Statistical Methods for Bridging Experimental Data and ... · Statistical Methods for Bridging...

Statistical Methods for Bridging ExperimentalData and Dynamic Models with Biomedical

Applications

Hulin Wu, Ph.D.Dr. D.R. Seth Family Professor & Associate Chair

Department of Biostatistics, School of Public Health

Professor, School of Biomedical InformaticsUniversity of Texas Health Science Center at Houston

Pittsburgh, March, 2017

Hulin Wu UTSPH March 2017 1 / 52

Outline

1 Introduction

2 Statistical estimation and inference methods for dynamic ODEmodels

I Naive Method: LS or MLE principleI Local solution and time-varying parameter problemsI Smoothing-based methodsI Sparse longitudinal data: mixed-effects ODE modelsI Bayesian methodsI High-dimensional ODE models: ODE model selection

3 Other dynamic models

4 Ongoing and future Work

5 Conclusions


Statistical Modeling Cultures

I Leo Breiman (Statistical Science, 2001): Two culturesI Data modeling (98% statisticians): What the data look like? e.g.,

regression modelsI Algorithmic modeling (2% statisticians): No models and for

prediction purpose, e.g., neural nets and decision treesI A third culture:

I Mechanistic modeling (<1% statisticians): Build mathematicalmodels based on the mechanisms behind the data

I How are the data generated?I Goal: Understand physics principles or biological mechanisms


Dynamic Systems/Models

Many engineering and biological systems can be described bydynamic models:

I Differential equations:I Ordinary differential equations (ODE)–simplestI Delay differential equations (DDE)I Hybrid differential equations (HDE)I Partial differential equations (PDE)I Stochastic differential equations (SDE)

I Difference equations and state-space modelsI Stochastic processes models: branching process etc.I Agent-based models and cellular automataI . . .


Modeling Goals

I Forward Problems: θ 7→ Pθ–Easier to doI PredictionsI Simulations

I Inverse Problems: Y 7→ θ ∈ Θ–More challengingI Determine model structures/formsI Estimate unknown parameters: θ


A Dynamic System: ODE Model

d

dtX(t) = G[X(t),θ], X(0) = X0 (1)

Y (ti) = H[X(ti),β] + e(ti), (2)e(ti) ∼ (0, σ2I), i = 1, . . . , n

whereI G(·): linear or nonlinear functionsI H(·): observation functionsI (θ,β): unknown parametersI e(ti): measurement error

The NLS method:

minθ,β,X0

n∑i=1

Y (ti)−H[X(ti,θ),β]T Y (ti)−H[X(ti,θ),β],

where X(ti) evaluated numerically from Eq (1).Hulin Wu UTSPH March 2017 6 / 52

Naive NLS Method: Challenging Problems

1 Identifiability problem

2 Local solutions

3 Time-varying parameters

4 Need to solve the forward problem numerically and many times:Numerical error vs. measurement error

5 Slow convergence and high computational cost

6 Sparse longitudinal data problem

7 Nonlinear optimization

8 High-dimensional parameter space

Motivate new statistical methods for dynamic models


Identifiability issues

I Theoretical identifiability: Mathematical identifiability

I Practical identifiability: Statistical and numerical identifiability

I Need to be investigated before the inverse problem

I How to deal with unidentifiable models?I Simplify or revise the modelI Lump some parameters togetherI Fixed some parametersI Bayesian approach: Use priors


Identifiability issues: References

I Wu, H., Zhu, H., Miao, H., and Perelson, A.S. (2008), ParameterIdentifiability and Estimation of HIV/AIDS Dynamic Models, Bulletin ofMathematical Biology, 70(3), 785-799.

I Miao, H., Dykes, C., Demeter, L.M., Cavenaugh, J., Park, S.Y., Perelson,A.S., and Wu, H. (2008), Modeling and Estimation of Kinetic Parametersand Replicative Fitness of HIV-1 from Flow-Cytometry-Based GrowthCompetition Experiments, Bulletin of Mathematical Biology, 70,1749-1771.

I Miao, H., Dykes, C., Demeter, L., Wu, H. (2009), Differential EquationModeling of HIV Viral Fitness Experiments: Model Identification, ModelSelection, and Multi-Model Inference, Biometrics, 65, 292-300.

I Liang, H., Miao, H., and Wu, H. (2010), Estimation of constant andtime-varying dynamic parameters of HIV infection in a nonlineardifferential equation model, Annals of Applied Statistics, 4, 460-483.

I Miao, H., Xia, X., Perelson, A.S., Wu, H. (2011), On Identifiability ofNonlinear ODE Models and Applications in Viral Dynamics, SIAMReview, 53(1): 3-39.


Naive NLS Method: Local solution and numerical errorproblems

I Local solution problem:I Global optimization methods: Differential evolution algorithms and

genetic algorithms (Storn et al 1997).I Mixture of stochastic global optimization method and deterministic

methods: scatter search method (Rodriguez-Fernandez et al.2006)

I Numerical error problem:I Xue, Miao and Wu (Annals of Statistics, 2010): theoretical results

on numerical error vs. measurement error


Naive NLS Method: Time-varying parameter problem

Xue, Miao and Wu, Annals of Statistics (2010)

dX(t)

dt= Ft,X(t),θ, η(t)

I The spline approach can be used to approximate thetime-varying parameter:

η(t) = π(t)Tα,

where π(t) = (B1(t), · · · , BN (t))T is a vector of basis functions.I The time-varying coefficient ODE model becomes an ODE

model with constant parameters:

dX(t)

dt= Ft,X(t),θ, π(t)Tα


Smoothing-Based Approaches: ODE ComputationalProblem

I Earlier ideas: Hemker (1972) and Varah (1982)

I Two-stage decoupling approaches: Chen and Wu (JASA 2008,Statistica Sinica 2008) and Liang and Wu (JASA, 2008)

I Parameter cascading method: Ramsay et al. JRSS-B (2007) andWang et al. Stat Comput 2014.


Smoothing-Based Approaches: Two-Stage Method

Chen and Wu (JASA 2008, Statistica Sinica 2008) and Liang and Wu(JASA, 2008):

X ′(ti) = F [X(ti),θ] (3)Y (ti) = X(ti) + e1(ti), e1(ti) ∼ (0, σ2I), (4)

I Step 1: Use a nonparametric smoothing to estimate X(t) andX ′(t) from model (4).

I Step 2: Substitute the estimate X(ti) into model (3) to obtain:

X ′(ti) = F [X(ti),θ] + e2(ti). (5)

Then fit the above regression model (5) to estimate θ.I F (·): Linear or nonlinear function


Smoothing-Based Approaches: Two-Step Methods

I Step 2 decoupled the system of ODEs: Fit the ODE one-by-one

I Convert ODE models to regression: Standard regressionsoftware tools can be used

I Avoid numerically solving the ODEs

I Computationally fast and efficient: Easy to deal withhigh-dimensional ODEs

I Price to pay:I The derivative estimate may not be accurateI The decoupled system: Some information lostI The “coupled" property: destroyed

Extension to higher-order numerical discretization-based algorithms:Wu, Xue and Kuman (Biometrics 2012)


Parameter Cascading or Profiling Method

Ramsay, Hooker, Campbell, Cao, JRSS-B, 2007

Fitting to dataI Observations: y(ti)

I Nonparametric function: f(t) = φ(t)′c

I Fitting to data: C1 =∑ni=1[y(ti)− f(ti)]

2

Fidelity to DE x′(t) = g(x|β)

I f ′(t) = φ′(t)c

I Difference between two sides of DE: Lf(t) = f ′(t)− g(f(t)|β)

I Fidelity to DE: C2 =∫

[Lf(t)]2dt

Criterion to estimate c: J(c|β) = C1 + λC2

Criterion to estimate β: H(β) =∑ni=1[y(ti)− φ(ti)

′c(β)]2


Numerical Comparisons: NLS, Profiling andTwo-Stage Estimates

Ding and Wu, Statistica Sinica, 2014I NLS: Not stable to get the global solution, computationally

expensive

I Profiling:I A 3-step iterative algorithmI More stable than NLS to get a better solutionI Computational efficiency: similar to NLS

I Two-Stage Method: Computationally fast, but not accurate.


Sparse Longitudinal Data Problem: Mixed-EffectsModeling Approaches

Deal with sparse data: Borrow information across subjectsI The MLE principle: Nonlinear Mixed-Effects Modeling (NLME)

I Treat the ODE solution as a nonlinear regression functionI Computational challenge: Stochastic Approximation EM (SAEM)

I Two-stage smoothing-based mixed-effects modeling approachesI Fang, Wu and Zhu, Statistica Sinica (2011)I Linear ODE: Linear mixed-effects model (LME)I Nonlinear ODE: NLME

I Bayes methodsI A three-stage hierarchical model: implemented by MCMCI Computation: expensive


Mixed-Effects ODE Model: NLME

I Within-subject variation:d

dtX(t) = G[X(t),θi], X(0) = Xi0 (6)

Y i(ti) = Hi[Xi(ti),θi] + ei(ti), i = 1, . . . , n

I Xi(ti): ODE solution for Subject i.I Y i = (yi1(t1), · · · , yimi(tmi))

T : Data from Subject iI ei = (ei(t1), · · · , ei(tmi))

T ∼ N (0, σ2Imi ): Measurement error

I Between-subject variation:

θi = µ+ bi, [bi|Σ] ∼ N (0,Σ)

I µ: population parameterI bi: random effects

I Estimation and inference: Stochastic Approximation EM (SAEM)I Delyon, Lavielle and Moulines (1999), Kuhn and Lavielle (2005)

Grenier, Louvet, Vigneaux (2014)


Smoothing-based Two-Stage Mixed-Effects Model

Fang, Wu and Zhu, Statistica Sinica (2011):

X ′(ti) = F [X(ti),θ] (7)Y (ti) = X(ti) + e1(ti), e1(ti) ∼ (0, σ2I), (8)

I Step 1: Use a nonparametric smoothing to estimate X(t) andX ′(t) from model (8).

I Step 2: Substitute the estimate X(ti) into model (7) to obtain:

X ′(ti) = F [X(ti),θ] + e2(ti). (9)

I Convert the model (9) into a LME or NLME if F (x) is linear ornonlinear.

I Fit the LME or NLME using a standard approach or SAEMmethod


Bayesian Methods: Borrow Information to Deal withSparse Data and Identifiability Problems

Huang, Liu and Wu, Biometrics (2006): Example

I A viral dynamic model: describe the population dynamics of HIVand its target cells in plasma

ddtT = λ− ρT − [1− γ(t)]kTVddtT∗ = [1− γ(t)]kTV − δT ∗

ddtV = NδT ∗ − cV

(10)

I T, T ∗, V : target uninfected cells, infected cells, virusI γ(t): time-varying antiviral drug efficacyI (λ, ρ, k, δ,N, c): unknown parameters to be estimatedI The equations (10): no closed-form solution


Antiviral Drug Efficacy Model

I A modified Emax (M-M) model for drug efficacy:

γ(t) =C(t)A(t)

φIC50(t) + C(t)A(t)=

IQ(t)A(t)

φ+ IQ(t)A(t), 0 ≤ γ(t) ≤ 1

(11)I C(t): the plasma drug concentrationI A(t): drug adherence measurementsI IC50: in vitro phenotype drug resistance markerI φ: a conversion factor parameterI IQ = C(t)

IC50(t): the Inhibitory Quotient (IQ)

I If γ(t) = 1, the drug: 100% effective

I If γ(t) = 0, the drug: no effect


Drug Susceptibility Model

I Phenotype marker IC50 is used to quantify agent-specific drugsensitivity

I The function: to describe changes overtime in IC50

IC50(t) =

I0 + Ir−I0

trt for 0 < t < tr,

Ir for t ≥ tr,(12)

I I0 and Ir: respective values of IC50(t) at baseline and time point trat which drug resistant mutations appear

I If Ir = I0, no resistance mutation developed during treatment


A Challenging Problem

How to estimate the unknown parameters in the complex dynamicmodel?

I Difficulties:I Identifiability problem: Too many parameters, (φ, λ, ρ, k, δ,N,C),

some of them are not identifiableI Data from individuals: sparse, only V (t) measuredI Nonlinear differential equations model: no closed-form solutions


Viral load data from a clinical trial

Real data up to day 112

Time (days)

log1

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NA

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Bayesian Modeling

I A three-stage Bayesian hierarchical modelI Stage 1. Within-subject variation:

yi = fi(θi) + ei, [ei|σ2,θi] ∼ N (0, σ2Imi )

I fi(θi) = (fi1(θi, t1), · · · , fimi(θi, tmi))T : ODE solutions.

I yi = (yi1(t1), · · · , yimi(tmi))T : Data from Subject i

I ei = (ei(t1), · · · , ei(tmi))T : Measurement error

I Stage 2. Between-subject variation:

θi = µ+ bi, [bi|Σ] ∼ N (0,Σ)

I Stage 3. Hyperprior distributions:

σ−2 ∼ Ga(a, b), µ ∼ N (η,Λ), Σ−1 ∼Wi(Ω, ν)

I Gamma (Ga), Normal (N ) and Wishart (Wi): independent distributionsI Hyper-parameters a, b,η,Λ,Ω and ν: known


Bayesian Estimation: Implementation

I Choose prior distributionsI Informative prior and non-informative priorI Rule of thumb: choose non-informative prior distributions for

parameters of interest

I Implement MCMC algorithmI Gibbs sampling step: closed form of conditional distributions forσ−2,µ, Σ−1

I Metropolis-Hastings step: no closed form of conditionaldistributions for θi

I Run a long chain: the number of iterations, initial “burn-in", everyfifth simulation samples

I Obtain posterior distributions (posterior means or credibleintervals) based on the final MCMC samples


A Clinical Study: A5055

I A study of HIV-1 infected patients failing PI-containing therapies.

I Two salvage regimens: 44 patientsI Arm A: IDV 800 mg q12h+RTV 200mg q12h+two NRTIsI Arm B: IDV 400 mg q12h+RTV 400mg q12h+two NRTIs

I Plasma HIV-1 RNA (viral load) measured at days 0, 7, 14, 28, 56,84, 112, 140 and 168 of follow-up


Clinical Data–Results of Population Parameters

Parameter PM SD 95% CIφ 2.1091 0.6354 (1.2143, 3.6392)c 2.9867 0.1466 (2.7139, 3.2881)δ 0.3729 0.0184 (0.3387, 0.4105)λ 100.645 4.9431 (91.497, 110.830)ρ 0.0997 0.0049 (0.0905, 0.1099)N 1004.988 49.795 (912.074, 1106.654)k 9.183× 10−6 0.290× 10−6 (8.632× 10−6, 9.774× 10−6)

I Posterior mean for the population parameter φ is 2.1091 with aSD of 0.6354 and the 95% CI of (1.2143, 3.6392)

I As φ plays a role of transforming the in vitro IC50 into in vivoIC50, our estimate shows that there is about 2-fold differencebetween in vitro IC50 and in vivo IC50


Clinical Data–Results of Individual Parameters

Patient φi ci δi λi ρi Ni ki e

1 0.447 2.254 0.270 410.462 0.024 456.757 8.33× 10−6 0.972 5.371 2.969 1.183 29.619 0.426 4795.813 10.84× 10−6 0.173 3.723 2.283 0.456 36.877 0.289 3258.347 8.66× 10−6 0.374 4.960 2.761 0.798 44.956 0.313 3051.988 9.09× 10−6 0.345 7.066 2.306 0.663 71.295 0.201 2735.239 6.54× 10−6 0.646 0.786 4.633 0.183 375.882 0.025 247.416 11.18× 10−6 0.897 0.091 7.008 0.299 4015.398 0.003 30.559 18.54× 10−6 0.988 8.484 2.280 0.663 32.722 0.416 4530.531 8.37× 10−6 0.24

I The individual-specific parameter estimates suggest a large inter-subject variationI The model provides a good fit to the clinical data


Patient 1

Time (day)

IC50

0 50 100 150 200

48

1216

IDVRTV

Patid= 1

Time (day)

Adh

eren

ce

0 50 100 150 200

0.80

0.90

1.00

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Patid= 1

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ficac

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oo

o oo

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log1

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Patid= 1

Fitted individual curves, drug efficacy, IC50 and adherence with IQ=c12h/IC50


Patient 2

Time (day)

IC50

0 50 100 150 200

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812

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Patid= 2

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eren

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Patid= 2

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log1

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Patid= 2


Patient 3

Time (day)

IC50

0 50 100 150 200

4.5

6.0

7.5

IDVRTV

Patid= 3

Time (day)

Adh

eren

ce

0 50 100 150 200

0.97

00.

990 IDV

RTV

Patid= 3

Time (day)

Dru

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0 50 100 150 200

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Patid= 3


Bayesian Methods: Pros & Cons

I ProsI Use prior to solve the identifiability problemI Deal with extremely complicated models such as nonlinear

differential equation modelsI Borrow information across subjects:

I Deal with sparse longitudinal dataI Estimate parameters for both population and individuals

I Always get reasonable estimatesI Use posterior distributions: Easy to quantify “uncertainty" for

inference

I ConsI Computation: complex and expensiveI Prior: dominate the results


High-Dimensional ODEs

I Require computationally fast and efficient methods

I Need to incorporate variable selection approaches: LASSO,SCAD etc.

I Easy to deal with longitudinal data: Mixed-effects models

I Two-stage smoothing-based method: good for this purpose


Linear ODEs

Time course gene expression data: Dynamic gene regulatorynetwork (GRN) reconstruction (Lu, Liang, Li and Wu, JASA 2011)

dxidt

=

n∑j=1

θijxj , i = 1, · · · , n, (13)

I When n is small, standard statistical inference and variableselection methods can be used

I When n is large, curse-of-dimensionality


High-Dimensional Linear ODE: Identifying SignificantRegulations

Two-Stage Method (Chen and Wu 2008a, 2008b; Liang and Wu2008):

I Obtain mean expression curves and their derivatives Mk(t) andM ′k(t) from Step II.

I Substitute Mk(t) and M ′k(t) into the ODE model to form aregression model

High Dimensional Linear Regression Model

yk(t) =∑pj=1 βkjxj(t) + εk(t),

k = 1, · · · , p; t = t1, t2, . . . , tN

yk(t) = M ′k(t) and xj(t) = Mj(t)


High Dimensional Model Selection

I Two-stage methodI Decouple the high-dimensional ODEsI Convert the ODE model into a simple linear modelI Computationally fast

I Stepwise selection and subset selectionI Bridge selection (Frank and Friedman 1993)I Least absolute shrinkage and selection operator (LASSO)

(Tibshirani 1996)I Smoothly Clipped Absolute Deviation (SCAD)(Fan and Li 2001;

Kim, Choi and Oh 2008)


Estimation Refinement: Stochastic Approximation EM(SAEM) Algorithm

Mixed-Effects ODE Model for Module k

dxkidt

=

mk∑j=1

βkijM[kj](t), i = 1, · · · , nk; k = 1, . . . , p, (14)

Longitudinal Measurement Model

yki(t) = xki(t) + εki(t) (15)

Random Effects Model

βki = βk + bki (16)

bki ∼ N (0,Dk)


Application: Identification of Dynamic GRN for YeastCell Cycle

DNA microarrays experiment: 18 equally spaced time points duringtwo cell cycles (Spellman 1998)

I Step I: 800 significant genes identified

I Step II: Cluster 800 genes into 41 functional modules

I Step III: Smoothing

I Step IV: Linear ODE model identification: SCAD variableselection

I Step V: Estimation Refinement

I Step VI: Function Enrichment Analysis


Yeast Cell Cycle Gene Expression Profile

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High-Dimensional Nonlinear/Nonparametric ODEs

I Generalized ODEs: Miao, Wu and Xue, Journal of the AmericanStatistical Association (2014)

I Sparse additive ODEs: Wu, Lu, Xue and Liang, Journal of theAmerican Statistical Association (2014)

I Additive nonlinear ODEs: Xue, Wu, Wu and Wu, a manuscript(2017)


Other Dynamic Models: State-Space Models (SSM)

Linear SSM:

Xt+1 = FtXt + Vt, Vt ∼ (0, Qt) (17)Yt = GtXt +Wt, Wt ∼ (0, Rt) (18)

whereI Vt and Wt: independent model noise and measurement noiseI Standard Kalman filter (Kalman, 1960): the core algorithm for

prediction and smoothing of state state vectors


Statistical Methods for State-Space Models

I Zhu and Wu, JCGS (2007)I Liu, Lu, Niu and Wu, Biometrics (2011)I Liu, Wu, Zhu, Miao, BMC Bioinformatics (2014)I Chen et al. PlusOne (2017), submitted


Extension to SDE and PDE: Possible but Challenging

I Theoretically difficult

I Computationally challenging

I Applications: Not common


Ongoing and Future Research

I High-dimensional ODEs: How to improve accuracy withoutsacrificing too much on computing?

I Extra-high dimensional ODE: 1000 ODEs with 1 million parameters(Wu, Qiu, Yuan and Wu, 2017, submitted).

I Characteristic analyses of large ODE systems: Controllabilityand stability analysis with uncertainty in parameter estimation

I Sun, Hu, Wu, Qiu, Linel, Wu, Infectious Disease Modelling 2016

I AI-driven ODE Model Builder


Conclusions

Dynamic Models:I Practically useful for both understanding associations and

predictions

I Both theoretically and computationally challenging

I Statistical methods for dynamic models: More work needed


Dr. Hulin Wu’s Publications on ODE Models by Topics

ODE identifiability 1. Wu, H.*, Zhu, H.+, Miao, H.+, and Perelson, A.S. (2008), Parameter Identifiability and

Estimation of HIV/AIDS Dynamic Models, Bulletin of Mathematical Biology, 70(3), 785-799.

2. Miao, H.+, Dykes, C., Demeter, L., Wu, H.* (2009), Differential Equation Modeling of HIV Viral Fitness Experiments: Model Identification, Model Selection, and Multi-Model Inference, Biometrics, 65, 292-300.

3. Liang, H., Miao, H., and Wu, H.* (2010), Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model, Annals of Applied Statistics, 4, 460-483.

4. Miao, H., Xia, X., Perelson, A.S., Wu, H.* (2011), On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics, SIAM Review, 53(1): 3-39.

5. Lee, Y.+ and Wu, H.* (2012), MARS Approach for Global Sensitivity Analysis of Differential Equation Models with Applications to Dynamics of Influenza Infection, Bulletin of Mathematical Biology, 74, 73-90.

6. Wu, H.*, Miao, H., Xue, H., Topham, D.J., Zand, M. (2015), Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters, Statistics in Biosciences, 7(1):147-166.

NLS Estimation of ODE parameters

1. Wu, H.*, Huang, Y.+, Dykes, C., Liu, D., Ma, J., Perelson, A.S., Demeter, L. (2006), Modeling and Estimation of Replication Fitness of HIV-1 in Vitro Experiments Using a Growth Competition Assay, Journal of Virology, 80, 2380-2389.

2. Xue, H.+, Miao, H., Wu, H.* (2010), Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error, Annals of Statistics, 38(4), 2351-87.

Two-stage methods for ODE models

1. Liang, H., Wu, H.*, (2008), Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models, Journal of the American Statistical Association, 103, 1570-1583.

2. Chen, J.+ and Wu, H.* (2008), Efficient Local Estimation for Time-varying Coefficients in Deterministic Dynamic Models with Applications to HIV-1 Dynamics, Journal of the American Statistical Association, 103, 369-384.

3. Fang, Y.+, Wu, H.*, Zhu, L. (2011), A Two-Stage Estimation Method for Random Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data, Statistica Sinica, 21, 1145-1170.

4. Lu, T.+, Liang, H., Li, H., Wu, H.* (2011), High Dimensional ODEs Coupled with Mixed-Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification, Journal of the American Statistical Association, 106, 1242-1258.

5. Wu, H.*, Xue, H., Kumar A.+ (2012), Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models via Penalized Spline Smoothing with Applications in Biomedical Research, Biometrics, 68(2), 344-353.

6. Ding, A.A. and Wu, H.* (2014), Estimation of ODE Parameters Using Constrained Local Polynomial Regression, Statistica Sinica, 24, 1613-1631.

7. Wu, H.*, Miao, H., Xue, H., Topham, D.J., Zand, M. (2015), Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters, Statistics in Biosciences, 7(1):147-166.

Time-varying parameter estimation in ODE Models

1. Chen, J.+ and Wu, H.* (2008), Efficient Local Estimation for Time-varying Coefficients in Deterministic Dynamic Models with Applications to HIV-1 Dynamics, Journal of the American Statistical Association, 103, 369-384.

2. Chen, J.+ and Wu, H.* (2008), Estimation of time-varying parameters in deterministic dynamic models, Statistica Sinica, 18, 987-1006.

3. Liang, H., Miao, H., and Wu, H.* (2010), Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model, Annals of Applied Statistics, 4, 460-483.

4. Xue, H.+, Miao, H., Wu, H.* (2010), Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error, Annals of Statistics, 38(4), 2351-87.

5. Cao, J., Huang, J.Z., Wu, H. (2012), Penalized Nonlinear Least Squares Estimation of Time-Varying Parameters in Ordinary Differential Equations, Journal of Computational and Graphical Statistics (JCGS), 21(1), 42-56.

Bayesian and mixed-effects ODE modeling approaches for longitudinal data 1. Wu, H.*, Ding, A. and DeGruttola. V. (1998), “Estimation of HIV Dynamic Parameters,"

Statistics in Medicine, 17, 2463-2485. 2. Wu, H.* and Ding, A. (1999), “Population HIV-1 Dynamics in Vivo: Applicable Models and

Inferential Tools for Virological Data from AIDS Clinical Trials," Biometrics, 55, 410-418. 3. Huang, Y.+ and Wu, H.* (2006), A Bayesian Approach for Estimating Antiviral Efficacy in

HIV Dynamic Models, Journal of Applied Statistics, 33, 155-174. 4. Huang, Y., Liu, D.+ and Wu, H.* (2006), Hierarchical Bayesian Methods for Estimation of

Parameters in a Longitudinal HIV Dynamic System, Biometrics, 62, 413-423. 5. Fang, Y.+, Wu, H.*, Zhu, L. (2011), A Two-Stage Estimation Method for Random

Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data, Statistica Sinica, 21, 1145-1170.

High-dimensional ODE models and model selections

1. Lu, T.+, Liang, H., Li, H., Wu, H.* (2011), High Dimensional ODEs Coupled with Mixed-Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification, Journal of the American Statistical Association, 106, 1242-1258.

2. Wu, H.*, Lu, T.+, Xue, H., and Liang, H. (2014), Sparse Additive ODEs for Dynamic Gene Regulatory Network Modeling, Journal of the American Statistical Association, 109:506, 700-716.

Nonlinear/nonparametric ODE models

1. Wu, H.*, Lu, T.+, Xue, H., and Liang, H. (2014), Sparse Additive ODEs for Dynamic Gene Regulatory Network Modeling, Journal of the American Statistical Association, 109:506, 700-716.

2. Miao, H., Wu, H., and Xue, H. (2014), Generalized Ordinary Differential Equation Models, Journal of the American Statistical Association, 109:508, 1672-1682.

Statistical methods for state-space models

1. Zhu, H.+ and Wu, H.* (2007), Estimation of Smoothing Time-Varying Parameters in State Space Models, Journal of Computational and Graphical Statistics (JCGS), 16(4), 813-832.

2. Liu, D.+, Lu, T.+, Niu, X.F., and Wu, H.* (2011), Mixed-Effects State Space Models for Analysis of Longitudinal Dynamic Systems, Biometrics, 67, 476-485.

ODE experimental design

1. Wu, H.* and Ding, A.A. (2002), “Design of Viral Dynamic Studies for Efficiently Assessing Anti-HIV Therapies in AIDS Clinical Trials," Biometrical Journal, 2, 175-196.

2. Huang, Y. and Wu, H.* (2008), Bayesian Experimental Design for Long-Term Longitudinal HIV Dynamic Studies, Journal of Statistical Planning and Inference, 138, 105-113.

3. Miao, H., Xia, X., Perelson, A.S., Wu, H.* (2011), On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics, SIAM Review, 53(1): 3-39.

Dynamic model property analysis with uncertainty

1. Sun, X.+, Hu, F.+, Wu, S., Qiu, X., Linel, P.+, Wu, H.* (2016), Controllability and Stability Analysis of Large Transcriptomic Dynamic Systems for Host Response to Influenza Infection in Human, Infectious Disease Modelling, 1(1), 52-70.

Our recent work in nonlinear/high-dimensional ODEmodels

I Lu, T., Liang, H., Li, H., Wu, H. (2011), High Dimensional ODEs Coupled withMixed-Effects Modeling Techniques for Dynamic Gene Regulatory NetworkIdentification, JASA, 106, 1242-1258.

I Wu, H., Xue, H., Kumar A. (2012), Numerical Discretization-Based EstimationMethods for Ordinary Differential Equation Models via Penalized SplineSmoothing with Applications in Biomedical Research, Biometrics, 68(2), 344-353.

I Miao, H., Wu, H., and Xue, H. (2014), Generalized Ordinary Differential EquationModels, JASA, 109:508, 1672-1682

I Wu, H., Lu, T., Xue, H., and Liang, H. (2014), Sparse Additive ODEs for DynamicGene Regulatory Network Modeling, JASA, 109:506, 700-716.

I Wu, S., Liu, Z.P., Qiu, X., and Wu, H. (2014), Modeling genome-wide dynamicregulatory network in mouse lungs with influenza infection usinghigh-dimensional ordinary differential equations, PLOS ONE, 9(5):e95276.

I Linel, P., Wu, S., Deng, N., Wu, H. (2014), Dynamic transcriptional signatures andnetwork responses for clinical symptoms in influenza-infected human subjectsusing systems biology approaches, Journal of PK/PD, 41, 509-521.

I Qiu, X. et al. (2015), Diversity in Compartmental Dynamics of Gene RegulatoryNetworks: The Immune Response in Primary Influenza A Infection in Mice, PLoSONE, 10(9).


Acknowledgement

I More than 30postdocs,students andcollaborators


Thank You!


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