Bayesian analysis for quantile smoothing spline

Figures & data

Figure 1. The curves are fitted when p = 0.5 and the true curves are the solid lines. Graph (a): Gamma prior, Ga($a_1,b_1$), for δ with $a_1=1$ and $b_1=0.01$ (dash-dotted), $b_1=0.1$ (dotted), $b_1=0.5$ (dashed); Graph (b): Scaled Pareto prior for η with c = 0.05 (dash-dotted), c = 0.1 (dashed) and c = 0.5 (dotted).

Table 1. ISEs for Gamma prior on δ and scaled Pareto prior on η.

	($a_1,b_1$)	(1,0.01)	(1,0.1)	(1,0.5)
Gamma($a_1,b_1)$	ISE	4.204$\times {10}^3$	2.189$\times {10}^3$	1.875$\times {10}^3$
Scaled Pareto$\left(c\right)$	c	0.05	0.1	0.5
	ISE	1.989$\times {10}^3$	2.142$\times {10}^3$	2.453$\times {10}^3$

Table 2. MIAEs and MISEs for Curve I with p = 0.1, 0.3 and 0.5.

		Error Distribution
		Normal		ALD		Cauchy
	Method	MIAE	MISE	MIAE	MISE	MIAE	MISE
p = 0.1	NBQSS	4.105$\times {10}^2$	2.607$\times {10}^3$	4.305$\times {10}^2$	3.495$\times {10}^3$	2.818$\times {10}^2$	1.355$\times {10}^3$
	OBQSS	1.796$\times {10}^1$	3.879$\times {10}^2$	1.328$\times {10}^1$	2.165$\times {10}^2$	2.403$\times {10}^1$	6.761$\times {10}^2$
	QSS	4.286$\times {10}^2$	2.998$\times {10}^3$	6.034$\times {10}^2$	5.916$\times {10}^3$	2.926$\times {10}^2$	1.383$\times {10}^3$
	BSS	2.774$\times {10}^2$	1.297$\times {10}^3$	9.654$\times {10}^2$	1.336$\times {10}^2$	3.512$\times {10}^2$	3.241$\times {10}^3$
	SS	2.557$\times {10}^2$	1.165$\times {10}^3$	1.038$\times {10}^1$	1.544$\times {10}^2$	4.543$\times {10}^2$	6.357$\times {10}^3$
p = 0.3	NBQSS	2.774$\times {10}^2$	1.476$\times {10}^3$	2.857$\times {10}^2$	1.609$\times {10}^3$	1.481$\times {10}^2$	8.329$\times {10}^4$
	OBQSS	5.697$\times {10}^2$	5.359$\times {10}^3$	5.622$\times {10}^2$	5.174$\times {10}^3$	6.593$\times {10}^2$	6.161$\times {10}^3$
	QSS	3.291$\times {10}^2$	1.733$\times {10}^3$	3.614$\times {10}^2$	2.309$\times {10}^3$	1.366$\times {10}^2$	5.372$\times {10}^4$
	BSS	2.417$\times {10}^2$	1.081$\times {10}^3$	4.143$\times {10}^2$	2.601$\times {10}^3$	3.239$\times {10}^2$	2.392$\times {10}^3$
	SS	2.522$\times {10}^2$	1.189$\times {10}^3$	3.981$\times {10}^2$	2.551$\times {10}^3$	3.231$\times {10}^2$	2.427$\times {10}^3$
p = 0.5	NBQSS	2.692$\times {10}^2$	1.152$\times {10}^3$	2.645$\times {10}^2$	1.284$\times {10}^3$	9.743$\times {10}^3$	3.461$\times {10}^4$
	OBQSS	4.373$\times {10}^2$	5.081$\times {10}^3$	4.296$\times {10}^2$	4.964$\times {10}^3$	3.718$\times {10}^2$	4.819$\times {10}^3$
	QSS	3.386$\times {10}^2$	1.906$\times {10}^3$	2.692$\times {10}^2$	1.308$\times {10}^3$	9.957$\times {10}^3$	3.538$\times {10}^4$
	BSS	2.667$\times {10}^2$	1.196$\times {10}^3$	3.318$\times {10}^2$	1.732$\times {10}^3$	2.677$\times {10}^2$	1.511$\times {10}^3$
	SS	2.559$\times {10}^2$	1.105$\times {10}^3$	3.341$\times {10}^2$	1.711$\times {10}^3$	2.756$\times {10}^2$	1.634$\times {10}^3$

Note: The bold numbers correspond to the smallest MIAE and MISE in each comparison.

Table 3. MIAEs and MISEs for Curve II with p = 0.1, 0.3 and 0.5.

		Error Distribution
		Normal		ALD		Cauchy
	Method	MIAE	MISE	MIAE	MISE	MIAE	MISE
p = 0.1	NBQSS	4.666$\times {10}^2$	3.269$\times {10}^3$	6.745$\times {10}^2$	6.844$\times {10}^3$	3.122$\times {10}^2$	2.118$\times {10}^3$
	OBQSS	2.351$\times {10}^1$	7.551$\times {10}^2$	1.764$\times {10}^1$	4.558$\times {10}^2$	3.017$\times {10}^1$	1.154$\times {10}^1$
	QSS	4.669$\times {10}^2$	3.282$\times {10}^3$	6.337$\times {10}^2$	6.351$\times {10}^3$	3.583$\times {10}^2$	2.379$\times {10}^3$
	BSS	2.832$\times {10}^2$	1.232$\times {10}^3$	1.144$\times {10}^1$	1.836$\times {10}^2$	5.098$\times {10}^2$	7.049$\times {10}^3$
	SS	3.051$\times {10}^2$	1.625$\times {10}^3$	1.105$\times {10}^1$	1.629$\times {10}^2$	5.198$\times {10}^2$	7.999$\times {10}^3$
p = 0.3	NBQSS	3.133$\times {10}^2$	1.501$\times {10}^3$	3.437$\times {10}^2$	1.951$\times {10}^3$	1.461$\times {10}^2$	5.264$\times {10}^4$
	OBQSS	1.309$\times {10}^1$	2.448$\times {10}^2$	1.261$\times {10}^1$	2.241$\times {10}^2$	1.444$\times {10}^1$	3.087$\times {10}^2$
	QSS	3.583$\times {10}^2$	1.984$\times {10}^3$	4.096$\times {10}^2$	2.555$\times {10}^3$	1.622$\times {10}^2$	6.108$\times {10}^4$
	BSS	2.761$\times {10}^2$	1.133$\times {10}^3$	4.799$\times {10}^2$	3.451$\times {10}^3$	3.353$\times {10}^2$	2.452$\times {10}^3$
	SS	2.925$\times {10}^2$	1.436$\times {10}^3$	4.889$\times {10}^2$	3.079$\times {10}^3$	3.721$\times {10}^2$	2.281$\times {10}^3$
p = 0.5	NBQSS	3.322$\times {10}^2$	1.775$\times {10}^3$	3.131$\times {10}^2$	1.588$\times {10}^3$	1.041$\times {10}^2$	2.712$\times {10}^4$
	OBQSS	1.169$\times {10}^1$	1.891$\times {10}^2$	1.175$\times {10}^1$	1.908$\times {10}^2$	1.165$\times {10}^1$	1.871$\times {10}^2$
	QSS	3.768$\times {10}^2$	2.364$\times {10}^3$	3.814$\times {10}^2$	2.369$\times {10}^3$	1.539$\times {10}^2$	5.329$\times {10}^4$
	BSS	2.795$\times {10}^2$	1.313$\times {10}^3$	3.587$\times {10}^2$	2.212$\times {10}^3$	3.189$\times {10}^2$	1.971$\times {10}^3$
	SS	3.006$\times {10}^2$	1.589$\times {10}^3$	3.612$\times {10}^2$	2.028$\times {10}^3$	3.847$\times {10}^2$	2.329$\times {10}^3$

Note: The bold numbers correspond to the smallest MIAE and MISE in each comparison.

Figure 2. The curves are fitted for Curve I under p = 0.3 and ALD error. Graph (a) NBQSS method; Graph (b) OBQSS method; Graph (c) QSS method; Graph (d) BSS method; Graph (e) SS method. The solid lines are  the true curve and the fitted curve. Two dashed lines are 2.5% and 97.5% pointwise empirical quantiles.

Figure 3. The curves are fitted for Curve II under p = 0.3 and ALD error. Graph (a) NBQSS method; Graph (b) OBQSS method; Graph (c) QSS method; Graph (d) BSS method; Graph (e) SS method. The solid lines are the true curve and the fitted curve. Two dashed lines are 2.5% and 97.5% pointwise empirical quantiles.

Figure 4. One illustration for Mechanisms I and II. Graphs (a)–(c) correspond to Mechanism I with $\alpha \mathrm{\%}=30\mathrm{\%},50\mathrm{\%}$ and 70%. Graphs (d) and (e) correspond to Mechanism II with $\alpha \mathrm{\%}=$30%, 50% and 70%.

Table 4. MIAEs and MISEs for Mechanism I with quantiles p = 0.1, 0.5, 0.9 and unobserved proportions $\alpha \mathrm{\%}=30\mathrm{\%},50\mathrm{\%},70\mathrm{\%}$.

			Error Distribution
			Normal		ALD		Cauchy
		Method	MIAE	MISE	MIAE	MISE	MIAE	MISE
30%	p = 0.1	NBQSS	3.985$\times {10}^2$	2.288$\times {10}^3$	4.422$\times {10}^2$	2.921$\times {10}^3$	4.147$\times {10}^2$	3.956$\times {10}^3$
		NBQSSK	3.786$\times {10}^2$	2.111$\times {10}^3$	4.463$\times {10}^2$	3.011$\times {10}^3$	3.865$\times {10}^2$	3.180$\times {10}^3$
	p = 0.5	NBQSS	2.759$\times {10}^2$	1.040$\times {10}^3$	2.733$\times {10}^2$	1.157$\times {10}^3$	7.713$\times {10}^3$	1.007$\times {10}^4$
		NBQSSK	2.688$\times {10}^2$	9.876$\times {10}^4$	2.561$\times {10}^2$	1.046$\times {10}^3$	7.566$\times {10}^3$	9.603$\times {10}^5$
	p = 0.9	NBQSS	4.686$\times {10}^2$	3.339$\times {10}^3$	4.647$\times {10}^2$	3.809$\times {10}^3$	4.007$\times {10}^2$	3.478$\times {10}^3$
		NBQSSK	4.482$\times {10}^2$	2.856$\times {10}^3$	4.705$\times {10}^2$	3.642$\times {10}^3$	4.314$\times {10}^2$	3.744$\times {10}^3$
50%	p = 0.1	NBQSS	6.968$\times {10}^2$	6.790$\times {10}^3$	7.258$\times {10}^2$	7.435$\times {10}^3$	5.464$\times {10}^2$	5.551$\times {10}^3$
		NBQSSK	6.336$\times {10}^2$	5.231$\times {10}^3$	7.223$\times {10}^2$	7.543$\times {10}^3$	4.878$\times {10}^2$	4.746$\times {10}^3$
	p = 0.5	NBQSS	4.344$\times {10}^2$	2.741$\times {10}^3$	4.526$\times {10}^2$	2.744$\times {10}^3$	1.530$\times {10}^2$	3.619$\times {10}^4$
		NBQSSK	4.235$\times {10}^2$	2.505$\times {10}^3$	4.308$\times {10}^2$	2.532$\times {10}^3$	1.494$\times {10}^2$	3.767$\times {10}^4$
	p = 0.9	NBQSS	7.627$\times {10}^2$	8.700$\times {10}^3$	6.108$\times {10}^2$	5.865$\times {10}^3$	7.548$\times {10}^2$	9.166$\times {10}^3$
		NBQSSK	6.983$\times {10}^2$	7.093$\times {10}^3$	5.440$\times {10}^2$	4.738$\times {10}^3$	6.487$\times {10}^2$	6.920$\times {10}^3$
70%	p = 0.1	NBQSS	1.134$\times {10}^1$	1.698$\times {10}^2$	2.174$\times {10}^1$	2.784$\times {10}^2$	1.487$\times {10}^1$	3.685$\times {10}^2$
		NBQSSK	1.008$\times {10}^1$	1.345$\times {10}^2$	1.582$\times {10}^1$	2.043$\times {10}^2$	1.222$\times {10}^1$	2.741$\times {10}^2$
	p = 0.5	NBQSS	8.685$\times {10}^2$	9.997$\times {10}^3$	7.782$\times {10}^2$	8.777$\times {10}^3$	4.316$\times {10}^2$	3.075$\times {10}^3$
		NBQSSK	8.377$\times {10}^2$	9.539$\times {10}^3$	6.850$\times {10}^2$	6.965$\times {10}^3$	4.349$\times {10}^2$	3.098$\times {10}^3$
	p = 0.9	NBQSS	1.156$\times {10}^1$	2.053$\times {10}^2$	1.593$\times {10}^1$	3.805$\times {10}^2$	1.592$\times {10}^1$	3.652$\times {10}^2$
		NBQSSK	1.117$\times {10}^1$	1.772$\times {10}^2$	1.467$\times {10}^1$	2.975$\times {10}^2$	1.416$\times {10}^1$	2.859$\times {10}^2$

Notes: The number of observations varies across different observed knots. The bold numbers correspond to the smallest MIAE and MISE in each comparison.

Table 5. MIAEs and MISEs for Mechanism II with quantiles p = 0.1, 0.5, 0.9 and unobserved proportions $\alpha \mathrm{\%}=30\mathrm{\%},50\mathrm{\%},70\mathrm{\%}$.

			Error Distribution
			Normal		ALD		Cauchy
		Method	MIAE	MISE	MIAE	MISE	MIAE	MISE
30%	p = 0.1	NBQSS	4.196$\times {10}^2$	2.607$\times {10}^3$	4.215$\times {10}^2$	2.575$\times {10}^3$	4.405$\times {10}^2$	3.849$\times {10}^3$
		NBQSSK	4.017$\times {10}^2$	2.299$\times {10}^3$	4.271$\times {10}^2$	2.595$\times {10}^3$	4.187$\times {10}^2$	3.336$\times {10}^3$
	p = 0.5	NBQSS	2.565$\times {10}^2$	9.338$\times {10}^4$	2.673$\times {10}^2$	1.040$\times {10}^3$	7.159$\times {10}^3$	8.339$\times {10}^5$
		NBQSSK	2.458$\times {10}^2$	9.094$\times {10}^4$	2.631$\times {10}^2$	1.037$\times {10}^3$	7.173$\times {10}^3$	7.987$\times {10}^5$
	p = 0.9	NBQSS	3.924$\times {10}^2$	2.235$\times {10}^3$	4.787$\times {10}^2$	3.176$\times {10}^3$	5.495$\times {10}^2$	5.160$\times {10}^3$
		NBQSSK	3.947$\times {10}^2$	2.166$\times {10}^3$	4.715$\times {10}^2$	3.098$\times {10}^3$	5.086$\times {10}^2$	4.271$\times {10}^3$
50%	p = 0.1	NBQSS	4.884$\times {10}^2$	3.485$\times {10}^3$	5.422$\times {10}^2$	4.058$\times {10}^3$	5.614$\times {10}^2$	5.494$\times {10}^3$
		NBQSSK	4.666$\times {10}^2$	3.186$\times {10}^3$	5.318$\times {10}^2$	3.658$\times {10}^3$	5.090$\times {10}^2$	5.160$\times {10}^3$
	p = 0.5	NBQSS	3.215$\times {10}^2$	1.548$\times {10}^3$	3.567$\times {10}^2$	1.912$\times {10}^3$	9.201$\times {10}^3$	1.250$\times {10}^4$
		NBQSSK	3.112$\times {10}^2$	1.449$\times {10}^3$	3.532$\times {10}^2$	1.819$\times {10}^3$	8.622$\times {10}^3$	1.147$\times {10}^4$
	p = 0.9	NBQSS	5.653$\times {10}^2$	4.493$\times {10}^3$	5.798$\times {10}^2$	4.931$\times {10}^3$	6.294$\times {10}^2$	7.644$\times {10}^3$
		NBQSSK	5.146$\times {10}^2$	3.487$\times {10}^3$	5.366$\times {10}^2$	4.437$\times {10}^3$	6.565$\times {10}^2$	8.191$\times {10}^3$
70%	p = 0.1	NBQSS	7.133$\times {10}^2$	7.174$\times {10}^3$	3.479$\times {10}^2$	1.779$\times {10}^3$	8.868$\times {10}^2$	1.204$\times {10}^2$
		NBQSSK	6.598$\times {10}^2$	5.856$\times {10}^3$	3.173$\times {10}^2$	1.449$\times {10}^3$	8.655$\times {10}^2$	1.096$\times {10}^2$
	p = 0.5	NBQSS	5.498$\times {10}^2$	4.099$\times {10}^3$	1.731$\times {10}^2$	4.062$\times {10}^4$	1.840$\times {10}^2$	5.022$\times {10}^4$
		NBQSSK	5.246$\times {10}^2$	3.654$\times {10}^3$	1.655$\times {10}^2$	3.733$\times {10}^4$	1.841$\times {10}^2$	5.010$\times {10}^4$
	p = 0.9	NBQSS	9.509$\times {10}^2$	1.329$\times {10}^2$	3.730$\times {10}^2$	2.211$\times {10}^3$	7.432$\times {10}^2$	7.884$\times {10}^3$
		NBQSSK	9.040$\times {10}^2$	1.182$\times {10}^2$	3.373$\times {10}^2$	1.787$\times {10}^3$	7.076$\times {10}^2$	6.899$\times {10}^3$

Notes: The number of observations varies across different observed knots. The bold numbers correspond to the smallest MIAE and MISE in each comparison.

Table 6. Summary of cases where NBQSSK outperforms NBQSS.

	Error Distribution
Mechanism I	Normal	ALD	Cauchy
30%	100%	50%	66.7%
50%	100%	83.3%	83.3%
70%	100%	100%	66.7%
	Error Distribution
Mechanism II	Normal	ALD	Cauchy
30%	83.3%	66.7%	83.3%
50%	100%	100%	67.6%
70%	100%	100%	83.3%

Note: Each cell is the proportion of the winning cases of NBQSSK out of six comparisons.

Figure 5. The solid lines are the true curve and fitted curves. The curves are fitted with p = 0.1, ALD random error. Dashed lines are empirical 95% credible interval corresponding to NBQSSK and dotted lines are empirical 95% credible interval corresponding to NBQSS. Graph (a): Mechanism I, $\alpha \mathrm{\%}=50\mathrm{\%}$; Graph (b): corresponds to Mechanism II, $\alpha \mathrm{\%}=50\mathrm{\%}$; Graph (c): Mechanism I, $\alpha \mathrm{\%}=70\mathrm{\%}$; Graph (d): Mechanism II, $\alpha \mathrm{\%}=70\mathrm{\%}$.

Figure 6. Graph (a): NBQSS method with p = 0.1, 0.5 and 0.9; Graph (b): OBQSS method with p = 0.5; Graph (c): BSS method; Graph (d): SS method.

Figure 7. BSS curve (solid) and NBQSS with $p=$0.1 (dashed), 0.5 (dotted) and 0.9 (dotted-dash) curves on 25 September 2018.