On Tracking Varying Bounds When Forecasting Bounded Time Series

Figures & data

Fig. 1 Sigmoid function $s_j(b)$ on the real line.

Fig. 2 A quasiconvex differentiable function on $\mathbb{R}$, the negative density of a normal variable, with plateau areas when going away from the global minimum.

Table 1 Hyperparameter values for each algorithm: step size η and minibatch size m (Algorithm 1) and forgetting factor α (Algorithm 2).

	η	m	α
Algorithm 1: ONGD	0.001	100	–
Algorithm 2: rMLE.b	–	–	0.975

Fig. 3 The extended time-dependent negative log-likelihood for the first MC simulation, t = 3000, $\alpha =0.975$, w.r.t. λ (top left), ω (top right), τ (bottom left) and b (bottom right). For each plot, the remaining parameters are set to their true values.

Fig. 4 Confidence intervals of the tracked parameters for ONGD (top) and rMLE.b (bottom) with coverage probabilities 0.9 and 0.5, along with the average estimates (solid lines) and the true parameters (dotted lines).

Table 2 1-step-ahead CRPS and respective improvement over persistence and rMLE.1.

	mean (sd)	Imp./persist.	Imp./rMLE.1
ideal forecaster	5.782% (0.095)	–	–
climatology	15.258% (0.237)	–	–
probabilistic persistence	6.279% (0.097)	–	–
rMLE.1	6.042% (0.087)	3.78%	–
Algorithm 1: ONGD	5.807% (0.095)	7.52%	3.89%
Algorithm 2: rMLE.b	6.035% (0.110)	3.89%	0.11%

NOTE: The CRPS is averaged over the MC samples, and the standard deviation is also provided.

Table 3 Hyperparameter values for each algorithm: order p of the AR process, step size η, minibatch size m and forgetting factor α.

	p	η	m	α
Algorithm 1: ONGD	4	0.03	1	–
Algorithm 2: rMLE.b	5	–	–	0.9982

Table 4 10-minute-ahead CRPS and respective improvements over probabilistic persistence and rMLE.1.

	CRPS	Imp./persist.	Imp./rMLE.1
climatology	22.033%	–	–
probabilistic persistence	1.350%	–	–
rMLE.1	1.082%	19.89%	–
Algorithm 1: ONGD	0.888%	34.22%	17.89%
Algorithm 2: rMLE.b	1.056%	21.83%	2.43%

NOTE: Best forecast bolded.

Fig. 5 Estimates ${\widehat{\Lambda }}_t,{\widehat{\sigma }}_t^2,{\widehat{\nu }}_t$ and projected estimate ${\tilde{b}}_t$ on a sub-sample of the test set for rMLE.1 (top), ONGD (center) and rMLE.b (bottom).

Fig. 6 Probabilistic forecasts from ONGD, based on prediction intervals with nominal coverage rates of 95% and 75%, along with power measurements (solid black line).