Fig. 1
The cost function and important parameters. The blue curve is the cost function of eq. (4). vertical black line denotes the solution at the minimum of the cost function. Vertical cyan line denotes the true posterior mean. Vertical green line denotes the BLUE from the Kalman smoother in eq. (8).
Fig. 2
Properties of TLMs and SLMs. (a) we show the value of the TLM (SLM) in blue (red) as a function of different reference states. Red dashed line is the estimate of the SLM from eq. (27). Blue line denotes the value of 1 below which the TLM decays perturbations. In (b) is the TLM in (blue) and its first (red) and second (green) derivative. In (c), we show the t=1 true variance (green) and the estimates from a TLM (blue) and (red) SLM.
Fig. 3
Convergence curves. The result of outer loop iterations in the incremental method using a TLM (blue), SLM derived using the true prior (red), and SLM derived using a reduced prior variance (red dashed). The horizontal black line is the posterior mode; horizontal cyan line is posterior mean; and horizontal green line is the state estimate obtained from the BLUE of Section 3.
Fig. 4
The joint prior and posterior are shown in (a) and (b), respectively. The black line shows how the model links x0 with x1. The colours represent the density. The horizontal and vertical lines denote the location of the mode of the joint posterior. The marginal prior and posterior are shown in (c) and (d), respectively. Blue (red) is t=0 (t=1). In (d), the vertical black (dashed) line is the mode at t=0 (t=1). The vertical green line is the estimate of the mode at t=1 using eq. (11). In (e) is plotted the TLM as a function of different reference states in blue and its inverse in red.