uncertainty quantification for ensemble models
TRANSCRIPT
Ensemble ModelingDAVID SWIGON
UNIVERSITY OF PITTSBURGH
Biomedical systemsGOAL: individualized medicine โ designing patient-specific treatment
Model of the disease
ODEPDESDE
Calibration with patient data
MLBootstrapBayesian
Risk assessment
CoxTreatment design
SimulationsAdjustment
Optimal control
HEALTH
Problems with biomedical model calibrationModel complexity
โฆ Large number of variables and parameters, no symmetries
โฆ Complex or unknown interactions
Data scarcityโฆ Unobserved variables
โฆ Insufficient time resolution
โฆ Small number of trajectories
Data inhomogeneityโฆ Variability of parameters across subjects
โฆ Multiple subjects combined in one data set
โฆ Data aggregation
Quality
QuantityHomogeneity
DATA TRADEOFF
ExamplesINFLUENZA BACTERIAL PNEUMONIA
Time courses for 16 human volunteers infected by Influenza A/Texas/91 (H1N1) [Hayden , JAMA, 96]
Data from MF1 mice subjects infected by D39 strain of pneumococcus, data are available for various other mice and pneumonia strains [Kadioglu et al., Inf. and Immun., 2000]:
INFLUENZA
Intranasal infection of 136 Balb/c mice by Influenza virus A/PR/8/34 (H1N1) [Toapanta, Ross, Morel]
20 variables, 97 parameters, ODE model
Dynamical model Data
Error minimization
Parameter estimates
IHVI
HVH
cVpIV
โ=
โ=
โ=
0.806)16.7,,101.33 0.0347,, 0.0373(),,,,( -50 =optcpV
0)0(
104)0(
)0(
8
0
=
=
=
I
H
VV kVk skVkV == )(,)(
=
โ8
12
2
,,,,
))((min
0 k k
k
cpV s
VkV
Traditional modeling
Questions
โข Is there a parameter set/model that reproduces the data exactly?
โข Is the optimal parameter set unique?
โข How sensitive is the optimal parameter set to data, their uncertainty, and assumptions about error distribution?
โข How well does the model fit the data? Is it even appropriate? Can the model be ruled out at the given data uncertainties?
โข How large is the region in data space for which the model is appropriate?
โข Since the data consist of multiple subjects, is the model appropriate and how do we know?
Answer
โข Analyze more closely the relation between the system (parameters) and its solutions.
โข Use probabilistic methods to describe the cohort of subjects or the parameter estimation uncertainty
Ensemble model
Distribution describesโฆ Variability of parameters across cohort of subjects
โฆ Uncertainty in determination of parameters from noisy data
Ensemble modelingDeterministic model
ParametersMechanismsTime points
Trajectory data Solution map
Distribution of data Parameters
Distribution of MechanismsTime points
ODE ensemble modelInitial value problem:
แถ๐ฅ = ๐ ๐ฅ, ๐ , ๐ฅ 0 = ๐
Solution: ๐ฅ = ๐ฅ ๐ก; ๐, ๐
Deterministic solution map: ๐ฆ = ๐น ๐ = (๐ฅ ๐ก1, ๐ , โฆ , ๐ฅ ๐ก๐, ๐ )
Parameter distribution - random variable ๐ด, density ๐(๐)
Data distribution - random variable ๐ = ๐น ๐ด , density ๐(๐ฆ)
Value of a variable at fixed time: ๐๐ก = ๐ฅ(๐ก; ๐ด)
0.806)16.7,,101.33 0.0347,, 0.0373(),,,,( -50 =optcpV
V H I
deterministic
ensemble
IHVI
HVH
cVpIV
โ=
โ=
โ=
๐ฆ = (๐ 1 , โฆ , ๐ 8 )
๐ = (๐0, ๐ฝ, ๐, ๐, ๐ฟ)
Influenza model
Constructing ensemble models
Distribution of data Distribution of Parameters
(1) Inverse map of data sampleโฆ Requires computation of ๐นโ1, solution of the ODE inverse problem
(2) Direct sampling of parameter distributionโฆ Requires formula for parameter density
Sample of data{๐ฆ๐}
Sample of Parameters {๐๐}
๐๐ = ๐นโ1(๐ฆ๐)
ODE modelsInitial value problem with matrix parameter ๐ด :
แถ๐ฅ = ๐ ๐ฅ, ๐ด , ๐ฅ 0 = ๐
INVERSE PROBLEM: Determine the function ๐ . , . and/or parameters ๐ด, ๐ from data about trajectory(ies) ๐ฅ(. ; ๐ด, ๐).
FORWARD PROBLEM๐ . , . , ๐ด, ๐ ๐ฅ(. ; ๐ด, ๐)
INVERSE PROBLEM
Inverse problem
Mathematical analysisโฆ Existence, uniqueness/identifiability
โฆ Robustness (existence/uniqueness on open sets)
โฆ Construction of the inverse map (numerical algorithm)
โฆ Maximal permissible uncertainty
โฆ Prediction uncertainty
๐ . , . , ๐ด, ๐ ๐ฅ(. ; ๐ด, ๐)INVERSE PROBLEM
Existing results on ODE inverse problemObservability theory
โฆ Search for conditions that guarantee identifiability of the system state (initial conditions) from observed trajectory (can be extended to parameters)
โฆ Kalman (1963) formulated observability conditions for linear dynamical systems
โฆ Griffith & Kumar (1971), Kou, Elliot, & Tarn (1973) formulated sufficient conditions for observability of nonlinear dynamical systems
โฆ Identification of analytic systems with ๐ parameters requires at most 2๐ +1 observations [Aeyels 1981; Sontag 2002]
Polynomial systemsโฆ Reconstruction of polynomial fields from known algebraic attractors
[Sverdlove 1980,81]
โฆ Analysis of all possible topologically invariant planar quadratic dynamical systems [Perko textbook]
Existing results on inverse problemAlgorithms for finding inverse
โฆ Numerical simulation combined with nonlinear least squares fitting
โฆ Multiple shooting combined with Gauss-Newton minimization [Bock 1983; Lee]
โฆ Collocation methods using basis function expansion, combined with NLS fitting [Varah 1982; Ramsey et al 2007]
โฆ Contraction map and โcollage methodโ [Kunze et al. 2004]
โฆ Parameter-free approach (using Takens map) [Sauer, Hamilton, โฆ]
โฆ Compressed sensing
Probabilistic approachesโฆ Transformation of measure
โฆ Numerical simulation combined with Bayesian inference
โฆ Bayesian integration of ODEs [Chkrebtii et al.]
Identifiability from single complete trajectorySystem: แถ๐ฅ = ๐(๐ฅ, ๐ด)
Initial condition: ๐ฅ 0 = ๐
Solution: ๐ฅ ๐ก; ๐ด, ๐
๐ฅ ๐ก; ๐ด, ๐
๐ฅ ๐ก; ๐ต, ๐
๐ด
๐ต
๐ฅ ๐ก; ๐ด, ๐ = ๐ฅ ๐ก; ๐ต, ๐๐ด
๐ต
๐ถ
๐ท
๐ฅ ๐ก; ๐ท, ๐๐ฅ ๐ก; ๐ถ, ๐
๐ฅ ๐ก; ๐ถ, เดค๐ = ๐ฅ ๐ก; ๐ท, เดค๐
๐ฅ ๐ก; ๐ด, ๐ = ๐ฅ ๐ก; ๐ต, ๐๐ด
๐ต
๐ถ
๐ท
๐ฅ ๐ก; ๐ท, ๐๐ฅ ๐ก; ๐ถ, ๐
๐ฅ ๐ก; ๐ถ, เดค๐ = ๐ฅ ๐ก; ๐ท, เดค๐
DefinitionsSystem แถ๐ฅ = ๐(๐ฅ, ๐ด) is
Identifiable in ฮฉ โ โ๐ร๐ iff for all ๐ด, ๐ต โ ฮฉ with ๐ด โ ๐ต there exists ๐ โ โ๐ such that ๐ฅ ๐ก; ๐ด, ๐ โ ๐ฅ(๐ก; ๐ต, ๐) for some ๐ก > 0.
Identifiable in ฮฉ โ โ๐ร๐ from ๐ iff for all ๐ด, ๐ต โ ฮฉ with ๐ด โ ๐ต it holds that ๐ฅ ๐ก; ๐ด, ๐ โ ๐ฅ(๐ก; ๐ต, ๐) for some ๐ก > 0.
Identifiable in ฮฉ โ โ๐ร๐ from ๐ฅ ๐ก; ๐ด, ๐ iff there exists no ๐ต โ ฮฉ with๐ด โ ๐ต such that ๐ฅ ๐ก; ๐ด, ๐ = ๐ฅ(๐ก; ๐ต, ๐) for all ๐ก.
Unconditionally identifiable in ฮฉ โ โ๐ร๐ iff for all ๐ด, ๐ต โ ฮฉ with ๐ด โ ๐ต and all ๐ โ โ๐ it holds that ๐ฅ ๐ก; ๐ด, ๐ โ ๐ฅ(๐ก; ๐ต, ๐) for some ๐ก > 0.
Results for linear system
Theorem [SRS14]: System แถ๐ฅ = ๐ด๐ฅ is identifiable in โ๐ร๐ from ๐ฅ ๐ก; ๐ด, ๐ if and only if the orbit ๐พ(๐ด, ๐) = {๐ฅ ๐ก; ๐ด, ๐ , ๐ก โ โ} is not confined to a proper subspace of โ๐.
Proof sketch: By theorem of Astrom, identifiability from b is equivalent to det[๐ ๐ด๐ โฆ |๐ด๐โ1๐] โ 0 . It suffices to show that this condition is equivalent to the orbit ๐พ(๐ด, ๐) not being confined to a proper subspace of โ๐.
[SRS14]
Red trajectories are confined to linear subspaces of the flux space => they do not identify the system
Examples: Inverse problem solution is unique only for certain trajectories/data.
แถ๐ฅแถ๐ฆแถ๐ง=
โ0.2 โ1 01 โ0.2 00 0 โ0.3
๐ฅ๐ฆ๐ง
แถ๐ฅแถ๐ฆแถ๐ง=
โ1 โ3 20 โ4 20 0 โ2
๐ฅ๐ฆ๐ง
แถ๐ฅแถ๐ฆแถ๐ง=
โ1 1 00 โ1 00 0 โ1
๐ฅ๐ฆ๐ง
๐ฅ๐ฆ
๐ง
Result for linear-in-parameter system
Trajectory is confined => System is not identifiable! System with the same trajectory:
โ
โ=
y
xy
x
y
x
310
012
=
1
1b
โโ
โ+โ=
y
xy
x
y
x
31
12
Theorem [SRS14]: System แถ๐ฅ = ๐ด๐(๐ฅ) is identifiable in โ๐ร๐ from ๐ฅ ๐ก; ๐ด, ๐ if and only if the image ๐(๐พ(๐ด, ๐)) of the orbit ๐พ(๐ด, ๐) is not confined to a proper subspace of โ๐.
๐ฅ
๐ฅ๐ฆ
๐ฆ
Result for confined trajectory
Theorem [SRS14]:Suppose that ๐ is a proper linear subspace โ๐ invariant under A. The following are equivalent(i) ๐ is the minimal A-invariant subspace such that ๐ โ ๐ .(ii) The orbit of ๐ฅ(๐ก; ๐ด, ๐) is not confined to a proper subspace of ๐(iii) There exists no ๐ต โ โ๐ร๐ such that ๐ต|๐ โ ๐ด|๐ and ๐ฅ ๐ก; ๐ด, ๐ = ๐ฅ(๐ก; ๐ต, ๐).
โ
โ
โโ
=
200
240
231
A
=
1
0
0
,
0
1
1
spanV
โ
โ
=
1
1
1
b
โ
โ
โโ
=
100
020
324
][ BA
=
0
1
0
,
1
0
0
,
0
1
1
B
โ
โ
=
13
12
11
00
20
24
][
BB
โโ
+โโ
โโ
=
2
24
24
2323
33133313
1313
B
๐ด แ๐
Numerical determination of confinementNumerical errors (roundoff) can obscure the confinement of trajectory represented as a collection of data points
Use Singular Value Decomposition
๐ = ๐๐๐๐
โฆ Dimension of confining subspace = ๐๐๐๐ ๐ = #{๐๐๐ > 0}
โฆ Numerical rank: ๐๐๐๐๐๐ข๐ ๐ = #{๐๐๐/max๐
๐๐๐ > ํ๐ก๐๐}
Identification from discrete data on a single trajectoryDiscrete data: ๐ = ๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐ , ๐ฅ๐ = ๐ฅ(๐ก๐; ๐ด, ๐)
Identifiability from complete trajectory does not imply identifiability from discrete data.
โ+
โ
โโ=
01
102
2/34
42/3rA
โ=
1458.01689.0
1689.01458.0Ae
Identification from discrete data on a single trajectoryGOAL:
Partition the data space into domains in which โฆ the system is identifiable from data
โฆ the system is robustly identifiable (on an open neighborhood)
โฆ the system is identifiable to within countable number of alternatives
โฆ the system has a continuous family of compatible parameters
โฆ there is no parameter set corresponding to the data
โฆ the system has a specific dynamical behavior (stable, saddle, โฆ)
Inverse map for linear systemsDiscrete data (uniformly spaced): ๐ = (๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐)
Solution map: ๐ฅ๐ = ๐๐ด๐๐
ALGORITHM
Step 1: Let ๐๐ = ๐ฅ๐ ๐ฅ๐+1 โฆ ๐ฅ๐+๐โ1 , ๐ = 0,1
Step 2: Compute ฮฆ = ๐1๐0โ1
Step 3: Solve ๐๐ด = ฮฆ
Existence and uniqueness of ๐ด derives from conditions for existence and uniqueness of real matrix logarithm [Culver 1966].
Robust existence & uniqueness
๐๐ด ฮฆ
๐
Robust existence & uniqueness
Theorem [SRS17]: There exists an open set ๐ โ โ๐ร๐ containing ฮฆ such that for any ฮจ โ ๐ the equation ๐๐ด = ฮจ has a solution ๐ด โ โ๐ร๐ iff(a) {๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐โ1} are linearly independent and (b) ฮฆ has only positive real or complex eigenvalues.
Theorem [SRS17]: There exists an open set ๐ โ โ๐ร๐ containing ฮฆ such that for any ฮจ โ ๐ the equation ๐๐ด = ฮจ has a unique solution ๐ด โ โ๐ร๐ iff(a) {๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐โ1} are linearly independent and (b) ฮฆ has ๐ distinct positive real eigenvalues.
Theorem [SRS17]: There exists an open set ๐ โ โ๐ร๐ containing ฮฆ such that for any ฮจ โ ๐ the equation ๐๐ด = ฮจ does not have a solution ๐ด โ โ๐ร๐ iff(a) {๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐โ1} are linearly independent and (b) ฮฆ has at least one negative real eigenvalue of odd multiplicity.
Region of unique inverses
Let
Eigenvalues of ฮฆ are determined by ๐ฆ = ๐0โ1๐ฅ๐
Conditions on y that guarantee positive real eigenvalues have been worked out for arbitrary dimension [Gantmacher; Yang]
Example in 2D (regions of data which give unique inverse)
ฮฆ = ๐0โ1ฮฆ๐0 = ๐0
โ1๐1
ฮฆ =
0 0 0 โฏ ๐ฆ11 0 0 โฏ ๐ฆ20 1 0 โฏ ๐ฆ3โฎ โฎ โฎ โฑ โฎ0 0 0 โฏ ๐ฆ๐
๐ฅ1
๐ฅ0
== det,tr DT
๐ฅ2
Regions of stability for 2x2 systems
๐ฅ2๐ฅ2
Numerical error of identificationExact formula
ฮฆ = ๐1๐0โ1
Numerical sensitivity of the inverse to errors in the data.
๐ฟฮฆ
ฮฆโค ๐ (๐0)
๐ฟ๐0๐0
+๐ฟ๐1๐1
โฆ Here โ is any matrix norm and ๐ ๐ถ = ๐ถ ๐ถโ1 is the condition number of the matrix ๐ถ.
โฆ Note that collinearity of vectors in ๐0 leads to a loss of accuracy.
kA
k xex =+1},...,,{10 njjj xxxd = bebAjxx k
k
Ajkj == ),;(
Identification from non-uniformly spaced data
Discrete data (non-uniformly spaced)
Lemma: Let ๐ฅ๐๐ = ฮฆ๐๐๐ where ๐๐ are integers such that 0 = ๐0 < ๐1 < โฏ < ๐๐.
Let ๐0 = ๐ฅ๐0 , ๐ฅ๐1 , โฆ , ๐ฅ๐๐โ1 be invertible. Let ๐ฆ = ๐0โ1๐ฅ๐๐ be a vector with
entries ๐ฆ1, ๐ฆ2, โฆ ๐ฆ๐. If ฮฆ is diagonalizable, then ๐ is an eigenvalue of ฮฆ only if it is a root of the polynomial
๐๐ฆ ๐ = ๐๐๐ โ ๐ฆ๐๐๐๐โ1 โโฏโ ๐ฆ2๐
๐1 โ ๐ฆ1
Identification from non-uniformly spaced dataALGORITHM
Step 1: Get ๐ฆ = ๐0โ1๐ฅ๐๐
Step 2: Find all roots of ๐๐ฆ ๐
Step 3: Choose a combination of n distinct roots and form ฮฆ.
Step 4: Compute vectors ๐ง๐0 , ๐ง๐1 , โฆ , ๐ง๐๐โ1 as ๐ง๐0 = ๐1 and ๐ง๐๐+1 =ฮฆ๐๐+1โ๐๐๐ง๐๐.
Step 5: Compute ฮฆ = ๐โ1ฮฆ๐ where P is such that ๐๐ฅ๐๐ = ๐ง๐๐ for ๐ =0,1, โฆ , ๐ โ 1.
Step 6: Solve ๐๐ด = ฮฆ .
Note: The variety of choices for ฮฆ may lead to non-uniqueness of parameter estimates.
Uncertainty analysisUncertain data: แ๐ = (๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐)
Bounds on uncertainty: ๐ถ ๐, ํ = { แ๐:max๐,๐
ฮ๐ฅ๐,๐ < ํ}
Neighborhood ๐ถ ๐, ํ is called permissible for property P iff P is shared by all systems corresponding to the data แ๐ โ ๐ถ ๐, ํ .
Value ํ๐ > 0 is maximal permissible uncertainty for property P iff๐ถ ๐, ํ is permissible for P for all 0 < ฮต < ํ๐, and ๐ถ ๐, วํ is not permissible for P for any วํ > ํ๐
P can beโฆ Existence of inverse (unique or nonunique)
โฆ Unique inverse
โฆ Unique stable inverse
ํ
๐
๐ถ ๐, ํ
Results for linear systems
Theorem (lower bound): Let d be such that ฮฆ has n distinct positive eigenvalues
๐1, ๐2, โฆ , ๐๐. Let ๐1 =1
2min๐<๐
๐๐ โ ๐๐ , ๐1 = min๐๐๐, and
๐ฟ๐ = min{๐1, ๐2}. If 0 < ํ โค ํ๐ where
ํ๐ =๐ฟ๐
๐(๐ฟ๐ + 1 + ฮ ) ๐โ1 ๐0โ1๐
with ฮฆ = ๐ฮ๐โ1, then for any แ๐ โ ๐ถ(๐, ํ), เทฉฮฆ has n distinct positive eigenvalues.
Results for linear systems
Theorem (upper bound): Let d be such that ฮฆ has n distinct positive eigenvalues ๐1, ๐2, โฆ , ๐๐, and let y be the last column vector of the companion
matrix form ฮฆ of ฮฆ. The smallest neighborhood ๐ถ(๐, ํ) that contains data แ๐ for
which companion matrix เทฉฮฆ has last column ๐ฆ has ํ = ํ๐ where
ํ๐ =๐0(๐ฆ โ ๐ฆ) โ
๐ฆ 1 + 1
Permissible data uncertainty: Amount of measurement error that can be tolerated without altering inference qualitatively
Upper and lower bounds for linear systems.
แถ๐ฅ = ๐ด๐ฅ, ๐ฅ 0 = ๐
[SRS17]
Prediction Uncertainty: In high dimensions inverse problem solution is very sensitive to small changes in data.
โ๐ฅ๐ =0.0049
โ๐ฅ๐ =0.0295 โ๐ฅ๐ =0.0005
[SRS17]
SUMMARYExistence and identifiability analysis provides regions on which the inverse problem for ODEs is well posed
Additional solution attributes (e.g., stability, spirality) can be included
OPEN PROBLEMS
Existence of inverse for linear-in-parameter systems
Maximal permissible uncertainty for linear-in-parameter systems
ODE Models with random effectsInitial value problem: แถ๐ฅ = ๐(๐ฅ, ๐), ๐ฅ 0 = ๐
Solution: ๐ฅ ๐ก; ๐, ๐ โ ๐ถ1 โ๐
Data vector: ๐ฆ = ๐ป(๐ฅ ๐ก0 , ๐ฅ ๐ก1 , โฆ , ๐ฅ ๐ก๐ )
Solution map: ๐ = ๐ญ ๐ invertible for ๐ โ ๐ช(เดฅ๐, ๐บ)
Random parameter model (RPM):
๐ = ๐น ๐ด ๐ด is a r. v. with density ๐ ๐
Random measurement error model (REM):
๐ = ๐น ๐ + ๐บ ๐บ is a r. v. with density ๐พ(๐)
Random-parameter model๐ = ๐น ๐ด
Inverse problem: Find parameter density ๐(๐) from the knowledge of data density ๐ ๐ฆ and ๐น.
Change of variables formula: for any set ฮ
เถฑฮ
๐ ๐ ๐๐ = เถฑ๐น(ฮ)
๐ ๐ฆ ๐๐ฆ = เถฑฮ
๐ ๐น ๐ ๐ฝ ๐ ๐๐
๐ฝ ๐ = | det๐ท๐๐น ๐ |
Parameter density: ๐ ๐ = ๐ ๐น ๐ ๐ฝ(๐)
Distribution of parameters: represents subject variability
๐น(ฮ)ฮ
Computational aspectsJacobian is expensive to compute.
Methods for computing ๐ฝ ๐โฆ Exact formula for linear models
For 2 ร 2 diagonalizable matrix ๐ด = ๐๐๐โ1:
๐ฝ ๐ =๐2๐ค11 โ ๐1๐ค21
2 ๐2๐ค12 โ ๐2๐ค222๐๐1+๐2 ๐๐1 โ ๐๐2 4
det๐ 2 ๐1 โ ๐22
โฆ Numerical differentiation๐๐น
๐๐๐เท๐ โ
1
ํ(๐น เท๐ + ํ๐๐ โ ๐น เท๐ )
Requires n additional integrations of the system in each step
[SSZR19]
Computational aspectsโฆ Numerical integration of the first variational equations (sensitivity
coefficients) for ODE models
Let ๐ ๐๐(๐ก) =๐๐ฅ(๐ก;๐,๐)
๐๐๐, ๐ ๐๐(๐ก) =
๐๐ฅ(๐ก;๐,๐)
๐๐๐
แถ๐ฅ = ๐ ๐ฅ, ๐ ๐ฅ 0 = ๐
แถ๐ ๐๐ ๐ก = แค๐๐ ๐ง, ๐
๐๐ง๐ง=๐ฅ ๐ก
๐ ๐๐ ๐ก + เธญ๐๐ ๐ง, ๐
๐๐๐๐ง=๐ฅ ๐ก
๐ ๐๐(0) = 0
แถ๐ ๐๐ ๐ก = แค๐๐ ๐ง, ๐
๐๐ง๐ง=๐ฅ ๐ก
๐ ๐๐ ๐ก ๐ ๐๐(0) = ๐๐
Dimension of the ODE system increases from ๐ to ๐(๐ + ๐ + 1)
[SSZR19]
Computational aspectsโฆ Broydenโs method (rank-1 update)
๐ท๐๐น เท๐ โ ๐ท๐๐น ๐๐ +๐น เท๐ โ ๐น ๐๐ โ ๐ท๐๐น ๐๐ ฮ๐๐
ฮ๐๐ ๐ฮ๐๐ฮ๐๐
๐
Only one computation of ๐น เท๐ (integration of the system) needed at every step
Update every 10 computations keeps relative error of ๐ท๐๐น เท๐ below 4%
Accuracy of the posterior is lower than the accuracy of Jacobian
[SSZR19]
ExamplesCommon problem:
โฆ Jacobian calculation is expensive, so can we replace ๐ฝ(๐) by a simpler distribution ๐(๐)?
๐(๐) ๐(๐ฆ)
Linear dynamical system, uniform data distribution ๐( าง๐ฅ๐๐ โ ํ, าง๐ฅ๐๐ + ํ)
โ
โ
=
3.2
5,
5624.1
2530.6,
2
10d
436.0=
xx =
โโ
โโ=
9519.04839.0
7491.05024.0
๐R
๐ด
๐J
๐U
),,,( 22211211 =a
๐ด is a sample of exact inverse density, ๐๐ is a sample of ๐ ๐ = ๐ ๐น ๐ ๐(๐), where ๐ ๐ = 1, ๐ ๐ = ฯ๐ ๐๐โ1
Linear dynamical system, uniform data distribution ๐( าง๐ฅ๐๐ โ ํ, าง๐ฅ๐๐ + ํ)
๐J
Broydenโs method (rank-1 update)
๐ท๐๐น เท๐ โ ๐ท๐๐น ๐๐ +๐น เท๐ โ ๐น ๐๐ โ ๐ท๐๐น ๐๐ ฮ๐๐
ฮ๐๐ ๐ฮ๐๐ฮ๐๐
๐
โข ๐ท๐๐น เท๐ recomputed exactly every N steps (101, 11, 2)
โ
=
15
30,
10
10,
15
5d 272.1=U
โ
โ
=
2.2
4,
3.2
5,
2
10d 134.0=U
โ
โ=
3694.04213.1
2187.01347.1
๐J
๐U
๐R
Reciprocal works well
Uniform works well
โโ
โโ=
21
5.11
Jacobian alternatives sometimes work well
[SSZR19]
xx =
xx =
๐R
๐J
๐U
๐R
IHVI
HVH
cVrIV
โ=
โ=
โ=
)0.4,0.3,012.0,107.2,104,093.0(),,,,,( 58
00
โ== crHVa
1.0=
}3,2{=t
Nonlinear influenza dynamics, uniform data distribution ))1(,)1(( ijij xxU +โ
๐R
๐ด
๐J
๐U
๐ด is a sample of exact inverse density, ๐๐ is a sample of ๐ ๐ = ๐ ๐น ๐ ๐(๐), where ๐ ๐ = 1, ๐ ๐ = ฯ๐ ๐๐โ1
[SSZR19]
Narrow data distribution, all ๐(๐) work well
ExamplesCommon problem:
โฆ Can ๐ ๐ฆ be approximated by aggregate Gaussian density ๐ ๐ฆ ?
๐(๐) ๐(๐ฆ)
๐(๐) ๐(๐ฆ)
Linear dynamical system, Gaussian parameter distribution, aggregate Gaussian data distribution.
โโ
โโ=
21
5.11
๐2 = 0.25
๐2 = 0.17
๐2 = 0.07
๐R
๐ด
๐J
๐U
๐ด is the exact source density, ๐๐ is a sample of ๐ ๐ =
๐ ๐น ๐ ๐(๐)
xx =
[SSZR19]
Only Jacobian produces localized distribution
Nonlinear influenza dynamics, Gaussian parameter distribution, aggregate Gaussian data distribution
IHVI
HVH
cVrIV
โ=
โ=
โ=
)0.4,0.3,012.0,107.2,104,093.0(),,,,,( 58
00
โ== crHVa
}2,1{=t
๐R
๐ด
๐J
๐U
[SSZR19]
Domain deformation
๐1
๐2
๐ฅ1
๐ฅ2
Summaryโข Accurate parameter inference of ensemble models requires
determination of the domain of invertibility of the solution map and the computation of Jacobian
โข Further work is required to design efficient computational procedures for Jacobian calculations
Random measurement error model๐ = ๐น ๐ + ๐บ
Inverse problem: Find ๐ from the knowledge of data ๐ท = (๐ฆ1, โฆ , ๐ฆ๐), ๐น(๐), and the error density ๐พ(๐).
Likelihood
๐ฟ ๐ ๐ฆ = ๐ ๐ฆ ๐ = ๐พ ๐ = ๐พ ๐ฆ โ ๐น ๐
Bayesian posterior ๐ ๐ ๐ฆ โ ๐ฟ ๐ ๐ฆ ๐ ๐
โฆ Prior ๐(๐) contains all information about parameters known before data are taken into account
Distribution of parameters: represents uncertainty in parameter inference
๐น ๐
๐ฆ๐
PriorsUniform prior: ๐ ๐ = 1
Reciprocal prior: ๐ ๐ = ฯ๐ ๐๐โ1
Jeffreys invariant prior, based on the Fisher information matrix:
๐ ๐ = det ๐ผ(๐)
๐ผ ๐ ๐๐ = เถฑ๐
๐๐๐ln ๐ฟ ๐ ๐ฆ
๐
๐๐๐ln ๐ฟ ๐ ๐ฆ ๐ฟ(๐|๐ฆ)
Jacobian prior: For model with likelihood ๐ฟ ๐ ๐ฆ = ๐พ ๐ฆ โ ๐น ๐ ,
๐ ๐ = ๐ฝ(๐) det๐ ๐ฝ ๐ = | det๐ท๐๐น ๐ |
Relation between Bayesian posterior for REM and parameter density for RPM
Let ๐ ๐ฆ = ๐พ(๐ฆ โ เดค๐ฆ) be the aggregate density obtained from data sample ๐ =๐ฆ1, ๐ฆ2, โฆ ๐ฆ๐ and let ๐ ๐ = ๐ ๐น ๐ ๐ฝ(๐)
Let ๐ ๐ เดค๐ฆ be the Bayesian posterior with likelihood เทจ๐ฟ ๐|เดค๐ฆ = ๐พ(เดค๐ฆ โ ๐น(๐)) and prior ๐(๐)
Theorem: ๐ ๐ = ๐(๐|เดค๐ฆ) whenever ๐พ(๐ฆ) is symmetric and ๐ ๐ = ๐ฝ(๐)
[SSZR19]
๐R
๐J
๐U
๐๐ is a sample of Bayesian posterior ๐ ๐|เดค๐ฆ = ๐พ เดค๐ฆ โ ๐น ๐ ๐(๐), where ๐ ๐ = 1, ๐ ๐ = ฯ๐ ๐๐โ1
Jacobian prior revisited๐ = ๐น ๐ + ๐บ
Likelihood: ๐ฟ ๐|๐ฆ = ๐พ ๐ฆ โ ๐น ๐
Posterior: ๐ ๐|๐ฆ = ๐ฟ ๐|๐ฆ ๐(๐)/๐(๐ฆ)
With Jacobian prior on parameters, the prior on data ๐ฆ is uniform
๐ ๐ฆ = ๐ฟ ๐|๐ฆ ๐ ๐ ๐๐
= เถฑ๐พ ๐ฆ โ ๐น ๐ ๐ฝ ๐ ๐๐ = เถฑ๐พ ๐ฆ โ ๐ง ๐๐ง = 1
The Jacobian prior is noninformative in the original sense proposed by Bayes โ it does not give preference to any observed data [Stigler, 1982].
[SSZR19]
๐น(๐)
Nonlinear model
Commonly used interpretation of noninformative prior
๐
๐(๐ฆ)
Prior on data
๐
๐(๐)
Prior on parameters
โข All parameters are equally likelyโข Observations can be anticipated
๐
Nonlinear model
Bayesโ understanding of noninformative prior
๐น(๐)
๐ฝ(๐)
๐Prior on parameters
๐(๐ฆ)
Prior on data
โข Any data vector is equally likelyโข Observations cannot be
anticipated
Relation to least squaresObjective function minimization
๐โ = arg min๐
๐๐ฅ๐ โ ๐ฅ ๐ก๐; ๐
2
Maximum likelihood estimate
๐๐๐ฟ๐ธ = arg max๐
๐ฟ(๐|๐ฆ) = arg max๐
๐พ ๐ฆ โ ๐น ๐ = ๐โ
The data ๐ฆ is the most likely for model with the parameter ๐๐๐ฟ๐ธ
Maximum a posteriori estimate
๐๐๐ด๐ = arg max๐
๐(๐|๐ฆ) = arg max๐
๐พ ๐ฆ โ ๐น ๐ ๐ฝ(๐)
The model with parameter ๐๐๐ด๐ is the most likely given observed data ๐ฆ
Applications to immunologyData on time courses of 16 volunteers infected by Influenza A/Texas/91 (H1N1) [Hayden , JAMA, 96]
6),16.7,0.80101.33 ,0.0347, 0.0373(),,,,( -50 =optcpV
Classical parameter estimates
แถ๐ = ๐๐ผ โ ๐๐แถ๐ป = โ๐ฝ๐ป๐แถ๐ผ = ๐ฝ๐ป๐ โ ๐ฟ๐ผ
Ensemble trajectory prediction
Ensemble parameter distribution
,3.2,3.2)102.7 ,0.014, 0.25(),,,,( -50 =basecpV
Parameter correlations โ parameter distribution is organized in clusters
Optimal trajectories reflect bimodality of the posterior
IHVI
HVH
cVpIV
โ=
โ=
โ=
Slow virus decayFast infected cell removal
Fast virus decaySlow infected cell removal
Treatment starting at 26 hrs
Treatment starting at 50 hrs
Data from [Hayden , JAMA, 96]
The effect of neuraminidase inhibitor GG167 is simulated by decreasing p to 1/20th of its value
Uncertainty quantification of antiviral treatment
Lower attrition of target cells
Viral phenotype characterization
PB1-F2 protein occurring in 1918 pandemic H1N1 strain increases apoptosis in monocytes, alters viral polymerase activity in vitro, enhances inflammation and increases secondary pneumonia in vivo.
Mice infected intranasally with 100 TCID50 of influenza A virus PR8 (squares) or PR8-PB1-F2(1918) (triangles).
[Smith et al., PloS ONE, 2011]
Comparison of model predictions and parameter distributions obtained using ensemble modeling.
PR8-PB1-F2PR8
Bacterial pneumonia
21
( )nl LL Ll L B L
L L
NPdP Pk P
dt P Pf bD a P bDP
= โ โ + + โ
+ +
2
(1 )1
nb BB B Bb B B L
B B
NPdP P Pk P a P bDP
dt P Pb
KD
โ = โ โ โ + +
+ +
(1 )L
dD
dtqP D cD= โ โ
1L
h L N
hPdN
t Pn
N
d
= โ +
+
Data from MF1 mice subjects infected by D39 pneumococcus strain [Kadioglu et al., Inf. and Immun., 2000]:
bacteria in lungs bacteria in blood
Lung bacteria
PL
Blood bacteria
PB
Phagocytic cellsN
DamageD
[MSELC14]
Bacterial pneumonia - strain dependenceResponse to D39 pneumococcus infection in four mice strains[Data from Kadioglu at al, I&I, 2000; Gingles et al., I&I, 2001; Kadioglu et al., JID, 2011]
[MSELC14]
severity
Bacterial pneumonia - strain dependenceResponse to D39 pneumococcus infection in four mice strains[Data from Kadioglu at al, I&I, 2000; Gingles et al., I&I, 2001; Kadioglu et al., JID, 2011]
[MSELC14]
Strain dependent parametersโ, ๐, ๐๐๐ , ๐๐๐
severity
Referencesโข [MSELC14] Mochan, E., Swigon, D., Ermentrout, G.B., Lukens, S., & Clermont, G., A mathematical
model of intrahost pneumococcal pneumonia infection dynamics in murine strains, J. Theor. Biol, 353, 44-54 (2014).
โข [SRS14] Stanhope, S., Rubin, J.E., & Swigon, D., Identifiability of linear dynamical systems from a single trajectory, SIADS, 13, 1792-1815 (2014).
โข [SRS17] Stanhope, S., Rubin, J. E., & Swigon, D., Robustness of solutions of the inverse problem for linear dynamical systems with uncertain data, SIAM/ASA JUQ, 5, 572โ597 (2017).
โข [S17] Swigon, D. Parameter estimation for nonsymmetric matrix Riccati differential equations, Proceedings of ENOC 2017, (2017).
โข [SSZR19] Swigon, D., Stanhope, S., Zenker, S., & Rubin, J. E., On the importance of the Jacobian determinant in parameter inference for random parameter and random measurement error models, to appear in SIAM JUQ.
โข [DRS19] Duan, X., Rubin, J.E., Swigon, D., Identification of Affine Dynamical Systems from Single Experiment, in preparation.
โข [S19] Swigon, D., Solution of inverse problem for linear-in-parameters dynamical systems with equally spaced data, in preparation.
CreditsInverse problems
โฆ Jon Rubin
โฆ Shelby Stanhope
โฆ Xiaoyu Duan
Influenza/pneumonia modelingโฆ Gilles Clermont
โฆ Bard Ermentrout
โฆ Ericka Mochan
โฆ Sarah Lukens
Fundingโฆ NIGMS
โฆ NSF-RTG
Appendix
Uniform prior
Reciprocal prior
Jacobian prior
,3.2,3.2)102.7 ,0.014, 0.25(),,,,( -50 =basecpV
Posterior distributions
Uniform prior
Reciprocal prior
Jacobian prior
Probabilistic trajectory predictions
VV
V
HH
H
II
I