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Application of Quasi-Newton Algorithms in Optimal Design . Sebastian Ueckert, Joakim Nyberg, Andrew C. Hooker. Pharmacometrics Research Group Department of Pharmaceutical Biosciences Uppsala University Sweden. Outline. Optimizing Designs Introduction: Quasi-Newton Methods (QNMs) - PowerPoint PPT Presentation
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Sebastian Ueckert, Joakim Nyberg, Andrew C. Hooker
Application of Quasi-Newton Algorithms in Optimal Design
Pharmacometrics Research GroupDepartment of Pharmaceutical BiosciencesUppsala UniversitySweden
Outline1. Optimizing Designs2. Introduction: Quasi-Newton Methods (QNMs)3. Performance QNMs4. Advantages QNMs5. Laplace Approximation for Global Optimal Design6. Using QNMs in Laplace Approximation
2
Optimizing a Design
( ) ( , )j x f F x
* arg max ( )x X
x j x
Model
e.g. D-Optimal Design ( ) ,j x F x
Parameters α Design variables x
3
DataEstimation
Optimization• Interval methods
True global optimizersHard to implementStill under development
• Stochastic methods – Simulated Annealing (SA), Ant colony optimization, Genetic Algorithm(GA) Easy to implement (SA)
Marketing effective (GA)SlowNo information about solutionHeuristic
4
Optimization• Derivative free methods
– Downhill Simplex MethodNo derivatives necessaryRobustSlowLocal
5
Gradient Based Methods
6
Gradient Based Methods
7
Gradient Based Methods
8
Gradient Based Methods
9
Gradient Based Methods
10
Gradient Based Methods
11
Gradient Based MethodsMathematically well understoodFast (if OFV calc not too expensive) Only localComplicated to implement
• Steepest Descent• Conjugate Gradient
12
1. Set xk=x0
2. Determine search direction
3. Do line search along p* to find minimal xk+1
4. Set xk=xk+1 and go to 2
Newton Method
212( ) : ( )T T
k k k k kf x p f p f p f p m p 2( ) 0k k km p f f p
* arg max ( )x R
x f x
* 2 1k kp f f
Goal:
Algorithm:
13
1. Set xk=x0
2. Determine search direction
3. Do line search along p* to find minimal xk+1
4. Set xk=xk+1 and go to 2
Newton Method
212( ) : ( )T T
k k k k kf x p f p f p f p m p 2( ) 0k k km p f f p
* arg max ( )x R
x f x
* 2 1k kp f f
Goal:
Algorithm:
Calculate Hessian
14
1. Set xk=x0, Bk=I2. Determine search direction
3. Do line search along p* to find minimal xk+1
4. Set xk=xk+1, Bk=Bk+Uk and go to 2
Quasi-Newton Methods
12( ) : ( )T T
k k k k k kf x p f p f p B f p m p
( ) 0k k km p f B p * 1
k kp B f
Algorithm:
Calculation of Hessian is computationally expensiveProblem:
Approach: Use approx. Hessian and build up during search
15
Quasi-Newton Methods• Different methods for different updating formulas
– Davidon–Fletcher–Powell (DFP)
– Broyden-Fletcher-Goldfarb-Shanno (BFGS) 1
TTk k k kk k
k k T Tk k k k k
B x B xy yB By x x B x
1
T T Tk k k k k k
k kT T T T Tk k k k k k
y x y x y yB I B I
y x y x y x
1k k ky f f 1k k ks x x
16
Constraints• Experiments usually come with practicality constraints e.g.:
– Administered dose has to be smaller than X mg– Sampling times can only be taken until 8 h after dosing
i i iu x b Box Constraints
BFGS-B 17
BFGS-B
1. Set xk=x0, Bk=I2. Determine search direction
3. Project search direction vector on feasible region4. Do line search along p* to find minimal xk+1 respecting bounds5. Set xk=xk+1, Bk=Bk+Uk and go to 2
Algorithm:
12( ) : ( )T T
k k k k k kf x p f p f p B f p m p
( ) 0k k km p f B p * 1
k kp B f
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Comparison• Test Scenario
– Model:• PKPD (1 cmp oral absorption; IMAX drug effect)• All parameters (ka,CL,V,IC50, E0, IMAX) with log-normal IIV 30% CV• PK parameters fixed• Combined error
– Design:• 3 groups (40,30,30 subjects)• 1 PK and 1 PD sample per subject
• Approach:– Generate random initial values – Optimize with steepest descent and BFGS
19
Results
BFGS Steepest Descent
010
2030
4050
60
15.03
60.84
Runti
me
[s]
Freq
uenc
y[%
]
Steepest Descent
BFGS 20
OFV0 2 4 6 8
x 1010
0
5
10
15
20
25
30
Design Sensitivity• Approximate Hessian matrix can be used to assess sensitivity
of design (at no additional computational costs)– Diagonal of the inverse of the Hessian– Use approximate efficiency
*
( )( ) j xEff xj x
* * * *12( ) T Tj x a j a j a B a
* * *12
*( )T Tj a j a B a
Eff aj
21
Design Sensitivity - Visual
7 7.5 8 8.5 9 9.50.9
0.92
0.94
0.96
0.98
1
1.02
6 6.5 7 7.5 8 8.50.9
0.92
0.94
0.96
0.98
1
1.02
1 1.5 2 2.5 30.9
0.92
0.94
0.96
0.98
1
1.02
Group 2 PD Group 1 PK Group 1 PD
22
Design Sensitivity - Numerical
PK Sample PD SampleGroup 1
7.12 [0.35;13.9]8.38[5.28;11.38]
Group 2
1.26 [0;3.74]1.79[1.03;2.55]
Group 3
9.22 [-1.31E;+1.31E] 0[0;0.0025]
0 20 40 60 80 10080
85
90
95
100
Group 2 PD
0 20 40 60 80 10080
85
90
95
100
0 20 40 60 80 10080
85
90
95
100
Group 1 PK Group 3 PK23
LAPLACE APPROXIMATION
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Global Optimal Design• Integral has to be evaluated• FIM occurs in integrand• For example ED optimal design:
• Usually evaluated with Monte-Carlo integrationComputationally intensive or imprecise
( ) ( ) ( , )EDj x p F x d
25
Laplace Approximation
,( ) ( ) ( , ) k xEDj x p F x d e d
, : log ( ) ( , )k x p F x
,
2
1
2 ,
mk x
me
k x
arg min ,m k x
26
Laplace Approximation
1. Minimize
2. Calculate the Hessian
3. Evaluate
Algorithm:
, : log ( ) ( , )k x p F x
2 ,mk x
,
2
1
2 ,
mk x
me
k x
B
27
Laplace-BFGS Approximation
1. Minimize using BFGS algorithm
2. Evaluate
Algorithm:
, : log ( ) ( , )k x p F x
,12
mk xe
B
28
Laplace-BFGS – Random Effects
arg min ,m k x
( )g e
Problem:
Approach:
For variance parameter α ≥ 0
Perform optimization on log-domain
1. Minimize using BFGS algorithm
2. Rescale approximate Hessian
3. Evaluate
, : log ( ( )) ( ( ), )k x p g F g x
( ),1
2
mk g xe
B
Algorithm:
1TB B g g
29
Comparison• Comparison of 4 algorithms:
1. Monte Carlo integration with random sampling (MC-RS)2. Monte Carlo integration with Latin hypercube sampling (MC-LHS)3. Laplace integral approximation (LAPLACE)4. Laplace integral approximation with BFGS Hessian (LAPLACE-BFGS)
• Testing MC methods with 50 and 500 random samples
30
Comparison• Test Scenario
– Model:• 1 cmp IV bolus• CL,V with log-normal IIV• Additive error
– Design:• 20 subjects• 2 samples per subject
– Parameter distribution:• Log-normal an all parameters (Fixed effect Var=0.05; Random Effect
Var=0.09)
31
Results - OFV
Method Mean OFV1021 [95% CI]MC-RS 100,000 3.24MC-RS 50 3.27[2.2-5.0]MC-RS 500 3.33[2.8-3.8]MC-LHS 50 3.24[2.2-4.6]MC-LHS 500 3.22[2.9-3.7]LAPLACE 2.95LAPLACE-BFGS 3.01
Mean OFV and non-parametric confidence intervals for different integration methods from 100 evaluations
32
Results - DesignMC-RS 50 MC-LHS 50 LAPLACE
LAPLACE-BFGSMC-RS 500 MC-LHS 500
33
Results – Runtimes
MC-LHS 50 MC-RS 50 LAPLACE-BFGS LAPLACE MC-LHS 500 MC-RS 500
01
23
4
0.35 0.37 0.460.63
3.533.67
Runt
ime
[s]
34
Conclusions• Quasi-Newton methods constitute fast alternative for
continuous design variable optimization • Information about design sensitivity can be obtained with no
additional cost• Global Optimal Design:
– Monte-Carlo methods are easy and flexible but need high number of samples to give stable results
– Laplace approximation constitutes fast alternative for priors with continuous probability distribution function
– Laplace integral approximation with BFGS Hessian gave same sampling times with approx. 30% shorter runtimes
35
THANK YOU!
36
References1) C.G. Broyden, “The Convergence of a Class of Double-rank Minimization Algorithms 1. General
Considerations,” IMA J Appl Math, vol. 6, Mar. 1970, pp. 76-90. 2) R. Fletcher, “A new approach to variable metric algorithms,” The Computer Journal, vol. 13,
1970, p. 317. 3) D. Goldfarb, “A family of variable-metric methods derived by variational means,” Mathematics
of Computation, 1970, pp. 23–26. 4) D.F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Mathematics
of Computation, 1970, pp. 647–656. 5) R.H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained
optimization,” SIAM J. Sci. Comput., vol. 16, 1995, pp. 1190-1208. 6) M. Dodds, A. Hooker, and P. Vicini, “Robust Population Pharmacokinetic Experiment Design,”
Journal of Pharmacokinetics and Pharmacodynamics, vol. 32, Feb. 2005, pp. 33-64.
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