Thank you Mooch for the opportunity to learn and share. Though I have long been an amateur thinker, the job title I have had for the past two years, "management analyst" asserts that I am now a professional thinker. Consequently, it is in my best professional interest to learn from my peers and avail myself to them, to the extent I am able. For now, I will not detail my background, because my current thought is that I want the ideas that I present to stand on their own. As you will see, they do require critique, maturing, and possibly expansion.
So, friends, without further introduction, here is one of my thoughts on "analysis." Analysis being defined as: seeking the truth of a particular matter with intent to understand in order to make useful decisions. To think productively, we need to understand the frameworks we use (one will not do, for reasons that appear intuitively obvious to me – let me know if you disagree; and to omit any one puts daylight between our analysis and reality/truth). Please let me know what other frameworks I have over looked. I will grant that they overlap, but I consider each a primary driver for how things work.
Frameworks:
Deterministic (because some things happen because another event caused them)
Random (because some things happen on a randomly distributed basis)
Chaotic (because some things happen on a non-random, non-periodic, unpredictable basis) Deliberate (because of free will)
Bias based (because we are human)
My goal is to increase in knowledge, understanding, and (some day) wisdom. Ogre
Ogre -- first if you are going to play in this sandbox you have to be practiced in the art of obfuscation. Why say "Random" when the almost perfect word "Stochastic" is available for your use?
ReplyDeleteI think you hit the fundamental ones...but remember additional frameworks can be a combination of these five...or in the real world all of them combined. I think these combinations can become their own framework we call Complex Systems. Also remember, each one has different sub-types. Deterministic can have linear and non-linear functions. Stochastic processes have many distributions to define their random nature, etc. With the Operations Research disciplines there are many models to deal with each of these...hopefully Mike will weigh in here on complex systems. But for linear and nonlinear deterministic systems we use optimization, for random systems we use probability theory and simulation (both continuous and discrete). When we try to model human subjectivity we use decision theory.
But these are just tools and most of the time they are misused. The folks you will meet in this forum are the analysts who will not substitute a tool for brain cells. Thinking first is what we believe to be key to any successes we might have.
Mooch,
ReplyDeleteHmmm…stochastic…
Okay, I looked it up – I am pretty underwhelmed. To me it seems a lot of the examples I found of stochastic processes behave chaotically in the real world. For example, gas under pressure. In my world, the atmosphere is gas under pressure, but we all acknowledge weather to be chaotic. Cancer is supposed to be stochastic, but it is not clear that we can control for all the thousands of variables we truly need to find the probabilistic ones. Just to read the social sciences example in Wikipedia it is hard to miss the chaotic nature (“The event creates its own conditions of possibility, rendering it unpredictable if simply for the amount of variables involved” – or is it unpredictable because of its chaotic nature?).
Yes all the frameworks play in the real world. So what tools do you use to model chaotic systems? Economic bubbles burst in a chaotic fashion, I suspect events in war may behave similarly. I also suspect that the human brain aided only with experience and proper education may be able to dope it out, but not stochastic models.
I hope this comment post works…
I like your sandbox, but I am not smart enough to be very good at obfuscation.
Ogre
@Ogre--I believe your sarcasm is a bit drier than mine. If not the rule of thumb for dealing with me is simply that everything I say has a brush of sarcasm and demands a sarcastic response.
ReplyDeleteYou are supposed to be underwhelmed...changing random to stochastic is like changing happy to glad. Subtle shades of gray are of no consequence to us engineers...the mathematicians among us however will beg to differ and maybe we will hear from them shortly.
But you've captured the essential difference...random is random. Stochastic is our attempt to capture seemingly random things thereby making the random predictable...which seems pretty silly when you think about it...but nevertheless some random things do behave and can be modeled in this fashion...leaving only the outliers to truly fit your definition.
Training the human brain to "dope it out", preferably with the assistance of the appropriate tools applied correctly...is what this blog is all about.
If you are no good at obfuscation you are welcome in this sandbox...I officially declare this a No Obfuscation zone.
I have met Bill O'Reilly, and you, Mr Muccio, are no Bill O'Reilly...
ReplyDeleteOne is enough...of each of you...
So if stochastic models are meant to make the random predictable, what do the modelers/mathematicians use to make the chaotic predictable? I vaguely recall having heard that computers can generate chaotic patterns - it doesn't seem too challenging. Something for my next Google search...
Mike and Merf can better describe chaotic patterns that fit no probability distribution and are the basis for some of Complexity Theory...I alas do not have much interest in the subject. With regard to outliers, which are not chaotic, but simply fall outside any predictive algorithm, Mandlebrot, according to Taleb, has produced some of the necessary math. Of course Taleb's books, "Fooled by Randomness" and "The Black Swan" are definitely recommended introductory texts on what's wrong with some of our current theories -- I can't pretend to understand the math. And it bores me...I get the point and can decide what distribution fits, or when no distribution fits...which in my mind gets me close enough. Again I hope Mike and Merf set up here and add a bit of commentary.
ReplyDeleteFinally - it is fairly easy to produce random noise. Go into your MS Excel and play around with the RAND function. Since I haven't figured out how to post pictures into comments yet I just went ahead and pasted a random sequence of numbers between 1 and 100 into your original blog post. It should look like noise.
Okay Mooch - I know chaoticness is tough to model and chaotic processes are by their nature unpredictable, but I assume that since the majority of reality is chaotic (weather, markets, war), it might figure into what you guys do more prominently. I do sense from the other big thread that the preference is to pick a predictable framework for our models and if we are right, we are geniuses and if wrong, we explain it away.
ReplyDeleteOuch...Tasha's comment was to highlight what we don't want to do..."Explain it away". Those comments are excuses for why their chosen framework did not work for sure, and why they got it wrong to cover their tails. Not as substitute for the proper thinking about problem that should have been done in the first place...which is what we are critical of in this blog...the lack of critical thinking.
ReplyDeleteThink about how we model the weather...in particular Hurricanes. Weather folks take a lot of heat but you can see on TV how they use up to seven separate models to predict a storms path. They are doing pretty pretty good job in my opinion. Now that's a far cry from being able to predict what the overall hurricane season year to year might look like.
You will not see those guys (the ones generating the paths not the guys reading the weather news) using any of Tasha's excuses because they leave very few stones unturned...if the Hurricane changes course they will have a pretty good understanding of why, what happened in the atmosphere, and the facts to demonstrate why the course change occurred.
Some of it's engineering, some of it's science, and some of it's art...but it all has to be there.
Finally, since prediction is very hard to do, comparative analysis is easier and when we make flexibility (for instance) an attribute of the analysis because of the known uncertainties...we know that there are things that we don't know...and we know there are things we think we know...that are not so...we weave flexibility in.
For instance...we know Boyd was thinking correctly about the F-16 from the standpoint of building the greatest dog fighting aircraft ever conceived and built...but we also know the type of dog fighting he was thinking about (knife fight in a phone booth) went away and in hind sight the Pentagon fighter wars probably hurt us more than helped us...and it set a precedence for how we think about our current fighter force structure...and that mind set is out dated now by 40 years.
Mooch,
ReplyDeleteThe F-22 does look pretty good in a phone booth.
That said, one of the main reasons for my hang-up with modeling chaotic systems is that they fascinate me. I figure they have a lot to do with why we are pretty bad at predicting the future needs of the defense establishment much more that 5-10 years out. I heard a talk by George Friedman for STRATFOR a few years ago in which he pretty well illustrated our failures to forecast the threat much more than ten years out. There are many reasons for this, but one of the things I seek out of these exchanges is to perhaps gain a better understanding of how things work. For example, I am convinced that one of the next major bubbles to burst is the entitlement bubble. The socio-political implications for our National security are substantial (both in implications for our ability to resource Defense spending, and our general political health). See you on another thread later.
With regard to the F-22, indeed those maneuvers are out of bounds, but do they exist for our ego or to gain the OODA loop advantage in air to air combat? Since close air to air combat does not seem like a significant threat with next generation radars and missiles what is the purpose of all that maneuverability besides wowing the crowd at air shows?
ReplyDeleteDon't get me wrong that's pretty important--the wow factor--and I also know those maneuvers are still important while in a missile/aircraft engagement as well. It's hard to jump out of the way of an incoming missile when it's moving at Mach "Fast" pulling "Butt-load" g's. The F-22 can almost jump backwards.
By the way I'm the biggest non-fighter pilot advocate for the F-22 in the Pentagon...I am not a fan of the F-35 however. The purpose of the F-22 is not to win dogfights...it's purpose is to be the biggest baddest aircraft in the sky...that ever flew ("Toruk Macto" in Na'vi). To do anything else, we have to have air supremacy...period. No matter when, no matter where. The few F-22's we are buying does not guarantee this in the future.
Which brings us to flexibility...if we cannot predict all the potential futures we might face we can strive to figure out a force structure that has as many flexibility features as possible. What if GPS suddenly goes away, for instance? That would surely burst our entitlement bubble.
...and flexibility is the key to airpower!
ReplyDeleteSo, when do we get to theories of evoluton and its role in analyzing force structure? Is it in our crosscheck?
Sorry I’ve been remiss in commenting on the last few topics…Much to say, not enough time to say it, but I’ll try to offer up some thoughts that may be of interest (to me if to no one else). All of this is a little disconnected, but I'm told that one must accept imperfection in the blogsphere...and inflict that imperfection on others. Here goes.
ReplyDeleteConsidering the original issue of “frameworks” I would probably add a couple of considerations, some of which overlap each other or those already given by Ogre or which supplement along another interesting axis. Since the goal here seem collective exhaustion rather than mutual exclusion--and food for thought rather than “the answer”--that may be OK.
Scale: The scale of resolution for the relevant problem is a critical, and will inform the frameworks that are relevant…and the relevant framework may be different for the same system viewed form a different scale. Are you interested in the interactions of individual elements? Or are you interested in the collective behavior in some aggregate or average sense? (I.e. are you measuring the energy of individual particles, or do you just care how hot the water is? Do you care about the biology of an individual fish, or the behavior of the entire school?)
Decomposability: Also driven by scale of resolution, this is the question of relative strength/import of interactions between and within elements. Systems can be decomposable (i.e., elements are independent of one another), nearly decomposable (there is data loss by considering them as independent systems, but the loss may be acceptable under certain conditions), and systems that are not decomposable (i.e., the interactions are fundamental to the behavior of the system). This isn't very far from the linearity/non-linearity discussion mentioned by Mooch. When we model the world linearly, we assume the system is decomposable, in some sense. This ALWAYS makes the math easier and is almost always a suspect assumption. (In my opinion.)
Complexity: This is "all about" emergence--not really "all" I suppose, but cartoon answers to complex questions sell tickets and buy votes, right?. One of the fathers of complex systems, H.A. Simon, wrote that one can naively define a complex system as “one made up of a large number of parts that interact in a non-simple way…the whole is more than the sum of the parts…in the sense that given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole.” (“Architecture of Complexity” 1962) Once again, implicit in the definition here is the criticality of scale.
These are just a couple of toss-ups, though the issue of complex systems is incredibly rich.
I might also have a (really) minor quibble with the characterization of chaotic behavior as " non-random, non-periodic, unpredictable." Chaotic behavior is wholly deterministic and predictable if the initial conditions are given. The critical issue is usually phrased as "sensitivity to initial conditions" a condition in which perturbations from a given position result in deviations from the previous outcome or trajectory. I know, this is REALLY picky, but there you go.
Speaking of "chaos" and coming to the question about the term "stochastic." Think for a moment about the set of "random" numbers offered up by Mooch. It's worth noting that the numbers given by Mooch--"random noise"--are not actually random. Technically, they're usually described as pseudo-random, but the "generative process" is actually a chaotic one (if you have access to the seed value...in the Analysis Tool Pack, for example). Given a "seed" value for the "generator" (an initial condition) the same sequence of values will emerge every time (assuming the code is correctly written)--the sequence is deterministic--and eventually--if you want a LOT of random numbers--it will repeat. But if you change the initial conditions (the seed), then the trajectory changes and is not predictably correlated with the sequence generated by the previous seed. It's an exploitation of chaotic behavior to produce "random" noise.
ReplyDeleteWhat else....Oh. Stochastic. Annotatively, I suppose there isn't necessarily a difference between "random" and "stochastic." Connotatively, there is in many minds, though, which is why Mooch suggested it way back when. First, the terms sounds "scientific," so it can establish bona fides in the minds of the non-technical (or, alternately, scare them into thinking you're about to do some crazy math shite.) Generally, though, I use the terms stochastic to describe a process over which I can claim some measure of understanding for the underlying distribution. That's just a personal preference, though.
Generating "chaotic patterns": Lots of (relatively) easy ways to do this. Consider, for example...
- The logistic map which x(n+1)=r*x(n)(1-x(n)) which generates the bifurcation diagram and is almost always a "first lesson" in chaos. (The Wolfram article on the logistic map is pretty good: http://mathworld.wolfram.com/LogisticMap.html).
- The Lorenz attractor is another famous example of a system that exhibits chaotic behavior. In this case, the system is a 3-d non-linear, dynamical system defined by 3 VERY simple first-order differential equations. The issue here is that given a set of defining parameters, trajectories of the dynamical system.
- Another favorite of mine is the Mandelbrot Set. If the recursive sequence z(n+1)=z(n)+c (where c is a complex number a+bi) is bounded, then the point c is in the Mandelbrot set. The boundary of the Mandelbrot set is a complex landscape in which small change is the value of c (can) result in radically different convergence behavior--chaos. You get very similar phenomenon in using Newton's method to compute the complex roots of polynomial equations--small changes in the initial condition (along the boundaries of the basins of attraction) can change the root computed.
Lots of really pretty pictures in the "chaos" world, but I'm not smart enough to figure out how to post them here.
I also wonder if our inability to "predict" the future more than a few years out is less about our ability to correctly quantify all of the variables and forecast the deterministic trajectory correctly (an interpretation of the failure rooted in chaos) than it is about a fundamental failure of the mental model we impose on the world.
ReplyDeleteThe world with which we’re interacting is a complex system involving partial information, competing objectives, volitional actors, a host of “stochastic” environmental influences, etc. We’re talking about a SERIOUSLY complex complex system, and human beings don’t conceive of and estimate probabilities (and forecast) well in this environment. Consider the simple Beer Game, for example.
The game consists of multiple players, each representing an element of a SIMPLE distribution process (e.g., supplier, wholesaler, retailer, etc.). Each player makes decisions about ordering, distribution, and sales. There is nothing random in the process, and inevitably we see chaotic (in the non-technical sense) behavior emerge, even in very small games. Just think of what this means in terms of our cartoon explanations for big, geopolitical trajectories.
Then there’s Mooch’s reference to Taleb and the concept of failed stochastic models. Basically, Taleb claims (correctly) that when we model stochastic processes, we tend to use distributions that make our lives easier—especially the normal or Gaussian distribution. Unfortunately, many interesting events are not distributed this way. “But,” you say, “that’s why we have other distributions (exponential, gamma, hypergeometric, etc.).” Right you are, and one of the reasons we think we can get away with using the normal distribution so often—which makes the math generally REALLY easy--is that when dealing with “averages” (or means) we can apply the central limit theorem, irrespective of the underlying distribution. Naively, the rule says that if the statistic of interest is a mean (arithmetic average) and the sample size is large enough, the average has a normal distribution with a mean equal to the underlying population mean and a variance equal to the population variance divided by the sample size. For a math geek, this theorem is a crown jewel among theorems, except…
ReplyDeleteI didn’t state the theorem technically, and I left out a HUGE assumption (as do most analysts when they mis-model real-world systems). For the central limit theorem to apply, the population mean and population variance must be finite. No problem, right? Well, if you’re dealing with many distributions, then there is, in fact no problem. But there’s one particularly nasty (and prevalent) distribution that causes us a REAL problem: the power distribution.
A power distribution is one which (asymptotically) acts like the function x^(-k) where k>1. (Technically, I need to throw in some additional constraints and a normalizing constant, but this is the basic form.) The problem here is that for k<2 the mean and all higher order moments (including the variance) are unbounded and for 2<k<3 the mean is bounded, but all higher order moments (including the variance) are not…and the central limit theorem does NOT hold.
So what? Well intensity of wars (with k=1.72), terrorist attack severity (k=2.4), and many other phenomenon follow power laws, and thinking of these events in a frequentist way with bad assumptions regarding underlying distributions is just plain dangerous. Just ask the quants involved in assessing risk n LTCM (1998 crash), for example, or some of the more recent fiscal disasters…It has been shown that stock market fluctuations follow a power law, so risk management through expected value is…”problematic.”
@ Merf - "Psuedo-random" like "quasi-wool"... basically like the seat covers I sold to my friends in high school. Almost wool, or wool like...but not wool...and definitely not random. Good clarification.
ReplyDeleteI posted a visual for the power law in the original blog.
Trust me to be the guy who picks a fight over Pseudo-random vs. random. I don't really have ny beef with either term (except in very technical usage). I was trying to make a point about the nature of deterministic trajectories with sensitivity to initial conditions, and you gave me a tailor-made opportunity.
ReplyDeleteNow, though, I'm left wondering where one harvests "quasi-wool." Is if the fur of the nauga, shorn in the process of creating naugahyde? Did the "wool" cover the seats of a Camaro driven by a man with a mullet?
I also posted an amended version of the "power law plot" with negative exponent. The "heaviness" in the tail is the point with power laws. That "heaviness" represents the relatively high probability of extreme events--a very different probability than in the center-heavy, thin-tailed Gaussian (normal) distribution.
Very nice...and it was a 67 Firebird belonging to a good friend...he may still have a mullet.
ReplyDelete