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Laplacian Method

The Laplacian approximation provides fast, efficient parameter estimation particularly suited for sparse data.

Overview

from neopkpd.estimation import estimate, EstimationConfig, LaplacianMethod

config = EstimationConfig(
    method=LaplacianMethod(),
    theta_init=[10.0, 50.0],
    omega_init=[[0.09, 0], [0, 0.04]],
    sigma_init=0.1
)

result = estimate(data, "OneCompIVBolus", config)

When to Use Laplacian

Situation Why Laplacian Helps
Sparse sampling Few observations per subject (1-3)
Large populations Fast computation scales well
Initial estimates Quick screening before FOCE/SAEM
Simple models One-compartment, few parameters

Configuration

LaplacianMethod Parameters

from neopkpd.estimation import LaplacianMethod

method = LaplacianMethod(
    max_inner_iter=50,     # Max iterations for eta mode
    inner_tol=1e-6,        # Inner optimization tolerance
    use_prior=True,        # Include prior on eta
    hessian_method="exact" # or "finite_diff"
)

Full Configuration

from neopkpd.estimation import (
    EstimationConfig, LaplacianMethod, BLQConfig, BLQMethod
)

config = EstimationConfig(
    method=LaplacianMethod(
        max_inner_iter=50,
        inner_tol=1e-6
    ),
    theta_init=[10.0, 50.0],
    omega_init=[[0.25, 0], [0, 0.16]],  # Wide initial IIV
    sigma_init={"additive": 0.5, "proportional": 0.2},
    blq_config=BLQConfig(method=BLQMethod.M3, lloq=0.1),
    max_iter=500,
    compute_se=True,
    verbose=True
)

Sparse Data Example

from neopkpd.estimation import (
    estimate, EstimationConfig, LaplacianMethod, EstimationData
)

# Very sparse data: 1-2 observations per subject
data = EstimationData(
    subject_ids=["1", "1", "2", "3", "3"],
    times=[1.0, 4.0, 2.0, 0.5, 8.0],
    observations=[8.5, 3.2, 6.1, 9.2, 1.1],
    doses=[
        {"time": 0.0, "amount": 500.0, "subject_id": "1"},
        {"time": 0.0, "amount": 500.0, "subject_id": "2"},
        {"time": 0.0, "amount": 500.0, "subject_id": "3"}
    ]
)

config = EstimationConfig(
    method=LaplacianMethod(),
    theta_init=[5.0, 30.0],
    omega_init=[[0.25, 0], [0, 0.16]],
    sigma_init=0.15,
    compute_se=True
)

result = estimate(data, "OneCompIVBolus", config)

print(f"CL = {result.theta[0]:.3f} ± {result.theta_se[0]:.3f}")
print(f"V  = {result.theta[1]:.3f} ± {result.theta_se[1]:.3f}")
print(f"OFV = {result.ofv:.2f}")

Comparison with FOCE-I

Aspect Laplacian FOCE-I
Interaction term No Yes
Computational cost Lower Higher
Accuracy (dense) Lower Higher
Accuracy (sparse) Good Good

See Also