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  • Welcome, everybody.

  • It's a great pleasure to welcome you to our CC Mei Distinguished

  • seminar series.

  • This is a series that is sponsored

  • by the Department of Civil and Environmental Engineering

  • and the CC Mei Fund, and this is our first Distinguished seminar

  • of the term.

  • It's a great pleasure to see it's a full house.

  • Hopefully for the people that will be late,

  • they will still find some seats.

  • And so for today's inauguration talk of the term,

  • we will be hearing from Professor George Sugihara,

  • and George Sugihara is a Professor

  • of Biological Oceanography at the Physical Oceanography

  • Research Division, Scripps Institute of Oceanography

  • at UC San Diego.

  • I'm co-hosting Professor George Sugihara with Professor Serguei

  • Saavedra here in CEE.

  • So professor Sugihara is a data-driven theoretician

  • whose work focuses on developing minimalist inductive theory,

  • extracting information from observational data

  • with minimal assumptions.

  • He has worked across many scientific domains,

  • including ecology, finance, climate science, medicine,

  • and fisheries.

  • He's most known for topological models in ecology,

  • empirical dynamic forecasting models, research

  • and genetic early warning signs of critical transitions,

  • methods of distinguishing correlation

  • from causal interaction time series,

  • and has championed the idea that causation

  • can occur without correlation.

  • He provided one of the earliest field demonstrations of chaos

  • in ecology and biology.

  • Professor Sugihara is the inaugural holder

  • of the McQuown Chair of Natural Science at the Scripps

  • Institute of Oceanography at UCSD.

  • He has won many other awards and recognitions,

  • including being member of National Academies

  • Board on Mathematical Sciences and their applications

  • for a few years.

  • And today, he will discuss understanding nature

  • holistically and without equations.

  • And that's extremely intriguing for all of us.

  • And so without further ado, please join me

  • in welcoming Professor Sugihara.

  • [APPLAUSE]

  • This is in my presenter notes, so I'm reading it

  • off of the screen here.

  • I want to make a disclaimer, however.

  • In the abstract, it says that these ideas are intuitive.

  • Are you good?

  • Are we good?

  • OK.

  • So the abstract says that the ideas that I'm going to present

  • are intuitive, but this is not entirely true.

  • In fact, for whatever reason, at one point,

  • the playwright Tom Stoppard approached me,

  • and he said that he was interested in writing something

  • about these ideas and wondered if it

  • would be possible to explain these to a theater audience.

  • And just read the dark black there.

  • His response was that if he tried to explain it

  • to a theater audience, they'd probably

  • be in the lobby drinking before he

  • got through the first sentence.

  • So the ideas are in fact decidedly counter-intuitively.

  • And this is a fact that in a sense

  • goes against how we usually try to understand things.

  • So I'll explain what that means in a second.

  • So we're all familiar with Berkeley's famous dictum,

  • but despite this warning, correlation

  • is very much at the core of Western science.

  • Untangling networks of cause and effect

  • is really how we try to understand nature.

  • It's essentially what the business of science

  • is all about.

  • And for the most part and very much

  • despite Berkeley's warning, correlation

  • is very much at the core of how we try to get a grasp on this.

  • It's an unspoken rule, in fact, that within science

  • and with how we normally operate,

  • it's a correlation is a reasonable thing to do.

  • It's innocent until it's proven guilty.

  • Thus, distinguishing this intuitive correlation

  • from the somewhat counter-intuitive causation

  • is at the crux, and it's the topic of this talk today.

  • So I'm going to develop a discussion

  • for making this distinction that hinges on two main elements.

  • First, the fact that the nature is

  • dynamic in the temporal sequence matters.

  • Meaning that nature is better understood as a movie than as

  • snapshots, OK?

  • And secondly is the fact that nature is nonlinear,

  • that it consists of interdependent parts that

  • are basically non-separable, that context really matters.

  • That nature can't be understood as independent pieces

  • but rather each piece needs to be studied

  • in the context surrounding it.

  • So let's start with a nice, simple example.

  • All right.

  • Consider these two time series.

  • One might be a species, or these might

  • be two species interacting, or one

  • might be an environmental driver and responding species,

  • or a driver and a physiological response,

  • or money supply and interest rates, something like that.

  • So if you look at 10 years of data,

  • you say your first hypothesis is that these things are

  • positively correlated.

  • You have this kind of working model for what's going on.

  • If you roll forward another dozen years,

  • you find your hypothesis holds, but then it

  • falls apart a little bit here and in the middle,

  • right in here.

  • And then it sort of flips back on here towards the end.

  • So out of 18 years of observations, actually

  • more like 22 years of observations,

  • we find that our hypothesis that these things are correlated

  • is a pretty good one.

  • If this was an ecology pattern, if this

  • was a pattern from ecology, we'd say that this

  • is a really good hypothesis.

  • So we might make an adaptive caveat here, kind of an excuse

  • for what happened when it became uncorrelated, but more or less,

  • this looks like a pretty good hypothesis.

  • This is, however, what we see if we roll forward

  • another couple of decades.

  • In fact, for very long periods of time,

  • these two variables are uncorrelated.

  • They're totally unrelated.

  • However, they appear from a statistical sense

  • to be unrelated, but they were actually

  • generated from a coupled two-species difference

  • equation.

  • So this is a simple example of nonlinear dynamics.

  • We see to two things can appear to be coupled

  • for short periods of time, uncoupled,

  • but for very long periods of time,

  • there's absolutely no correlation.

  • So not only does correlation not imply causation,

  • but with simple nonlinear dynamics, lack of correlation

  • does not imply lack of causation.

  • That's actually something that I think is fairly important.

  • In retrospect, what I just showed you,

  • you might think this is obvious, but apparently this

  • is not well known, and it contradicts a currently held

  • view that correlation is a necessary condition

  • for causation.

  • So this was Edward Tufte who said

  • that empirically observed variation is

  • a necessary condition for causation.

  • OK.

  • So the activity of correlation, I think,

  • reflects the physiology of how we learn.

  • And one can argue that it's almost wired

  • into our cognitive apparatus.

  • So the basic notion beyond Hebbian learning

  • is that cells that fire together wire together.

  • So the mechanism of how we learn is really

  • very sort of supportive of the whole notion of correlation.

  • So I think it's very fundamental to how we perceive things

  • as human beings.

  • OK.

  • The picture that emerges is not only

  • that correlation does not necessarily imply causation,

  • but that you can have causation without correlation.

  • OK, and this is the realm of nonlinear systems.

  • This is interesting, because this is also

  • the realm of biological systems.

  • So within this realm, there's a further consequence

  • of non-linearity that was demonstrated in the model

  • example, and that's this phenomenon

  • of mirage correlation.

  • So correlations that come and go and that even change sign.

  • So here is a nice, simple example of mirage correlation.

  • This is an example not from finance but from ecology.

  • This is a study by John McGowan, and it was an attempt

  • to try to explain harmful algal blooms at Scripps,

  • these red tides.

  • So these spikes here are spikes in chlorophyll

  • found at Scripps Pier.

  • And what we see in the blue at the bottom

  • are sea surface temperature anomalies.

  • And so the idea was that the spikes in chlorophyll

  • were really caused by the sea surface temperature anomalies.

  • This is about a decade's worth of observations.

  • They were about to publish it, but they

  • were kind of slow in doing so.

  • And in the meantime, this correlation reversed itself.

  • And not only did it reverse itself,

  • it then became completely uncorrelated.

  • So I think this is a classic example

  • of a mirage correlation.

  • OK.

  • So here's another example from Southern California.

  • Using data up to 1991, there's a very significant relationship

  • between sea surface temperature here,

  • and this is a measure of sardine production,

  • so-called recruitment.

  • So this was reported in '94 and was subsequently written

  • into state law for managing harvest.

  • So if you are above 17 degrees, the harvest levels are higher.

  • If you're below 17 degrees, they were lower.

  • However, when data-- if you add it to this existing data,

  • data from '94 up to 2010, this is what you find.

  • The correlation seemed to disappear in both cases.

  • So these are two different ways of measuring productivity,

  • and the correlation disappeared in both of them.

  • So this statute that was written into state law

  • has now been suspended.

  • And this is where it now stands.

  • All right, so another famous example from fisheries

  • was this meta-analysis on 74 environment recruitment

  • correlations that were reported in the literature.

  • So these correlations were tested

  • subsequent to the publication of each original paper

  • by adding additional data to see if they were upheld.

  • And only 28 out of the 74 were.

  • And among the 28 that were upheld

  • was the sardine, so we know what happened there.

  • OK, so relationships that we thought we understood

  • seemed to disappear.

  • This sort of thing is familiar in finance

  • where relationships are uncovered but often disappear

  • even before we try to exploit them.

  • OK.

  • So how do we address this?

  • The approach that I'm going to present today

  • is based on not only your state space reconstruction, which

  • I refer to here with a little less technical

  • but I think more descriptive name,

  • which is empirical dynamics.

  • So EDM, Empirical Dynamic Modeling,

  • is basically a holistic data-driven approach

  • for studying complex systems from their attractors.

  • It's designed to address nonlinear issues

  • such as mirage correlation.

  • I'm now going to play a brief video that I

  • think is going to explain all.

  • This is something that my son actually made for me when

  • I tried to explain it to him.

  • And he said, no, no, no, you can do this--

  • it doesn't take three hours to explain this to someone.

  • You can do this in like two minutes

  • with a reasonable video.

  • So he made this nice video for me.

  • The narration is by Robert May.

  • [VIDEO PLAYBACK]

  • - This animation illustrates the Lorentz attractor.

  • The Lorentz is an example of a coupled dynamic system

  • consisting of three differential equations, where each--

  • [END PLAYBACK]

  • Oh, technical difficulties.

  • Sorry.

  • Let me start it again.

  • Hold on.

  • [VIDEO PLAYBACK]

  • - This animation illustrates the Lorentz attractor.

  • The Lorentz is an example of a coupled dynamic system

  • consisting of three differential equations

  • where each component depends on the state and dynamics

  • of the other two components.

  • Think of each component, for example, as being species--

  • foxes, rabbits, grasses.

  • And each one changes depending on the state of the other two.

  • So these components shown here as the axes

  • are actually the state variables or the Cartesian coordinates

  • that form the state space.

  • Notice that when the system is in one lobe,

  • X and Z are positively correlated.

  • And when the system is in the lobe,

  • X and Z are negatively correlated.

  • The other wing of the butterfly.

  • We can view a time series thus as a projection

  • from that manifold onto a coordinate axis of the state

  • space.

  • Here we see the projection onto axis X and the resulting time

  • series recording displacement of X.

  • This can be repeated on the other coordinate axes

  • to generate other simultaneous time series.

  • And so these time series are really

  • just projections of the manifold dynamics

  • on the coordinate axes.

  • Conversely, we can recreate the manifold

  • by projecting the individual time series back into the state

  • space to create the flow.

  • On this panel, we can see the three time series, X, Y,

  • and Z, each of which is really a projection

  • of the motion on that manifold.

  • And what we're doing is the opposite here.

  • We are taking a time series and projecting them back

  • into the original three-dimensional state space

  • to recreate the manifold.

  • It's a butterfly attractor.

  • [END PLAYBACK]

  • OK.

  • To summarize, these time series are really observations

  • of motion on an attractor.

  • Indeed, the jargon term in dynamical systems

  • is to call a time series an observation function.

  • Conversely, you can actually create attractors

  • by taking the appropriate time series,

  • plotting them in the right space,

  • and generating some kind of a shape.

  • OK, this is really the basis of this empirical dynamic

  • approach.

  • What is important, I think, to understand here

  • is that the attractor and the equations

  • are actually equivalent.

  • Both contain identical information,

  • and both represent the rules governing the relationships

  • among variables.

  • And depending on when they are viewed,

  • these relationships can appear to change.

  • And this is what can give rise to mirage correlations.

  • So over the short term here, there might be correlations.

  • But over a longer term--

  • so for example, if it's in this lobe--

  • I'm very bad with machines.

  • All right.

  • If it's in that lobe, you'll get a positive relationship.

  • If it's in the lobe on this side,

  • you'll get a negative correlation.

  • If you sample the system sparsely

  • over long periods of time, you'd find no apparent correlation

  • at all, OK?

  • OK, let's look at another real example of this.

  • So this is an application that I was initially skeptical about,

  • mainly because I couldn't see how to get time series.

  • But luckily, I was wrong here.

  • These are experimental data obtained

  • by Gerald Pao from the Salk Institute

  • on expression levels of transcription factor SWI4

  • and cyclin CLN3.

  • This is in yeast.

  • If you view it statistically, so this is viewed statistically,

  • the relationship between these two variables,

  • there's absolutely no statistical relationship.

  • There's no cross-correlation.

  • However, if you connect these observations in time,

  • they're clearly inter-related.

  • So we see the skeleton of an attractor emerging.

  • So the way that they generated this data, actually, which--

  • so when I was originally approached about this,

  • and they said, well, we want to apply these methods

  • to gene expression.

  • And I said, but you can't make a time series

  • for gene expression.

  • And they said, oh, yes, we can.

  • And what they did in this case, because it was yeast,

  • they were able to shock cells, which synchronizes them

  • in their cell cycle, and then sample them

  • every 30 minutes for two days.

  • And so at each sample, they would

  • sequence several thousands of genes

  • and do this every 30 minutes for two days.

  • You can do a lot if you have post-docs and graduate

  • students, all right?

  • OK.

  • So we were able to get this thing to actually reflect

  • an attractor.

  • Very interesting.

  • Of course, if you randomize these observations in time,

  • you get absolutely nothing.

  • You still get singularities.

  • So you get these crossings in two dimensions.

  • However, if you include the cyclin CLB2,

  • the crossing disappear, OK?

  • So we have this nice cluster of three things,

  • that actually if you looked at them statistically,

  • appear to be uncorrelated, or essentially invisible

  • to bioinformatics techniques that are, in fact, dynamically

  • interacting.

  • So here is another short video clip

  • that I think presents what I consider

  • to be a really important basic theorem that

  • supports a lot of this empirical dynamics work.

  • [VIDEO PLAYBACK]

  • - There's a very powerful theorem proven by [INAUDIBLE]..

  • It shows generically that one can reconstruct a shadow

  • version of the original manifold simply by looking at one

  • of its time series projections.

  • For example, consider the three times series shown her.

  • These are all copies of each other.

  • They are all copies of variable eggs.

  • Each is displaced by an amount tau.

  • So the top one is unlagged, the second one is lag by tau,

  • and the blue one at the bottom is lag by two tau.

  • Takens' theorem then says that we

  • should be able to use these three time

  • series as new coordinates and reconstruct

  • a shadow of the original butterfly manifold.

  • This is the reconstructed manifold produced

  • from lags of a single variable, and you

  • can see that it actually does look

  • very similar to the butterfly attractor.

  • Each point in the three-dimensional

  • reconstruction can be thought of as a time segment

  • with different points capturing different signals

  • of [INAUDIBLE] of variable eggs.

  • This method represents a one-to-one map

  • between the original manifold, butterfly attractor,

  • and the reconstruction, allowing us

  • to recover states of the original dynamic system

  • by using lags of just a single time series.

  • [END PLAYBACK]

  • OK.

  • So to recap, the attractor really

  • describes how the variables relate to each other

  • through time.

  • And Takens' theorem says quite powerfully

  • that any one variable contains information about the others.

  • This fact allows us to use a single variable basically

  • to construct a shadow manifold using

  • time lags as proxy coordinates that has

  • a one-to-one relationship with the original manifold.

  • So constructing attractors, again, from time series data

  • is the real basis of the empirical dynamic approach.

  • And as we see, we can do this univariately

  • by taking time lags of one variable.

  • We can do this multivariately with a set

  • of native coordinates, and we can also

  • make mixed embeddings that have some time lags as well as

  • some multivariate coordinates.

  • So let's look at some examples.

  • So this is an example of using lags with the expression time

  • series.

  • This is a mammalian model.

  • Mouse fibroblast production of an insulin-like growth factor

  • binding protein.

  • And again, this is the case of synchronizing and then sampling

  • over a number of days.

  • So clearly gene expression is a dynamic process, which

  • is quite a radical departure, I think,

  • from normal bioinformatics approaches,

  • which are essentially static

  • OK.

  • Here we have another ecological example.

  • These are attractors constructed for sockeye salmon returns,

  • and this is for the Fraser River in Canada, which is

  • like the iconic salmon fishery.

  • And you can see for each one of these different spawning lakes,

  • you get an attractor that looks relatively similar.

  • They all look like Pringle chips, basically.

  • And what's interesting about this--

  • and I'll talk about this a little bit more later--

  • is that you can use these attractors

  • that you construct from data to make very good predictions.

  • And the fact that you can make predictions and make

  • these predictions out of sample, I think,

  • should give you some confidence that this is reasonable.

  • So again, I'm talking about a kind of modeling

  • where there really are almost no free parameters.

  • There's one in this case, right?

  • I'm assuming that I can't adjust the fact that I'm

  • observing this once a year.

  • So that's given.

  • Tau is given.

  • The time lag is given.

  • The only variable that I'm using here

  • that I need to kind of estimate is the number

  • of dimensions, so the number of embedding dimensions

  • that we need for this.

  • In this case, I'm showing it in three dimensions.

  • Not all of these attractors, of course,

  • are going to be three-dimensionals.

  • The ones that I'll show you tend to be,

  • only because you can see them and they're

  • easy to understand what's going on.

  • So the basic process is really involving very few

  • assumptions and with only one fitted parameter,

  • with that fitted parameter being the embedding dimension.

  • OK.

  • So the fact that I'm able to get to using--

  • this is again, just using lags--

  • something coherent in three dimensions

  • means that I might be able to construct a mechanistic model

  • that has three variables.

  • So maybe sea surface temperature, river discharge,

  • maybe spawning, smolts going into the ocean, something

  • like that.

  • OK.

  • So again, one of the most compelling features, I think,

  • of this general set of techniques

  • is that it can be used to forecast.

  • And the fact that you could forecast

  • was something that originally got

  • me interested in this area or this set of techniques.

  • And it kind of led me into finance,

  • so I worked for like half a decade as a managing

  • director for Deutsche Bank.

  • And things like this were used to manage

  • on the order of $2 billion a day in notional risk.

  • So it's very bottom line, it's very pragmatic, and verifiable

  • with prediction, all of which I find--

  • plus it's extremely economical.

  • There are very few moving parts.

  • OK.

  • So I'm going to quickly show you two

  • basic methods for forecasting.

  • There are many other possibilities that exist,

  • but these are just two very simple ones, simplex projection

  • and S-maps.

  • So simplex projection is basically a nearest neighbor

  • forecasting technique.

  • Now you can imagine having the number of nearest neighbors

  • to be a tunable parameter, but the idea here is to be minimal,

  • and the nearest neighbors are essentially determined

  • by the embedding dimension.

  • So if you have an embedding dimension of e,

  • you can always--

  • a point in an e dimensional space

  • can be an interior point in e plus one dimensions,

  • which means you just need e plus one neighbors.

  • And so e plus one-- so the number of neighbors

  • is determined.

  • It's not a free variable in this, OK?

  • So the idea then is to take these nearest neighbors

  • in this space, which are analogs,

  • project them forward, and see where they went,

  • and that'll give you an idea for where the system is headed.

  • OK.

  • So again, each point on this attractor

  • is a history vector or a history fragment, basically.

  • And so here is this point that I'm trying to predict from.

  • And I look at the nearest neighbors, and then I--

  • these are points in the past, right?

  • And now I say, where do they go next?

  • And so I get a spread of points going forward,

  • and I take the center of mass of that spread,

  • the exponentially weighted center of mass,

  • and that gives me a prediction.

  • So how do you predict the future?

  • You do it by looking at similar points in the past.

  • But what do you mean by similar?

  • What you mean by similar is that the points

  • have to be in the correct dimensionality.

  • So for example, if I'm trying to predict the temperature

  • at the end of Scripps Pier tomorrow,

  • the sea surface temperature, and it's

  • a three-dimensional process, and let's say the right lag should

  • be a week, then I'm not just going

  • to look at temperatures that are similar to today's temperature.

  • I'm going to look at temperatures where today's

  • temperature, the temperature a week ago,

  • and the temperature two weeks ago are most similar, right?

  • And so the knowing the dimensionality

  • is quite important for determining what the nearest

  • neighbors are, all right?

  • So you take the weighted average and that

  • becomes your prediction.

  • Here's an example.

  • This looks like white noise.

  • What I'm going to do is cut this data in half,

  • and I'm going to use the first half to build a model,

  • I'm going to predict on the second half.

  • So if I take time lag coordinates, and in this case,

  • again, I'm choosing on purpose three three-dimensional things,

  • because they're easy to show.

  • This is like taking a fork with three prongs,

  • laying it down on the time series,

  • and calling one x, the other one y, the other one z.

  • So I'm going to plot all those points going forward,

  • and this is the shape I get.

  • So you actually get what looked like white noise,

  • and it totally random actually was not.

  • In fact, I generated it from first differences

  • of [INAUDIBLE], OK?

  • So if we now use this simple zeroth order technique

  • and we try to predict that second half of the time series

  • that looked totally noisy, you can do quite well.

  • This is actually predicting to two points

  • into the future, two steps into the future.

  • OK.

  • So again, how did I know to choose three dimensions?

  • Basically you do this by trial and error.

  • You try like one, two, three.

  • And it peaks So this is, again, how well you can predict.

  • This is the Pearson correlation coefficient.

  • And this is trying different embedding dimensions,

  • trying a two-pronged fork, a three-pronged fork, so on.

  • And again, so the embedding with the best predictability

  • is the one that best unfolds the attractor, the one that best

  • resolves the singularities.

  • And this relies basically on the Whitney embedding theorem.

  • So if the attractor actually was a ball of thread, OK,

  • and I tried to embed this ball of thread in one dimension,

  • that would be like shining a light down across over a line.

  • Then at any point, I could be going right or left.

  • So there's singularities everywhere.

  • If I shine it down on two dimensions, I now have a disk.

  • At any point I can go right, left, up, down, so forth.

  • Everywhere is a singularity.

  • If I know embed it in three dimensions--

  • so the thread is one-dimensional, right?

  • If I embed it in three dimensions, all of a sudden,

  • I can see that I have individual threads.

  • And if you have these individual threads,

  • that allows you to make better predictions, right?

  • So this is how you can tell how well you've

  • embedded the attractor, how well you

  • can predict with the attractor.

  • OK.

  • All right.

  • So the other-- sort of the next order of complexity

  • is basically a first-order map, which

  • is a weighted autoregressive model where you're effectively

  • computing a plane along the manifold along this attractor

  • and using the coefficients of the Jacobian matrix

  • that you compute for this hyperplane,

  • basically, to give you predictions.

  • But when you're computing this plane,

  • there's a weighting function.

  • It's this weighting function that we're calling theta here.

  • And that weighting function determines how heavily you

  • weight points that are nearby on the attractor versus points

  • that are far away, OK?

  • So if theta is equal to zero, then all points

  • are equally weighted.

  • That's just like fitting a standard AR model

  • to a cloud of points, right?

  • All points are equally valid.

  • But if the attractor really matters,

  • then points nearby should be weighted more heavily

  • than points far away, OK?

  • So if there's actual curvature in there,

  • then if you weight more heavily, you're

  • taking advantage of that information, OK?

  • So this is if you crank theta up to 0.5,

  • your weighting points nearby more heavily,

  • so forth and so on.

  • OK.

  • This is a really simple test for non-linearity.

  • You can actually try increasing that theta,

  • the tuning parameter.

  • And if as you increase it the predictability goes up,

  • then that's an indication that you get an advantage

  • by acknowledging the fact that the function is

  • different at different parts on the attractor, which

  • is another way of saying the dynamics are state dependent,

  • which is another way of saying the manifold

  • has curvature to it, OK?

  • So curvature is actually ubiquitous in nature.

  • This is a study that my student [? Zach ?] [? Shee ?] did.

  • And if you look at 20th century records

  • for specific biological populations,

  • you find all of them exhibit non-linearity.

  • We didn't find non-linearity, actually,

  • for some of the physical measurements.

  • But again, we were just looking at the 20th century,

  • and it might've been too short to pick that up.

  • Other examples include other fish species, sheep, diatoms,

  • and an assortment of many other kinds of phenomena.

  • All show this kind of non-linearity.

  • It seems to be ubiquitous.

  • Wherever you look for it, it's actually rare

  • that you don't find it, OK?

  • So the fact that things are nonlinear is pretty important,

  • I think.

  • It affects the way that you should think about the problem

  • and analyze it.

  • And in fact, the non-linearity is a property

  • that I believe can be exploited.

  • This is an example of doing just that.

  • So this paper appeared last year in PRSB,

  • and it used S-maps, this technique that we just

  • saw, to show how species interactions vary

  • in time depending on where on the attractor they are, OK?

  • So it really showed how we can take

  • real-time measurements of the interactions that

  • are state dependent, OK?

  • And the basic idea is as follows.

  • So the S-map involves calculating a hyperplane

  • or a surface at each point as the system travels

  • along its attractor.

  • So this involves calculating the Jacobian matrix, whose elements

  • are partial derivatives that measure the effect of one

  • species on another.

  • So note that the embeddings here are multi-variate.

  • So these aren't lags of one variable,

  • but they're native variables, right?

  • So I want to know how the relationship

  • of each native variable affects the other variable

  • and how that changes through time.

  • So what I do is at each point, I compute a Jacobian matrix.

  • If this was an equilibrium system,

  • there would just be one point, and I

  • would be looking at the-- it's like

  • the standard linear stability analysis for an equilibrium

  • system.

  • But what I'm doing is I'm taking that analysis,

  • but I'm applying it to each as the system travels successively

  • along each point on the attractor.

  • So the coefficients are in effect

  • fit sequentially as the system travels along its attractor.

  • And they vary, therefore, according to the location

  • on the attractor.

  • So what's really nice about this is that it's something

  • that you can actually accomplish very easily on real data.

  • And here's an example.

  • This is data from a marine mesocosm that

  • was collected by Huisman, and what you want to focus on

  • is the competition between copepods and rotifers.

  • These are the two main consumers in this.

  • So these are both zooplanktons that eat phytoplankton.

  • And this is basically the partial

  • of how the callenoids vary with the rotifers.

  • And so you can see that the competition--

  • so this shows how the coefficients

  • are changing as you computed along as the system is

  • traveling along its attractor.

  • So what's the interesting thing, what

  • I think is interesting here is that I was totally surprised.

  • Competition is not a fairly smooth and long-term

  • relationship, right?

  • In classical ecology, it's regarded as a constant.

  • So two species compete, you compute their alpha.ij,

  • and that's the constant.

  • In fact, it's very episodic.

  • It seems to only occur like in these little bottlenecks, which

  • I think is-- so I mean, this is nature.

  • This is not my model.

  • This is what nature is telling me,

  • that you get competition in these little bottlenecks.

  • So that fact I found fairly surprising.

  • But what's even more interesting is

  • to ask the question, what is it about the system when

  • it does occur that causes this competition?

  • And it turns out that what you can do

  • is make a graph basically of how that coefficient--

  • this is terrible.

  • I think I got this when I talked at Stanford last fall.

  • OK.

  • All right.

  • All right, it's broken.

  • So you can make a plot of what the competition coefficient--

  • how the competition coefficient varies as a function of food

  • abundance.

  • And the obvious thing that you get here

  • is that when do you get competition?

  • When food is scarce.

  • I mean, duh.

  • That seems like it should be obvious.

  • But what wasn't clear before is how episodic this all is.

  • It's not sort of a gradual constant affair.

  • It's something that happens in these sudden bottlenecks.

  • So what we have then is a pretty good tool for probing changing

  • interactions.

  • And I can see other potential for this

  • in terms of looking for--

  • you can compute the matrix and maybe

  • compute something like an eigenvalue for the matrix

  • as it changes to look for changes where--

  • to look for instances where you were

  • about to enter a critical transition.

  • So this stuff really hasn't been written up yet.

  • You should go ahead and do it.

  • But I see a lot of potential for just

  • using this fairly simple approach, which again,

  • is very empirical, and it allows the data to tell you

  • what's actually happening.

  • OK.

  • So let's see how EDM deals with causation.

  • OK.

  • This is the formal statement of Granger causality.

  • So basically he's saying, I'm going

  • to try to predict Y2 from the universe

  • of all possible variables.

  • And this is the variance, my uncertainty in my prediction.

  • And it says that if however I remove Y1

  • and I'm trying to predict Y2, and this variance is greater,

  • than I know that Y1 was causal.

  • So it says if I exclude a variable

  • and I don't do as well at predicting, then

  • that variable was causal.

  • That's the formal definition of Granger causality.

  • The problem, however, is that this seems

  • to contradict Takens' theorem.

  • So Takens' theorem says the information

  • about other variables in the system

  • are contained in each other variable, OK?

  • So how can you remove a variable if that variable's information

  • is contained in the others?

  • So there is a little bit of a problem.

  • What's interesting is if you look at Granger's '68 paper

  • where he describes this, he says explicitly,

  • this may not work for dynamic systems.

  • So--

  • [LAUGHTER]

  • He was covered.

  • OK.

  • So I think this is a useful criterion

  • sort of as a kind of a rule of thumb, practical rule of thumb.

  • But it really is intended more for stochastic systems rather

  • than dynamic systems.

  • OK.

  • So in dynamic systems, time series variables

  • are causally related again if they're coupled and belong

  • to the same dynamic system.

  • If X causes Y, then information about X

  • must be encoded in this shadow manifold of Y.

  • And this is something that you can test with cross-mapping.

  • This was the paper that was published at the end of 2012

  • that describes the idea.

  • And I have one final video clip.

  • It's not narrated by Bob May.

  • I had my student [? Hal ?] [? Yee ?] do the narration

  • on this one.

  • But it'll explain it.

  • [VIDEO PLAYBACK]

  • - Takens' theorem gives us a one-to-one mapping between

  • the original manifold and reconstructed shadow manifolds.

  • Here we will explain how this important aspect of attractor

  • reconstruction can be used to [INAUDIBLE] two time series

  • variables belong to the same dynamic system

  • and are thus causally related.

  • This particular reconstruction is based on lags of variable x.

  • If we now do the same for variable y,

  • we find something similar.

  • Here we see the original manifold M, as well as

  • the shadow manifolds, Mx and My, created from lags of x and y

  • respectively.

  • Because both Mx and My map one-to-one

  • to the original manifold M, they also

  • map one-to-one to each other.

  • This implies that the points that are nearby

  • on the manifold My correspond to points that are also nearby

  • on Mx.

  • We can demonstrate this principle

  • by finding the nearest neighbors in My

  • and using their time indices to find

  • the corresponding points in Mx.

  • These points will be nearest neighbors on Mx

  • only if x and y are causally related.

  • Thus, we can use nearby points on My

  • to identify nearby points on Mx.

  • This allows us to use the historical record of y

  • to estimate the states of x and vice versa,

  • a technique we call cross-mapping.

  • With longer time series, the reconstructed manifolds

  • are denser, nearest neighbors are closer,

  • and a cross-map estimates increase in precision.

  • We call this phenomenon convergent cross-mapping

  • and use this convergence as a practical criterion

  • for detecting causation.

  • [END PLAYBACK]

  • OK.

  • So with convergent cross-mapping,

  • what we're trying to do is we're trying to recover states

  • of the affected variable--

  • we're trying to recover states of the causal variable

  • from the affected variable.

  • And so this is basic.

  • Let's see.

  • The idea is that instead of looking specifically

  • at the cause, we're looking at the effect

  • to try to infer what the cause was.

  • So basically from the victim, we can find something

  • about the aggressor or the perpetrator, right?

  • OK.

  • This little piece, I think, will give you

  • a little bit of intuition.

  • So these two time series are what you get if alpha is zero.

  • So this is y is red and x is blue.

  • And you can see that with alpha equal to zero,

  • they're independent.

  • If I crank up alpha, and then this is what I get.

  • So again, you can see that the blues time series is not

  • altered, but the red one, but y actually is.

  • And it's in this alteration of the time series

  • that I'm able, from the reconstructed manifold,

  • to be able to backtrack the values of the blue time series.

  • And so that shows that x was causal on y.

  • OK.

  • A necessary condition for a cross-map estimate for--

  • a necessary condition for a convergence

  • is to show that the cross-map estimate improves

  • with data length.

  • And so that's basically what we see here.

  • So as points get closer in the attractor,

  • your estimates should get better,

  • and so predictions should get better.

  • So let's look at some examples.

  • This is a classic predator/prey experiment

  • that Gauss made famous.

  • So didinium is the rotifer predator,

  • paramecium is the prey.

  • And you can see, you can get cross-mapping

  • in both directions, sure.

  • The predator is affecting the prey,

  • the prey is affecting the predator.

  • This sort of looks like maybe the predator

  • is affecting the prey more than the prey is

  • affecting the predator.

  • But if you look at this in a time lag way,

  • so this is looking at different prediction lags

  • for doing the cross-mapping, you find

  • that the effect of the predator on the prey

  • is almost instantaneous, which you kind of expect.

  • These are rotifers eating paramecia.

  • But the effect of the paramecia itself on the predator

  • is delayed, and it's delayed looks

  • like by about a day or so.

  • So you get sort of a sensible time delay here.

  • OK.

  • This is a field example.

  • These are sardines and anchovies that

  • have been sort of a mystery for quite a while.

  • They show reciprocal abundance patterns.

  • And it was thought that maybe they compete.

  • These are data for Southern California.

  • It may well be that they are competitive in other areas,

  • maybe the Japan sea.

  • But not in Southern California.

  • There's absolutely no evidence for mutual effect to sardines

  • and anchovies there.

  • However, if you look at sea surface temperatures,

  • you find that they're both influenced

  • by sea surface temperature, but probably

  • in slightly opposite ways.

  • So that's kind of a nice result for that problem.

  • OK, now final ecological example are these red tides.

  • Episodic red tides are a classic example

  • that no one has been able to predict.

  • They've been thought to be regime-like,

  • and the mechanism for this rapid transition

  • has remained a mystery for over a century.

  • So despite about a dozen or so Scripps theses

  • all showing by experiment that certain factors should

  • be important, none of them show a correlation.

  • So if you look at the field data,

  • you actually don't see the correlation

  • that you would expect if you had done the experiments.

  • So this was exactly the case that we saw, for example,

  • with sea surface temperature anomaly and chlorophyll.

  • So you get these little temporal correlations

  • that then disappear.

  • So the absence of environmental correlations

  • suggests that these events can't be

  • explained by linear dynamics.

  • And you can confirm this by doing an S-map test.

  • You find, in fact, chlorophyll is very nonlinear.

  • If you increase theta, it improves.

  • But the most convincing thing is that you can actually

  • find very good predictability using

  • a simple manifold constructed in forecasting using an S-map.

  • So the univariate construction, because you're just

  • looking at the one variable, is really

  • summarizing the internal dynamics,

  • the intrinsic dynamics.

  • And so if you just focus now on the red tides,

  • the correlation goes down.

  • So we actually can't predict these red tides quite as well

  • from just the simple internal dynamics, which

  • suggests that there may be stochastic variables coming in

  • to force the system, OK?

  • So we then did the obvious thing,

  • which was to apply CCM to these environmental variables that

  • were thought to be important but that showed no correlation.

  • And so that's what we did.

  • And these candidate variables fall

  • into two groups, those that describe nutrient history

  • and those that describe stratification.

  • If you look at correlation, you actually

  • find very little correlation in there at all.

  • But if you do cross-mapping, just about all of them

  • show a good cross-map scale, OK?

  • So just about all of them contained some information.

  • And so this was very encouraging.

  • This is actually a class project,

  • and there were eight of us involved.

  • And we had data.

  • We did all this analysis, and we had data going up to 2010.

  • The data from 2010 onward had not yet been analyzed.

  • We had all the samples, but they hadn't been analyzed.

  • And so we came up with our set of models,

  • and then we were able to process the data, and we all sort of--

  • there were 16 fingers being crossed.

  • And we did the out-of-sample test,

  • and this is the result, which was very good.

  • We actually found very good predictability

  • with a correlation coefficient of about 0.6.

  • So this is a really nice example of out-of-sample forecasting

  • using these methods.

  • So we've learned something about the mechanism,

  • that the mechanism has something to do

  • with water stratification, stability of the water column.

  • And on top of that, we're actually

  • able to forecast these red tides with some accuracy.

  • All right, so this is potentially

  • the most exciting application of these ideas,

  • and this is the last big piece that I want to talk about.

  • So this experiment, this is the experimental work

  • being done in the Verma Lab at the Salk Institute.

  • And this is the attractor that we saw earlier.

  • Remember, these things were all mutually

  • uncorrelated in a statistical sense,

  • but we found were causally related.

  • And so if you make an attractor using all three,

  • this is what you get.

  • These things are also uncorrelated, but very strongly

  • causally linked to the transcription regulator WHI5.

  • So this suggested that one could do an experiment with WHI5

  • to see how well this method of CCM

  • actually does at identifying uncorrelated causal links.

  • So this is an example showing the uncorrelated linkage.

  • You can see that WHI5 and SWI4 are completely uncorrelated.

  • Those are the original time series.

  • But if you do cross-mapping, you find in fact there's

  • a significant signal.

  • You can recover values of WHI5 from values of SWI4

  • on the attractor.

  • OK, so the experiment that this suggests

  • is if you alter the value of WHI5

  • artificially, if you experimentally

  • enhance WHI5, because it's causally related,

  • it should produce an effect on the dynamics

  • of these other genes.

  • And so that's what we did.

  • And so the black is while type, and the purple dotted one

  • is the manipulation.

  • So the manipulation clearly deformed the attractor.

  • And this is something that you can actually

  • quantify pretty easily.

  • OK, so if you repeat this procedure

  • for other genes showing a low correlation with WHI5--

  • this is the money panel right here--

  • you can find that 82% of the genes that were identified

  • by CCM to be causal--

  • these are all uncorrelated-- to be causal,

  • were actually verified by experiment

  • to be causal, which is really good,

  • because the industry standard for [INAUDIBLE] is 3%.

  • So this is better.

  • This is actually better.

  • The other thing that I think makes

  • this interesting is that these non-correlated genes that

  • are also causal are thought to be signal integrators,

  • and signal integrators may be really, really important

  • for gene expression.

  • So we'll see how this all goes.

  • So I think that this could have immediate practical importance,

  • because the networks that you generate

  • this way can provide good guidance for the experiments

  • that need to be done.

  • So you have 25,000 genes, and so you can do 25,000 [INAUDIBLE]

  • experiments.

  • That's just too much.

  • And so you need something to kind of narrow down

  • what to focus on, and this may be a reasonable thing.

  • All right.

  • So this is a mammalian example of the same sort of thing.

  • This is a group of genes that Verma has

  • studied for about 30 years--

  • so a very well studied group--

  • that have to do with the immune response.

  • And this is the network that you would get if you just

  • looked at cross correlations.

  • But this network turns out to be entirely wrong.

  • There's a very well known bi-directional feedback

  • between I kappa, B alpha, and relA.

  • And this is the network that you get with CCM.

  • So what's interesting is that this CCM network actually

  • identifies another link that looks

  • interesting between relA in June that was not previously known.

  • And so this link, because you have

  • this bi-directional feedback, should produce some kind

  • of limit cycle-like behavior.

  • And so if you make a phase portrait of these two,

  • you should see something that looks kind of limit cycle-like.

  • The same should be true here, OK?

  • I'm almost done.

  • All right.

  • So if you do this, this is the known link,

  • and we get something that looks kind of limit cycle-like.

  • This was the previously unknown link, OK,

  • and you do get this behavior.

  • And this was actually the incorrect link that

  • was suggested by correlation.

  • So kind of interesting.

  • All right.

  • So there are a bunch of recent studies

  • that have looked at this.

  • I'll just go through them really fast.

  • This one was focused on forecasting.

  • OK, hold on.

  • Let me go--

  • OK.

  • So this one had to do with the incidents of cosmic rays

  • in 20th century that's been used to suggest that climate warming

  • is natural and not due to man.

  • And what we did that was interesting

  • is that we found that if you look at over the 20th century,

  • there is no causal relationship between cosmic rays

  • and global warming.

  • However, if you look at the time scale year-to-year,

  • you find a causal signal.

  • So in fact, it does have a very short-term effect

  • on inter-year dynamics, but it doesn't explain the trend.

  • OK.

  • So this was a study on the Vostok ice core

  • to see if there is a direct observational--

  • if we get direct observational evidence for the effect

  • of greenhouse gases on warming.

  • And we found it, of course.

  • But the other thing that we found

  • that was kind of interesting was you actually

  • have a link in the other direction as well,

  • but it's delayed.

  • And so this is a more immediate effect.

  • This one takes hundreds of years to occur, OK?

  • And then this one focused on forecasting.

  • It was a great success story, because the models that we were

  • able to produce got some interest in the Canadian press,

  • and we made forecasts for 2014, 2015, and 2016

  • that were all pretty good.

  • So, so far, so good.

  • I don't know what's going to happen in 2017.

  • So this is a nice example of out-of-sample sample

  • forecasting.

  • The classical models, if you actually

  • try to include environmental variables,

  • do worse if you do that.

  • With these models, it does better.

  • OK, and then this one appeared last fall.

  • It was an application of these ideas to look at flu epidemics.

  • And what's interesting here is that we were actually

  • able to find a particular temperature threshold, 75

  • degrees, below which, absolute humidity

  • has a negative effect on flu incidence, above which,

  • absolute humidity has a positive influence.

  • And I think the hypothesized mechanism is below 75 degrees,

  • the main environmental stressor is viral envelope disruption

  • due to excess water, right?

  • Above 75 degrees, desiccation becomes

  • the main environmental stressor.

  • And so higher humidity helps actually

  • flu incidence at higher temperatures,

  • but it inhibits flu incidence at lower temperatures.

  • And so, of course there are many other factors

  • than absolute humidity, but this was one that came out.

  • And it may actually be that the proximal driver is

  • relative humidity, but you're asking, what's the--

  • but relative humidity varies depending on whether you're

  • inside or outside.

  • Absolute humidity is much more robust.

  • Absolute humidity outside is going to be about the same

  • as it is inside.

  • So yeah, an interesting nuance.

  • All right.

  • This paper won the 2016 William James Prize in Consciousness.

  • We'll stop at that.

  • All right.

  • So I'm just going to stop there.

  • All right.

  • And so these are my tasteless thematic closing slides.

  • This is a little politically incorrect.

  • In fact, all of them are politically incorrect.

  • My wife told me this was cute, so you can blame her.

  • All right.

  • And so this is with particular reference

  • to the fisheries models that are built

  • on assumptions of equilibrium.

  • There we go.

  • Yeah.

  • And then, as we all know, this is true.

  • Thank you.

  • [APPLAUSE]

  • All right, thank you very much for a great talk.

  • Questions.

  • Thank you for the nice talk.

  • I would like to ask you, what is your feeling

  • about the applicability of data-driven methods in general

  • in systems with high intrinsic dimensionality?

  • Let's say fluid flows, climate models.

  • In this case, how do you choose the variables to model?

  • And what is the effect of sparse data in phase space?

  • Have you considered such issues?

  • Yeah.

  • Well, I think that--

  • well, I've chosen examples of opportunity.

  • So the things that I've chosen have all

  • shown attractors that are relatively low dimensional.

  • They were taken from problems that you may not necessarily

  • have thought in advance should be low-dimensional,

  • so like gene expression I figured

  • should be very high-dimensional, but it turns out

  • that there are facets of it that certainly look relatively

  • low-dimensional.

  • So this is kind of a copout answer,

  • but really you just have to try.

  • You have to see if you have data.

  • So the place to start is just with data.

  • You say, well, maybe I don't have the perfect observations.

  • That's fine.

  • But you need to start with some set of observations,

  • and then you can build from there.

  • And you might find that there are maybe two or three

  • variables that are very tightly related

  • and something interesting might come out of that.

  • So it really is, it's kind of like following your nose.

  • I mean, you don't necessarily have the whole plan in advance,

  • but what it requires is an initial set

  • of data, the willingness to actually give it a try.

  • So again, the dimensionality that we're

  • getting in these models is not an absolute number.

  • So surely any one of these things, even

  • the fisheries models, probably really, really

  • high-dimensional in principle.

  • However, for all practical purposes,

  • you can do quite well, and you can

  • measure how much variance you can explain by how

  • much predictability you have.

  • You can do quite well with about four dimensions.

  • Four is not a bad number of dimensions

  • to use in some of these salmon returns models.

  • So you can think of a problem hypothetically

  • as being very high-dimensional, but if you have data,

  • and that data actually shows that in maybe six or seven

  • dimensions you can get some degree of predictability,

  • then I think you have a little bit of insight

  • into that problem.

  • You've gained a little bit of advantage on that problem.

  • Yeah.

  • OK.

  • So it looks as if there is a bit of an assumption underlying

  • some of this where you kind of have to assume the underlying

  • manifold remains stable while you're

  • collecting these data to populate this shadow manifold.

  • So are you working on methods for detecting shifts

  • in that underlying attractor, whether you're

  • on a non-stationary, nonlinear regime?

  • Yeah, I mean, that's a great question.

  • So whether something is stationary or not

  • can depend on how long you observe it.

  • So for example, if you have motion

  • in one lobe of the attractor and then it

  • flips to the other lobe, do you say this

  • is an unstationary process?

  • No.

  • It just depends on how long you've looked at it.

  • You're asking a really important question,

  • and it's something that you can answer practically pretty

  • much by windowing, sort values in the past

  • to see if you're getting essentially the same dynamics

  • as you go forward.

  • The danger with that, though, is that you

  • can have problems where the essential dynamics-- let's

  • say it's an annual cycle of an epidemic,

  • for example, were the essential dynamics,

  • say, during the outbreak are five-dimensional,

  • but as the thing recovers, it collapses down

  • to zero-dimensional, becomes stable for a period of time.

  • And so what you're asking, I mean,

  • it really is an important question,

  • but I believe there are some practical ways of addressing,

  • but there is no simple universal way of doing it.

  • So maybe windowing is one way of doing it.

  • But again, you have to be careful that by windowing you

  • haven't artificially created what

  • looks like a non-stationary process that actually

  • is stationary.

  • And in the end, the way that you judge how well you've done

  • this is how well you can predict.

  • So if your predictions actually start

  • to degrade as you're going forward,

  • then you have some reason to be suspicious.

  • Yeah, yeah.

  • Oh, thanks for a nice talk.

  • I have a question.

  • Have you ever tried that the methods fail in some cases,

  • like if you use some mathematical principles,

  • like in what kind of system this method will be successful,

  • and then in what kind of system this method will not

  • be successful?

  • Can you describe it in using some mathematical patterns?

  • Yeah.

  • So one kind of system where they would may not be successful

  • is where you don't really--

  • a system that's really fundamentally stochastic,

  • right, where, in fact, there are no deterministic rules

  • of any kind.

  • But those are systems that as scientists we

  • like to stay away from, right, because what is there

  • to explain?

  • So my answer to this would be that I tend to like problems

  • where I start looking at them, and they're

  • giving me little sugar tablets of results.

  • And so I keep going in that direction.

  • And personally, that's how I operate.

  • So I would stay away from a problem.

  • And maybe that's why I'm not encountering as many problems

  • that are totally intractable is that they haven't--

  • I'm like an ant following a gradient of sugar.

  • They haven't kind of led me in that direction.

  • But it's a good question.

  • I don't think this is going to work for everything obviously,

  • right?

  • But I've just had pretty good luck so far sort of.

  • But it's not just luck, because I'm actually

  • following a gradient.

  • So I'm attracted to the problems where it seems to be working.

  • Yeah.

  • So the gene expression problem, I had basically written off.

  • So when I was initially approached and I thought,

  • this could not possibly work, they walked away,

  • and I thought, oh, OK.

  • I won't see these people again.

  • But then they came back with data,

  • and they showed that it did work.

  • And then we have this really good collaboration

  • going right now.

  • Yeah.

  • Just along those lines, it's pretty obvious

  • that it won't work in cases where your observations simply

  • aren't sufficient to fully describe the system.

  • So yeah.

  • It was also on the tip of my tongue to ask if you were

  • working at Deutsche Bank in 2008, but I won't.

  • What?

  • If you were working at Deutsche Bank in 2008,

  • but I won't ask that.

  • Oh, no, no, no, no.

  • No.

  • No.

  • So yeah.

  • No, I was there from '96 to 2002.

  • OK, that was safe, then.

  • Got out in time.

  • Last question then is, has any of the stuff

  • been extended to [INAUDIBLE] categorical types of data?

  • I think that it's possible.

  • We are working on something right now

  • that we're calling static cross-mapping, which is trying

  • to do that sort of thing.

  • We have some initial results that look pretty good.

  • But no, I think that's a really important area.

  • So we don't always have time series.

  • And it's much harder to get that kind of data

  • than it is to get--

  • [INAUDIBLE]

  • Exactly.

  • But there's another kind of ordering, of course,

  • that you can put on this data.

  • And I think that like in ecology,

  • it's much harder to find time series

  • than it is to find cross-sectional studies where

  • you have lots of samples of lots of species.

  • And there is a method that was in the end

  • that I had to flip through and not show

  • that basically allows you to--

  • if you've observed a system for a short period of time,

  • so if you're limited by the length of your time series,

  • but if there are many interacting variables,

  • you have an advantage, and that advantage grows factorially

  • as you have more variables, which is strange,

  • because it goes counter to our idea

  • that complex systems should be a problem.

  • So the curse of complexity we can actually exploit.

  • So the fact that these things are interconnected basically

  • means that each variable provides

  • another view of the system.

  • And so if you have many interconnected variables,

  • you have many different views of the system.

  • [INAUDIBLE]

  • Well, yeah.

  • What this is saying is that this is

  • kind of a way to counter the problem of high dimensionality,

  • that if you have a lot of interacting dimensions,

  • you have the potential for many alternative views

  • of the problem.

  • So if you did an embedding using--

  • [INAUDIBLE] embedding using each dimension,

  • each one gives you another view of the problem.

  • You can then actually do these mixed embeddings that

  • combine to take lags plus combine other dimensions,

  • and you end up then with a factorial number

  • of representations of the system.

  • And so this is actually a good way to reduce noise in systems.

  • There's a paper that came out last summer in Science,

  • it's called "Multi-View Embedding,"

  • that tries to exploit this.

  • Yeah.

  • So a couple times during your talk,

  • you alluded to having found the right lag,

  • i.e. the right value of tau.

  • Are their values of tau that perform better than others

  • in practice, and why is this?

  • Yeah.

  • So there no doubt are.

  • In ecology, we never do--

  • we rarely have the luxury of having oversampled data.

  • And so by default, the lag is typically

  • just one, whatever the sampling interval was.

  • So in the limit of continuous data, the tau shouldn't matter?

  • Oh, no, no, no, the tau will matter.

  • So if you are a physicist recording

  • something almost continuously in time, then the tau does matter.

  • So now you have two free variables.

  • You have to fit tau and you have to fit e.

  • And what you're doing, you want to choose

  • tau that allows you to unfold the attractor maximally.

  • And the way that you can determine that maximal

  • unfolding is by prediction, simple prediction.

  • Thank you.

  • Yeah.

  • OK, one last question if anybody has one.

  • No?

  • So this might be a bit of a naive question,

  • but where can one learn more about this?

  • Because it seems like it's relatively new.

  • There's advancements all the time

  • in our understanding of the world with this tool.

  • What realm is it under?

  • Is it statistics or biology?

  • Or what are the places that are doing research with this?

  • Yeah, so it is relatively new.

  • My lab has produced a bunch of papers dealing with this.

  • There is a software package now, it's

  • called rEDM that's on CRAN that has a tutorial,

  • and it discusses some of this.

  • But no, I need to write a review paper or a book or something

  • that puts it all in one place, and that hasn't been done yet.

  • So yeah.

  • But the software package is good.

  • My student put it together.

  • So I had all this horrible research software

  • that really was not intended for human consumption.

  • It was like for me in my pajamas at home.

  • But he rewrote it in R, and we put

  • in a very nice sort of tutorial with it

  • to kind of explain some of the basics.

  • But it's amazingly easy to use, and the ideas

  • are actually quite intuitive.

  • And it's something actually that I

  • think a number-- it is gaining.

  • It seems to be accelerating in usefulness.

  • And the citations, for example, are just like doing that.

  • So I think, again, having something that looks good

  • or that sounds good or that seems interesting

  • is very different from having something that actually works.

  • My lab is pretty pragmatic.

  • I say, these things actually have to work.

  • To make it easy for people to understand how they work,

  • we have to provide our markdown so that everything

  • can be exactly reproduced, and the code has to be there.

  • I would encourage you to check out the rEDM code.

  • Yeah.

  • Yeah.

  • All right.

  • Thank you very much.

  • Please join me in thanking your speaker.

  • [APPLAUSE]

Welcome, everybody.

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B1 中級

C.C. C. Mei Distinguished Speaker Series:杉原治博士 (C. C. Mei Distinguished Speaker Series: Dr. George Sugihara)

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    Josh 發佈於 2021 年 01 月 14 日
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