0:00:13using window so maybe you can should we have from the
0:00:38thank you so uh the title of my talk is cooperative operative maximum likelihood estimation for food flow dynamics you
0:00:45know by a sensor array and uh
0:00:48maybe somewhat unlike the previous talks in the section this is actually a real system with built so so we
0:00:53actually trying to
0:00:55see how we can model the very complicated system
0:00:58uh so for so to actually describe you what this real system is and it's kind of quite and uh
0:01:02interesting thing in it's own right
0:01:04so uh let's focus first on the diagram of the
0:01:07right hand side of the slide and
0:01:09oh what what happened was that a colleague of whose name is bruce cornell uh in
0:01:15in in the nineteen nineties an actually he's continue to work on that
0:01:18he's build a a remarkable now don't machine
0:01:21which can actually be uses as a by sensor and
0:01:24the goal of our work is to try and mortal that how does that work and how can be used
0:01:29that to do useful things
0:01:32it's first important to to understand how this thing works because then we can model the dynamics of that
0:01:38so this by sensors actually build
0:01:41our of a
0:01:42synthetic cell number eight i mean all of the know what a cell membrane as it's on to sell and
0:01:47it can of two lose of fact basic bits score than that that by layer
0:01:51mathematically the tool is a fact walk around
0:01:55move around according to a random walk that's that's typically what people to
0:01:58so what what we do next is they insert
0:02:02in this slip by there
0:02:04protein now don't two
0:02:06these two are very easy to synthesise an approach you make that
0:02:10oh what you can imagine now is you have two layers moving around according to a random walk
0:02:15when two two
0:02:18they form of conducting or and a car can go through
0:02:22when to to to do not combine there is a car going to
0:02:25of course are several thousands of such do and so the probability of
0:02:28a couple of dupes combining is quite large and what you see some sort of car which goes up and
0:02:34so that's really what's shown there
0:02:36that's fairly easy to synthesise that the main idea behind this of course as the next stage where
0:02:42you attached to the top layer
0:02:44pacific and bodies which can detect
0:02:47molecules you wanted to do
0:02:49so suppose to interesting in to a interested in detecting H I V or H one and one or or
0:02:53or expose a molecule
0:02:55you can build an approach you makes flat
0:02:57specific antibodies which latch on to this
0:03:01talk talk Q
0:03:02so what happens is when you target molecule comes
0:03:06these antibodies bodies score to point to them
0:03:09and this stock
0:03:10so what happens is a top player cannot move anymore
0:03:13they graph
0:03:15so this changes the dynamics of the system
0:03:17previously we or on hindered random what what things were moving around
0:03:21you are on and off
0:03:25now all the top is
0:03:27and so the current dramatically decreases one other words that impedance
0:03:31increases substantially
0:03:33to this was the by sensor they build he publishes paper and ninety ninety seven in nature
0:03:37and uh it's it's quite a remarkable sense so because this can detect
0:03:42a low concentrations
0:03:44we can detect up a fan till more lower concentrations and if you think about that that's that's pretty surprising
0:03:49oh one more are as as you know from high school chemistry is one have a cat was number roughly
0:03:53ten to the twenty three a ten to the three molecules in one or water
0:03:57once stand to model lower
0:03:59multiply by the by ten to the minus fifty
0:04:01so what can to the eight
0:04:03molecules molecule lead or more and that is extremely low
0:04:07and that these things work
0:04:09remarkably we for that
0:04:10okay so that this is a system that field
0:04:12oh goal has been to try to see
0:04:15how can be more the system how can we predict how before
0:04:18was if we can do that we could possibly fine the system and make it work better
0:04:23and we can also maybe
0:04:25extend the system to work in the scenarios
0:04:28so we actually done a lot of work in this in the past and we have a couple of people
0:04:31which came out of the transaction that of technology just last year
0:04:34which dealt with more in the specific system
0:04:37what wanna talk about today is
0:04:39just ongoing work is
0:04:41suppose that you take this by sensor when you build a a of such five sensors
0:04:45how can you model that
0:04:47and it turns are be highly nontrivial problem tell you why know
0:04:51okay so that's for start with the individual electrodes to this is a signal by sensor
0:04:55what happens is you of the fluid
0:04:57which is still a word
0:04:58to this by a the
0:05:00you know food would delivery system
0:05:02so you basically have a
0:05:03some liquid
0:05:05such as sodium chloride
0:05:07containing the molecules tools you wish to detect
0:05:10and that is flowing
0:05:12this by sense
0:05:14so this equation is the part of the to equation of of fluid flow
0:05:17it's the parabolic pde
0:05:19with the diffusion constant and so on
0:05:22now what happens is where
0:05:25the molecules in the fluid
0:05:27and come to this and electrode
0:05:30the trade off a chemical reaction which is a bunch of non than your or be different role questions
0:05:34so i here is the concentration
0:05:38the stuff stuff you want estimate
0:05:40such as a H be you whatever
0:05:42these are the chemical reactions to trade off
0:05:44and what you measure eventually a some noisy version
0:05:47of a specific chemical in this chemical reaction
0:05:52so these are the dynamics of the system
0:05:54a there pretty dirty for several reasons you see this guy and it's all is pretty nice this is just
0:05:59a straightforward forward flow P D to still be Q
0:06:03the back part lot of things we conditions
0:06:05you see that select lies at the bottom of the food chamber
0:06:08and this is where the stuff happens
0:06:10this is where the molecules which which we should detect
0:06:13re yeah with this by sensor
0:06:16to give you your measurement which is a increase in impedance
0:06:19this is a a boundary condition only and one location so it's not smooth
0:06:23it's not a it it's just a that location
0:06:26uh so when you have these complicated
0:06:29do rate it is that your boundary should these of
0:06:31uh one alignment boundary conditions that and the fairly difficult to deal with
0:06:35uh in addition you have noise you okay so this this system is is quite sophisticated it are so that
0:06:40you can construct a very nice models for this and and we've done that of the part
0:06:44now let's look at what we're trying to do here
0:06:46so now we have a a rate of such things
0:06:48see what is really nonstandard standard in this is but you have an array of such a electrode
0:06:53and particular if your concentration a is very small
0:06:56you of fluid goes pasta for select role
0:06:59the electrode graph
0:07:00some some of the molecules to react with
0:07:03that means when you measure the system you actually changing the system
0:07:06and that's likely non standard and signal processing
0:07:08in most signal processing we do typically menu measure the system you don't change
0:07:13so as to four flows but here
0:07:15the first electrode brat the molecules so the second electrode has fewer Q
0:07:20to detect
0:07:20so if you placed the second electrode very close to the first select road
0:07:24there's a depletion layer and no you don't a measure and fig
0:07:27if you pay a second were very far away from the first electrode
0:07:30it takes a while for the food to reach their and it means that you're detection times very low
0:07:34so so actually designing
0:07:36where you place you electrodes is is
0:07:38it itself also an interesting problem
0:07:40so anyway a goal at the stage is per some to be able to model this and come up with
0:07:44some tractable approximations as to how how this stay
0:07:47and i one wanna describe some of those two
0:07:49so you can be this is a problem of saying that given this
0:07:52fairly complex the
0:07:54would multiple measuring devices
0:07:56how do like estimate the concentration at the initial concentration
0:08:00that's that's what we wish to
0:08:01to me
0:08:03so i let's go a little bit of it you should before we start so P D's activity i mean
0:08:07so the first step you can think of is
0:08:09can be construct some sort of
0:08:11time scale approximation to replace this by or we differ to question
0:08:15in that sense we would simply we have a nonlinear D ease which which is a more tractable problem
0:08:19then you can deal with the non linear regression and and and a signal processing like that it for
0:08:24not not not reveal but least it's it's
0:08:27okay so the way one can do it is
0:08:29you can use actually multi time scale dynamics
0:08:32it turns so that as flow goes by
0:08:35you see stuff at the top of the chamber which is very far away for the electrode
0:08:40it very quickly
0:08:41a G station stationarity in other words
0:08:43stuff at the top
0:08:45at of T reaches at infinity
0:08:47very quick
0:08:49oh it turns out you can really segment
0:08:51spatially this
0:08:52in to several compartment
0:08:55regions which uh far away from the electrode
0:08:58are are basically constant you don't need the speedy
0:09:00that you just have to sort
0:09:02regions which are very close
0:09:04of course you have to speed D but you make for the approximation to as described in a minute
0:09:08so the idea you you to use something called averaging two you with some of you may be familiar with
0:09:12but when you D with the filters
0:09:15one at time scale where things happen slow and fast
0:09:18on the slow time scale you can replace the fast i i'd average which in this case is a constant
0:09:23away from the true
0:09:24and a man the stuff the electrode you have to be but a bit more careful to to see what
0:09:28to be done
0:09:29oh case that's roughly didn't you should another vehicle want to
0:09:31how you construct such more
0:09:35so uh this is actually the as that mentioned before you have this fluid would flow and then these are
0:09:39the equations of the chemical reactions to the sensors
0:09:42you don't so actually these chemical reactions of cells at two time scales and so you can for the simplified
0:09:47you can actually average job the fast times deal with a so that's good
0:09:51and as i set a goal is to estimate the concentration at in the in light of this
0:09:55through chamber
0:09:58this is what we don't to do is we are place this distributed parameter system or P D if you
0:10:02like by multiple compartment
0:10:04and that becomes a bunch of more than your own to use it and then you can do of a
0:10:12so uh the way you do this this this idea of using multiple compartment model that is widely studied by
0:10:17people who to be on with fluid flow chemistry and and they they analyse some fairly complex devices
0:10:23for that sort of thing
0:10:24the main idea is this
0:10:25so if you could can our say a single electrode
0:10:28what you can do is
0:10:30i and this is that in and you can actually make this quite rigorous mathematical you uh so
0:10:34if you know that you P
0:10:36you could view
0:10:37heuristic lead a P well with the spatial dimension as an infinite system of all the east but you want
0:10:43you space
0:10:45so you to think of a great over space
0:10:47where at each grade you have a or T E and of course all these all these are interacting you
0:10:52of infinite
0:10:52system a to use based
0:10:55what what you think that show is that the only D which is very close to here
0:10:59is really the boundary condition
0:11:01so that's a chemical reaction
0:11:04the only which is a way from this
0:11:06is essentially the one which is a power with this chamber here with things happen very fast
0:11:11and that there's no P D it's just a that because
0:11:14it's a it's fluid flow
0:11:16approximation of right
0:11:18so it's you have to boundary conditions
0:11:20and this chamber just a bunch of than you these that's that's really that you
0:11:24so what you and it up with is a two compartment model here
0:11:28where here you have a constant
0:11:29here you have a nonlinear only
0:11:32and you have the boundary condition which is of the all that would be
0:11:35so you now down to a finite dimensional system
0:11:37which is which is tractable
0:11:38now as a said this was a heuristic thing because
0:11:41to be really uh
0:11:42is you have to kind of quantized of course this up sub step i learn
0:11:46and then you have to
0:11:47proof that the errors are bounded in so
0:11:50this is a deterministic system so it's not that difficult to do things i mean it's of a stochastic could
0:11:54be slightly harder
0:11:55so actually it it's it's not not that difficult if if you assumes so the regularity the system then then
0:11:59you can do one
0:12:00okay now so that's the to compartment model which is easy to do
0:12:04now you can imagine conceptually uh extended this robotic a model
0:12:07and it's roughly the same thing so in the first compartment is identical to to what you have to you
0:12:12for the first like true
0:12:13i'm and stuff was fast post to to the second electronic can you have a to keep up with the
0:12:18and and of course you have to keep in mind that because the first electrode is grad
0:12:22some of the and a like
0:12:23some of the ball was to try to better
0:12:26the second electrode has different boundary conditions which are vertical boundary conditions you
0:12:31but it's it's quite easy to take care of conceptual
0:12:33so that you have it to basically have a whole system of
0:12:37norm that you're or ease which is conceptually at least easier than dealing with a much more complex as to
0:12:43so once you
0:12:44now as i mentioned that i'm not gonna give you the details of these all these it doesn't really serve
0:12:48any purpose
0:12:49but just to kind of go back and you roughly what these ordinary we differential equations do
0:12:56you see they really model the chemical reactions you
0:12:59the model the movement of these
0:13:03now i tubes in the
0:13:05a number eight
0:13:06they model how these chemical reactions happen
0:13:09the chemical bonds
0:13:11and how when things couple
0:13:13they don't move so really there about seven chemical species it you know which you you model by of order
0:13:18to control question
0:13:20these are quite tractable in in fact
0:13:21the reaction rate constants and the one by by a of so the the
0:13:25all the parameters of those equations are actually
0:13:29or we so once to end up with this as a said you know all the reaction rate parameters you
0:13:33only unknown quantity is your input
0:13:35and like
0:13:36the creation which which is what we wish to be
0:13:39so that was really the main idea uh and
0:13:42just a can give you now some it you should be how this would work
0:13:46so if you think about
0:13:47even if this was a of now a bunch of ordinary differential equations questions or if you like discrete as
0:13:51a what time it's a bunch of difference equations
0:13:54a you have this fluid flow coming in here
0:13:57so as you think that this guy remote say some proportion of the
0:14:03see alpha
0:14:05that means you're signal power here is reduced by but for because you grab some of the guys
0:14:09so you can think that the second electrode what has a little or signal to noise ratio because you less
0:14:15now of course the next electrode will be out of the squared and so becomes a geometric matrix V still
0:14:19it's obvious that you won't get
0:14:22dramatic improvement as you have multiple electrodes because it dies of as a geometric matrix series
0:14:26i mean that's gonna a used
0:14:28even at the linear case now of course is the non linear regression you have to be slightly more careful
0:14:32uh in any case one could actually
0:14:34workout out
0:14:35the asymptotic covariance of this
0:14:37okay and it's it's a fairly complex question
0:14:39and one could actually show that
0:14:42in as you have more more electrodes you your performance that's really from frame
0:14:47to you'd expect if you and N electrodes everything was
0:14:49i i D you had no interaction between things you get one of and improved
0:14:53sure you don't
0:14:54so that's that's something which
0:14:56so be the case
0:14:59okay so this is the actual system be but we've tested this on on
0:15:03so it to test this several things you can do
0:15:05you not to run of the real experimental system
0:15:08and compare it with without a of different look should model that's to kind of to a model that's the
0:15:12first step
0:15:13and those that things we've done it quick you come the pa that's that that were
0:15:16a second stage is how can you actually
0:15:19if the multi component model is a good approximation because that's the we we want to estimate of constant
0:15:25and it are actually they work extremely well as
0:15:27we can see these diagrams you
0:15:29the actually approximate the
0:15:31P E extremely well
0:15:33uh the arrow it use that to be equally between six to eight percent
0:15:37and even maybe go to
0:15:39concentrations which are very very low
0:15:41concentrations of
0:15:42almost almost
0:15:44well below an animal are still
0:15:45so these these that she work quite well
0:15:48uh and and therefore it means that you can apply standard a question analysis to solve for these concentrations
0:15:54i want make a few other
0:15:56comments before finish the all
0:15:57this is still work in progress i mean it's a very nice to come up with these approximations but you
0:16:02don't the sort of things we have done which would not please people are rigorous not my
0:16:06we haven't even shown that this system of equations
0:16:09has a unique solution it's really down hard to show that
0:16:12so i mean if you think about P D easy we bad this in a function space like a stop
0:16:16let's space you wanna show that you it's of a solution
0:16:19highly nontrivial
0:16:21i mean although the system works other real axis system
0:16:24sure you you this is really hard
0:16:26it's something which we working on of the moment
0:16:29the are the issues are it would be really useful to come up with a nice
0:16:36for this pde itself
0:16:39it's still of just using a multi compartment model we've done
0:16:42because we still ending up with a bunch of all we different role equations which
0:16:46we don't have a
0:16:47so form solution but eventually do you man
0:16:50we much nice if we can come up with further there were approximations which allows to get
0:16:53some inside
0:16:55as to how the system works
0:16:56so there are still a couple of
0:16:58a that's to our approach
0:16:59but i think this is a in the sense that
0:17:02given the complexity of the system we can model it
0:17:05it works pretty good
0:17:06we can approximate it and we can actually estimate using elementary nonlinear linear regression
0:17:11the sort of concentrations
0:17:12we want to estimate
0:17:14so that's really all of wanna say
0:17:16uh if you're interested in any of the stuff this was the original paper we by a colleague
0:17:20uh and these are a couple as we did where we dealt with the signal electrode case maybe be model
0:17:24it that and actually did
0:17:26a not than you're question on that
0:17:28oh can thank you very much
0:17:37yeah right are you know
0:17:42oh okay if gram so a a really interesting problem uh uh of course um
0:17:46is a lot of comments i guess for instance with the pay T um
0:17:51you've got a linear spatial up right the first to terms so there's a brings function that's an an
0:17:57i i you could generate a one approximations and stuff like that a you've been looking in that direction we
0:18:01have quite a bit what really kills us is cool
0:18:05is is this
0:18:07i a gonna this is a horrible but we could dish of the that
0:18:10this a are could if should which likes to the P
0:18:13right and and
0:18:15and this this really is a a a i was not a at the takes the all of
0:18:20you know the concentration respect to spatial axes and sets to one of the things so then one is thing
0:18:24of boundary element methods because uh that's the use a to the with that of stuff yeah
0:18:29i numerically what we done actually is because you have a a solution are obvious so you don't have strong
0:18:33so should we use a finite element method which takes to functions automatically
0:18:36but again
0:18:37oh for our approach is completely and hearing in the sense that we have a solution it works
0:18:42but from a mathematical point to we we still have even shown any structure proper okay but good of array
0:18:47element method is different from finite element you i don't know if you familiar menu with a a a a
0:18:53type find so
0:18:54a any whites aims to make that might worth looking at
0:18:56that are thing is on not entirely clear that the regression analysis because the use is i'd time varying quantity
0:19:04so um
0:19:07i great so for instance if you nate to
0:19:10in house some regularization on the a temporal times of the use signals
0:19:15okay so let's a go to it so a is the concentration which changes over time yes we interested a
0:19:20zero which is a initial concentration wrong
0:19:23a a is coupled to these sceptical reactions which are you which you also over time
0:19:28these are pretty easy to show for example that these are always non-negative and so the they just basic chemical
0:19:33reactions a can i the only difficulty in this is a non linearity of that
0:19:37but you've got gotta a construct you of tape right
0:19:40well one specific um of that is a what be measure in noise
0:19:45okay i but when you do the regression what do us to my just i it's a so one one
0:19:49gram use of that
0:19:50you of taste he is in there must be dealt with a the why something is it so what you
0:19:55have is but you re braces by multi can pop and model you have a a bunch of or very
0:19:58different role questions here
0:20:00and here
0:20:01which are in you
0:20:02and the interested in one specific component of you
0:20:05observed in
0:20:07which is a concentration of the diners basically
0:20:14the number of couple guys you
0:20:16because the cover is proportional to the
0:20:18no i i understand all that but i guess again but in the middle somewhere in that regression you are
0:20:22as a function of tape must some have a being with carpet
0:20:25in order to waste one one this upon it
0:20:27one can hold diffuse for so i just one where the in a some temporal regularization or or or or
0:20:32the that of
0:20:33a us we have applied in such thing so far so are
0:20:36yeah my help
0:20:37it it back
0:20:39or a