The Inverse Problem in Random Dynamical Systems

Dynamical system used in local weather prediction (see proper right here)

We are dealing proper right here with random variables recursively outlined by Xn+1 = g(Xn), with X1 being the preliminary state of affairs. The examples talked about listed under are simple, discrete and one-dimensional: the intention is for example the concepts in order that it might be understood and useful to an enormous viewers, not merely to mathematicians. I wrote many articles about dynamical packages, see as an example proper right here. The originality in this textual content is that the packages talked about in the intervening time are random, as X1 is a random variable. Applications embrace the design of non-periodic pseudorandom amount mills, and cryptography. Also, such packages, notably further difficult ones equivalent to completely stochastic dynamical packages, are routinely used in financial modeling of commodity prices.

We focus on mappings g on the mounted interval [0, 1]. That is, the assistance space of Xn is [0, 1], and g is a many-to-one mapping onto [0,1]. The most trivial occasion, typically often called the dyadic or Bernoulli map, is when g(x) = 2x – INT(2x) = { 2x } the place the curly brackets symbolize the fractional half carry out (see proper right here). This is often denoted as g(x) = 2x mod 1. The most well-known and doubtless oldest occasion is the logistic map (see proper right here) with g(x) = 4x(1 – x).

We start with a simple practice that requires little or no mathematical information, nevertheless an outstanding amount of out-of-the-box contemplating. The reply is equipped. The dialogue is a number of specific, distinctive downside, often called the inverse downside, and launched in half 2. The causes for being in the inverse downside are moreover talked about. Finally, I current an Excel spreadsheet with all my simulations, for replication capabilities. Before discussing the inverse downside, we discuss in regards to the standard downside in half 1.

1. The regular downside

One of the precept points in dynamical packages is to hunt out if the distribution of Xn converges, and uncover the limit, known as invariant measure, invariant distribution, fixed-point distribution, or attractor. The attractor, counting on g, is often the equivalent regardless of the preliminary state of affairs X1, aside from some specific preliminary circumstances inflicting points (this set of harmful preliminary circumstances has Lebesgue measure zero, and we ignore it proper right here). As an occasion, with the Bernoulli map g(x) = { 2x }, all rational numbers (and loads of totally different numbers) are harmful preliminary circumstances. They are nonetheless far outnumbered by good preliminary circumstances. It is often very powerful to search out out if a specific preliminary state of affairs is an environment friendly one. Proving that π/4 is an environment friendly preliminary state of affairs for the Bernoulli map may very well be a big accomplishment, making you instantly well-known in the mathematical group, and proving that the digits of π in base 2, behave exactly like independently and identically distributed Bernoulli random variables. Good preliminary circumstances for the Bernoulli map are known as common numbers in base 2.

It will also be assumed that the dynamical system is ergodic: all packages investigated listed under are ergodic; I can’t elaborate on this concept, nevertheless the curious, math-savvy reader can look at the which means on Wikipedia. Finding the attractor is a troublesome downside, and it typically requires fixing a stochastic integral equation. Except in unusual occasions (talked about proper right here and in my e-book, proper right here), no exact reply is known, and one desires to utilize numerical methods to hunt out an approximation. This is illustrated in half 1.1., with the attractor found (roughly) using simulations in Excel. In half 2., we focus on the quite a bit easier inverse downside, which is the precept topic of this textual content.

1.1. Standard downside: occasion

Let’s start with X1 outlined as follows: X1 = U / (1 – U)^α, the place U is a uniform deviate on [0, 1], α = 0.25, and ^ denotes the ability operator (2^3 = 8). We use g(x) = { 4x(1 – x) }, the place { } denotes the fractional half carry out. Essentially, that’s the logistic map. I produced 10,000 deviates for X1, after which utilized the mapping g iteratively to each of these deviates, as a lot as Xn with n = 53. The scatterplot beneath represents the empirical percentile distribution carry out (PDF), respectively for X3 in blue, and X53 in orange. These PDF’s, for X2, X3, and so forth, slowly converge to a limit, much like the attractor. The orange S-curve (n = 53) is very close to the limiting PDF, and additional iterations (that is, rising n) barely current any change. So we found the limit (roughly) using simulations. Note that the cumulative distribution carry out (CDF) is the inverse of the PDF. All this was achieved with Excel alone.

2. The inverse downside

The inverse downside consists of discovering g, assuming the attractor distribution (the orange curve in the above decide) is known. Typically, there are plenty of attainable choices. One of the attainable causes for fixing the inverse downside is to get a sequence of random variables X1, X2, and so forth, that shows little or no auto-correlations. For event, the lag-1 auto-correlation (between Xn and Xn+1) for the Bernoulli map, is 1/2, which is method too extreme counting on the needs you’ll have in ideas. It is important in cryptography functions to remove these auto-correlations. The reply proposed proper right here moreover satisfies the subsequent property: X2 = g(X1), X3 = g(X2), X4 = g(X3) and so forth, all have the equivalent pre-specified attractor distribution, regardless of the (non-singular) distribution of X1. 

2.1. Exercise

Before diving into a solution, if in case you have got time, I ask you to resolve the subsequent simple inverse downside. 

Find a mapping g such that if Xn+1 = g(Xn), the attractor distribution is uniform on [0, 1]. Can you uncover one yielding very low auto-correlations between the successive Xn‘s? Hint: g might be not regular. 

2.2. A standard reply to the inverse downside

A potential reply to the problem in half 2.1 is g(x) = { bx } the place b is an integer larger than 1. This is because of the uniform distribution on [0, 1] is the attractor for this map. The case b = 2 corresponds to the Bernoulli map talked about earlier. Regardless of b, INT(bXn) represents the n-th digit of X1, in base b. The lag-1 autocorrelation between Xn and Xn+1, is then equal to 1 / b. Thus, the higher b, the upper. Note that ought to you employ Excel for simulations, steer clear of even integer values for b, as Excel has an inside glitch which will make your simulations meaningless after n = 45 iterations or so. 

Now, a standard reply equipped proper right here, for any pre-specified attractor and any non-singular distribution for X1, depends on a consequence proved proper right here. If g is the reply in question, then all Xn (with n  >  1) have the equivalent distribution as a result of the pre-specified attractor. I current an Excel spreadsheet displaying the way in which it really works for a specific occasion.

First, let’s assume that g* is a solution when the attractor is the uniform distribution on [0, 1]. For event g*(x) = { bx } as talked about earlier. Let F be the CDF of the aim attractor, and assume its assist space is [0, 1]. Then a solution g is given by

For event, if F(x) = x^2, with x in [0, 1], then g(x) = SQRT( { bx^2 } ) works, assuming b is an integer larger than 1. The scatterplot beneath displays the empirical CDF of X2 (blue dots, based totally on 10,000 deviates) versus the CDF of the aim attractor with distribution F (purple curve): they’re practically indistinguishable. I used b = 3, and for X1, I used the equivalent distribution as in half 1.1. The detailed computations will be discovered in my spreadsheet, proper right here (13 MB receive).

The summary statistics and the above plot are found in columns BD to BH, in my spreadsheet.

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About the author:  Vincent Granville is a data science pioneer, mathematician, e-book author (Wiley), patent proprietor, former post-doc at Cambridge University, former VC-funded govt, with 20+ years of firm experience along with CNET, NBC, Visa, Wells Fargo, Microsoft, eBay. Vincent will also be self-publisher at DataShaping.com, and based mostly and co-founded quite a lot of start-ups, along with one with a worthwhile exit (Data Science Central acquired by Tech Target). You can entry Vincent’s articles and books, proper right here.