Structural identifiability: An Introduction Mike Chappell & Neil - - PowerPoint PPT Presentation

Motivation Structural identifiability Techniques for nonlinear models

Structural identifiability: An Introduction

Mike Chappell & Neil Evans

m.j.chappell@warwick.ac.uk

AMR Summer School, University of Warwick, July 2016

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

Motivation Structural identifiability Techniques for nonlinear models

Outline

1

Motivation Skeletal tracer kinetics Infectious disease modelling

2

Structural identifiability Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

3

Techniques for nonlinear models Taylor series approach Observable normal form

MJ Chappell University of Warwick July 2016 Structural identifiability 2/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious 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 a14 a15 a21 a22 a23 a32 −a23 a41 −a14 a51 a55      aii = −

5

• j=0,i=j

aji

I II III a05 0.612 0.612 0.612 a12 0.908 0.524 0.671 a14 0.567 1.518 0.012 a15 0.388 0.388 0.388 a21 0.246 1.291 1.337 a23 0.020 0.013 1.283 a32 0.602 0.042 0.131 a41 1.191 0.146 0.100 a51 0.024 0.024 0.024

MJ Chappell University of Warwick July 2016 Structural identifiability 3/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

Model simulations

10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set I t y1 10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set II t y1 10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set III t y1

MJ Chappell University of Warwick July 2016 Structural identifiability 4/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

Model simulations

10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set I t y1 10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set II t y1 10 20 30 40 50 0.2 0.4 0.6 0.8 1 Parameter Set III t y1 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Compartment 2 t x2 I II III 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Compartment 3 t x3 I II III 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Compartment 4 t x4 I II III

MJ Chappell University of Warwick July 2016 Structural identifiability 4/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

SIR Model

SIR infectious disease model:

S I

y(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 N XY ˙ Y = β N XY − (µ + γ) Y y = kY

MJ Chappell University of Warwick July 2016 Structural identifiability 5/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

SIR model

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I 2 4 6 8 10 2250 2300 2350 2400 2450 2500 2550 Time (yrs) Susceptibles, S

MJ Chappell University of Warwick July 2016 Structural identifiability 6/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

SIR model

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I 2 4 6 8 10 2250 2300 2350 2400 2450 2500 2550 Time (yrs) Susceptibles, S

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

MJ Chappell University of Warwick July 2016 Structural identifiability 6/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

SIR model

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I 2 4 6 8 10 2250 2300 2350 2400 2450 2500 2550 Time (yrs) Susceptibles, S

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I

MJ Chappell University of Warwick July 2016 Structural identifiability 6/49

Motivation Structural identifiability Techniques for nonlinear models Skeletal tracer kinetics Infectious disease modelling

SIR model

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I 2 4 6 8 10 2250 2300 2350 2400 2450 2500 2550 Time (yrs) Susceptibles, S

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

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10 Time (yrs) Model output, k I 2 4 6 8 10 4500 4600 4700 4800 4900 5000 5100 Time (yrs) Susceptibles, S

MJ Chappell University of Warwick July 2016 Structural identifiability 6/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Structural identifiability

input

• utput

system?

Given postulated state-space models for a given biological or biomedical process: Structural Identifiability Are the unknown parameters uniquely determined by the input-output behaviour?

MJ Chappell University of Warwick July 2016 Structural identifiability 7/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Structural identifiability

input

• utput

system Given postulated state-space model, are the unknown parameters uniquely determined by the output (ie, perfect, continuous, noise-free data)? Necessary theoretical prerequisite to: experiment design system identification parameter estimation

MJ Chappell University of Warwick July 2016 Structural identifiability 8/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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 is assumed to lie in a set of feasible vectors: p ∈ Ω. n dimensional vector q(t, p) is state vector, such that q0(p) is the initial state (may depend on the unknown parameters) m dimensional vector u(t) is input/control vector (our influence on system); what inputs are available depends on experiment to be performed, so u(·) ∈ U, a set of admissible inputs (might be empty). y(t, p) is the l-dimensional output/observation vector (what we can measure in the system). In the following we make explicit that output y depends on p ∈ Ω and u ∈ U by writing y(t, p; u).

MJ Chappell University of Warwick July 2016 Structural identifiability 9/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Parameter identifiability

For generic p ∈ Ω, the parameter pi is said to be locally identifiable 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 the previous definition, then the parameter pi is globally/uniquely identifiable. If the parameter pi is not locally identifiable, i.e., there is no suitable neighbourhood N(p), then it is said to be unidentifiable.

MJ Chappell University of Warwick July 2016 Structural identifiability 10/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Structural identifiability

Structurally globally/uniquely identifiable A parameterised state space model is structurally globally/uniquely identifiable (SGI) if all of the unknown parameters pi are globally/uniquely identifiable. Structurally locally identifiable A state space model is structurally locally identifiable (SLI) if all

• f the unknown parameters pi are locally identifiable and at

least one of these parameters is not globally identifiable. Unidentifiable A state space model is unidentifiable if at least one of the unknown parameters pi is unidentifiable.

MJ Chappell University of Warwick July 2016 Structural identifiability 11/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Remarks

Necessary condition for parameter estimation

Essential for parameters with practical significance Prerequisite to experiment design

Identifiability does not guarantee

Good fit to experimental data Good fit only with unique vector of parameters

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

Laplace transform or transfer function Taylor series of output Similarity transformation (exhaustive modelling)

Taylor series and similarity transformation approaches are applicable for nonlinear systems Differential algebra

Rational systems with differentiable inputs/outputs Heavily dependent on symbolic computation

MJ Chappell University of Warwick July 2016 Structural identifiability 12/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Laplace Transform Approach

MJ Chappell University of Warwick July 2016 Structural identifiability 13/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling 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), where A(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)

MJ Chappell University of Warwick July 2016 Structural identifiability 14/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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 & denominators uniquely determined by input-output relationship

MJ Chappell University of Warwick July 2016 Structural identifiability 15/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: Transfer function: G(s) =

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) =

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) = b1c1 s + a01

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) = b1c1 s + a01 So b1c1 and a01 globally identifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) = b1c1 s + a01 So b1c1 and a01 globally identifiable But b1 and c1 unidentifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

b1u(t) a01 y = c1q1

Input: impulse: b1u(t) = b1n0δ(t); b1 unknown, n0 known Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1 + b1u(t), q1(0) = 0, y = c1q1 Transfer function: G(s) = C(p) (sIn − A(p))−1 B(p) = b1c1 s + a01 So b1c1 and a01 globally identifiable But b1 and c1 unidentifiable So model is unidentifiable unless b1 or c1 known (then SGI)

MJ Chappell University of Warwick July 2016 Structural identifiability 16/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 2 Compartments

1 2

I = bu(t) a01 y = c1x1 a12

Model is: ˙ x1 ˙ x2

• =

−a01 a12 −a12 x1 x2

• +

b

• u(t)

y =

• c

x1 x2

• Transfer function:

G(s) =

• c

s + a01 −a12 s + a12 −1 0 b

• =

bca12 (s + a01)(s + a12)

MJ Chappell University of Warwick July 2016 Structural identifiability 17/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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) hence locally identifiable If neither b nor c known then unidentifiable If both b and c known then globally identifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 18/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Taylor series approach

MJ Chappell University of Warwick July 2016 Structural identifiability 19/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling 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! +. . . 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 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 equations needed

MJ Chappell University of Warwick July 2016 Structural identifiability 20/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: First coefficient: y(0, p) = Second coefficient: ˙ y(0, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: First coefficient: y(0, p) = Second coefficient: ˙ y(0, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = Second coefficient: ˙ y(0, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) = − a01b1c1n0

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) = − a01b1c1n0 So b1c1 & b1c1a01 unique

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) = − a01b1c1n0 So b1c1 & b1c1a01 unique (ie b1c1 & a01 globally identifiable)

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) = − a01b1c1n0 So b1c1 & b1c1a01 unique (ie b1c1 & a01 globally identifiable) But b1 and c1 unidentifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 1 Compartment

1

a01 y = c1q1

Input: impulse in I.C.s: q1(0) = b1n0; b1 unknown, n0 known. Output: y = c1q1, where c1 unknown. System equations: ˙ q1 = −a01q1, q1(0) = b1n0, y = c1q1 First coefficient: y(0, p) = b1c1n0 Second coefficient: ˙ y(0, p) = − a01b1c1n0 So b1c1 & b1c1a01 unique (ie b1c1 & a01 globally identifiable) But b1 and c1 unidentifiable So model unidentifiable unless b1 &/or c1 known (then SGI)

MJ Chappell University of Warwick July 2016 Structural identifiability 21/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 2 Compartments

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) =

MJ Chappell University of Warwick July 2016 Structural identifiability 22/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 2 Compartments

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) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ q2(t, p) = y(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 22/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 2 Compartments

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) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ q2(t, p) = a21q1(t, p) − a12q2(t, p), q2(0, p) = 0 y(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 22/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Example: 2 Compartments

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) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ q2(t, p) = a21q1(t, p) − a12q2(t, p), q2(0, p) = 0 y(t, p) = c1q1(t, p)

MJ Chappell University of Warwick July 2016 Structural identifiability 22/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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

MJ Chappell University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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

MJ Chappell University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

˙ q1(t, p) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ 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: ˙ y1(0, p) = −c1 (a01 + a21) b1 Third coefficient: Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

˙ q1(t, p) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ 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: ˙ y1(0, p) = −c1 (a01 + a21) b1 Third coefficient: y(2)

1 (t, p) = c1 (− (a01 + a21) ˙

q1(t, p) + a12 ˙ q2(t, p)) Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

˙ q1(t, p) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ 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: ˙ y1(0, p) = −c1 (a01 + a21) b1 Third coefficient: y(2)

1 (t, p) = c1 (− (a01 + a21) ˙

q1(t, p) + a12 ˙ q2(t, p)) = ⇒ y(2)

1 (0, p) = c1

• (a01 + a21)2 b1 + a12a21b1
• Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

˙ q1(t, p) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ 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: ˙ y1(0, p) = −c1 (a01 + a21) b1 Third coefficient: y(2)

1 (t, p) = c1 (− (a01 + a21) ˙

q1(t, p) + a12 ˙ q2(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 + a12a21 ˙ q1 − a2

12 ˙

q2

• MJ Chappell

University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

˙ q1(t, p) = − (a01 + a21) q1(t, p) + a12q2(t, p), q1(0, p) = b1 ˙ 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: ˙ y1(0, p) = −c1 (a01 + a21) b1 Third coefficient: y(2)

1 (t, p) = c1 (− (a01 + a21) ˙

q1(t, p) + a12 ˙ q2(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 + a12a21 ˙ q1 − a2

12 ˙

q2

• ⇒ y(3)

1 (0, p) = b1c1

• (a01 + a21)
• −(a01 + a21)2 − 2a12a21
• − a2

12a21

• MJ Chappell

University of Warwick July 2016 Structural identifiability 23/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient:

Second coefficient: Third coefficient: Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: Third coefficient: Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: a01 + a21 unique Third coefficient: Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: a01 + a21 unique Third coefficient: a12a21 unique Fourth coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: a01 + a21 unique Third coefficient: a12a21 unique Fourth coefficient: a12 unique

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: a01 + a21 unique Third coefficient: a12a21 unique Fourth coefficient: a12 unique So a21 and then a01 unique

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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
• − a2

12a21

• First coefficient: b1c1 unique

Second coefficient: a01 + a21 unique Third coefficient: a12a21 unique Fourth coefficient: a12 unique So a21 and then a01 unique Same result as before

MJ Chappell University of Warwick July 2016 Structural identifiability 24/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Similarity transformation/exhaustive modelling approach

MJ Chappell University of Warwick July 2016 Structural identifiability 25/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach 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

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ 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), then A(p) = B(p) = C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = 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), then A(p) = B(p) = C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = TA(p)T −1z(t) + TB(p)u(t), 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), then A(p) = B(p) = C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = TA(p)T −1z(t) + TB(p)u(t), z(0) = Tq0(p), y(t, p) = C(p)q(t, p) = C(p)T −1z(t). 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), then A(p) = B(p) = C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = TA(p)T −1z(t) + TB(p)u(t), z(0) = Tq0(p), y(t, p) = C(p)q(t, p) = C(p)T −1z(t). 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), then A(p) = TA(p)T −1, B(p) = C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = TA(p)T −1z(t) + TB(p)u(t), z(0) = Tq0(p), y(t, p) = C(p)q(t, p) = C(p)T −1z(t). 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), then A(p) = TA(p)T −1, B(p) = TB(p), C(p) = for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

Then there exists invertible n × n matrix T such that, if z = Tq: ˙ z(t) = T ˙ q(t, p) = TA(p)q(t, p) + TB(p)u(t) = TA(p)T −1z(t) + TB(p)u(t), z(0) = Tq0(p), y(t, p) = C(p)q(t, p) = C(p)T −1z(t). 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), then A(p) = TA(p)T −1, B(p) = TB(p), C(p) = C(p)T −1, for some invertible n × n matrix T.

MJ Chappell University of Warwick July 2016 Structural identifiability 27/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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)

• f possibilities, then the system is SLI

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

MJ Chappell University of Warwick July 2016 Structural identifiability 28/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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) =

MJ Chappell University of Warwick July 2016 Structural identifiability 29/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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) = − (a01 + a21) a12 a21 −a12

• , B(p) =

, C(p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 29/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 29/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

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) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• MJ Chappell

University of Warwick July 2016 Structural identifiability 29/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =
• Observability:
• C(p)

C(p)A(p)

• =
• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1

• Observability:
• C(p)

C(p)A(p)

• =
• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• Observability:
• C(p)

C(p)A(p)

• =
• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1
• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• Equation (3):

B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

Equation (3): B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

So model is minimal Equation (3): B(p) = TB(p) and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

So model is minimal Equation (3): B(p) =

• b1
• = TB(p) =

t11 t12 t21 t22 b1

• and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

So model is minimal Equation (3): B(p) =

• b1
• = TB(p) =

t11 t12 t21 t22 b1

• =

t11b1 t21b1

• and so

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

So model is minimal Equation (3): B(p) =

• b1
• = TB(p) =

t11 t12 t21 t22 b1

• =

t11b1 t21b1

• and so t21 = 0 and

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Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , B(p) =

b1

• , C(p) =
• c1
• Controllability:
• B(p) A(p)B(p)
• =

b1 − b1 (a01 + a21) b1a21

• rank 2

Observability:

• C(p)

C(p)A(p)

• =
• c1

− c1 (a01 + a21) c1a12

• rank 2

So model is minimal Equation (3): B(p) =

• b1
• = TB(p) =

t11 t12 t21 t22 b1

• =

t11b1 t21b1

• and so t21 = 0 and t11 = b1/b1

MJ Chappell University of Warwick July 2016 Structural identifiability 30/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• c1

b1/b1 t12 t22

• =
• c1
• = C(p)

and so Equation (2): A(p)T = = TA(p) = =

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Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so Equation (2): A(p)T = = TA(p) = =

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and Equation (2): A(p)T = = TA(p) = =

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and b1c1 = b1c1 Equation (2): A(p)T = = TA(p) = =

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and b1c1 = b1c1 Equation (2): A(p)T = − (a01 + a21) a12 a21 −a12 b1/b1 t22

• = TA(p) =

=

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and b1c1 = b1c1 Equation (2): A(p)T = − (a01 + a21) a12 a21 −a12 b1/b1 t22

• = TA(p) =
• b1/b1

t22 − (a01 + a21) a12 a21 −a12

• =

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and b1c1 = b1c1 Equation (2): A(p)T = − (a01 + a21) a12 a21 −a12 b1/b1 t22

• = TA(p) =
• b1/b1

t22 − (a01 + a21) a12 a21 −a12

• =
• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =

MJ Chappell University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

A(p) = − (a01 + a21) a12 a21 −a12

• , C(p) =
• c1 0
• , T =
• b1/b1

t12 t22

• Equation (4):

C(p)T =

• b1c1/b1

c1t12

• =
• c1
• = C(p)

and so t12 = 0 and b1c1 = b1c1 Equation (2): A(p)T = − (a01 + a21) a12 a21 −a12 b1/b1 t22

• = TA(p) =
• b1/b1

t22 − (a01 + a21) a12 a21 −a12

• =
• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• MJ Chappell

University of Warwick July 2016 Structural identifiability 31/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

so (1,2) component: (2,1) component: (1,1) component: So:

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Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: (2,1) component: (1,1) component: So:

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Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: (1,1) component: So:

MJ Chappell University of Warwick July 2016 Structural identifiability 32/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: a21 = a21 (1,1) component: So:

MJ Chappell University of Warwick July 2016 Structural identifiability 32/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: a21 = a21 (1,1) component: a01 = a01 So:

MJ Chappell University of Warwick July 2016 Structural identifiability 32/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: a21 = a21 (1,1) component: a01 = a01 So: a01, a12 and a21 all globally identifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 32/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: a21 = a21 (1,1) component: a01 = a01 So: a01, a12 and a21 all globally identifiable combination b1c1 globally identifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 32/49

Motivation Structural identifiability Techniques for nonlinear models Laplace transform approach Taylor series approach Similarity transformation/exhaustive modelling approach

• − b1

b1 (a01 + a21)

t22a12

b1 b1 a21

−a12t22

• =
• − b1

b1 (a01 + a21) b1 b1 a12

a21t22 −a12t22

• (2,2) component:

a12 = a12 so (1,2) component: t22 = b1/b1 (2,1) component: a21 = a21 (1,1) component: a01 = a01 So: a01, a12 and a21 all globally identifiable combination b1c1 globally identifiable individual b1 and c1 unidentifiable

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Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Techniques for nonlinear models

MJ Chappell University of Warwick July 2016 Structural identifiability 33/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable 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

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Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Taylor series approach

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Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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 + r If any coefficient rational: n + r + 1 Quite difficult to use TSA to prove model is unidentifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 36/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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) = d dt

• − c1Vmx1

Km + x1

• MJ Chappell

University of Warwick July 2016 Structural identifiability 37/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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) = d dt

• − c1Vmx1

Km + x1

• Use symbolic tools such as MATHEMATICA, MAPLE

MJ Chappell University of Warwick July 2016 Structural identifiability 37/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: First coefficient: y(0, p) = Second coefficient: ˙ y(0, p) = Third coefficient: y(2)(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: ˙ q1 = −αq1(β − q1), q1(0) = 1 y = c1q1 First coefficient: y(0, p) = Second coefficient: ˙ y(0, p) = Third coefficient: y(2)(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: ˙ q1 = −αq1(β − q1), q1(0) = 1 y = c1q1 First coefficient: y(0, p) = c1 Second coefficient: ˙ y(0, p) = Third coefficient: y(2)(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: ˙ q1 = −αq1(β − q1), q1(0) = 1 y = c1q1 First coefficient: y(0, p) = c1 Second coefficient: ˙ y(0, p) = − c1α(β − 1) Third coefficient: y(2)(t, p) =

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: ˙ q1 = −αq1(β − q1), q1(0) = 1 y = c1q1 First coefficient: y(0, p) = c1 Second coefficient: ˙ y(0, p) = − c1α(β − 1) Third coefficient: y(2)(t, p) = − c1α (β ˙ q1 − 2q1 ˙ q1)

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: One compartment with Langmuir elimination:

1

b1u(t) y = c1q1 α(β − q1)

Model equations: ˙ q1 = −αq1(β − q1), q1(0) = 1 y = c1q1 First coefficient: y(0, p) = c1 Second coefficient: ˙ y(0, p) = − c1α(β − 1) Third coefficient: y(2)(t, p) = − c1α (β ˙ q1 − 2q1 ˙ q1) = ⇒ y(2)(0, p) = c1α2(β − 1) (β − 2)

MJ Chappell University of Warwick July 2016 Structural identifiability 38/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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:

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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:

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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: α(β − 1) unique Third coefficient:

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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: α(β − 1) unique Third coefficient: α unique (globally identifiable) And so β globally identifiable

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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: α(β − 1) unique Third coefficient: α unique (globally identifiable) And so β globally identifiable All parameters globally identifiable so model is SGI

MJ Chappell University of Warwick July 2016 Structural identifiability 39/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Now for something a little more advanced . . .

MJ Chappell University of Warwick July 2016 Structural identifiability 40/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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 . . .

MJ Chappell University of Warwick July 2016 Structural identifiability 41/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Previous approach

Coordinate transformation between models that are indistinguishable via available output

Hp (λ(x)) = Hp(x)

Σ(p) Σ(p) ˆ Σ(p) ˆ Σ(p) λ Hp Hp id

Determine S(p) set of all parameters p s.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))

MJ Chappell University of Warwick July 2016 Structural identifiability 42/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Observability normal form

System ˆ Σ is the observability normal form, z = Hp(x): ˙ z1 = z2 . . . ˙ zn−1 = zn ˙ zn = µn+1(H−1

p (z), p)

y = z1 Last equation gives input-output equation for system and so, for all p ∈ S(p), have µn+1(H−1

p (z(t)), p) = µn+1(H−1 p (z(t)), p)

∀t ≥ 0

MJ Chappell University of Warwick July 2016 Structural identifiability 43/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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 so σi(p) = σi(p) i = 1, . . . , m

MJ Chappell University of Warwick July 2016 Structural identifiability 44/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example

Consider two-compartment model: 1 2

y = 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

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Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: Observability normal form

Observability condition met provided p2 = 0 (ie, for all p) so can transform 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

MJ Chappell University of Warwick July 2016 Structural identifiability 46/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Example: Output equation

Output equation: z2

1 ˙

z2 + p2

4 ˙

z2 + 2p4z1 ˙ z2 + p2p3p4z1 + p2p3z2

1

+

• p3p4 + p2

4(p1 + p2)

• z2 + 2p4(p1 + p2)z1z2

+ (p1 + p2)z2

1z2 = φ0(z, ˙

zn) + 7

i=1 σi(p)φi(z, ˙

zn) = 0 Linear independence of terms guaranteed by checking the Wronskian, or can use constructive algebra methods (in MAPLE):

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 was based on PhD by Forsman (1991) Constructive Commutative Algebra in Nonlinear Control Theory

MJ Chappell University of Warwick July 2016 Structural identifiability 47/49

Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

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 + p2

4(p1 + p2)

= ⇒ p3 = p3 Solving these shows that p = p, ie S(p) = {p} Therefore model is structurally globally identifiable

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Motivation Structural identifiability Techniques for nonlinear models Taylor series approach Observable normal form

Summary

Structural identifiability is an important step in modelling process

Theoretical prerequisite to experiment design, system identification, and parameter estimation Techniques involve generation, manipulation & solution of nonlinear algebraic equations

Observability normal form highly appropriate for both analyses

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

Structural indistinguishability similarly important

More general framework but exact Generally pairwise comparison of schemes

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