The hedgehog, the fox, and the chaos they've created.

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  • "The fox knows many things, but the hedgehog knows one big thing."

    This is one of the reoccurring themes of work dealing with the methods involved in what makes our minds tick when formulating ideas about the world. There are those with more singular beliefs and those with understandings that are diverse to the point where they may seem without certainty. I view this concept not so much as a direct classification but a spectrum where people can be both foxes and hedgehogs to different degrees. Nevertheless, it has some serious truth to it and we might just be witnessing it with our football team.

    Pete Carroll is the consummate hedgehog. His singular belief above all else is competition. He's a competitor, his players will be competitors, his practices are competitive, and come game day, our players aren't so much playing a game as they are following their refined competitive instincts. He's conformed his idea to what should constitute the proper organization to support his teams. This is to have an entrusting owner, a general manager that if all else fails must abide by his decisions, and to place himself at the head of the table to instill his singular belief throughout the organization. This isn't to say PC is a man of few ideas. He's bounced around the league, gathered his own ideas, and even incorporated influences from the late great John Wooden. But these ideas have been honed under the scope of competition and "Win Forever". The man doesn't just speak or act in cliches formed of his own prerogatives and ideals, he is them.

    John Schneider, on the other hand, is the consummate fox. "No stone unturned" is commonplace when describing his approach to obtaining players. CFL, character issues, too short, too long, or hasn't even played the position? I imagine he would shrug assuming he even cared enough with the various schools of conventional wisdom to begin with. However, checking every rock is a product of something occurring in his intellect. It's the realization that for him to best fulfill his role and his potential as a GM and FO figurehead, almost no idea will be dismissed (unless it's you, Tebow). He may be a product of Green Bay's remarkable FO where they hold the belief in building through the draft as singularly as PC believes in competition, but, as we've seen with his signings of Rice, Miller, and Harvin, he holds no loyalty to the GB method. His only loyalty is to the opportunities of the market and the multitude of ways he may use that for our franchise's benefit.

    The potential of what happens when a fox like JS is teamed under the scope of a powerful singular belief like PC's is difficult to frame. Many of our poster's struggle calculating this very thing when attempting to predict what our future will hold with mulling over the products of this relationship. Bad ass defense + bad ass run game x possibly great QB + sick off-season acquisitions = many titles? #1 in DVOA + sick off-season acquisitions = #1 in DVOA with authority....? Homerism x homerism^homerism = EVERY TITLE TIL RW RETIRES??!?! But there is a theory in place that can map out a basis for why our current success has been cultivated so suddenly and what our future potential may be even beyond the current squad.

    Enter Chaos Theory.

    Chaos theory involves a definition of chaos not necessarily meaning destruction or the like. The severe range of possibilities of this theory is best described by the expression: "the flap of a butterfly’s wings in Brazil can set off a tornado in Texas". It's a concept about how even small and seemingly completely unrelated events early enough in a process can culminate and cultivate even drastic events if it is capable of and allowed to grow exponentially. Nate Silver features a story in his book, The Signal and the Noise, where a group of meteorologists discovered they had predicted both a sunny day and a storm with seemingly the same data. They discovered that someone had rounded the 4th decimal of a single, seemingly unimportant number up and that fraction of a fraction was the difference between sun and storm in this particular situation. That's what a 0.0001 difference in barometric pressure in a small section of the atmosphere can potentially do.

    What I wish to hypothesize for the members of our forum is aspects of chaos theory, mainly those about dynamic environments and exponential growth, very much apply to the growth of an organization like our Seahawks. It's not just something that can happen but it may have been a phenomenon that has happened and may still be happening. Pete Carroll's singular belief of competition may seem like a minuscule, unequivocal concept when he's not using it to push out holdover veterans from a defunct regime. John Schneider's prudence in evaluating not just every player seemingly known to man but even his own methods of evaluation may occasionally seem like exercises in futility when he's looking at players with seemingly no chance of making our team let alone any team in the NFL. However, with chaos theory in mind, I ponder about how the "Always Compete" mantra and these revolving nobodies taking their only shot at making it to the big leagues may contribute to just a small increase in work ethic and output. I ponder about how this small increase, even as little as a 0.0001 increase, is just the sort of thing that culminates a bottom-feeder into one of the quickest and total rebuilds in the NFL - and possibly even a championship.
    Last edited by BirdsCommaAngry on Sun May 12, 2013 9:58 pm, edited 2 times in total.
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  • Interesting take. Dr. Ian Malcom approves.

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  • as a person who follows the weather ridiculously close, i approve. however. watch the clouds boil in the spring, it dont seem so chaotic...



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  • "Let the Chaos reign down from the mountaintop...", thus spake Carrollthustra...
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  • Brain hurts bad, but interesting take on Chaos Theory related to John and Pete.
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  • A butterfly in Brazil can cause a tornado in Texas?
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  • BirdsCommaAngry wrote:This isn't to say PC is a man of few ideas. He's bounced around the league, gathered his own ideas, and even incorporated influences from the late great John Wooden. But these ideas have been honed under the scope of competition and "Win Forever". The man doesn't just speak or act in cliches formed of his own prerogatives and ideals, he is them.


    A little bit about what lead him to form his philosophy from Pete's own mouth.

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  • Throwdown wrote:A butterfly in Brazil can cause a tornado in Texas?

    Yes, according to Chaos Theory. Basically everything is connected and the smallest change can lead to the largest change or most improbable change if left unchecked or tried again under the exact same circumstances. Modern Magic is the very definition of such thinking.
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  • I dig it! Very very nice description. Chaos theory actually captures a lot of that je ne sais quoi that has all of us über Pumped and Jacked. Woo!
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  • My first though was Ian Malcolm as well. Haha
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  • BirdsCommaAngry wrote:What I wish to hypothesize for the members of our forum is aspects of chaos theory, mainly those about dynamic environments and exponential growth, very much apply to the growth of an organization like our Seahawks. It's not just something that can happen but it may have been a phenomenon that has happened and may still be happening. Pete Carroll's singular belief of competition may seem like a minuscule, unequivocal concept when he's not using it to push out holdover veterans from a defunct regime. John Schneider's prudence in evaluating not just every player seemingly known to man but even his own methods of evaluation may occasionally seem like exercises in futility when he's looking at players with seemingly no chance of making our team let alone any team in the NFL. However, with chaos theory in mind, I ponder about how the "Always Compete" mantra and these revolving nobodies taking their only shot at making it to the big leagues may contribute to just a small increase in work ethic and output. I ponder about how this small increase, even as little as a 0.0001 increase, is just the sort of thing that culminates a bottom-feeder into one of the quickest and total rebuilds in the NFL - and possibly even a championship.


    While the effects described are there, I don't really see the chaos theory link so strongly. To me what Pete and John do simply seems practical and wise, and looking at things without being artificially constrained by "conventional wisdom". (I see it more along Kaizen lines than Chaos Theory lines) Pete has put a lot of effort into learning and applying Bill Walsh's ideas, in particular, and has brought a lot of great ideas with him.

    As an example, I believe it wasn't the players or the coaches that lost Super Bowl XL, or even the refs crappy and biased work. The real difference was the Seahawks front office, and their failure to create competitive depth throughout the roster. (Just the thing PC/JS excel at, "next man up".) From the middle of the 2nd quarter on, the Seahawks safety was a 3rd string practice squad player, essentially a man off the street a few weeks before. Marquand Manuel, our 2nd stringer (after Ken Hamlin, but better than most 1st stringers in the game) pulled a hamstring, out for the game. Enter Etric Pruitt, a HUGE dropoff from Marquand Manuel. Pruitt was taken advantage of for 2 of the 3 Steelers TDs. The Pittsburgh coaches identified the weakness the moment he went in the game, and a little luck (the 75 yard Willie Parker run, with Pruitt missing his assignment) and a little cleverness (the Steelers came up with that trick play that went for a 40 yard TD to Pruitt's guy).

    You could say the real reason the Seahawks lost Super Bowl XL was a random bar fight in October, the one Ken Hamlin got his skull fractured in and was out for the rest of the season. The guy that knocked him out was a random dude that Hamlin got in a shoving match with because they were in the same physical space in a club. Any random event (one gets a phone call or sees a hot chick) that doesn't have them in the same physical space, and the fight doesn't happen and the Seahawks win the Super Bowl, so the theory would go.

    I don't really buy that the outcome would have been different; more than likely, the thinness of the roster would have popped out at some other time in some other way. It was an achilles heel that got exploited by a capable opponent.

    This front office and coaching staff will never lose a big game that way due to a huge dropoff from the next man up. They will have a more capable next man up, without such a huge dropoff; they will rapidly adjust their scheme during the game to fit the personnel they have available. Because they have created the competitive depth, they will nearly always have good and viable options for on-the-fly scheme and personnel adjustments.
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  • First off: Very cool post. I would much rather read something like this than another thread of "RICHARD SHERMAN TALKS TOO MUCH AND HAVE YOU SEEN OUR UNIFORMS?" and the like that dominates the board in the offseason.

    But I do have a problem* with it: *pushes up nerd glasses*. You refer to chaotic systems as those having "dynamic environments and exponential growth". While they are most certainly dynamic, they are not necessarily exponential. There is nothing in chaos theory that stipulates that a chaotic system grows exponentially, or even grows at all. I think you may be confusing "exponential" with "non-linear", as non-linearity is the reason that chaos arises.

    I'm not a mathematician, so I might be sticking my foot in my mouth and someone who actually knows what they are talking about might come on here and yell at me, but a non-linear system is one where you can't simply determine the values of all the variables that influence an outcome, plug them into an equation, and come up with a predictive result. Chaos arises because the equation behaves differently depending on the state of the system, making predictability difficult or impossible. Just ask an engineer.

    If you take the "exponential growth" idea, and apply it to a population of animals, that is by definition a linear process: if you have 8 offensive tackles, and their population doubles every day (somehow, don't ask), you will have 16 offensive tackles after 1 day, 64 offensive tackles after 2 days, and so on. This is a deterministic and therefore linear equation (despite not looking like a "line" if you draw it on a graph.

    But other variables exist that affect the growth of offensive tackles: Coaches will come and cut them if they get too populous, limiting the growth. Maybe an offensive tackle that gets cut pulls a reverse Sweezy, switches to the defensive side of the ball, and starts handing out concussions to Offensive Tackles, dwindling the population even further. This gets the coach fired, and a NEW coach that isn't so quick to fire offensive tackles gets hired, so the population starts growing again.

    This is a system that is NOT deterministic. It is too complex to look at and say "there are 12 offensive tackles, so there will be 18 tomorrow". That's why it is chaotic.

    So if PC and JS have REALLY created a chaotic system, where the output we measure is "success". Is that a good thing? A little bit TOO much competition and suddenly we tip the success graph in the wrong direction, and instead of being Super Bowl contenders we get so bad that we get relegated to the CFL.

    Here is where I should note that I don't think that this is the case. Competition is fantastic. That's why the Seahawks have BECOME fantastic in such a short time. I'm just having fun with the whole chaos theory idea.

    But really.....success IS chaotic in the NFL. A team's talent level relative to another team DOES follow nonlinear dynamics. My admittedly silly example of Offensive Tackles really isn't that far off from what actually happens. So maybe my problem* with the post is that I know you're right.....the Seahawks, like every other NFL team, is just an incremental change away from dropping back into last place.



    *note: I don't have an actual problem with the post
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  • olyfan63 wrote:the 75 yard Willie Parker run, with Pruitt missing his assignment


    The rest of your post was fantastic, however, I fail to see how Pruitt who was responsible for the other side of the field while Boulware was on the side that the run actually went to, and took a horrible angle on, equates to Pruitt missing his assignment.
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  • Cool post, interesting food for thought, thanks.

    Incremental improvement that is fuelled by competition adds up eventually to exponential change quickly. It is a hallmark of this regime and something other teams are now envious about and beginning to attempt to emulate.

    Chaos theory may be pushing the envelope some but it is an interesting concept added to an incredible unremitting amount of plain hard work in rolling of the rocks to find and capture any available incremental improvement while making the team stronger in terms of it's depth speed size strength and toughness. In short before long chaos theory may soon not have as much application to the Seahawks team. Put another way our front office is doing their job well and trying to compete there to be the best at it while putting together a deep young cap smart physically dominating big strong fast football team that won't back down to any amount of competition regardless of the time of day or location. The evidence is starting to clearly point to the quantifiable proof they are succeeding as the team that was here in 2009 isn't any more. The coaching isn't too shabby either.

    Since 1976 I haven't seen a FO that has done more in such a short time to instil and begin the perpetuation of a team mentality that is developing into a championship team.
    Last edited by jammerhawk on Mon May 13, 2013 2:34 pm, edited 1 time in total.
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  • olyfan63 wrote:While the effects described are there, I don't really see the chaos theory link so strongly. To me what Pete and John do simply seems practical and wise, and looking at things without being artificially constrained by "conventional wisdom". (I see it more along Kaizen lines than Chaos Theory lines) Pete has put a lot of effort into learning and applying Bill Walsh's ideas, in particular, and has brought a lot of great ideas with him.


    Their process is a certainly a very Kaizen process focusing on linear growth. However, to me the results of these linear processes thus far appear to have been much less linear than they are exponential and perhaps even chaotic. Team DVOA might offer the best generalization about this effect. We were 30th in 2010, 19th in 2011, and then spiking to 1st in 2012. This growth may appear to be somewhat linear since we improved from year to year and it does fit into a presumable curve, but my instincts say this type of leap into 1st is very uncommon. The curve we're seeing is more happenstance because the curve is an average and many single examples shy away from that average. I imagine leaping ~18 places from a ranking in the 20's to the top 10 may be fairly common but not into the pinnacle of the highest tier of the league as we did according to their website. We could attribute this spike largely to RW stepping up but we were ranking well in DVOA even when he was struggling. We seemed to have learned to embody our potential and then some in a single off-season. If the result isn't truly linear, then the equation, or process, can't logically truly be linear either.

    Excellent points, DaveyP, and like you I'm no mathematician either. However, the growth we're seeing may not be the kind that operates like a traditionally noted chaotic ecosystem like your example based on the population of animals. With our players and FO, it'd be more like growth of the amount of expertise our guys possess, the amount of habits/routines conducive to winning in the NFL we have, and the amount of critical thinking the members of our franchise are able to summon seemingly on the fly. The population we're referring to would then be a population of cognitive functions which abide by different rules of potential than the nature-like parts of nature. If it's chaos, it's a ideally a disciplined form of chaos, which will hopefully prevent the same methods of insane growth from sending us crashing down to from whence we came while guys like PC and JS are still here.

    If it's even possible, we still may never know enough to be able to understand an organization like the Seahawks well enough to deem it a deterministic system but what the Seahawks do can be broken down into what it is that increases our probabilities of winning. I reckon the effect of what "Always Compete" and "No stone unturned" represent are a few of the principles at the sources of what most greatly increases our odds, and chaos theory, or at least parts of it, may provide somewhat of a road map for how this occurs.
    Last edited by BirdsCommaAngry on Mon May 13, 2013 2:44 pm, edited 2 times in total.
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  • DaveyP wrote:
    I'm not a mathematician, so I might be sticking my foot in my mouth and someone who actually knows what they are talking about might come on here and yell at me, but a non-linear system is one where you can't simply determine the values of all the variables that influence an outcome, plug them into an equation, and come up with a predictive result. Chaos arises because the equation behaves differently depending on the state of the system, making predictability difficult or impossible. Just ask an engineer.

    If you take the "exponential growth" idea, and apply it to a population of animals, that is by definition a linear process: if you have 8 offensive tackles, and their population doubles every day (somehow, don't ask), you will have 16 offensive tackles after 1 day, 64 offensive tackles after 2 days, and so on. This is a deterministic and therefore linear equation (despite not looking like a "line" if you draw it on a graph.

    But other variables exist that affect the growth of offensive tackles: Coaches will come and cut them if they get too populous, limiting the growth. Maybe an offensive tackle that gets cut pulls a reverse Sweezy, switches to the defensive side of the ball, and starts handing out concussions to Offensive Tackles, dwindling the population even further. This gets the coach fired, and a NEW coach that isn't so quick to fire offensive tackles gets hired, so the population starts growing again.

    This is a system that is NOT deterministic. It is too complex to look at and say "there are 12 offensive tackles, so there will be 18 tomorrow". That's why it is chaotic.



    No yelling, but the idea of chaos theory originates from the usage of differential equations when predicting events in unstable equilibriums. An unstable equilibrium, is as an example, a pendulum at 'rest' if it were straight up. The game plinko on The Price is Right is predicated on this same idea, so you can't just drop the disc and aim. (A stable equilibrium would be a pendulum that is resting while straight down.) Neither outcome (pendulum either swings left or swings right from its position) is predictable and so the starting conditions of the solution to the differential equation change. Over time, these build up into unexpected and unpredictable solutions which will be non-linear as they'll contain gaps, apexes, and so on, but it's not so much that we can't enter all the starting conditions as much as we can't guarantee which starting condition will be chosen in the system. So in this case, I'm not sure chaos theory actually applies.

    Edit: Population growth is also predicted by differential equations, for the record. I don't think I've ever heard anyone describe it as chaotic until you introduce outside forces.
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  • olyfan63

    I would say that the drop off when we lost Clemons was extremely significant.
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  • Throwdown wrote:A butterfly in Brazil can cause a tornado in Texas?

    No. There are millions more butterflies in Brazil than tornadoes in Texas. If a single wing flap of a butterfly in Brazil could cause a tornado in Texas, there would be nothing left of the entire southern half of the country.

    And why would a butterfly wing flap in Brazil be more likely to cause a tornado in Texas than a butterfly wing flap in Texas?
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  • Throwdown wrote:A butterfly in Brazil can cause a tornado in Texas?

    Yes grasshopper. :P
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  • BlueTalon wrote:
    Throwdown wrote:A butterfly in Brazil can cause a tornado in Texas?

    No. There are millions more butterflies in Brazil than tornadoes in Texas. If a single wing flap of a butterfly in Brazil could cause a tornado in Texas, there would be nothing left of the entire southern half of the country.

    And why would a butterfly wing flap in Brazil be more likely to cause a tornado in Texas than a butterfly wing flap in Texas?

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  • BlueTalon wrote:
    Throwdown wrote:A butterfly in Brazil can cause a tornado in Texas?

    No. There are millions more butterflies in Brazil than tornadoes in Texas. If a single wing flap of a butterfly in Brazil could cause a tornado in Texas, there would be nothing left of the entire southern half of the country.

    And why would a butterfly wing flap in Brazil be more likely to cause a tornado in Texas than a butterfly wing flap in Texas?


    People take the butterfly statement waaaaaay too literally. It's a metaphor for how a small deviation in a chaotic system causes huge variance at the other end of the spectrum when working with the math. As usual, people didn't bother to check the meaning because the analogy sounds cool.
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  • Buzzkill. I was having fun with it. I know it's a metaphor. And I know people take it way too seriously. People are idiots.

    But just imagine the havoc we could create in other countries if we could harness the climate-altering power of butterflies!
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  • kigenzun wrote:"Let the Chaos reign down from the mountaintop...", thus spake Carrollthustra...
    :th2thumbs:


    now that is nice right there. . . .
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  • Great post. Ultimately we are talking about the big show. Winning the Super Bowl. And sometimes the best team doesn't always win. With the age of our team that does decrease the likelihood of injury, but really you have to get lucky. Those couple calls that could have gone your way, that holding call that got missed, that questionable pass interference. The good news is- that after this year, with the popularity of our team, those highly questionable ref calls (wether for or against) will get more media coverage, and that is a good thing. The reason is because we are a great team, and if officiated correctly (05 SB) we will win!

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  • bestfightstory wrote:olyfan63

    I would say that the drop off when we lost Clemons was extremely significant.


    Excellent point. In its own way, the loss of Clemons was pretty similar (to the loss of Marquand Manuel in SB XL) in its devastating impact on the Seahawks overall game. With Clemons in the game, we beat Atlanta, no question. We get either a crucial sack, and/or a crucial, pressure-related pick, or both. Our run defense is stronger, and Atlanta doesn't go out of their way to run at Clemons, and our run D doesn't allow a washed-up Michael Turner to look quite so much like Emmett Smith in his prime. Losing Clemons = similar impact, but very different reasons.

    Clemons was the guy whose play partially covered up for the lack of interior pass rush from our DT's. Losing Clemons highlighted the Seahawks lack of pass rush from the DT spots. It highlighted Bruce Irvin's overall inexperience, and tendency to get locked up with, and mowed over by, big O-linemen. Losing Clemons exposed that too much of the Seattle D was held together by chewing gum and baling twine. Read: Jason Jones(out), Alan Branch(very little pass rush), Red Bryant (with a bad wheel), and toss in Bruce Irvin and Gregg Scruggs (rookies still getting schooled), then toss in Leroy Hill (old and slowing)

    The difference is that the PC/JS front office isn't taking it lying down. They have aggressively addressed each of those issues. Clemons out? No problem, we got Cliff Avril, a near Clemons clone. We got Michael Bennett, who also can pick up a lot of that slack as a pass rushing DE or even rush from a DT spot in some packages. PC/JS addressed the DT issues by drafting Jordan Hill and Jesse Williams, and we can hope Greg Scruggs and/or Jaye Howard show us something this year. Marcus Trufant slowing/getting beaten? OK, bring in Antoine Winfield, an upgrade at slot corner.
    PC/JS seem to have a similar incremental improvement program going in the secondary, and, with TC, in the O-line area as well.

    Another huge, huge risk area is Marshawn Lynch, and the overall beating he takes. I love, love, love the Christine Michael pick. Lynch goes down, and other teams still need to fear and adjust to our featured running back. They even addressed the MRob aging dropoff/injury risk with the Spencer Ware pick.

    Right now, the biggest risks to this team are: 1) Losing Russell Wilson; 2) Losing Earl Thomas. In both cases, the dropoff would be huge.

    I'm not seeing the chaos theory as much as shrewd, probability-theory-based gamblers, looking to improve the percentage of 1-on-1 battles their side wins, with whatever combination of players are on the field, knowing that on any one given play during a game, one player winning his 1-on-1 battle could make the difference between winning or losing the game. (And yes, more like combination of scheme and player executing within that scheme)

    I'm not really trying to put the chaos theory explanation down... in truth, I can't say I really understand chaos theory or how and when it can be useful, and especially its usefulness in predictions and in guiding decisions. I'm seeing it more as an after-the-fact, gee-whiz-that's-fascinating explanation of what happened. I acknowledge my ignorance in that area, and if someone can shed more light on how the chaos theory model is predictively useful, I'm all ears. For now, applying Occam's Razor, the simpler probability-based explanation is one I can make sense of.
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  • This is perhaps the best piece of sports psychology/analytics I have ever read.

    Thank you.
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  • BlueTalon wrote:Buzzkill. I was having fun with it. I know it's a metaphor. And I know people take it way too seriously. People are idiots.

    But just imagine the havoc we could create in other countries if we could harness the climate-altering power of butterflies!


    I was adding onto your statement, but I'm glad to see I harshed your vibe. :thirishdrinkers:
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  • A public service announcement for anyone who might possibly care. [I am a mathematician by education only.]

    A linear equation always "looks like" a line if it's graphed. If you graph the thing and it doesn't look like a line, it's not linear. Guaranteed.

    The confusion in the general public arises because mathematicians and scientists often use the term "non-linear" to mean "not simple". If you can describe something with a linear equation, it's dead simple -- and boring -- so "non-linear" has come to mean "more interesting to math and science geeks".

    Exponential functions are certainly non-linear, but are actually pretty simple. As an example, consider the classic story about the smart Chinese guy who asked the Emperor for one grain of rice on the first square on the chessboard, two grains of rice on the second, four on the third, eight on the fourth, and so on. By the time you get to the 64th square, you're at something close to 2 quintillion grains of rice. Exponential functions grow (or decay) very quickly -- but once you understand how they work, they're entirely predictable.

    A chaotic process is one where (as someone else pointed out) small changes in the initial state can result in a state that is difficult to predict down the road. This is the "butterfly effect" also mentioned, where killing one butterfly might result in tornadoes in Texas. However, chaotic processes aren't non-deterministic: in theory, we can calculate the weather with 100% accuracy. It's just difficult to do. Chaotic processes typically cannot be represented in what we would call "closed form", which means they evade analysis and have to be simulated using software. Climate change models are chaotic in the sense that they're very sensitive to small changes in initial conditions and assumptions, which is one reason why they still generate controversy.
    HawkHack
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  • In other words, there are so many variables, that by the time you get done identifying and analyzing and measuring them, and putting them into a predictive model, whatever event they were to be used to predict is long over. And not only that, but in the meantime, it's become clear that only half the actual variables have been identified.
    So I think you are saying chaos theory is not practical or useful as a reliable predictive tool. (Though I think you are allowing for models that simulate many variables to be *useful* for analyzing *potential* outcomes of different choices)

    It seems "injuries" are a big part of the chaos of NFL football that make things so hard to predict.

    Speaking of additional variables, in my writeup above, I completely neglected to mention the free agent signing of DT Tony McDaniel. Maybe he will turn out to be the X-factor that turns a few games. Impossible to predict at this time... but he gives us a few more chances.
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    olyfan63
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  • HawkHack wrote:A public service announcement for anyone who might possibly care. [I am a mathematician by education only.]

    A linear equation always "looks like" a line if it's graphed. If you graph the thing and it doesn't look like a line, it's not linear. Guaranteed.

    The confusion in the general public arises because mathematicians and scientists often use the term "non-linear" to mean "not simple". If you can describe something with a linear equation, it's dead simple -- and boring -- so "non-linear" has come to mean "more interesting to math and science geeks".

    Exponential functions are certainly non-linear, but are actually pretty simple. As an example, consider the classic story about the smart Chinese guy who asked the Emperor for one grain of rice on the first square on the chessboard, two grains of rice on the second, four on the third, eight on the fourth, and so on. By the time you get to the 64th square, you're at something close to 2 quintillion grains of rice. Exponential functions grow (or decay) very quickly -- but once you understand how they work, they're entirely predictable.

    A chaotic process is one where (as someone else pointed out) small changes in the initial state can result in a state that is difficult to predict down the road. This is the "butterfly effect" also mentioned, where killing one butterfly might result in tornadoes in Texas. However, chaotic processes aren't non-deterministic: in theory, we can calculate the weather with 100% accuracy. It's just difficult to do. Chaotic processes typically cannot be represented in what we would call "closed form", which means they evade analysis and have to be simulated using software. Climate change models are chaotic in the sense that they're very sensitive to small changes in initial conditions and assumptions, which is one reason why they still generate controversy.



    Thanks!! I figured a mathematician that actually knew what they were talking about would be lurking on this board someplace.
    DaveyP
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