
			}
		},
		steps: 20
	});
	
	ran.pipe(process.stdout);  // stdout evs used in R3.3
		

Example

R3.3 - supervised learning with R3.2 evs using sync pipe to store:

	var 
		evs = [
				{ at: 'jump', t: 1, s: 1, index: 3, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 1, s: 1, index: 5, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 1, s: 1, index: 6, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 1, s: 1, index: 7, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 1, s: 1, index: 8, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 1, s: 1, index: 9, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 2, s: 2, index: 2, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 2, s: 2, index: 3, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 3, s: 3, index: 2, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 3, s: 3, index: 6, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 4, s: 4, index: 6, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 5, s: 5, index: 1, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 6, s: 6, index: 1, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 8, s: 8, index: 9, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 9, s: 9, index: 3, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 9, s: 9, index: 8, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 9, s: 9, index: 9, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 10, s: 10, index: 3, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 10, s: 10, index: 8, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 11, s: 11, index: 4, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 13, s: 13, index: 4, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 13, s: 13, index: 8, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 14, s: 14, index: 8, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 15, s: 15, index: 0, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 16, s: 16, index: 0, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 16, s: 16, index: 1, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 16, s: 16, index: 3, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 17, s: 17, index: 3, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 18, s: 18, index: 1, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 18, s: 18, index: 3, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 18, s: 18, index: 6, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 18, s: 18, index: 9, state: 0, hold: 0, obs: null },
				{ at: 'jump', t: 19, s: 19, index: 3, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 19, s: 19, index: 6, state: 1, hold: 0, obs: null },
				{ at: 'jump', t: 19, s: 19, index: 9, state: 1, hold: 0, obs: null }
			],
			
		ran = new RAN({

			learn: function (supercb) {
				evs.$( true, (evs) => {
					Trace( evs ? ` supervising ${evs.length} events` : " supervised" );

					if (evs) // feed supervisor
						supercb(evs);

					else // terminate supervisor
						supercb(null);
				});
			},			

			batch: 50,  // supervised learning every 50 steps

			filter: function (str, ev) {  
				switch (ev.at) {
					case "config":
						Trace(ev);
						str.push(ev);
						break;

					case "batch":
						//Trace(ev.s,ev.rel_txpr_error);
						Trace(ev);
						break;

					case "end":
						//Trace(ev);
						str.push(ev);
						break;
				}
			},

			markov: {},  
			//keys: {state:"u", index: "n"},
			K: 2,  // assume 2-state process
			N: 50  // assume 50 members in ensemble
		});

	ran.pipe( function (store) {
		Trace(store);
	});

RANDPR~filter()

Output event filter filter: function (str, ev, ran) { // event ev for stream/store str switch ( ev.at ) { // streaming plugins provide an "at" to filter events on case "...": case "...": str.push(ev); // return the event } }

Kind: inner method of RANDPR

Contacting, Contributing, Following

Feel free to

• submit and status TOTEM issues
• contribute to TOTEM notebooks
• revise TOTEM requirements
• browse TOTEM holdings
• or follow TOTEM milestones

License

MIT

© 2012 ACMESDS