Wavelet Feature Screening

Figures & data

Fig. 1 Test functions of Donoho and Johnstone (1994).

Table 1 Average number of true features (#feat) selected among the 15 most relevant according to each screening method, and average runtime in seconds (time).

	WWS		gen. corr./splines		SIRS		DC-SIS
n	#feat	time	#feat	time	#feat	time	#feat	time
Blocks
128	1.1(0.93)	3.48	1.4(0.97)	1.57	0(0.23)	1.09	0.1(0.41)	2.75
256	3.3(0.95)	3.86	3.5(1.12)	1.83	0.1(0.29)	2.39	0.4(0.71)	11.86
512	5.4(0.8)	4.3	5.5(0.89)	2.36	0.2(0.53)	6.04	1.4(1.08)	74.76
Heavisine
128	3.3(0.88)	3.47	3.4(0.96)	1.57	0.2(0.55)	1.08	0.5(0.81)	2.71
256	5.2(0.86)	3.84	5.3(0.92)	1.8	0.7(0.92)	2.36	2.2(1.01)	11.77
512	6.5(0.77)	4.47	6.7(0.77)	2.46	2.1(1.13)	6.09	4.3(0.85)	76.74
Bumps
128	0.2(0.43)	3.47	0.1(0.23)	1.57	0.1(0.27)	1.08	0.1(0.28)	2.74
256	0.9(0.84)	3.87	0.2(0.48)	1.83	0.2(0.52)	2.38	0.3(0.59)	11.84
512	3.0(1.02)	4.33	0.7(0.88)	2.35	0.5(0.81)	5.78	0.8(1.01)	74.54
Doppler
128	0.2(0.45)	3.46	0.1(0.37)	1.56	0.1(0.41)	1.08	0.1(0.44)	2.72
256	0.8(0.84)	3.85	0.4(0.66)	1.81	0.2(0.57)	2.34	0.3(0.63)	11.62
512	2.3(1.05)	4.23	1.6(1.12)	2.3	0.5(0.87)	5.54	1(1.03)	71.22
Identity
128	2.2(1.11)	3.55	2.3(1.16)	1.6	3.1(1.15)	1.11	3.2(1.17)	2.77
256	4.1(1.05)	3.83	4.5(1.0)	1.81	5.1(0.98)	2.33	5.1(0.98)	11.67
512	5.4(0.89)	4.26	6(0.78)	2.3	6.3(0.75)	5.5	6.3(0.7)	71.2

NOTE: The standard deviation of #feat is presented in parentheses. The functions of true covariates are the four test functions of Donoho and Johnstone (1994) and the identity function. WWS denotes the warped wavelet feature screening.

Fig. 2 Time series of the meteorological variables recorded at Pinheiros air quality station in São Paulo. The abbreviations wd and gwd refer to wind direction and global wind direction, respectively.

Table 2 The five most relevant variables according to different feature screening methods.

Method	1st	2nd	3rd	4th	5th
WWS	Humidity	Temperature	Wind speed	$\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}$ (wd)	$\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}$ (gwd)
gen. corr.	Humidity	Temperature	Wind speed	$\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}$ (wd)	${\text{NO}}_2$
SIS	Humidity	$\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}$ (wd)	Wind speed	$\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}$ (wd)	${\text{NO}}_2$
SIRS	Humidity	$\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}$ (wd)	$\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}$ (wd)	Wind speed	NO
DC-SIS	Humidity	$\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}$ (wd)	Wind speed	$\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}$ (wd)	NO

NOTE: WWS denotes the warped wavelet feature screening. Variables are ordered in decreasing order of importance.

Fig. 3 Plots of the response variable Y (differenced and centered ozone concentrations) and each covariate (differenced and normalized to the unit interval). The curves are warped wavelet regression estimates. The observations are considerably noisy, and only a few features display a noticeable effect on Y, like humidity and wind speed.

Fig. 4 Boxplot of the observed ages for each group in the epilepsy study.

Table 3 The five most relevant variables according to different feature screening methods for each group of participants in the epilepsy study.

Group	Method	1st	2nd	3rd	4th	5th
	WWS	BODY_FA	Genu_FA	Genu_MD	Genu_AD	Genu_RD
	gen. corr.	BODY_FA	BODY_MD	BODY_AD	BODY_RD	Genu_FA
Control	SIS	BODY_FA	BODY_RD	Genu_FA	Genu_MD	Genu_RD
	SIRS	BODY_FA	BODY_RD	Genu_FA	Genu_MD	Genu_RD
	DC-SIS	BODY_FA	BODY_RD	Genu_FA	Genu_MD	Genu_RD
	WWS	BODY_FA	Genu_FA	Genu_RD	SPLEN_AD	SPLEN_RD
	gen. corr.	BODY_FA	Genu_FA	Genu_MD	Genu_RD	SPLEN_MD
RTLE	SIS	BODY_FA	BODY_RD	Genu_FA	Genu_MD	Genu_RD
	SIRS	BODY_FA	BODY_RD	Genu_FA	Genu_MD	Genu_RD
	DC-SIS	BODY_FA	Genu_FA	Genu_MD	Genu_RD	SPLEN_RD
	WWS	BODY_FA	BODY_MD	BODY_AD	BODY_RD	Genu_FA
	gen. corr.	BODY_FA	BODY_MD	BODY_RD	Genu_FA	Genu_MD
LTLE	SIS	BODY_FA	BODY_MD	BODY_AD	BODY_RD	Genu_FA
	SIRS	BODY_FA	BODY_MD	BODY_AD	BODY_RD	Genu_FA
	DC-SIS	BODY_FA	BODY_MD	BODY_AD	BODY_RD	Genu_FA

NOTE: WWS denotes the warped wavelet feature screening. Variables are ordered in decreasing order of importance.

Fig. 5 Scatterplot of observed ages versus the covariates BODY_FA (right) and Genu_FA (left). The symbols $·$, ×, and + correspond to control, LTLE, and RTLE groups, respectively. The solid curves were estimated via warped wavelet regression for each group of patients.

Donoho, D. L., and Johnstone, J. M. (1994), “Ideal Spatial Adaptation by Wavelet Shrinkage,” Biometrika, 81, 425–455. DOI: 10.1093/biomet/81.3.425.

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