structural identifiability: an introduction...motivation structural identiﬁability techniques for...

MotivationStructural identifiability

Techniques for nonlinear models

Structural identifiability: An Introduction

Mike Chappell & Neil [email protected]

AMR Summer School, University of Warwick, July 2016

MJ Chappell University of Warwick July 2016 Structural identifiability 1/49



Outline

1 MotivationSkeletal tracer kineticsInfectious disease modelling

2 Structural identifiabilityLaplace transform approachTaylor series approachSimilarity transformation/exhaustive modelling approach

3 Techniques for nonlinear modelsTaylor series approachObservable normal form




Skeletal tracer kineticsInfectious disease modelling

Skeletal tracer kinetics model

1

blood

(EF: extracellular fluid)

bone EF

bone

nonbone EF

tubular urine

5

4

2 3

a21 a32

a51

a41

a05

a12 a23

a15

a14

y1

y2

x = Ax + bu

y = Cx A =

a11 a12 0 a14 a15a21 a22 a23 0 00 a32 −a23 0 0

a41 0 0 −a14 0a51 0 0 0 a55

aii = −

5∑

j=0,i 6=j

aji

I II IIIa05 0.612 0.612 0.612a12 0.908 0.524 0.671a14 0.567 1.518 0.012a15 0.388 0.388 0.388a21 0.246 1.291 1.337a23 0.020 0.013 1.283a32 0.602 0.042 0.131a41 1.191 0.146 0.100a51 0.024 0.024 0.024





Model simulations

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set I

t

y 1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set II

t

y 1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set III

t

y 1





Model simulations

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set I

t

y 1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set II

t

y 1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1Parameter Set III

t

y 1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Compartment 2

t

x 2

I II III

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Compartment 3

t

x 3

I II III

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7Compartment 4

t

x 4

I II III





SIR Model

SIR infectious disease model:

S Iy(t ,p) = kY (t ,p)µN

µ µ+ γ

λ

Proportion of prevalence measured: y(t ,p) = kY (t ,p)

Model equations:

X = µN − µX −β

NXY

Y =β

NXY − (µ+ γ)Y

y = kY





SIR model

µ = 0.0125, γ = 12N = 10000β = 50, k = 0.5X (0) = 2400Y (0) = 20

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I

0 2 4 6 8 102250

2300

2350

2400

2450

2500

2550

Time (yrs)

Susceptibles, S





SIR model

µ = 0.0125, γ = 12N = 10000β = 50, k = 0.5X (0) = 2400Y (0) = 20

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I

0 2 4 6 8 102250

2300

2350

2400

2450

2500

2550

Time (yrs)

Susceptibles, S

µ = 0.0125, γ = 12N = 20000β = 50, k = 0.25X (0) = 4800Y (0) = 40





SIR model

µ = 0.0125, γ = 12N = 10000β = 50, k = 0.5X (0) = 2400Y (0) = 20

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I

0 2 4 6 8 102250

2300

2350

2400

2450

2500

2550

Time (yrs)

Susceptibles, S

µ = 0.0125, γ = 12N = 20000β = 50, k = 0.25X (0) = 4800Y (0) = 40

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I





SIR model

µ = 0.0125, γ = 12N = 10000β = 50, k = 0.5X (0) = 2400Y (0) = 20

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I

0 2 4 6 8 102250

2300

2350

2400

2450

2500

2550

Time (yrs)

Susceptibles, S

µ = 0.0125, γ = 12N = 20000β = 50, k = 0.25X (0) = 4800Y (0) = 40

0 2 4 6 8 101

2

3

4

5

6

7

8

9

10

Time (yrs)

Model output, k I

0 2 4 6 8 104500

4600

4700

4800

4900

5000

5100

Time (yrs)

Susceptibles, S




Laplace transform approachTaylor series approachSimilarity transformation/exhaustive modelling approach

Structural identifiability

input outputsystem?

Given postulated state-space models for a given biological orbiomedical process:

Structural Identifiability

Are the unknown parameters uniquely determined by theinput-output behaviour?






input outputsystem

Given postulated state-space model, are the unknownparameters uniquely determined by the output (ie, perfect,continuous, noise-free data)?

Necessary theoretical prerequisite to:

experiment design

system identification

parameter estimation





Formal definition

Consider following general parameterised state-space model:

x(t ,p) = f (x(t ,p),u(t),p), x(0,p) = x0(p),

y(t ,p) = h(x(t ,p),p),

where p is the r -dimensional vector of unknown parameters , and isassumed to lie in a set of feasible vectors: p ∈ Ω.

n dimensional vector q(t ,p) is state vector, such that q0(p) is theinitial state (may depend on the unknown parameters)

m dimensional vector u(t) is input/control vector (our influence onsystem); what inputs are available depends on experiment to beperformed, so u(·) ∈ U , a set of admissible inputs (might be empty).

y(t ,p) is the l-dimensional output/observation vector (what we canmeasure in the system). In the following we make explicit that outputy depends on p ∈ Ω and u ∈ U by writing y(t ,p;u).





Parameter identifiability

For generic p ∈ Ω, the parameter pi is said to be locallyidentifiable if there exists a neighbourhood of vectors around p,N (p), such that if p ∈ N (p) ⊆ Ω and:

for every input u ∈ U and t ≥ 0, y(t ,p;u) = y(t ,p;u)

then pi = pi .

In particular, if the neighbourhood N (p) = Ω can be used in theprevious definition, then the parameter pi is globally/uniquelyidentifiable.

If the parameter pi is not locally identifiable , i.e., there is nosuitable neighbourhood N (p), then it is said to beunidentifiable.






Structurally globally/uniquely identifiable

A parameterised state space model is structurallyglobally/uniquely identifiable (SGI) if all of the unknownparameters pi are globally/uniquely identifiable.

Structurally locally identifiable

A state space model is structurally locally identifiable (SLI) if allof the unknown parameters pi are locally identifiable and atleast one of these parameters is not globally identifiable.

Unidentifiable

A state space model is unidentifiable if at least one of theunknown parameters pi is unidentifiable.





Remarks

Necessary condition for parameter estimationEssential for parameters with practical significancePrerequisite to experiment design

Identifiability does not guaranteeGood fit to experimental dataGood fit only with unique vector of parameters

Unidentifiable implies infinite number of parameter vectors willgive same fit (even for perfect data)Many techniques for linear systems

Laplace transform or transfer functionTaylor series of outputSimilarity transformation (exhaustive modelling)

Taylor series and similarity transformation approaches areapplicable for nonlinear systemsDifferential algebra

Rational systems with differentiable inputs/outputsHeavily dependent on symbolic computation





Laplace Transform Approach





General linear system

x(t ,p) = A(p)x(t ,p) + B(p)u(t), x(0,p) = x0(p),

y(t ,p) = C(p)x(t ,p),

whereA(p) is an n × n matrix of rate constants

B(p) is an n × m input matrix

C(p) is an l × n output matrix

Assume that x0 = 0 (not essential) & take Laplace transforms:

sQ(s) = A(p)Q(s) + B(p)U(s)

Y (s) = C(p)Q(s)

= C(p) (sIn − A(p))−1 B(p)U(s)





Laplace Transform Approach

This gives relationship between LTs of input & output:

Y (s) = G(s)U(s),

where the matrix

G(s) = C(p) (sIn − A(p))−1 B(p)

is the transfer (function) matrix

Measurements for G(s) assumed known

Coefficients of powers of s in numerators & denominatorsuniquely determined by input-output relationship





Example: 1 Compartment

1

b1u(t)

a01y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known

Output: y = c1q1, where c1 unknown.

System equations:

Transfer function: G(s) =






1

b1u(t)

a01y = c1q1



System equations:q1 = −a01q1 + b1u(t), q1(0) = 0,

y = c1q1

Transfer function: G(s) =






1

b1u(t)

a01y = c1q1




y = c1q1

Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) =






1

b1u(t)

a01y = c1q1




y = c1q1

Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) =b1c1

s + a01






1

b1u(t)

a01y = c1q1




y = c1q1


s + a01So b1c1 and a01 globally identifiable






1

b1u(t)

a01y = c1q1




y = c1q1


s + a01So b1c1 and a01 globally identifiableBut b1 and c1 unidentifiable






1

b1u(t)

a01y = c1q1




y = c1q1


s + a01So b1c1 and a01 globally identifiableBut b1 and c1 unidentifiableSo model is unidentifiable unless b1 or c1 known (then SGI)





Example: 2 Compartments

1 2

I = bu(t)

a01

y = c1x1a12

Model is:[

x1

x2

]

=

[

−a01 a12

0 −a12

] [

x1

x2

]

+

[

0b

]

u(t)

y =[

c 0]

[

x1

x2

]

Transfer function:

G(s) =[

c 0]

[

s + a01 −a12

0 s + a12

]−1 [0b

]

=bca12

(s + a01)(s + a12)





Locally identifiable example

Transfer function:

G(s) =bca12

(s + a01)(s + a12)

and so the following are unique:

bca12, a01 + a12 and a01a12

Yields two possible solutions for a01 and a21

If b (or c) known then two possible solutions for c (or b) hencelocally identifiable

If neither b nor c known then unidentifiable

If both b and c known then globally identifiable





Taylor series approach





Generally applied when there is a single input (eg, 0 or impulse)

Outputs yi(t ,p) expanded as Taylor series about t = 0:

yi(t ,p) = yi(0,p)+ yi(0,p)t + yi(0,p)t2

2!+ · · ·+y (k)

i (0,p)tk

k !+ . . .

wherey (k)

i (0,p) =dkyi

dtk

t=0

(k = 1, 2, . . . ).

Taylor series coefficients y (k)i (0,p) unique for particular output

Approach reduces to determining solutions for p that give:

yi(0,p), y (k)i (0,p) (1 ≤ i ≤ l , k ≥ 1).

Notice that we have a possibly infinite list of coefficients:

y1(0,p), . . . , yl(0,p), y1(0,p), . . . , yl(0,p), y1(0,p), . . . , yl(0,p), . . .

For linear systems: at most 2n − 1 independent equationsneeded






1a01

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known.


System equations:

First coefficient: y(0,p) =

Second coefficient: y(0,p) =






1a01y = c1q1



System equations:








1a01y = c1q1



System equations:q1 = −a01q1, q1(0) = b1n0,

y = c1q1








1a01y = c1q1




y = c1q1

First coefficient: y(0,p) = b1c1n0







1a01y = c1q1




y = c1q1


Second coefficient: y(0,p) = − a01b1c1n0






1a01y = c1q1




y = c1q1



So b1c1 & b1c1a01 unique






1a01y = c1q1




y = c1q1



So b1c1 & b1c1a01 unique (ie b1c1 & a01 globallyidentifiable)






1a01y = c1q1




y = c1q1



So b1c1 & b1c1a01 unique (ie b1c1 & a01 globallyidentifiable)But b1 and c1 unidentifiable






1a01y = c1q1




y = c1q1



So b1c1 & b1c1a01 unique (ie b1c1 & a01 globallyidentifiable)But b1 and c1 unidentifiableSo model unidentifiable unless b1 &/or c1 known (then SGI)






1 2

a01

y = c1q1

a21

a12

Input: bolus intravenous injection of drug (unknown amount)

Output: concentration of drug in the plasma

System equations:

q1(t ,p) =

q2(t ,p) =

y(t ,p) =






1 2

a01

y = c1q1

a21

a12



System equations:

q1(t ,p) = − (a01 + a21) q1(t ,p) + a12q2(t ,p), q1(0,p) = b1

q2(t ,p) =

y(t ,p) =






1 2

a01

y = c1q1

a21

a12



System equations:


q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) =






1 2

a01

y = c1q1

a21

a12



System equations:


q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)

First coefficient:

Second coefficient:

Third coefficient:

Fourth coefficient:






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)

First coefficient: y1(0,p) = c1b1

Second coefficient:

Third coefficient:

Fourth coefficient:






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)


Second coefficient: y1(0,p) = −c1 (a01 + a21) b1

Third coefficient:

Fourth coefficient:






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)



Third coefficient:y (2)

1 (t ,p) = c1 (− (a01 + a21) q1(t ,p) + a12q2(t ,p))

Fourth coefficient:






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)




1 (t ,p) = c1 (− (a01 + a21) q1(t ,p) + a12q2(t ,p))

=⇒ y (2)1 (0,p) = c1

(

(a01 + a21)2 b1 + a12a21b1

)

Fourth coefficient:






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)




1 (t ,p) = c1 (− (a01 + a21) q1(t ,p) + a12q2(t ,p))

=⇒ y (2)1 (0,p) = c1

(

(a01 + a21)2 b1 + a12a21b1

)

Fourth coefficient: y (3)1 (t ,p) =

c1

(

(a01 + a21)2 q1 − a12 (a01 + a21) q2 + a12a21q1 − a2

12q2

)






q2(t ,p) = a21q1(t ,p)− a12q2(t ,p), q2(0,p) = 0

y(t ,p) = c1q1(t ,p)




1 (t ,p) = c1 (− (a01 + a21) q1(t ,p) + a12q2(t ,p))

=⇒ y (2)1 (0,p) = c1

(

(a01 + a21)2 b1 + a12a21b1

)

Fourth coefficient: y (3)1 (t ,p) =

c1

(

(a01 + a21)2 q1 − a12 (a01 + a21) q2 + a12a21q1 − a2

12q2

)

⇒ y (3)1 (0,p) = b1c1

[

(a01 + a21)[

−(a01 + a21)2 − 2a12a21

]

− a212a21

]





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)

First coefficient:

Second coefficient:

Third coefficient:

Fourth coefficient:





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)

First coefficient: b1c1 unique

Second coefficient:

Third coefficient:

Fourth coefficient:





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)


Second coefficient: a01 + a21 unique

Third coefficient:

Fourth coefficient:





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)



Third coefficient: a12a21 unique

Fourth coefficient:





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)




Fourth coefficient: a12 unique





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)





So a21 and then a01 unique





y1(0,p) = c1b1

y1(0,p) = −b1c1 (a01 + a21)

y (2)1 (0,p) = b1c1

(

(a01 + a21)2 + a12a21

)

y (3)1 (0,p) = b1c1

(

(a01 + a21)(

− (a01 + a21)2 − 2a12a21

)

− a212a21

)





So a21 and then a01 unique

Same result as before





Similarity transformation/exhaustive modelling approach





Generates set of all possible linear models: (A(p),B(p),C(p))

with same I/O behaviour as given one: (A(p),B(p),C(p))

Consider the model given by

q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = q0(p),

y(t ,p) = C(p)q(t ,p),(1)

and suppose that following are satisfied:

Controllability rank condition:

rank(

B(p) A(p)B(p) . . . A(p)n−1B(p))

= n

Observability rank condition: rank

C(p)C(p)A(p)

...C(p)A(p)n−1

= n

If both are satisfied model is minimal.MJ Chappell University of Warwick July 2016 Structural identifiability 26/49




Then there exists invertible n × n matrix T such that, if z = T q:z(t) = T q(t ,p) =

= z(0) = Tq0(p),

y(t ,p) = C(p)q(t ,p) =

has identical input-output behaviour.

Therefore, if p ∈ Ω gives rise to a model:q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = q0(p),

y(t ,p) = C(p)q(t ,p),

with identical input-output behaviour as the initial one (1), thenA(p) =

B(p) =

C(p) =

for some invertible n × n matrix T .MJ Chappell University of Warwick July 2016 Structural identifiability 27/49




Then there exists invertible n × n matrix T such that, if z = T q:z(t) = T q(t ,p) = TA(p)q(t ,p) + T B(p)u(t)

= z(0) = Tq0(p),

y(t ,p) = C(p)q(t ,p) =



y(t ,p) = C(p)q(t ,p),


B(p) =

C(p) =






= TA(p)T−1z(t) + T B(p)u(t), z(0) = Tq0(p),

y(t ,p) = C(p)q(t ,p) =



y(t ,p) = C(p)q(t ,p),


B(p) =

C(p) =







y(t ,p) = C(p)q(t ,p) = C(p)T−1z(t).



y(t ,p) = C(p)q(t ,p),


B(p) =

C(p) =







y(t ,p) = C(p)q(t ,p) = C(p)T−1z(t).



y(t ,p) = C(p)q(t ,p),

with identical input-output behaviour as the initial one (1), thenA(p) = T A(p)T−1,

B(p) =

C(p) =







y(t ,p) = C(p)q(t ,p) = C(p)T−1z(t).



y(t ,p) = C(p)q(t ,p),


B(p) = T B(p),

C(p) =







y(t ,p) = C(p)q(t ,p) = C(p)T−1z(t).



y(t ,p) = C(p)q(t ,p),


B(p) = T B(p),

C(p) = C(p)T−1,





Sometimes easier to deal with:

A(p)T = TA(p), (2)

B(p) = TB(p), (3)

C(p)T = C(p). (4)

If only solution is T = In then p = p and the system is SGI

If T can take any of a finite set (with more than 1 element)of possibilities, then the system is SLI

Otherwise, (T can take any of a infinite set of possibilities)then the system is unidentifiable





Example: Two-compartment model.

1 2

I = b1u(t)

a01

y = c1q1

a21

a12

System equations:

q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = 0

y(t ,p) = C(p)q(t ,p)

where

A(p) = , B(p) = , C(p) =






1 2

I = b1u(t)

a01

y = c1q1

a21

a12

System equations:

q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = 0

y(t ,p) = C(p)q(t ,p)

where

A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) = , C(p) =






1 2

I = b1u(t)

a01

y = c1q1

a21

a12

System equations:

q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = 0

y(t ,p) = C(p)q(t ,p)

where

A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =






1 2

I = b1u(t)

a01

y = c1q1

a21

a12

System equations:

q(t ,p) = A(p)q(t ,p) + B(p)u(t), q(0,p) = 0

y(t ,p) = C(p)q(t ,p)

where

A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[ ]

Observability:[

C(p)C(p)A(p)

]

=

[ ]

Equation (3):

B(p) = T B(p)

and soMJ Chappell University of Warwick July 2016 Structural identifiability 30/49




A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1

0

]

Observability:[

C(p)C(p)A(p)

]

=

[ ]

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

Observability:[

C(p)C(p)A(p)

]

=

[ ]

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[ ]

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0]

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

So model is minimal

Equation (3):

B(p) = T B(p)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

So model is minimal

Equation (3):

B(p) =(

b1

0

)

= T B(p) =(

t11 t12

t21 t22

)(

b1

0

)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

So model is minimal

Equation (3):

B(p) =(

b1

0

)

= T B(p) =(

t11 t12

t21 t22

)(

b1

0

)

=

(

t11b1

t21b1

)





A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

So model is minimal

Equation (3):

B(p) =(

b1

0

)

= T B(p) =(

t11 t12

t21 t22

)(

b1

0

)

=

(

t11b1

t21b1

)

and so t21 = 0 andMJ Chappell University of Warwick July 2016 Structural identifiability 30/49




A(p) =(

− (a01 + a21) a12

a21 −a12

)

, B(p) =(

b1

0

)

, C(p) =(

c1 0)

Controllability:[

B(p) A(p)B(p)]

=

[

b1 − b1 (a01 + a21)0 b1a21

]

rank 2

Observability:[

C(p)C(p)A(p)

]

=

[

c1 0− c1 (a01 + a21) c1a12

]

rank 2

So model is minimal

Equation (3):

B(p) =(

b1

0

)

= T B(p) =(

t11 t12

t21 t22

)(

b1

0

)

=

(

t11b1

t21b1

)

and so t21 = 0 and t11 = b1/b1





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

c1 0)(

b1/b1 t12

0 t22

)

=

(

c1 0)

= C(p)

and so

Equation (2):

A(p)T =

= T A(p) =

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)

and so

Equation (2):

A(p)T =

= T A(p) =

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)

and so t12 = 0 and

Equation (2):

A(p)T =

= T A(p) =

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)

and so t12 = 0 and b1c1 = b1c1

Equation (2):

A(p)T =

= T A(p) =

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)


Equation (2):

A(p)T =

(

− (a01 + a21) a12

a21 −a12

)(

b1/b1 00 t22

)

= T A(p) =

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)


Equation (2):

A(p)T =

(

− (a01 + a21) a12

a21 −a12

)(

b1/b1 00 t22

)

= T A(p) =(

b1/b1 00 t22

)(

− (a01 + a21) a12

a21 −a12

)

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)


Equation (2):

A(p)T =

(

− (a01 + a21) a12

a21 −a12

)(

b1/b1 00 t22

)

= T A(p) =(

b1/b1 00 t22

)(

− (a01 + a21) a12

a21 −a12

)

=

[

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

]

=





A(p) =[

− (a01 + a21) a12

a21 −a12

]

, C(p) =[

c1 0]

, T =

[

b1/b1 t12

0 t22

]

Equation (4):

C(p)T =

(

b1c1/b1 c1t12

)

=

(

c1 0)

= C(p)


Equation (2):

A(p)T =

(

− (a01 + a21) a12

a21 −a12

)(

b1/b1 00 t22

)

= T A(p) =(

b1/b1 00 t22

)(

− (a01 + a21) a12

a21 −a12

)

=

[

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

]

=

[

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

]





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)

(2,2) component:

so (1,2) component:

(2,1) component:

(1,1) component:

So:





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)

(2,2) component:a12 = a12

so (1,2) component:

(2,1) component:

(1,1) component:

So:





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)


so (1,2) component:t22 = b1/b1

(2,1) component:

(1,1) component:

So:





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)




(1,1) component:

So:





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)





So:





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)





So:a01, a12 and a21 all globally identifiable





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)





So:a01, a12 and a21 all globally identifiablecombination b1c1 globally identifiable





(

−b1b1

(a01 + a21) t22a12b1b1

a21 −a12t22

)

=

(

−b1b1

(a01 + a21)b1b1

a12

a21t22 −a12t22

)





So:a01, a12 and a21 all globally identifiablecombination b1c1 globally identifiableindividual b1 and c1 unidentifiable




Taylor series approachObservable normal form






Techniques for nonlinear models:

generally more difficult to apply

can be less systematic

do not always yield full information concerning identifiability

must be careful about what inputs there are to the system

Dealing with state space models of form:

x(t ,p) = f (x(t ,p),p,u(t)), x(0,p) = x0(p),

y(t ,p) = h(x(t ,p),p),(5)

where

p ∈ Ω is an r dimensional (parameter) vector

x(t ,p) is an n dimensional (state) vector

u(t) is an m dimensional (input) vector

y(t ,p) is an l dimensional (output) vector





Taylor series approach





This approach for linear models also works for nonlinear ones:

yi(t ,p) = yi(0,p)+ yi(0,p)t + yi(0,p)t2

2!+ · · ·+y (k)

i (0,p)tk

k !+ . . .

where y (k)i (0,p) =

dkyi

dtk

t=0

(k = 1, 2, . . . ).

Taylor series coefficients y (k)i (0,p) unique for particular output

Notice that we have a possibly infinite list of coefficients:

yi(0,p), yi(0,p), yi(0,p), . . . i = 1, . . . , l

& upper bound on number of coefficients needed more difficult

If model is autonomous, single output (m = 1), upper bound is:Transfer coefficients all polynomial: n + rIf any coefficient rational: n + r + 1

Quite difficult to use TSA to prove model is unidentifiableMJ Chappell University of Warwick July 2016 Structural identifiability 36/49




Example: 1 compartment

1

b1u(t)

y = c1x1 Vm

Km + x1

Model equations:

x1 = −Vmx1

Km + x1, x1(0) = b1

y = c1x1

First coefficient: y(0,p) = b1c1

Second coefficient: y(0,p) = −c1Vmb1

Km + b1

Third coefficient: y (2)(t ,p) =ddt

(

−c1Vmx1

Km + x1

)





Example: 1 compartment

1

b1u(t)

y = c1x1 Vm

Km + x1

Model equations:

x1 = −Vmx1

Km + x1, x1(0) = b1

y = c1x1

First coefficient: y(0,p) = b1c1

Second coefficient: y(0,p) = −c1Vmb1

Km + b1

Third coefficient: y (2)(t ,p) =ddt

(

−c1Vmx1

Km + x1

)

Use symbolic tools such as MATHEMATICA, MAPLE





Example: One compartment with Langmuir elimination:

1

b1u(t)

y = c1q1α(β − q1)

Model equations:



Third coefficient: y (2)(t ,p) =






1

b1u(t)

y = c1q1α(β − q1)

Model equations:

q1 = −αq1(β − q1), q1(0) = 1

y = c1q1









1

b1u(t)

y = c1q1α(β − q1)

Model equations:

q1 = −αq1(β − q1), q1(0) = 1

y = c1q1

First coefficient: y(0,p) = c1








1

b1u(t)

y = c1q1α(β − q1)

Model equations:

q1 = −αq1(β − q1), q1(0) = 1

y = c1q1


Second coefficient: y(0,p) = − c1α(β − 1)







1

b1u(t)

y = c1q1α(β − q1)

Model equations:

q1 = −αq1(β − q1), q1(0) = 1

y = c1q1



Third coefficient: y (2)(t ,p) = − c1α (βq1 − 2q1q1)






1

b1u(t)

y = c1q1α(β − q1)

Model equations:

q1 = −αq1(β − q1), q1(0) = 1

y = c1q1



Third coefficient: y (2)(t ,p) = − c1α (βq1 − 2q1q1)

=⇒ y (2)(0,p) = c1α2(β − 1) (β − 2)





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)

First coefficient:

Second coefficient:

Third coefficient:





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)

First coefficient: c1 unique (globally identifiable)

Second coefficient:

Third coefficient:





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)


Second coefficient: α(β − 1) unique

Third coefficient:





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)



Third coefficient: α2(β − 1) (β − 2) unique





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)



Third coefficient: α (β − 2) unique





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)



Third coefficient: α (β − 1)− α unique





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)



Third coefficient: α unique (globally identifiable)





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)




And so β globally identifiable





y(0,p) = c1

y(0,p) = −c1α(β − 1)

y (2)(0,p) = c1α2(β − 1) (β − 2)




And so β globally identifiable

All parameters globally identifiable so model is SGI





Now for something a little more advanced . . .





Observable normal form approach

Single output, no (or single) input

For generic parameter vector p:Check an observability criterion

Define µ1(x ,p) = h(x ,p) and

µi+1(x ,p) =∂µi

∂x(x ,p)f (x ,p) i = 1, . . . ,n − 1

Define Hp(x) = (µ1(x ,p), . . . , µn(x ,p))T

Rank of∂Hp

∂x(x0(p)) is n

So Hp(·) diffeomorphism on neighbourhood of x0(p)Hence is a coordinate transformation . . .





Previous approach

Coordinate transformation between models that areindistinguishable via available output

Hp (λ(x)) = Hp(x)

Σ(p) Σ(p)

Σ(p) Σ(p)

λ

Hp Hp

idDetermine S(p) set of all parameters ps.t.

λ(x0(p)) = x0(p)

f (λ(x(t)),p) =∂λ

∂x(x(t))f (x(t),p)

h(λ(x(t)),p) = h(x(t),p)

(x(t) = x(t ,p))





Observability normal form

System Σ is the observability normal form, z = Hp(x):

z1 = z2

...

zn−1 = zn

zn = µn+1(H−1p (z),p)

y = z1

Last equation gives input-output equation for system and so, forall p ∈ S(p), have

µn+1(H−1p (z(t)),p) = µn+1(H

−1p (z(t)),p) ∀t ≥ 0





Using output equation

Now rewrite output equation in form:

φ0(z(t), zn(t)) +m∑

i=1

σi(p)φi(z(t), zn(t)) = 0

where φi(z(t), zn(t)) are linearly independent

Then if p ∈ S(p)

m∑

i=1

(σi(p)− σi(p))φi(z(t), zn(t)) = 0

and soσi(p) = σi(p) i = 1, . . . ,m





Example

Consider two-compartment model:

1 2y = x1

p = p1, p2, p3, p4

I(t) = Dδ(t)

p3

p4 + x1

p1

p2

µ1(x ,p) = x1

µ2(x ,p) = −p1x1 + p2x2 −p3x1

p4 + x1





Example: Observability normal form

Observability condition met provided p2 6= 0 (ie, for all p) so cantransform into:

z1(t ,p) = z2(t ,p)

z2(t ,p) = −(p1 + p2)z2(t ,p)−p2p3z1(t ,p)p4 + z1(t ,p)

−p3p4z2(t ,p)

(p4 + z1(t ,p))2

where

z1(0,p) = D and z2(0,p) = −p2D −p3D

p4 + D





Example: Output equation

Output equation:

z21 z2 + p2

4z2 + 2p4z1z2 + p2p3p4z1 + p2p3z21

+(

p3p4 + p24(p1 + p2)

)

z2 + 2p4(p1 + p2)z1z2

+ (p1 + p2)z21z2 = φ0(z , zn) +

∑7i=1 σi(p)φi(z , zn) = 0

Linear independence of terms guaranteed by checking theWronskian, or can use constructive algebra methods (inMAPLE):F := Vector([-p[1]*x[1]+p[2]*x[2]-p[3]*x[1]/(p[4]+x[1]),p[1]*x[1]-p[2]*x[2]]);H := x[1];io := iorel(F,H)

Code modified from Evans et al Automatica 49:48-57, 2013, which wasbased on PhD by Forsman (1991) Constructive Commutative Algebra inNonlinear Control Theory





Example: Identifiability

σi(p) = σi(p) i = 1, . . . , 7

for any p ∈ S(p).

σ2(p) = p4 =⇒ p4 = p4

σ4(p) = p2p3 =⇒ p2p3 = p2p3

σ7(p) = p1 + p2 =⇒ p1 + p2 = p1 + p2

σ5(p) = p3p4 + p24(p1 + p2) =⇒ p3 = p3

Solving these shows that p = p, ie S(p) = p

Therefore model is structurally globally identifiable





Summary

Structural identifiability is an important step in modellingprocess

Theoretical prerequisite to experiment design, systemidentification, and parameter estimationTechniques involve generation, manipulation & solution ofnonlinear algebraic equations

Observability normal form highly appropriate for bothanalyses

Previously unsolved example (for identifiability) now solved!Some computational difficulties remainGenerates input-output relations

Structural indistinguishability similarly importantMore general framework but exactGenerally pairwise comparison of schemes


structural identifiability: an introduction...motivation structural identiﬁability techniques for...

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