Fig 1.
Profile-Wise Analysis: A profile likelihood-based workflow illustrating: (A) statistical framework; (B) estimation and prediction using the full likelihood function; and, (C) estimation and prediction using profile-wise, profile likelihood approximations.
Fig 2.
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Gaussian noise with σ = 10. The MLE (cyan) solution is superimposed, . (B) Univariate profiles for λ, K and C(0), respectively. Each profile is superimposed with the MLE (vertical green) and the 95% threshold
is shown (horizontal orange). Points of intersection of each profile and the threshold define asymptotic confidence intervals:
;
; and,
. (C)-(E) C(t) trajectories associated with the λ, K and C(0) univariate profiles in (b), respectively. In each case we uniformly discretise the interest parameter using N = 100 points along the 95% confidence interval identified in (B), and solve the model forward with θ = (ψ, ω) and plot the family of solutions (grey), superimposing the MLE (cyan), and we identify the prediction interval defined by these solutions (solid red). (F) Compares approximate prediction intervals obtained by computing the union of the univariate profiles with the ‘exact’ prediction intervals computed from the full likelihood. Trajectories (grey) are constructed by considering N = 104 choices of θ and we plot solutions with
only. These solutions define an ‘exact’ (or gold-standard) prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by taking the union of the three univariate intervals (red).
Fig 3.
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Gaussian noise with σ = 10. The MLE solution (cyan) is superimposed, . (B)-(D) Bivariate profiles for (λ, K), (λ, C(0)) and (C(0), K), respectively. In (B)-(D) the left-most panel shows the bivariate profile uniformly discretised on a 20 × 20 grid with the
contour and the MLE (pink disc) superimposed. The right-most panels in (B)-(D) show C(t) predictions (grey) for each grid point contained within the
contour, together with the profile-wise prediction interval (solid red). (E) Compares approximate prediction intervals obtained by computing the union of the profile-wise prediction intervals with the ‘exact’ (more precisely, conservative, assuming the parameter confidence set has proper coverage) prediction intervals from the full likelihood function. Predictions (grey) are constructed by considering N = 104 choices of θ and we plot solutions with
only. These solutions define a gold-standard prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by the union of the three bivariate trajectories (red).
Fig 4.
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Gaussian noise with σ = 10. The MLE solution (cyan) is superimposed, . (b)-(d) Bivariate profiles for (λ, K), (λ, C(0)) and (C(0), K), respectively. In (B)-(D) the left-most panel shows 200 points along the
contours (blue discs) superimposed with the MLE (pink disc). The right-most panels shows 200 C(t) predictions (grey) that are associated with the 200 points along the
boundary, together with the profile-wise prediction interval (solid red). (E) Compares approximate prediction intervals obtained by computing the union of the profile-wise prediction intervals with the prediction intervals from the full likelihood function. Predictions (grey) are constructed by considering N = 104 choices of θ and we plot solutions with
only. These solutions define a gold-standard prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by the union of the three bivariate trajectories (red).
Fig 5.
(A) Data obtained by solving Eqs (25) and (26), with θ = (α, β, x(0), y(0)) = (0.9, 1.1, 0.8, 0.3), at t = 0, 0.5, 1.0, …, 10 is corrupted with Gaussian noise with σ = 0.2. The MLE solution (solid curves) is superimposed, . (B)-(G) Bivariate profiles for (α, β), (α, x(0)), (α, y(0)), (β, x(0)), (β, y(0)) and (x(0), y(0)), respectively. In (B)-(G) the left-most panel shows 100 points along the
contour (blue discs) and the MLE (pink disc). The middle- and right-most panels show 100 predictions of a(t) and b(t), respectively, associated with the 100 points along the
contour, together with the profile-wise prediction interval (solid red). (H) Compares approximate prediction intervals obtained by computing the union of the profile-wise prediction intervals with the prediction intervals from the full likelihood function. Predictions (grey) are constructed by considering N = 104 choices of θ and we plot solutions with
only. These solutions define a gold-standard prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by the union of the three bivariate trajectories (red).
Fig 6.
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Poisson noise. The MLE solution (cyan) is superimposed, . (B) Univariate profiles for λ, K and C(0), respectively. Each profile is superimposed with the MLE (vertical green) and the 95% threshold
is shown (horizontal orange). Points of intersection of each profile and the threshold define asymptotic confidence intervals:
;
; and,
. (C)-(E) C(t) predictions associated with the λ, K and C(0) univariate profiles in (B), respectively. In each case we uniformly discretise the interest parameter using N = 100 points along the 95% confidence interval identified in (B), and solve the model forward with θ = (ψ, ω) and plot the family of solutions (grey), superimposing the MLE (cyan), and we identify the prediction interval defined by these solutions (solid red). (F) Compares approximate prediction intervals obtained by computing the union of the three univariate profiles with the prediction intervals computed from the full likelihood. Trajectories (grey) are constructed by considering N = 104 choices of θ and we plot solutions with ℓ ≥ ℓ* = −Δ0.95,3/2 only. These solutions define a prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by taking the union of the three univariate intervals (red).
Fig 7.
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Poisson noise. The MLE solution (cyan) is superimposed, . (B)-(D) Bivariate profiles for (λ, K), (λ, C(0)) and (C(0), K), respectively. The left-most panel in (B)-(D) shows 200 points along the
contour (blue discs) and the MLE (pink disc). The right-most panel in (B)-(D) shows 200 C(t) predictions associated with the points along the
contour that define a prediction interval (solid red). (E) Compares approximate prediction intervals obtained by computing the union of the three bivariate profiles with the exact prediction interval computed from the full likelihood. Trajectories (grey) are constructed by considering N = 104 choices of θ and we plot solutions with
only. These solutions define an exact prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by taking the union of the three univariate intervals (red).