Don't just do something, stand there! Reflections on the counterintuitive behavior of complex systems, seen through the eyes of System Dynamics, Systems Thinking and simulation.
E, a.k.a. Euler’s number or the base of the natural logarithm, is near and dear to dynamic modelers. It’s not just the root of exponential growth and decay; thanks to Euler’s Formula it encompasses oscillation, and therefore all things dynamic.
E is approximately 2.718, and today is 2/7/18, at least to Americans, so this is the biggest e day for a while. (NASA has the next 1,999,996 digits, should you need them.) Unlike π, e has not been contested in any state legislature that I know of.
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
This post should be required reading for all modelers. And no, I’m not going to reproach sloppy modeling practices. This is much more interesting than that.
Sloppy models are an idea that formalizes a statement Jay Forrester made long ago, in Industrial Dynamics (13.5):
The third and least important aspect of a model to be considered in judging its validity concerns the values for its parameters (constant coefficients). The system dynamics will be found to be relatively insensitive to many of them. They may be chosen anywhere within a plausible range. The few sensitive parameters will be identified by model tests, and it is not so important to know their past values as it is to control their future values in a system redesign.
This remains true when you’re interested in estimation of parameters from data. At Ventana, we rely on the fact that structure and parameters for which you have no measurements will typically reveal themselves in the dynamics, if they’re dynamically important. (There are always pathological cases, where a nonlinearity makes something irrelevant in the past important in the future, but that’s why we don’t base models solely on formal data.)
Now, the required part. Continue reading “Sloppy System Dynamics”
Vi Hart’s interesting comments on the dynamics of political polarization, following the release of an innocuous video:
I wonder what made those commenters think we have opposite views; surely it couldn’t just be that I suggest people consider the consequences of their words and actions. My working theory is that other markers have placed me on the opposite side of a cultural divide that they feel exists, and they are in the habit of demonizing the people they’ve put on this side of their imaginary divide with whatever moral outrage sounds irreproachable to them. It’s a rather common tool in the rhetorical toolset, because it’s easy to make the perceived good outweigh the perceived harm if you add fear to the equation.
Many groups have grown their numbers through this feedback loop: have a charismatic leader convince people there’s a big risk that group x will do y, therefore it seems worth the cost of being divisive with those who think that risk is not worth acting on, and that divisiveness cuts out those who think that risk is lower, which then increases the perceived risk, which lowers the cost of being increasingly divisive, and so on.
The above feedback loop works great when the divide cuts off a trust of the institutions of science, or glorifies a distrust of data. It breaks the feedback loop if you act on science’s best knowledge of the risk, which trends towards staying constant, rather than perceived risk, which can easily grow exponentially, especially when someone is stoking your fear and distrust.
If a group believes that there’s too much risk in trusting outsiders about where the real risk and harm are, then, well, of course I’ll get distrustful people afraid that my mathematical views on risk/benefit are in danger of creating a fascist state. The risk/benefit calculation demands it be so.
I’ve been working with pharma brand tracking data, used to calibrate a part of an integrated model of prescriptions in a disease class. Understanding docs’ perceptions of drugs is pretty important, because it’s the major driver of rx. Drug companies spend a lot of money collecting this data; vendors work hard to collect it by conducting quarterly interviews with doctors in a variety of specialties.
Unfortunately, most of the data is poorly targeted for dynamic modeling. It seems to be collected to track and guide ad messaging, but that leads to turbulence that prevents drawing any long term conclusions from the data. That’s likely to lead to reactive decision making. Here’s how to minimize strategic information content:
On the other hand, please don’t! A few consistent, well-quantified questions are pure gold if you want to untangle causality that plays out over more than a quarter.
Project management has been one of the most productive and successful areas of system dynamics. And yet, when I recently looked at project management tools and advice, I couldn’t find a hint of SD dynamic insights into product management. Lists of reasons for project failure almost entirely neglect endogenous explanations.
Nothing about rework, late change orders, design/implementation balance, schedule pressure effects on quality and productivity, overtime burnout and turnover, Brooks’ Law, multiphase resource allocation, firefighting or tipping points.
I think there’s an insight and a puzzle here. The insight is that mismanaged dynamics and misperceptions of feedback aren’t the only way to screw up. There are exogenous and single-cause failure modes, like hiring people with the wrong skill set for a job, building something no one wants, or just failing to keep in touch with your team.
However, I’m pretty sure the dominant cause of execution failure is dynamic. Large projects are like sleeping monsters. They are full of positive feedback loops that, when triggered cause increasing delays and overruns, perhaps explaining the heavy-tailed distribution of massive project failures. So, the puzzle is, how could there be so little mention, and so few tools, for management of the internal causes of project success?
Not coincidentally, this problem is one of the major reasons we built Ventity. We’re currently working on project models that are entirely data driven, so you can switch from building a house to building a power plant just by changing some tables of input. We think this will be the missing link between data-oriented tools that manage projects statically in exquisite detail and dynamic models that realistically describe projects, but have traditionally been hard to build, calibrate and reuse.
DARPA put out a request for a BS detector for science. I responded with a strategy for combining the results of multiple models (using Rahmandad, Jalali & Paynabar’s generalized meta-analysis with some supporting infrastructure like data archiving) to establish whether new findings are consistent with an existing body of knowledge.
DARPA didn’t bite. I have no idea why, but could speculate from the RFC that they had in mind something more like a big data approach that would use text analysis to evaluate claims. Hopefully not, because a text-only approach will have limited power. Here’s why.
The word “quizzaciously” was literally absent from the web until Vsauce mentioned it in this cool video on Zipf’s law.
Google Trends reflects this. The week of the video’s release, there was a huge spike in interest, followed by a rapid decay, not all the way to zero, but to a slow simmer of interest.
The video discusses power laws as a model of memory. So … has the internet remembered the video according to a power law? Not exactly, but it certainly has a hint of one:
My guess is that the trajectory is modified by word-of-mouth processes that create sustained interest.
Quite a while back, I posted about the dynamics of the thyroid and its interactions with other systems.
That was a conceptual model; this is a mathematical model. This is a Vensim replication of:
Marisa Eisenberg, Mary Samuels, and Joseph J. DiStefano III
Extensions, Validation, and Clinical Applications of a Feedback Control System Simulator of the Hypothalamo-Pituitary-Thyroid Axis
Background:We upgraded our recent feedback control system (FBCS) simulation model of human thyroid hormone (TH) regulation to include explicit representation of hypothalamic and pituitary dynamics, and up-dated TH distribution and elimination (D&E) parameters. This new model greatly expands the range of clinical and basic science scenarios explorable by computer simulation.
Methods: We quantified the model from pharmacokinetic (PK) and physiological human data and validated it comparatively against several independent clinical data sets. We then explored three contemporary clinical issues with the new model: …
… These results highlight how highly nonlinear feedback in the hypothalamic-pituitary-thyroid axis acts to maintain normal hormone levels, even with severely reduced TSH secretion.
THYROID
Volume 18, Number 10, 2008
DOI: 10.1089=thy.2007.0388
This version is a superset of the authors’ earlier 2006 model, and closely reproduces that with a few parameter changes.
L-T4 Bioequivalence and Hormone Replacement Studies via Feedback Control Simulations
THYROID
Volume 16, Number 12, 2006
The model is used in:
TSH-Based Protocol, Tablet Instability, and Absorption Effects on L-T4 Bioequivalence
THYROID
Volume 19, Number 2, 2009
DOI: 10.1089=thy.2008.0148
This works with any Vensim version: