okay so today I'm going to speak about checking global identifiability this is
joint work with Hoon Hong, Alexey Ovchinnikov, and Chee Yap so before I introduce
the identifiability itself let me show you just one example of a
biological model this is predator prey model so what does it mean it means that
I can have a forest there are rabbits and wolves and there are some natural
interactions between them and how is this model constructed so let x1 is
number of rabbits x2 is number of wolves and you think that okay rabbit
they bore new rabbits and this is this right is proportional to the number of
rabbits and each time that rather than the rabbit means wolf okay rabbit might
disappear so this is this is this is the second term okay so then like modeler
constructs this model he has this sort of reasoning but he or she doesn't know
the values of corresponding parameters and if you want to really apply this
model to any predictions of future you have to know the parameters because
otherwise it doesn't make sense so the question is can we get them from some
observations and the situation is might be a bit more complicated if for example you
cannot measure X 1 or X 2 for example rabbits are too fast and too small to
count them actually so so it for example it might be that you can observe only X
1 or only X 2 and then you have a question can I get these all numbers or
no chance so this is actually more or less the general identifiability question
so you have kind of model like give an example of the answers I will
tell later argumentation suspense so okay so the main questions you have this sort
of model and you know that you can measure some things and some you cannot
example the answer is always yes for example the answer is not obvious and
I will discuss it so it's even a bit non-intuitive so I will talk about
that so and you really want to somehow understand which parameters can you in
principle find if you have noise-free excellent measurements and which
parameters you just have any hope to find okay so and in this session when I
say word parameters as you have seen in previous example you have two kinds of
parameter one type is parameters that appear in equations and govern the
dynamic and another type is initial values so if you want to model the situation
you have to know them all okay so why if you want to model the
whole situation I mean you just want to know where to start
if you depends what you want to find out yes of course maybe you are just in some
certain parameter it's the situation might be even but you might be
interested in both then and then in principle so but sometimes your interest
are somehow restricted because like you don't care about some things that you
care more about other things yeah we'll just enough are you saying that there
are frequent situations in which everything has to be known yes there are
such situations not always but not always of this how important situations
yes yes then you want to select the whole thing yes yes correct
okay so just to give some very simple toy examples then like you
have this equation X dot is equal to X plus K then of course if you can observe K if
you can observe X you can find K obviously because let's write this and
measure X and X dot at every point that you like and that's it and to have
example it of the model is not identifiable it's a bit stupid but if you
K is actually sum of two different K's then you cannot just distinguish me from
them so this is just because there are infinitely many possibilities so I will
give formal definition with more non-trivial examples later actually our
first example is much less trivial so ok so yeah so in this situation the second
model I can find from many for some measurements K 1 plus K 2 as one whole
entity but I can't distinguish them I can just take something from K 1 to K 2 and
observer will not notice that so last one here minus 1 here this is the same
same observed K and so you're claiming that you can deduce that by
just looking at this but what you're going to present will address cases in which you cannot say yes yes so this is kind of artificial example just to first
yes receive the notion a simpler situation but there are very entangled systems there
it's hard to predict what is actually in its final
variable so you can't be tell me K 1 K 2 but you can even the sum and it's their
sum that affects the outcome of the equation probably that's correct yes why
do you want to separately you tell me K 1 and K 2 if you only interested about
the systems yeah this is like see first we want to deter
ah I wish you yeah I know yes yes it would it was actually discussed
yesterday but well but in other examples there are cases where you would want to
distinguish but in this toy example yeah yeah so but anyway before try to find
any remedy you have to diagnose that you have a problem right so first you have
to find that you have such situation and then you do change variables reparametrization
whatsoever but first you have to find that okay this is also when
did one of the easiest examples that can be explained without technicalities so this
is the reason but it turns out as you point out this example serves
several purposes also really good example especially if it's for several
purposes yeah I would write k1 plus k2 equals k sum okay so before your formal
definition I want so how to introduce with what kinds of equations do I work
because in the models that we have seen usually you don't have kind of
complicated differential polynomials or such things you have equations of very
very specific but actually very wide and express expressive form of this form so
you have variables X again X Y and mu and mu are all vectors if so they're
bold so X is actually state variables that somehow define the dynamic of the
system and in general you don't know them what you know are Y's that are
observable output variables so they might be some functions of the state yes this is
what you can really measure and u is an input so maybe first before doing
any measurements you can maybe apply some force
on the to the system the force you know and you choose within some restrictions
so y's and u's our friends know them and x is in principle our unknown and so
total vector parameters it consists of two parts of new parameters that appear
in dynamics here here and of initial values so I will not speak about input that
much almost everything okay everything what I will talk about works
with presence of input just to simplify their presentation I will immediately
control theory is one of possible applications of this technology no I
mean to me I mean the control theory people have studied this kind of system yeah sure sure yeah
but yes theory yeah I mean I have used it already by looking at transcendence
degrees yeah yeah sure I will mention these
people from control theory this is yeah but in some sense this
language is it's it's very expressed if you can speak about control theory is in
this language you can speak with biology gets me so this language some general
setup there's no question I'm just seeing the connection that people have
studied these mathematicians about it don't actively control system so the
nus are the inputs the y's are the outputs the X is in terms of state that
inside the black box so they were expressed by this kind of equation the Y
equation is some kind
so you put in this
can you are you gonna do example to show like given a system how when each of
these variables are in that system next slide I hope oh yeah so okay so they have a
systems that is exactly of this form we kept only one x one y and three parameters so
mu 1 mu 2 and initial value of x so this is kind of when x linear then Y is also
linear and this model can be solved exactly just you can write formula
because then your function linear function and then you see that this
example is the elder brother of our earlier unidentifiable example because
in your observed variable Y again sum of two parameters occur and again you
don't have any way to split them so this is a little bit more obscure
identifiability it might be even more obscure much more so right this is this equation
is of the form we discussed so okay state variables differential equations in
terms of state variables and then we have output Y written in terms of state
variables and parameters so you were proposing
in the sort of easier example replace q1 plus q2 with a new thing yeah how would
you do it now i will follow the so okay let me repeat the question
do it doesn't help okay thanks for the question
so well do you remember the examples k1 plus k2 yeah sure
okay and you made the proposal to replace k 1 plus k 2 with something new
really yeah okay okay so this example has some similarities it also has a sum of parameters
how could you act in this case now in this case it actually explained why you don't want to
make them as an aggregate even though even though the system behavior won't
change if mu 2 plus X star don't change but since mu two and also XY has
different physical meaning then you do want to separate my next time is that
the initial condition condition initial how do you start
whereas mu1 is actually a rate in a sense technically speaking the two shouldn't
be added together because they're different dimensions numerically fine I
mean even you can always have 3 plus 4 but if 3 is a rate and 4 is a value
initial my population that you should be able to add them it doesn't make sense I
mean if the initial system makes sense that the solution also have to make to
mu 2 is not a rate this is added to they are given mu 2 is x dot that's the rate
such device or okay so they could be added so yeah they could be so is it a
puzzle or is how is that yeah that you can add them
together it's not possible to rewrite the system in such a way that yeah
because you try to translate the variable you just translate the variable by mu 2
and put them into initial conditions as usual yep so yeah who's right if you replace X plus 2 with X
tilde then the system will be the same but this is different change of purpose
that was previous different kind of different kind of change purpose a lot
of systems the whole idea is to use dimensional analysis to find as few
every variable to express the same system it took up we hate it but tell me
how many variables there are and in terms of the algebra you're really
saying what generates the solution yeah I would do exactly that
do it new change of variable with X plus mu2 but mu2 is still unidentifiable
ok thank you for discussion so and now a positive example that actually a let
I will use this example throughout the next slide so in this situation X grows
exponentially and you observe X plus something plus some shift so in
this situation again you can write the solution because okay you know that this
X exponential function and this is how does while loop and then actually using
just this formula you can find formulas for all three parameters I'll do it for
you so you can differentiate three times and then you see that y dot and
y double do different just by mu1
so the ratio at every point isn't one and then you can find X star but if by
measuring the derivative at zero and then you can find yes I mean since this
identifiability question raised in different fields they have several names
in principle so I will stick to the word identifiability
IIIi have looked and I will cite them in several slides so far just explain the
definition I didn't have chance to mention anybody yeah I will mention
please and and deal with these people so what I want to point out right now is
that all these formulas have denominators so in principle this nice
scheme of finding parameters might fail if some of the denominators vanish so what we'll
talk today about is identifiability in generic case so generally they don't
vanish so if you have like mu1 is equal to zero it's very very special
degenerate case into exponential growth is actually constant so today we will
speak about what happens generically because with this degenerate situation
in principle has probability zero but I don't want to say that this is not
interesting not important problem I mean sometimes you just set some initial
conditions and you just set them degenerate and this is another challenging
thing I just do not talk about today I will talk about generic situation
all you need to determine these three
parameters is that at one single time one point one data point yeah
it's more subtle than that
if you can calculate the derivative at a particular point
you see so if one y dot which is the right yeah y double dot y part okay
yes one y dot at certain time is zero but
another time is not yes yes yes I mean as a function yeah actually I mean if mu
1 is zero then the system is really identifiable because then X is constant
and mu2 are constants and you cannot separate them so this is really an issue in
the degenerate case station really changes but however in practical problems
knowing this generically is sufficient interest yeah it's actually usually the
case because all all quantities are not actually not not exact and they usually
don't vanish your certain polynomial no which is the denominator so this this is what
I said what probability 0 principle so yeah so today I will speak about
identifiability for generic parameters so this is exactly the reason why I will do that
because otherwise things really changed this is really a bit different problem
so and here we are we're at it for almost formal definition so I have this
system I take one of its parameters theta sub I and I will have actually two
notions of identifiability global and local so I will most use global so you
can you want you can ignore local but it's also interesting so it's globally
identifiable if
I have these two polynomials let me explain what what do they mean these P
of theta means actually some polynomial that vanishes on all degenerate cases and non
zero polynomials two nonzero polynomials so like if I can identify such degeneracy loss
loss I for thetas and some differential polynomials or polynomials in u's and
they're variables that somehow captures degenerate inputs so if I can find
such things yes yes I just I I just I I don't want to distract you by the second
part
yes I want okay so yeah so if I can find such two polynomials such that for every
parameter vector that is not degenerate and every input vector that
is not degenerate in this sense so value of this function at zero is not zero then
if for every such choice my Y corresponding Y I will say what this means a few
seconds determines the value of this parameter uniquely so because why does it mean
then you pick a parameter vector and you pick input then the behavior of state
variables is defined because of the theorem from differential equations
solution exist in unique analytic functions at the point and then Y is
also uniquely defined when you fix parameters and input in some neighborhood
everybody is uniquely defined so you have unique function function y and now if this
function determines back the value of theta I
uniquely then theta I is globally identifiable if it determines up to finite number
of options then it's locally identifiable so by the way so F and G have or
F is has constant coefficients yes yes yes F and F and G have constant
coefficients and if you have non constant coefficients you can hide them
in different place actually you can in x in principle okay so yes I
assume constant coefficients right now so why do you choose this as a definition
for I wouldn't say that this is my choice IIIi didn't fight I mean you
presented this so it's yeah yeah so yeah so yeah okay it's yeah the one is I
don't see intuitively why this definition says that these this
particular parameter satisfying these conditions is actually identifiable yeah
but it'll take a theorem to prove that but you see when you see I'm
sure you see your defining identifiable using this this definition so I want to
see that that is intuitive so explain slowly pointing out precisely what is measured
does more example maybe I will just emphasis so very simple examples earlier
where some variables are identifiable in one case that the other one where they're not now we've got
we've got to satisfy this kind of condition yeah yes yes yeah first first may may be for phrase what's
written here okay would you go back to examples
should be a question okay so this was once like I want to able to use the
formulas okay so so yeah this is the system okay which one do you want to
want to prove first or second well obviously if you prove something is not identifiable
it's very hard according to that definition no no which for every p and every q
some of them failed yes yes so o show something is identifiable is to produce a p and a q
right yes which is in some sense easier which one do you want to discuss now I'd like to first of all to
see the one that is identifiable okay okay okay so here in this example we
don't without have input okay we only have parameters and for P from
definition I choose can you right here this was is y from the definition
y from the definition is y no no no the one that you used in the definition ah
so yeah every set of parameters mu1 mu2 and because the ex ex yes ex star they
define y yeah so I have a map from parameters to y and I want to know if I
can go back so if I can see in y understand what the parameters were so they
do confirm that the y that is written in the last line is the same y that appears yes yes
the problem is identifiable I mean the reason why it
is identifiable is because of the linear dependence of the exponential function
and one yes sometimes there isn't right I mean you you get two points and then
you can solve them okay so uneasy so but how does this being identifiable for mu1
and mu2 fit into your definition that's what I want to know yes for my definition I take the
polynomial P to be just mu1 so why it was P that that says what a choice of
parameters are degenerate so for every choice of parameters such that mu 1 is
not 0 you will use so yes so u theta in this situation is just mu mu 1
so for every choice of parameters then
whenever mu1 is not 0
I claim so yeah yeah yeah so I claim that known function y
function y is enough to find values mu1 mu2 and x star not just this is the case it is what a modeler
would actually accept as a definition yes
the question is based on measurements again we need to find mu1 mu2 x star
why isn't that the definition this is precisely what's written there
no I example I think now in a while you know knowing why this what is measured
yes I know y can you find mu1 and I just wanted it out then it was because
of independent of the two former they wanted
there is no why this is correct why you can find is a mathematical question
about algorithms or about some theory you asking about the definition so I
guess you must address proportionable definition right your question about the
definition was why is this a natural definition to confirm that this was
your question yeah okay so he justified this by well he explained it by giving
a concrete example and we so what you're saying is that the definition is a
sufficient condition for that we can identify when I question is is there an
alternate definition test equipment Onegin 4500 there are plenty of
alternative definitions - okay talking a few points and you can find it to
devalue all these estimates just so there is enough that's what you mean so
there isn't that will even on use this course will provide after some time
actual expressions of the parameters on the promise each other in terms of so
that's why I believe yes correct okay sometimes however as far as we
understand from the modelers point of view this is the main definition and not
a definition through an algorithm so the invisible tyrannous even only it applies
then why do you status the theorem and then we find that state condition one in
the village are equivalent and he says like that we call that in fact
because it's not stated that way because psychologically it's more correct to say
to this great why it doesn't make any sense to me it doesn't have to make
sense to you to decide what you can agree and to be beneficial for the
community but but what is wrong if you do have an even know me of theorem to
birth date then what do people find my equivalent and then give you cells like
any of this it's going in the file okay that's the best way you put in correct
question so it looks like in this definition say you want to show the
primary data identifiable mm-hmm it it doesn't rely on any of the other
parameters is that true yes yes yes so you can identify it regardless of what
you can you define other parts so then so then in the previous example in order
to say identify x star or mu2 to say and that example you needed to
first know what mu1 was I mean the way how do you proceed I mean so during
their sense right that you asked why might the way I wrote formulas relied on
the previous parameters right yes yeah no this is just a way I showed that
integrity is here so it's super so there's a way yes oh sure yeah there is
a way to assess directly me to rain right now and let me just so recent
raise a very good point because in for some of this example the parameter V 1
is identifiable so the whole model is a bit ill but the parameter mu 1 is
healthy it's you can't find it and the definition applies to the situation so
it says that nothing is that bad I can Richard actually prompt me for
another question I'm sorry let's go back to your question okay okay now from what
I see here well they went up whether ad this definition is not sort of
rigorously finished not really written with all details he doesn't again say
what whyis or on the slide so let me see they didn't say what Y is
yes he's talked through it but he didn't explain it on the slide
my name is David what Y is a solution of crisp on differential equation that
exists because of the theorem of practicing Snickers lies
is that a conclusion or is it final definition
it's about intonation if the corresponding Y determines the when you
uniquely then I say that this global 85 or so
so the such step has to one that will have three points you know how I should
we do what what's not suffice the these bullets are quantifiers and this is
formula so for every for every so why are not which one is that see the
viability suppose it's supposed to depend on the system and if you if I
just look at the first two conditions that does not then what you're saying
that that the hypothesis so that the conclusion and the whole thing is yeah
so this is mr. disappoint parts for every every solo something so so that
means
William I never saw so many different ways of painted no no no no I won't
understand what it means yes well now you understand thanks that's great
that's great that's great so I mean see I was missing the tie if I
just look at the look on the Sun stack what but they did such that if for every
day it's just like no okay no I know I think the way this point yeah just maybe
maybe the structure of slide was a bit misleading so I just wanted to avoid
large mass of plain text so I want to structure so then what's really written
is that for almost every for every almost every depth or almost every you
then and then what happens did I and can you read this for me with the pieces you
can be archival have to reduce this the icon can be read from and was no measure
host Elizabeth of overstated there so this is business agenda Pedro this is a
may be perform phrase of course wait in the paper stated rigorously just just to
avoid the explosion was officially it's possible and it's not super complicated
there is just some words about functions mean
and which was the base type of definition to the west of our search
through discouraged his is what a mother would intuitively
also think that this is what it is and from mathematician point of view and
from the algorithm important once we see that we get stuck using the definition
is what develops in Syria into addresses now I understand I understand why makes
a little sense of why is the terminal alright then if I do because the last
box resembles in such that that's also property no two points are just print
cards all right so okay there are different ways to establish
identifiability and nonidentifiability so for local identifiability that means
that you cannot make maybe you cannot find the value of the parameter uniquely
but you have just finite number of options there are fairly efficient
algorithms based on computation of the jacobian matrix they go back to some late
70s by Krener and later they were refined implemented by Sedoglavic and
some other authors so the I will not speak a lot about
local instability because algorithmically this question is more or
less solved and because this magic computation is quite cheap yeah just in
this function theory so you can do it without for large systems out much
problems so instant thing is global instability and this differs
dramatically so there are actually a bunch of methods was when that addressed this
problem for in some situations in cases so historically one of the oldest
methods Taylor series method I will not go into
details it somehow looks and powers series expansions of all these functions and
the issue is that in general we don't know how far should you expand them so
method relies on certain bounds and to the best of our knowledge there are no
general bounds there are bounds only in some cases and in this case if they are
sometimes exponential this is a bit painful for computation so it's we
haven't seen any implementation of this method so it's as well so it's realizing
these bounds that are not that complete and there is a whole family of
algorithms based on differential elimination and this was actually
initiated by these control theory people do Diop Fliess and their coauthors and then brought
to the other communities and developed further so this is I could feel the whole
slide people contributing this this area of research so I mentioned some just the earliest yes
yeah yeah this is this is what are called finding options this is checked
by jacobian matrix okay yes yes if it's full or from here okay so so I want to
give a small overview of these differential elimination approaches so
roughly speaking you can divide them in two different parts depending on how do
you think about travelers so first part and historically would be it is the
first is the following so let's just return to our system with
exponent and shift that we already have seen what to say we say that let mu 1
and mu 2 be also functions but constant functions so we say they are functions and
add two new equations mu1 go to zero mu2 go to zero
so in this situation we have just okay one two three four differential
equations and now we can do differential elimination compute characteristic set
I will not go into details this some way some sort of algorithm rosenfeld groebner
algorithm computes certain representation of an ideal defined by
these equations and it's even that Y dot is nonzero it's not the zero function
so it satisfies the non degeneracy condition we have only one
characteristic set and I wrote down the part of and this part actually gives almost
the same formulas I wrote already so you can just see these three polynomials as
linear equations in X mu one and mu2 and you can express them so and now you
can deduce global identifiability because you have formulas then it's globally identifiable
if you don't have formulas it's a bit more tricky to explain why it is not
globally identifiable and we haven't seen full rigorous proof yet
so didn't find only sketched proof so but okay this method works like that and
there are some possibilities to tweak to speed up it but the point is that it is
still very slow even for modest sized equations there are I will say that no
by the way maybe possibly explain why we look at this problem
yes yes I will give some reasons right now yeah I was going to
explain this yeah yeah so one reason why it's slow is that you in
you quite recently in large number of variables so all the parameters are not
variables it's not very convenient for rosenfeld groebner algorithm at all
another reason is that rosenfeld groebner algorithm actually computes much more
than that you will need because I have this dot dot dot here I will not use it
and this this is the rest of their characteristic set and it might be actually much
larger than the part I will really use not just that there are extra terms but
also the first terms give you expressions yes yes and sometimes the
customer doesn't ask for expressions it's interesting to know the expressions
for some problems for some problems they are just not needed so question is maybe
can that be avoided so you see so that this automatically computes
more more and more so this is this is good if you really want this results
but sometimes and quite often you really don't care much in general
you probably can get formulas if the mu-1 mu-2 and the parameters appear
nonlinear okay but I'm asking about global identifiability so then they will be linear
if they're globally identifiable if instead of mu2 in the Y equation I
say I have mu1 times mu2 whether it makes sense or not when you have a system of differential
equations and you computer calculate the set there's no guarantee that they will
appear linearly no there is theorem that says that global identifiability is
equivalent to having to linear appearance of variables it's it's I mean
okay I would say the definition required that I can solve
but it turns out to be equivalent
we haven't we haven't haven't seen full proof of this theory but
we know that fifth degree equation can't have a formula to solve it right
so how can you tell me that you can always solve it
you can read the proof and see
but if you can't explain how you can actually solve a 5th degree question that's okay but it doesn't apply here that's why
so now I like to ask why why does the 5th degree equation
well we'll invite him for another talk in which he will explain the proof
no no I I don't want to read your proof I want to understand why
a 5th degree equation will occur if you can give us example that will occur we will gladly consider it
no the burden is on you well decline to answer
your question okay fine that's okay so this is this is the first first sub
family of this differential elimination algorithm is to treat parameters
as variables as differential variables another approach is to say that they are
actually coefficients so we do some differential algebra over the field
of rational functions in mu1 mu2
>>I hope you don't mind if I interrupt you previous slide suppose I replace mu2
and write mu mu 3 to the fifth power >> it will not be globally identifiable >>oh why not
because you have to take a 5th root right? >>yes >>but there are only 5 roots >>but this is locally identifiable you want
unique >>see that explains why you see okay okay thank you
because is unique so we can't have 5th degree equation >> yes this is what's
written like >>or even if it's 5th degree it should factor so not only that you have no I
can it is still unique this repeatable button it's repeated good and I'm you
free to the first power the fingers are still unique right
where's first know it it's unique and only way in
situation than this new three to the fifth power will be equal to zero
no no no mu3 where's my saw okay all right you three to the fifth would
be me but not you three years oh but so was there our situation in which that
happens so right I mean you got a formula for mu 3 to the fifth
power equal to something >>yeah yeah but this is not globally identifiable case this is
case that that can be made global identifiable after a reparametrization but this is not the
story >>all right okay >>so that so then you can determine then if parameters are not
globally identifiable using that but also by seeing if you get them nonlinearly >>yes yes so if you
don't have this linear polynomials then it's not global identifiable
>>so when you say global what kind of rings do you allow your
parameter space be >>C to the n >>the whole thing >>the whole space of complex numbers >> subtracted a closed subset
>>good so and the second idea is to consider parameters as
coefficients and again not now if it had just two equations and can compute
characteristic set again and I will not write the whole thing I will just write one polynomial there
elements of characteristic set that doesn't contain state variables in
this approach is called input output equation okay because we don't have
input this example is actually output equation so this formula doesn't give
you a way to immediately write what is mu1 equal to because so yes so but then
you can say that assume that I can measure these
two coefficients like if y dot and Y are somehow independent in some sense then I
maybe can measure these coefficients and if I can measure them I know mu one and
I can express mu2 so >>can you explain so first of all what you are now
presenting is not your result >> yes yes yes I explained
some family of approaches that compute characteristic set or something similar there
are also options >>can you not explain in detail but just vaguely >>yeah so how does this method proceed we compute such
a polynomial that doesn't include access doesn't include state variables then they
assume that coefficients of this polynomial can be measured and so if
they can be measured then the next step is to check if you can find formulas for
for mus from the coefficients like in this situation I can find formulas from
mu one mu two from coefficients because mu1 is a coefficient itself
and mu two is the ratio of two coefficients so two things I input here is
that I use certain assumption and I don't touch initial conditions so I now
speak only about mu1 mu2 so yeah so then this is this is the outline the
compute equation and then take the coefficients and try to express your
parameters in terms of these coefficients
>>In the example here you do have nonlinear terms
but each parameter is linear >> no Ican't express parameters as if I can
express mu one and mu two as a rational function in this polynomials >>and you talk
about triangular systems then?
>>You can do it using triangular systems
yeah so you just check that and you have certain freedom you can use triangular
systems you can use groebner bases maybe you can use something else it's it details and
they they different different for different authors it is a family of
algorithms not just one ok so this approach is in many cases
faster than previous one because you take less variables in rosenfeld
groebner and you take give less equations so this this is supposed to
work work faster there is the second stage but you somehow divide the problem
into stages usually helps in computation however then there are still some issues
here one is that rosenfeld groebner still computes more than we want so we still
get certain formulas and you still get other elements of characteristic set that
you were not interested in and the second issue is that this assumption that
coefficients can be really measured fails in some situations I just want to
show you one of them this will look a bit artificial I will explain later what
what does this example mean in a differential algebra context so this is a system of for
the to consider a company one state variable this is constant and we can
observe this constant and you can observe some linear combination of
something a function of this constant so then this ended up with equation will be
of this form because you can just substitute Y 1 instead of X here and
it'll get rid of X's okay so and now the assumption is that you can measure
coefficients and coefficients are actually our parameters so the method would yield
that mu1 and mu2 are identifiable but
actually they are not because let's just analyze the system because this is
they're not function but just constants they are not
identifiable because you can take any value a for mu1 and then for mu2
you can just compute this number so you take different a's from your one
this formula gives you different values of mu2 and you can infinity many of them
so so this assumption about the observability of coefficients of input
output equation is a bit subtle point for this method and so what does example
mean what kind of situations does it capture
>>where does your definition of identifiable involve any choice of y1 and y2?
>>could you repeat please? >>where in your definition on identifiable parameters
you say for any choice of y >>I say for any choice of parameter
>>so he's commenting using some equivalent statement he would also express this non
identifiability using his definition perhaps this will point to expresses the
expression would amount to solving the system and
>>you're already inverting the input/output is that correct? >>he will now write the definition why this doesn't work
is it's possible to write in different ways but since we didn't state the theorem
and all we have is the definition let's just argue by definition
so this is intuitive argument okay so far because there was no theorem >> yes
yes so check this by definition let's say that if you have fixed some
parameters so what is y? so in terms of
parameters y 1 and y 2 can be expressed because we can just solve this equation
y 1 is just X star its constant and Y 2 is mu1 okay so now I see that we picked some
values of x star mu1 and mu2 okay so we have values of y1 and y2 and now I take any
other value a for mu1
and I take value y2 minus a y1 so here is still
so I place this as some value a I place this with y2 minus a y1 and
yeah so these actually will give you okay y2 is this expression y1 is and
actually x start so they cancel and you see that it's actually
y2 the same y2 think much of that and since I didn't
change x star y1 is unchanged so for any for every a you'll get you new set
of parameters so and here we also have had say some words for any this set to
degenerate parameters you can avoid it >>so you're saying this example violates uniqueness?
>>yes because it's not unique fix something and then you can take a new one
>>I think the idea is that here you're saying for any choice of y1 y2 but
really what you mean is you're given Y 1 and Y2 that's part of the
that's observed >>no what he's saying is that y1 and y2 as
functions of the parameters is given like that so think of mu1 and mu2
and X star as variables okay now you can pick a particular X star 0 and mu 1 0
and mu 2 0 that determines y1 and y for it
and then if you put it in there but in that formula you actually
get a different mu1 and a different mu2 and different things so they're
not uniquely determined by the y1 and y2 >>yes yes this is how
the definition should be checked yes right okay if I just almost like it's
amazing that you violate uniqueness in a linear situation >>so this like yeah somehow underlying
reason for this violation is that our system has some conserved quantities so
we have we have use in language of differential algebra we have new
constants and like from differential Galois theory we know that new constants is
always kind of headache and and this is what were this headache appears in this
context actually quite quite unexpectedly so so this is the
assumption might violate in the presence of pens conservation laws in the system
so actually this okay so actually the reason why is because you have two
equations say y1 and y2 is given but we have three unknowns that's why you don't have a unique solution you don't have
have enough equation >>yes I don't have okay yeah this is another way to check
but this is actually not identifiable >>oh so because of the uniqueness condition
you must have as many equations in y as you have parameters >>no I mean in principle in principle you have
infinitely many equations in Y because you can differentiate this until you're
completely tired >> no no no I don't mean under unique system I mean in every
system where you can use the Y's to determine the parameters they have to be as
many I mean it has to be how do we express the fact that the system has
you need to find a unique solution you said this thing anyway eventually so in some
okay so you do not have any question >>yeah but it is nonlinear I don't
know how to express the fact that a nonlinear system has a unique solution which
means that the variety you find in the parameter space
is one single point >>yes it happens >>so what's
the condition like that you always zero but it's also far more than that >>it's a
hard competition problem to check that I will speak about it okay I hope so okay
so in what situation are we right now all right now we don't have a general
algorithm that works with guarantee and with reasonable speed for global
identifiability I and my coauthors or person okay so like a couple of months ago we didn't
have yeah didn't have so and now I would like to present how do we approach this
problem so I will use still yes I was roughly because I will omit some del I
will omit all inputs they don't really make situation much more complicated and
I will show you the how it works on some example this one actually so this is
this example that we already have seen exponent exponential and and shift and
you're going to take the equation for y and start differentiating
it and each time like differentiate it one time mu2 disappears and have X dot and say hmm
I have an equation for x dot express X dot in terms of music and X so I don't
allow X dot to appear in some sense I'd immediately replace it then I
differentiate one more time again X dot appears again replace it so after I do
this I will have infinitely many equations of the form certain derivative
of y is equal to a polynomial in mu1 mu2 X so dramatically this is a map
from c3 to see to the infinity okay so I map every Triple mu1 mu2 x to the whole
bunch of derivatives of y in general you have s parameters and you have several
outputs so so in the same way you can obtain map from C to the a three
parameter space to certain infinite dimensional affine space so this is
almost algebraic contexts so you have polynomial maps polynomiall maps are
algebraic objects so there it is infinite but we'll deal with it later
so what I want to say that we want to reformulate to give an equivalent definition in
terms of this map result involving analytic functions in differential
equations and so on so and there is so this is equivalent definition in some sense
so or is a way to check the global identifiability using this geometric picture so
the parameters globally identifiable if all elements of this set have the same ith
component what is this set
so phi is my map from parameters to y's and this set inverse phi of phi of P
is actually the set of parameters that will generate the same y's roughly
speaking so and what I say that for every such parameters >>like those >>yes
like those so I have this this this different choices of mu1 mu2
will be actually this this inverse through this range so and I said
a parameter is globally identifiable if for every element in this preimage I have the
same value of this parameter
so this is this is the reformulation in
this geometric language of the our initial definition and this is equivalent the
problem is that we have infinite dimensional spaces we can't work with
them constructively okay so we would like somehow to make the situation
finite dimensional so how can do that if you have such map to infinite dimensional
space we can truncate it we can throw away components that correspond to too
high derivatives of Y yeah we can construct starting from this map and taking some
tuple of non-negative integers you can construct a polynomial map with in finite
dimension spaces from our initial map just ignoring some coordinates of of the
image >>I don't understand the quantifiers here
how does h quantified? >>>for every phi and every h
no I don't find for every age I can construct
such a map and this is kind of finite reduction of my big map and
what I want I want to find h such that this reduction will not lose any
important information so that I could use this age instead of the whole infinity
and this is this is actually next theorem so I find h any maximal h
such that this truncation is surjective and I do one more step and then make one
more step and then this truncation carries enough information to make a
decision about identifiability so I can replace phi in the theorem of the previous slide
by this one >>and there is an algorithm calculating this >> yeah yeah there's
so then there's
nothing so first they can find such h it's not hard this all some Jacobian
business then pick random point parameter space and compute the
preimage so and okay so there are two things random point and computation of pre
image for computation of preimage you can do
computing of pre image you can use any polynomial computation technique calculates
groebner bases this random point we did did some analysis so you can
guarantee any probability of success that you want less than one so so for by
based on the input data of degrees and orders and number variables we're going
to build some some way of doing this random choice that this result will be
guaranteed to be correct with probability
you want like you said so it's not just equation that would take something
random yeah yeah it's not like high probability oh no just say me you want
99% I give you 99% really good thing and okay just finish
this is an example some chemical reaction and this is okay it's
reasonably large and if you will set probability 99.9 percent the algorithm
says that everything is globally identifiable in one and a half minutes so and this is your
kind of kind of deal with this with rosenfeld groebner it's just one of
each so and yeah this is this is I think the end of the talk
aren't you still sort of computing and input output
equation in the sense that you're taking derivatives and then you
know plugging information back in isn't it >>I don't eliminate x >>but in this example
X was y minus mu2 but you don't know mu2 >> - right but
>>if you want if you can replace X with Y minus mu2 here and get him to
talk with equation but I don't do that I do different thing ok i don't don't--i
don't eliminate x's I keep them all the way so but of course in any any method
you will differentiate at some point and there's several things which
do that >>you actually rule out any singularities because you talk about global identifiability?
>>Is your question does global identifiability imply implies local?
>>I'm saying that in your calculations because you are aiming at global identifiability do you ever
encounter any singularities? >> Maps that we construct can have singularities in
what sense I mean I I compute the preimage i I don't need to think about
singularities if I compute preimage of a point
okay let me put other way do you ask what will I do if my pre image will
touch a singularity? I avoid this with high probability making enough making
right choice of trouble of the parameters and I can give you this
probability
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