Polynomials & Interpolating Functions for Decision Rules

Sometimes it’s useful to have a way to express a variable as a flexible function of time, so that you can find the trajectory that maximizes some quantity like profit or fit to data. A caveat: this is not generally the best thing to do. A simple feedback rule will be more robust to rescaling and uncertainty and more informative than a function of time. However, there are times when it’s useful for testing or data approximation to have an open-loop decision rule. The attached models illustrate some options.

If you have access to arrays in Vensim, the simplest is to use the VECTOR LOOKUP function, which reads a subscripted table of values with interpolation. However, that has two limitations: a uniform time axis, and linear interpolation.

If you want a smooth function, a natural option is to pick a polynomial, like

y = a + b*t + c*t^2 + d*t^3 …

However, it can be a little fiddly to interpret the coefficients or get them to produce a desired behavior. The Legendre polynomials provide a basis with nicer scaling, which still recovers the basic linear, quadratic, cubic (etc.) terms when needed. (In terms of my last post, their improved properties make them less sloppy.)

 

You can generalize these to 2 dimensions by taking tensor products of the 1D series. Another option is to pick the first n terms of Pascal’s triangle. These yield essentially the same result, and either way, things get complex fast.

Back to 1D series, what if you want to express the values as a sequence of x-y points, with smooth interpolation, rather than arcane coefficients? One option is the Lagrange interpolating polynomial. It’s simple to implement, and has continuous derivatives, but it’s an N^2 problem and therefore potentially compute-intensive. It might also behave badly outside its interval, or inside due to ringing.

Probably the best choice for a smooth trajectory specified by x-y points (and optionally, the slope at each point) is a cubic spline or Bezier curve.

Polynomials1.mdl – simple smooth functions, Legendre, Lagrange and spline, runs in any version of Vensim

InterpolatingArrays.mdl InterpolatingArrays.vpm – array functions, VECTOR LOOKUP, Lagrange and spline, requires Pro/DSS or the free Reader

Path Dependence, Competition, and Succession in the Dynamics of Scientific Revolution

This is a very interesting model, both because it tackles ‘soft’ dynamics of paradigm formation in ‘hard’ science, and because it is an aggregate approach to an agent problem. Unfortunately, until now, the model was only available in DYNAMO, which limited access severely. It turns out to be fairly easy to translate to Vensim using the dyn2ven utility, once you know how to map the DYNAMO array FOR loops to Vensim subscripts.

Path Dependence, Competition, and Succession in the Dynamics of Scientific Revolution

J. Wittenberg and J. D. Sterman, 1999

Abstract

What is the relative importance of structural versus contextual forces in the birth and death of scientific theories? We describe a dynamic model of the birth, evolution, and death of scientific paradigms based on Kuhn’s Structure of Scientific Revolutions. The model creates a simulated ecology of interacting paradigms in which the creation of new theories is stochastic and endogenous. The model captures the sociological dynamics of paradigms as they compete against one another for members. Puzzle solving and anomaly recognition are also endogenous. We specify various regression models to examine the role of intrinsic versus contextual factors in determining paradigm success. We find that situational factors attending the birth of a paradigm largely determine its probability of rising to dominance, while the intrinsic explanatory power of a paradigm is only weakly related to the likelihood of success. For those paradigms that do survive the emergence phase, greater explanatory power is significantly related to longevity. However, the relationship between a paradigm’s ‘strength’ and the duration of normal science is also contingent on the competitive environment during the emergence phase. Analysis of the model shows the dynamics of competition and succession among paradigms to be conditioned by many positive feedback loops. These self-reinforcing processes amplify intrinsically unobservable micro-level perturbations in the environment – the local conditions of science, society, and self faced by the creators of a new theory – until they reach macroscopic significance. Such dynamics are the hallmark of self-organizing evolutionary systems.

We consider the implications of these results for the rise and fall of new ideas in contexts outside the natural sciences such as management fads.

Cite as: J. Wittenberg and J. D. Sterman (1999) Path Dependence, Competition, and Succession in the Dynamics of Scientific Revolution. Organization Science, 10.

I believe that this version is faithful to the original, but it’s difficult to be sure because the model is stochastic, so the results differ due to differences in the random number streams. For the moment, this model should be regarded as a beta release.

Continue reading “Path Dependence, Competition, and Succession in the Dynamics of Scientific Revolution”

Rental car stochastic dynamics

This is a little experimental model that I developed to investigate stochastic allocation of rental cars, in response to a Vensim forum question.

There’s a single fleet of rental cars distributed around 50 cities, connected by a random distance matrix (probably not physically realizable on a 2D manifold, but good enough for test purposes). In each city, customers arrive at random, rent a car if available, and return it locally or in another city. Along the way, the dawdle a bit, so returns are essentially a 2nd order delay of rentals: a combination of transit time and idle time.

The two interesting features here are:

  • Proper use of Poisson arrivals within each time step, so that car flows are dimensionally consistent and preserve the integer constraint (no fractional cars)
  • Use of Vensim’s ALLOC_P/MARKETP functions to constrain rentals when car availability is low. The usual approach, setting actual = MIN(desired, available/TIME STEP), doesn’t work because available is subscripted by 50 cities, while desired has 50 x 50 origin-destination pairs. Therefore the constrained allocation could result in fractional cars. The alternative approach is to set up a randomized first-come, first-served queue, so that any shortfall preserves the integer constraint.

The interesting experiment with this model is to lower the fleet until it becomes a constraint (at around 10,000 cars).

Documentation is sparse, but units balance.

Requires an advanced Vensim version (for arrays) or the free Model Reader.

carRental.vpm carRental.vmf

Update, with improved distribution choice and smaller array dimensions for convenience:

carRental2.mdl carRental2.vpm