0:00:13 | using window so maybe you can should we have from the |
---|---|

0:00:16 | so |

0:00:24 | yeah |

0:00:38 | thank you so uh the title of my talk is cooperative operative maximum likelihood estimation for food flow dynamics you |

0:00:45 | know by a sensor array and uh |

0:00:48 | maybe somewhat unlike the previous talks in the section this is actually a real system with built so so we |

0:00:53 | actually trying to |

0:00:55 | see how we can model the very complicated system |

0:00:58 | uh so for so to actually describe you what this real system is and it's kind of quite and uh |

0:01:02 | interesting thing in it's own right |

0:01:04 | so uh let's focus first on the diagram of the |

0:01:07 | right hand side of the slide and |

0:01:09 | oh what what happened was that a colleague of whose name is bruce cornell uh in |

0:01:15 | in in the nineteen nineties an actually he's continue to work on that |

0:01:18 | he's build a a remarkable now don't machine |

0:01:21 | which can actually be uses as a by sensor and |

0:01:24 | the goal of our work is to try and mortal that how does that work and how can be used |

0:01:29 | that to do useful things |

0:01:31 | so |

0:01:32 | it's first important to to understand how this thing works because then we can model the dynamics of that |

0:01:37 | uh |

0:01:38 | so this by sensors actually build |

0:01:41 | our of a |

0:01:42 | synthetic cell number eight i mean all of the know what a cell membrane as it's on to sell and |

0:01:47 | it can of two lose of fact basic bits score than that that by layer |

0:01:50 | uh |

0:01:51 | mathematically the tool is a fact walk around |

0:01:55 | move around according to a random walk that's that's typically what people to |

0:01:58 | so what what we do next is they insert |

0:02:02 | in this slip by there |

0:02:04 | protein now don't two |

0:02:06 | these two are very easy to synthesise an approach you make that |

0:02:10 | oh what you can imagine now is you have two layers moving around according to a random walk |

0:02:15 | when two two |

0:02:17 | combine |

0:02:18 | they form of conducting or and a car can go through |

0:02:22 | when to to to do not combine there is a car going to |

0:02:25 | of course are several thousands of such do and so the probability of |

0:02:28 | a couple of dupes combining is quite large and what you see some sort of car which goes up and |

0:02:32 | down |

0:02:34 | so that's really what's shown there |

0:02:36 | now |

0:02:36 | that's fairly easy to synthesise that the main idea behind this of course as the next stage where |

0:02:42 | you attached to the top layer |

0:02:44 | pacific and bodies which can detect |

0:02:47 | molecules you wanted to do |

0:02:49 | so suppose to interesting in to a interested in detecting H I V or H one and one or or |

0:02:53 | or expose a molecule |

0:02:55 | you can build an approach you makes flat |

0:02:57 | specific antibodies which latch on to this |

0:03:01 | talk talk Q |

0:03:02 | so what happens is when you target molecule comes |

0:03:06 | these antibodies bodies score to point to them |

0:03:09 | and this stock |

0:03:10 | so what happens is a top player cannot move anymore |

0:03:13 | they graph |

0:03:15 | so this changes the dynamics of the system |

0:03:17 | previously we or on hindered random what what things were moving around |

0:03:21 | you are on and off |

0:03:23 | basically |

0:03:24 | card |

0:03:25 | now all the top is |

0:03:27 | stuff |

0:03:27 | and so the current dramatically decreases one other words that impedance |

0:03:31 | increases substantially |

0:03:33 | to this was the by sensor they build he publishes paper and ninety ninety seven in nature |

0:03:37 | and uh it's it's quite a remarkable sense so because this can detect |

0:03:42 | a low concentrations |

0:03:44 | we can detect up a fan till more lower concentrations and if you think about that that's that's pretty surprising |

0:03:48 | because |

0:03:49 | oh one more are as as you know from high school chemistry is one have a cat was number roughly |

0:03:53 | ten to the twenty three a ten to the three molecules in one or water |

0:03:57 | once stand to model lower |

0:03:59 | multiply by the by ten to the minus fifty |

0:04:01 | so what can to the eight |

0:04:03 | molecules molecule lead or more and that is extremely low |

0:04:06 | concentration |

0:04:07 | and that these things work |

0:04:09 | remarkably we for that |

0:04:10 | okay so that this is a system that field |

0:04:12 | oh goal has been to try to see |

0:04:15 | how can be more the system how can we predict how before |

0:04:18 | was if we can do that we could possibly fine the system and make it work better |

0:04:23 | and we can also maybe |

0:04:25 | extend the system to work in the scenarios |

0:04:28 | so we actually done a lot of work in this in the past and we have a couple of people |

0:04:31 | which came out of the transaction that of technology just last year |

0:04:34 | which dealt with more in the specific system |

0:04:37 | what wanna talk about today is |

0:04:39 | just ongoing work is |

0:04:41 | suppose that you take this by sensor when you build a a of such five sensors |

0:04:45 | how can you model that |

0:04:47 | and it turns are be highly nontrivial problem tell you why know |

0:04:51 | okay so that's for start with the individual electrodes to this is a signal by sensor |

0:04:55 | what happens is you of the fluid |

0:04:57 | which is still a word |

0:04:58 | to this by a the |

0:05:00 | you know food would delivery system |

0:05:02 | so you basically have a |

0:05:03 | some liquid |

0:05:05 | such as sodium chloride |

0:05:07 | containing the molecules tools you wish to detect |

0:05:10 | and that is flowing |

0:05:11 | how |

0:05:12 | this by sense |

0:05:14 | so this equation is the part of the to equation of of fluid flow |

0:05:17 | it's the parabolic pde |

0:05:19 | with the diffusion constant and so on |

0:05:22 | now what happens is where |

0:05:25 | the molecules in the fluid |

0:05:27 | and come to this and electrode |

0:05:30 | the trade off a chemical reaction which is a bunch of non than your or be different role questions |

0:05:34 | so i here is the concentration |

0:05:37 | both |

0:05:38 | the stuff stuff you want estimate |

0:05:40 | such as a H be you whatever |

0:05:42 | these are the chemical reactions to trade off |

0:05:44 | and what you measure eventually a some noisy version |

0:05:47 | of a specific chemical in this chemical reaction |

0:05:50 | plus |

0:05:51 | but |

0:05:52 | so these are the dynamics of the system |

0:05:54 | a there pretty dirty for several reasons you see this guy and it's all is pretty nice this is just |

0:05:59 | a straightforward forward flow P D to still be Q |

0:06:03 | the back part lot of things we conditions |

0:06:05 | you see that select lies at the bottom of the food chamber |

0:06:08 | and this is where the stuff happens |

0:06:10 | this is where the molecules which which we should detect |

0:06:13 | re yeah with this by sensor |

0:06:16 | to give you your measurement which is a increase in impedance |

0:06:19 | this is a a boundary condition only and one location so it's not smooth |

0:06:23 | it's not a it it's just a that location |

0:06:26 | uh so when you have these complicated |

0:06:29 | do rate it is that your boundary should these of |

0:06:31 | uh one alignment boundary conditions that and the fairly difficult to deal with |

0:06:35 | uh in addition you have noise you okay so this this system is is quite sophisticated it are so that |

0:06:40 | you can construct a very nice models for this and and we've done that of the part |

0:06:44 | now let's look at what we're trying to do here |

0:06:46 | so now we have a a rate of such things |

0:06:48 | see what is really nonstandard standard in this is but you have an array of such a electrode |

0:06:53 | and particular if your concentration a is very small |

0:06:56 | you of fluid goes pasta for select role |

0:06:59 | the electrode graph |

0:07:00 | some some of the molecules to react with |

0:07:03 | that means when you measure the system you actually changing the system |

0:07:06 | and that's likely non standard and signal processing |

0:07:08 | in most signal processing we do typically menu measure the system you don't change |

0:07:13 | so as to four flows but here |

0:07:15 | the first electrode brat the molecules so the second electrode has fewer Q |

0:07:20 | to detect |

0:07:20 | so if you placed the second electrode very close to the first select road |

0:07:24 | there's a depletion layer and no you don't a measure and fig |

0:07:27 | if you pay a second were very far away from the first electrode |

0:07:30 | it takes a while for the food to reach their and it means that you're detection times very low |

0:07:34 | so so actually designing |

0:07:36 | where you place you electrodes is is |

0:07:38 | it itself also an interesting problem |

0:07:40 | so anyway a goal at the stage is per some to be able to model this and come up with |

0:07:44 | some tractable approximations as to how how this stay |

0:07:47 | and i one wanna describe some of those two |

0:07:49 | so you can be this is a problem of saying that given this |

0:07:52 | fairly complex the |

0:07:54 | would multiple measuring devices |

0:07:56 | how do like estimate the concentration at the initial concentration |

0:08:00 | that's that's what we wish to |

0:08:01 | to me |

0:08:03 | so i let's go a little bit of it you should before we start so P D's activity i mean |

0:08:07 | so the first step you can think of is |

0:08:09 | can be construct some sort of |

0:08:11 | time scale approximation to replace this by or we differ to question |

0:08:15 | in that sense we would simply we have a nonlinear D ease which which is a more tractable problem |

0:08:19 | then you can deal with the non linear regression and and and a signal processing like that it for |

0:08:24 | not not not reveal but least it's it's |

0:08:26 | manageable |

0:08:27 | okay so the way one can do it is |

0:08:29 | you can use actually multi time scale dynamics |

0:08:32 | it turns so that as flow goes by |

0:08:35 | you see stuff at the top of the chamber which is very far away for the electrode |

0:08:39 | uh |

0:08:40 | it very quickly |

0:08:41 | a G station stationarity in other words |

0:08:43 | stuff at the top |

0:08:45 | at of T reaches at infinity |

0:08:47 | very quick |

0:08:49 | oh it turns out you can really segment |

0:08:51 | spatially this |

0:08:52 | in to several compartment |

0:08:55 | regions which uh far away from the electrode |

0:08:58 | are are basically constant you don't need the speedy |

0:09:00 | that you just have to sort |

0:09:02 | regions which are very close |

0:09:04 | of course you have to speed D but you make for the approximation to as described in a minute |

0:09:08 | so the idea you you to use something called averaging two you with some of you may be familiar with |

0:09:12 | but when you D with the filters |

0:09:14 | that |

0:09:15 | one at time scale where things happen slow and fast |

0:09:18 | on the slow time scale you can replace the fast i i'd average which in this case is a constant |

0:09:23 | away from the true |

0:09:24 | and a man the stuff the electrode you have to be but a bit more careful to to see what |

0:09:28 | to be done |

0:09:29 | oh case that's roughly didn't you should another vehicle want to |

0:09:31 | how you construct such more |

0:09:35 | so uh this is actually the as that mentioned before you have this fluid would flow and then these are |

0:09:39 | the equations of the chemical reactions to the sensors |

0:09:42 | you don't so actually these chemical reactions of cells at two time scales and so you can for the simplified |

0:09:46 | things |

0:09:47 | you can actually average job the fast times deal with a so that's good |

0:09:51 | and as i set a goal is to estimate the concentration at in the in light of this |

0:09:55 | through chamber |

0:09:57 | so |

0:09:58 | this is what we don't to do is we are place this distributed parameter system or P D if you |

0:10:02 | like by multiple compartment |

0:10:04 | and that becomes a bunch of more than your own to use it and then you can do of a |

0:10:07 | question |

0:10:08 | questions |

0:10:12 | so uh the way you do this this this idea of using multiple compartment model that is widely studied by |

0:10:17 | people who to be on with fluid flow chemistry and and they they analyse some fairly complex devices |

0:10:23 | for that sort of thing |

0:10:24 | the main idea is this |

0:10:25 | so if you could can our say a single electrode |

0:10:28 | what you can do is |

0:10:30 | i and this is that in and you can actually make this quite rigorous mathematical you uh so |

0:10:34 | if you know that you P |

0:10:36 | you could view |

0:10:37 | heuristic lead a P well with the spatial dimension as an infinite system of all the east but you want |

0:10:43 | you space |

0:10:44 | okay |

0:10:45 | so you to think of a great over space |

0:10:47 | where at each grade you have a or T E and of course all these all these are interacting you |

0:10:52 | of infinite |

0:10:52 | system a to use based |

0:10:55 | what what you think that show is that the only D which is very close to here |

0:10:59 | is really the boundary condition |

0:11:01 | so that's a chemical reaction |

0:11:04 | the only which is a way from this |

0:11:06 | is essentially the one which is a power with this chamber here with things happen very fast |

0:11:11 | and that there's no P D it's just a that because |

0:11:14 | it's a it's fluid flow |

0:11:16 | approximation of right |

0:11:18 | so it's you have to boundary conditions |

0:11:20 | and this chamber just a bunch of than you these that's that's really that you |

0:11:24 | so what you and it up with is a two compartment model here |

0:11:28 | where here you have a constant |

0:11:29 | here you have a nonlinear only |

0:11:32 | and you have the boundary condition which is of the all that would be |

0:11:35 | so you now down to a finite dimensional system |

0:11:37 | which is which is tractable |

0:11:38 | now as a said this was a heuristic thing because |

0:11:41 | to be really uh |

0:11:42 | is you have to kind of quantized of course this up sub step i learn |

0:11:46 | and then you have to |

0:11:47 | proof that the errors are bounded in so |

0:11:50 | this is a deterministic system so it's not that difficult to do things i mean it's of a stochastic could |

0:11:54 | be slightly harder |

0:11:55 | so actually it it's it's not not that difficult if if you assumes so the regularity the system then then |

0:11:59 | you can do one |

0:12:00 | okay now so that's the to compartment model which is easy to do |

0:12:04 | now you can imagine conceptually uh extended this robotic a model |

0:12:07 | and it's roughly the same thing so in the first compartment is identical to to what you have to you |

0:12:12 | for the first like true |

0:12:13 | i'm and stuff was fast post to to the second electronic can you have a to keep up with the |

0:12:18 | and and of course you have to keep in mind that because the first electrode is grad |

0:12:22 | some of the and a like |

0:12:23 | some of the ball was to try to better |

0:12:26 | the second electrode has different boundary conditions which are vertical boundary conditions you |

0:12:31 | but it's it's quite easy to take care of conceptual |

0:12:33 | so that you have it to basically have a whole system of |

0:12:37 | norm that you're or ease which is conceptually at least easier than dealing with a much more complex as to |

0:12:43 | so once you |

0:12:44 | now as i mentioned that i'm not gonna give you the details of these all these it doesn't really serve |

0:12:48 | any purpose |

0:12:49 | but just to kind of go back and you roughly what these ordinary we differential equations do |

0:12:54 | uh |

0:12:56 | you see they really model the chemical reactions you |

0:12:59 | the model the movement of these |

0:13:03 | now i tubes in the |

0:13:05 | a number eight |

0:13:06 | they model how these chemical reactions happen |

0:13:09 | the chemical bonds |

0:13:11 | and how when things couple |

0:13:13 | they don't move so really there about seven chemical species it you know which you you model by of order |

0:13:18 | to control question |

0:13:20 | these are quite tractable in in fact |

0:13:21 | the reaction rate constants and the one by by a of so the the |

0:13:25 | all the parameters of those equations are actually |

0:13:29 | or we so once to end up with this as a said you know all the reaction rate parameters you |

0:13:33 | only unknown quantity is your input |

0:13:35 | and like |

0:13:36 | the creation which which is what we wish to be |

0:13:39 | so that was really the main idea uh and |

0:13:42 | just a can give you now some it you should be how this would work |

0:13:46 | so if you think about |

0:13:47 | even if this was a of now a bunch of ordinary differential equations questions or if you like discrete as |

0:13:51 | a what time it's a bunch of difference equations |

0:13:54 | a you have this fluid flow coming in here |

0:13:57 | so as you think that this guy remote say some proportion of the |

0:14:02 | molecules |

0:14:03 | see alpha |

0:14:05 | that means you're signal power here is reduced by but for because you grab some of the guys |

0:14:09 | so you can think that the second electrode what has a little or signal to noise ratio because you less |

0:14:14 | signal |

0:14:15 | now of course the next electrode will be out of the squared and so becomes a geometric matrix V still |

0:14:19 | it's obvious that you won't get |

0:14:22 | dramatic improvement as you have multiple electrodes because it dies of as a geometric matrix series |

0:14:26 | i mean that's gonna a used |

0:14:28 | even at the linear case now of course is the non linear regression you have to be slightly more careful |

0:14:32 | uh in any case one could actually |

0:14:34 | workout out |

0:14:35 | the asymptotic covariance of this |

0:14:37 | okay and it's it's a fairly complex question |

0:14:39 | and one could actually show that |

0:14:42 | in as you have more more electrodes you your performance that's really from frame |

0:14:47 | to you'd expect if you and N electrodes everything was |

0:14:49 | i i D you had no interaction between things you get one of and improved |

0:14:53 | sure you don't |

0:14:54 | so that's that's something which |

0:14:56 | so be the case |

0:14:57 | uh |

0:14:59 | okay so this is the actual system be but we've tested this on on |

0:15:03 | so it to test this several things you can do |

0:15:05 | you not to run of the real experimental system |

0:15:08 | and compare it with without a of different look should model that's to kind of to a model that's the |

0:15:12 | first step |

0:15:13 | and those that things we've done it quick you come the pa that's that that were |

0:15:16 | a second stage is how can you actually |

0:15:18 | shall |

0:15:19 | if the multi component model is a good approximation because that's the we we want to estimate of constant |

0:15:25 | and it are actually they work extremely well as |

0:15:27 | we can see these diagrams you |

0:15:29 | the actually approximate the |

0:15:31 | P E extremely well |

0:15:33 | uh the arrow it use that to be equally between six to eight percent |

0:15:37 | and even maybe go to |

0:15:39 | concentrations which are very very low |

0:15:41 | concentrations of |

0:15:42 | almost almost |

0:15:44 | well below an animal are still |

0:15:45 | so these these that she work quite well |

0:15:48 | uh and and therefore it means that you can apply standard a question analysis to solve for these concentrations |

0:15:54 | i want make a few other |

0:15:56 | comments before finish the all |

0:15:57 | this is still work in progress i mean it's a very nice to come up with these approximations but you |

0:16:02 | don't the sort of things we have done which would not please people are rigorous not my |

0:16:06 | we haven't even shown that this system of equations |

0:16:09 | has a unique solution it's really down hard to show that |

0:16:12 | so i mean if you think about P D easy we bad this in a function space like a stop |

0:16:16 | let's space you wanna show that you it's of a solution |

0:16:19 | highly nontrivial |

0:16:20 | so |

0:16:21 | i mean although the system works other real axis system |

0:16:24 | sure you you this is really hard |

0:16:26 | it's something which we working on of the moment |

0:16:28 | uh |

0:16:29 | the are the issues are it would be really useful to come up with a nice |

0:16:34 | approximations |

0:16:36 | for this pde itself |

0:16:39 | it's still of just using a multi compartment model we've done |

0:16:42 | because we still ending up with a bunch of all we different role equations which |

0:16:46 | we don't have a |

0:16:47 | so form solution but eventually do you man |

0:16:50 | we much nice if we can come up with further there were approximations which allows to get |

0:16:53 | some inside |

0:16:55 | as to how the system works |

0:16:56 | so there are still a couple of |

0:16:58 | a that's to our approach |

0:16:59 | but i think this is a in the sense that |

0:17:02 | given the complexity of the system we can model it |

0:17:05 | it works pretty good |

0:17:06 | we can approximate it and we can actually estimate using elementary nonlinear linear regression |

0:17:11 | the sort of concentrations |

0:17:12 | we want to estimate |

0:17:14 | so that's really all of wanna say |

0:17:16 | uh if you're interested in any of the stuff this was the original paper we by a colleague |

0:17:20 | uh and these are a couple as we did where we dealt with the signal electrode case maybe be model |

0:17:24 | it that and actually did |

0:17:26 | a not than you're question on that |

0:17:28 | oh can thank you very much |

0:17:37 | yeah right are you know |

0:17:42 | oh okay if gram so a a really interesting problem uh uh of course um |

0:17:46 | is a lot of comments i guess for instance with the pay T um |

0:17:51 | you've got a linear spatial up right the first to terms so there's a brings function that's an an |

0:17:57 | i i you could generate a one approximations and stuff like that a you've been looking in that direction we |

0:18:01 | have quite a bit what really kills us is cool |

0:18:04 | so |

0:18:05 | is is this |

0:18:07 | i a gonna this is a horrible but we could dish of the that |

0:18:10 | this a are could if should which likes to the P |

0:18:13 | right and and |

0:18:15 | and this this really is a a a i was not a at the takes the all of |

0:18:20 | you know the concentration respect to spatial axes and sets to one of the things so then one is thing |

0:18:24 | of boundary element methods because uh that's the use a to the with that of stuff yeah |

0:18:29 | i numerically what we done actually is because you have a a solution are obvious so you don't have strong |

0:18:33 | so should we use a finite element method which takes to functions automatically |

0:18:36 | but again |

0:18:37 | oh for our approach is completely and hearing in the sense that we have a solution it works |

0:18:42 | but from a mathematical point to we we still have even shown any structure proper okay but good of array |

0:18:47 | element method is different from finite element you i don't know if you familiar menu with a a a a |

0:18:53 | type find so |

0:18:54 | a any whites aims to make that might worth looking at |

0:18:56 | that are thing is on not entirely clear that the regression analysis because the use is i'd time varying quantity |

0:19:04 | so um |

0:19:05 | uh |

0:19:07 | i great so for instance if you nate to |

0:19:10 | in house some regularization on the a temporal times of the use signals |

0:19:15 | okay so let's a go to it so a is the concentration which changes over time yes we interested a |

0:19:20 | zero which is a initial concentration wrong |

0:19:23 | a a is coupled to these sceptical reactions which are you which you also over time |

0:19:28 | these are pretty easy to show for example that these are always non-negative and so the they just basic chemical |

0:19:33 | reactions a can i the only difficulty in this is a non linearity of that |

0:19:37 | but you've got gotta a construct you of tape right |

0:19:40 | well one specific um of that is a what be measure in noise |

0:19:45 | okay i but when you do the regression what do us to my just i it's a so one one |

0:19:49 | gram use of that |

0:19:50 | you of taste he is in there must be dealt with a the why something is it so what you |

0:19:55 | have is but you re braces by multi can pop and model you have a a bunch of or very |

0:19:58 | different role questions here |

0:20:00 | and here |

0:20:01 | which are in you |

0:20:02 | and the interested in one specific component of you |

0:20:05 | observed in |

0:20:07 | which is a concentration of the diners basically |

0:20:10 | uh |

0:20:13 | basically |

0:20:14 | the number of couple guys you |

0:20:16 | because the cover is proportional to the |

0:20:18 | no i i understand all that but i guess again but in the middle somewhere in that regression you are |

0:20:22 | as a function of tape must some have a being with carpet |

0:20:25 | in order to waste one one this upon it |

0:20:27 | one can hold diffuse for so i just one where the in a some temporal regularization or or or or |

0:20:32 | the that of |

0:20:33 | a us we have applied in such thing so far so are |

0:20:36 | yeah my help |

0:20:37 | it it back |

0:20:39 | or a |