using window so maybe you can should we have from the

so

yeah

thank you so uh the title of my talk is cooperative operative maximum likelihood estimation for food flow dynamics you

know by a sensor array and uh

maybe somewhat unlike the previous talks in the section this is actually a real system with built so so we

actually trying to

see how we can model the very complicated system

uh so for so to actually describe you what this real system is and it's kind of quite and uh

interesting thing in it's own right

so uh let's focus first on the diagram of the

right hand side of the slide and

oh what what happened was that a colleague of whose name is bruce cornell uh in

in in the nineteen nineties an actually he's continue to work on that

he's build a a remarkable now don't machine

which can actually be uses as a by sensor and

the goal of our work is to try and mortal that how does that work and how can be used

that to do useful things

so

it's first important to to understand how this thing works because then we can model the dynamics of that

uh

so this by sensors actually build

our of a

synthetic cell number eight i mean all of the know what a cell membrane as it's on to sell and

it can of two lose of fact basic bits score than that that by layer

uh

mathematically the tool is a fact walk around

move around according to a random walk that's that's typically what people to

so what what we do next is they insert

in this slip by there

protein now don't two

these two are very easy to synthesise an approach you make that

oh what you can imagine now is you have two layers moving around according to a random walk

when two two

combine

they form of conducting or and a car can go through

when to to to do not combine there is a car going to

of course are several thousands of such do and so the probability of

a couple of dupes combining is quite large and what you see some sort of car which goes up and

down

so that's really what's shown there

now

that's fairly easy to synthesise that the main idea behind this of course as the next stage where

you attached to the top layer

pacific and bodies which can detect

molecules you wanted to do

so suppose to interesting in to a interested in detecting H I V or H one and one or or

or expose a molecule

you can build an approach you makes flat

specific antibodies which latch on to this

talk talk Q

so what happens is when you target molecule comes

these antibodies bodies score to point to them

and this stock

so what happens is a top player cannot move anymore

they graph

so this changes the dynamics of the system

previously we or on hindered random what what things were moving around

you are on and off

basically

card

now all the top is

stuff

and so the current dramatically decreases one other words that impedance

increases substantially

to this was the by sensor they build he publishes paper and ninety ninety seven in nature

and uh it's it's quite a remarkable sense so because this can detect

a low concentrations

we can detect up a fan till more lower concentrations and if you think about that that's that's pretty surprising

because

oh one more are as as you know from high school chemistry is one have a cat was number roughly

ten to the twenty three a ten to the three molecules in one or water

once stand to model lower

multiply by the by ten to the minus fifty

so what can to the eight

molecules molecule lead or more and that is extremely low

concentration

and that these things work

remarkably we for that

okay so that this is a system that field

oh goal has been to try to see

how can be more the system how can we predict how before

was if we can do that we could possibly fine the system and make it work better

and we can also maybe

extend the system to work in the scenarios

so we actually done a lot of work in this in the past and we have a couple of people

which came out of the transaction that of technology just last year

which dealt with more in the specific system

what wanna talk about today is

just ongoing work is

suppose that you take this by sensor when you build a a of such five sensors

how can you model that

and it turns are be highly nontrivial problem tell you why know

okay so that's for start with the individual electrodes to this is a signal by sensor

what happens is you of the fluid

which is still a word

to this by a the

you know food would delivery system

so you basically have a

some liquid

such as sodium chloride

containing the molecules tools you wish to detect

and that is flowing

how

this by sense

so this equation is the part of the to equation of of fluid flow

it's the parabolic pde

with the diffusion constant and so on

now what happens is where

the molecules in the fluid

and come to this and electrode

the trade off a chemical reaction which is a bunch of non than your or be different role questions

so i here is the concentration

both

the stuff stuff you want estimate

such as a H be you whatever

these are the chemical reactions to trade off

and what you measure eventually a some noisy version

of a specific chemical in this chemical reaction

plus

but

so these are the dynamics of the system

a there pretty dirty for several reasons you see this guy and it's all is pretty nice this is just

a straightforward forward flow P D to still be Q

the back part lot of things we conditions

you see that select lies at the bottom of the food chamber

and this is where the stuff happens

this is where the molecules which which we should detect

re yeah with this by sensor

to give you your measurement which is a increase in impedance

this is a a boundary condition only and one location so it's not smooth

it's not a it it's just a that location

uh so when you have these complicated

do rate it is that your boundary should these of

uh one alignment boundary conditions that and the fairly difficult to deal with

uh in addition you have noise you okay so this this system is is quite sophisticated it are so that

you can construct a very nice models for this and and we've done that of the part

now let's look at what we're trying to do here

so now we have a a rate of such things

see what is really nonstandard standard in this is but you have an array of such a electrode

and particular if your concentration a is very small

you of fluid goes pasta for select role

the electrode graph

some some of the molecules to react with

that means when you measure the system you actually changing the system

and that's likely non standard and signal processing

in most signal processing we do typically menu measure the system you don't change

so as to four flows but here

the first electrode brat the molecules so the second electrode has fewer Q

to detect

so if you placed the second electrode very close to the first select road

there's a depletion layer and no you don't a measure and fig

if you pay a second were very far away from the first electrode

it takes a while for the food to reach their and it means that you're detection times very low

so so actually designing

where you place you electrodes is is

it itself also an interesting problem

so anyway a goal at the stage is per some to be able to model this and come up with

some tractable approximations as to how how this stay

and i one wanna describe some of those two

so you can be this is a problem of saying that given this

fairly complex the

would multiple measuring devices

how do like estimate the concentration at the initial concentration

that's that's what we wish to

to me

so i let's go a little bit of it you should before we start so P D's activity i mean

so the first step you can think of is

can be construct some sort of

time scale approximation to replace this by or we differ to question

in that sense we would simply we have a nonlinear D ease which which is a more tractable problem

then you can deal with the non linear regression and and and a signal processing like that it for

not not not reveal but least it's it's

manageable

okay so the way one can do it is

you can use actually multi time scale dynamics

it turns so that as flow goes by

you see stuff at the top of the chamber which is very far away for the electrode

uh

it very quickly

a G station stationarity in other words

stuff at the top

at of T reaches at infinity

very quick

oh it turns out you can really segment

spatially this

in to several compartment

regions which uh far away from the electrode

are are basically constant you don't need the speedy

that you just have to sort

regions which are very close

of course you have to speed D but you make for the approximation to as described in a minute

so the idea you you to use something called averaging two you with some of you may be familiar with

but when you D with the filters

that

one at time scale where things happen slow and fast

on the slow time scale you can replace the fast i i'd average which in this case is a constant

away from the true

and a man the stuff the electrode you have to be but a bit more careful to to see what

to be done

oh case that's roughly didn't you should another vehicle want to

how you construct such more

so uh this is actually the as that mentioned before you have this fluid would flow and then these are

the equations of the chemical reactions to the sensors

you don't so actually these chemical reactions of cells at two time scales and so you can for the simplified

things

you can actually average job the fast times deal with a so that's good

and as i set a goal is to estimate the concentration at in the in light of this

through chamber

so

this is what we don't to do is we are place this distributed parameter system or P D if you

like by multiple compartment

and that becomes a bunch of more than your own to use it and then you can do of a

question

questions

so uh the way you do this this this idea of using multiple compartment model that is widely studied by

people who to be on with fluid flow chemistry and and they they analyse some fairly complex devices

for that sort of thing

the main idea is this

so if you could can our say a single electrode

what you can do is

i and this is that in and you can actually make this quite rigorous mathematical you uh so

if you know that you P

you could view

heuristic lead a P well with the spatial dimension as an infinite system of all the east but you want

you space

okay

so you to think of a great over space

where at each grade you have a or T E and of course all these all these are interacting you

of infinite

system a to use based

what what you think that show is that the only D which is very close to here

is really the boundary condition

so that's a chemical reaction

the only which is a way from this

is essentially the one which is a power with this chamber here with things happen very fast

and that there's no P D it's just a that because

it's a it's fluid flow

approximation of right

so it's you have to boundary conditions

and this chamber just a bunch of than you these that's that's really that you

so what you and it up with is a two compartment model here

where here you have a constant

here you have a nonlinear only

and you have the boundary condition which is of the all that would be

so you now down to a finite dimensional system

which is which is tractable

now as a said this was a heuristic thing because

to be really uh

is you have to kind of quantized of course this up sub step i learn

and then you have to

proof that the errors are bounded in so

this is a deterministic system so it's not that difficult to do things i mean it's of a stochastic could

be slightly harder

so actually it it's it's not not that difficult if if you assumes so the regularity the system then then

you can do one

okay now so that's the to compartment model which is easy to do

now you can imagine conceptually uh extended this robotic a model

and it's roughly the same thing so in the first compartment is identical to to what you have to you

for the first like true

i'm and stuff was fast post to to the second electronic can you have a to keep up with the

and and of course you have to keep in mind that because the first electrode is grad

some of the and a like

some of the ball was to try to better

the second electrode has different boundary conditions which are vertical boundary conditions you

but it's it's quite easy to take care of conceptual

so that you have it to basically have a whole system of

norm that you're or ease which is conceptually at least easier than dealing with a much more complex as to

so once you

now as i mentioned that i'm not gonna give you the details of these all these it doesn't really serve

any purpose

but just to kind of go back and you roughly what these ordinary we differential equations do

uh

you see they really model the chemical reactions you

the model the movement of these

now i tubes in the

a number eight

they model how these chemical reactions happen

the chemical bonds

and how when things couple

they don't move so really there about seven chemical species it you know which you you model by of order

to control question

these are quite tractable in in fact

the reaction rate constants and the one by by a of so the the

all the parameters of those equations are actually

or we so once to end up with this as a said you know all the reaction rate parameters you

only unknown quantity is your input

and like

the creation which which is what we wish to be

so that was really the main idea uh and

just a can give you now some it you should be how this would work

so if you think about

even if this was a of now a bunch of ordinary differential equations questions or if you like discrete as

a what time it's a bunch of difference equations

a you have this fluid flow coming in here

so as you think that this guy remote say some proportion of the

molecules

see alpha

that means you're signal power here is reduced by but for because you grab some of the guys

so you can think that the second electrode what has a little or signal to noise ratio because you less

signal

now of course the next electrode will be out of the squared and so becomes a geometric matrix V still

it's obvious that you won't get

dramatic improvement as you have multiple electrodes because it dies of as a geometric matrix series

i mean that's gonna a used

even at the linear case now of course is the non linear regression you have to be slightly more careful

uh in any case one could actually

workout out

the asymptotic covariance of this

okay and it's it's a fairly complex question

and one could actually show that

in as you have more more electrodes you your performance that's really from frame

to you'd expect if you and N electrodes everything was

i i D you had no interaction between things you get one of and improved

sure you don't

so that's that's something which

so be the case

uh

okay so this is the actual system be but we've tested this on on

so it to test this several things you can do

you not to run of the real experimental system

and compare it with without a of different look should model that's to kind of to a model that's the

first step

and those that things we've done it quick you come the pa that's that that were

a second stage is how can you actually

shall

if the multi component model is a good approximation because that's the we we want to estimate of constant

and it are actually they work extremely well as

we can see these diagrams you

the actually approximate the

P E extremely well

uh the arrow it use that to be equally between six to eight percent

and even maybe go to

concentrations which are very very low

concentrations of

almost almost

well below an animal are still

so these these that she work quite well

uh and and therefore it means that you can apply standard a question analysis to solve for these concentrations

i want make a few other

comments before finish the all

this is still work in progress i mean it's a very nice to come up with these approximations but you

don't the sort of things we have done which would not please people are rigorous not my

we haven't even shown that this system of equations

has a unique solution it's really down hard to show that

so i mean if you think about P D easy we bad this in a function space like a stop

let's space you wanna show that you it's of a solution

highly nontrivial

so

i mean although the system works other real axis system

sure you you this is really hard

it's something which we working on of the moment

uh

the are the issues are it would be really useful to come up with a nice

approximations

for this pde itself

it's still of just using a multi compartment model we've done

because we still ending up with a bunch of all we different role equations which

we don't have a

so form solution but eventually do you man

we much nice if we can come up with further there were approximations which allows to get

some inside

as to how the system works

so there are still a couple of

a that's to our approach

but i think this is a in the sense that

given the complexity of the system we can model it

it works pretty good

we can approximate it and we can actually estimate using elementary nonlinear linear regression

the sort of concentrations

we want to estimate

so that's really all of wanna say

uh if you're interested in any of the stuff this was the original paper we by a colleague

uh and these are a couple as we did where we dealt with the signal electrode case maybe be model

it that and actually did

a not than you're question on that

oh can thank you very much

yeah right are you know

oh okay if gram so a a really interesting problem uh uh of course um

is a lot of comments i guess for instance with the pay T um

you've got a linear spatial up right the first to terms so there's a brings function that's an an

i i you could generate a one approximations and stuff like that a you've been looking in that direction we

have quite a bit what really kills us is cool

so

is is this

i a gonna this is a horrible but we could dish of the that

this a are could if should which likes to the P

right and and

and this this really is a a a i was not a at the takes the all of

you know the concentration respect to spatial axes and sets to one of the things so then one is thing

of boundary element methods because uh that's the use a to the with that of stuff yeah

i numerically what we done actually is because you have a a solution are obvious so you don't have strong

so should we use a finite element method which takes to functions automatically

but again

oh for our approach is completely and hearing in the sense that we have a solution it works

but from a mathematical point to we we still have even shown any structure proper okay but good of array

element method is different from finite element you i don't know if you familiar menu with a a a a

type find so

a any whites aims to make that might worth looking at

that are thing is on not entirely clear that the regression analysis because the use is i'd time varying quantity

so um

uh

i great so for instance if you nate to

in house some regularization on the a temporal times of the use signals

okay so let's a go to it so a is the concentration which changes over time yes we interested a

zero which is a initial concentration wrong

a a is coupled to these sceptical reactions which are you which you also over time

these are pretty easy to show for example that these are always non-negative and so the they just basic chemical

reactions a can i the only difficulty in this is a non linearity of that

but you've got gotta a construct you of tape right

well one specific um of that is a what be measure in noise

okay i but when you do the regression what do us to my just i it's a so one one

gram use of that

you of taste he is in there must be dealt with a the why something is it so what you

have is but you re braces by multi can pop and model you have a a bunch of or very

different role questions here

and here

which are in you

and the interested in one specific component of you

observed in

which is a concentration of the diners basically

uh

basically

the number of couple guys you

because the cover is proportional to the

no i i understand all that but i guess again but in the middle somewhere in that regression you are

as a function of tape must some have a being with carpet

in order to waste one one this upon it

one can hold diffuse for so i just one where the in a some temporal regularization or or or or

the that of

a us we have applied in such thing so far so are

yeah my help

it it back

or a