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Received: October 12, 2020

Revised: December 29, 2020

Accept: January 6, 2021

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Table 1: Results on 100 repetitions in simulation studies for n = 50.

σe = 0.05, δ = 0.25

MSE

X1

X2

X3

X4

X5

MLE

2.97 (0.14)

1.00 1.00 1.00 1.00 1.00

SCAD

3.02 (0.18) 1.00 × 10−2 0.91 1.00 1.00 0.26 0.31

Elastic net 11.12 (0.95) 3.13 × 10−5 0.04 1.00 1.00 0.00 0.00

MCP

3.16 (0.30) 6.44 × 10−3 0.60 1.00 1.00 0.43 0.44

σe = 0.1, δ = 0.25

MSE

X1

X2

X3

X4

X5

MLE

3.18 (0.14)

1.00 1.00 1.00 1.00 1.00

SCAD

3.19 (0.22) 1.02 × 10−2 0.92 1.00 1.00 0.43 0.43

Elastic net 11.15 (1.14) 3.63 × 10−5 0.02 1.00 1.00 0.02 0.02

MCP

3.48 (0.34) 8.22 × 10−3 0.42 1.00 1.00 0.27 0.28

σe = 0.05, δ = 0.50

X2

X3

X4

X5

MSE

X1

MLE

2.14 (0.12)

1.00 1.00 1.00 1.00 1.00

SCAD

2.07 (0.16) 6.31 × 10−3 0.96 1.00 1.00 0.37 0.45

Elastic net 11.16 (1.05) 3.16 × 10−5 0.07 1.00 1.00 0.00 0.03

MCP

2.07 (0.17) 3.09 × 10−3 0.94 1.00 1.00 0.27 0.33

σe = 0.1, δ = 0.50

MSE

X1

X2

X3

X4

X5

MLE

2.51 (0.15)

1.00 1.00 1.00 1.00 1.00

SCAD

2.55 (0.20) 4.46 × 10−3 0.89 1.00 1.00 0.44 0.45

Elastic net 11.10 (0.92) 3.06 × 10−5 0.03 1.00 1.00 0.00 0.01

MCP

2.43 (0.19) 3.56 × 10−2 0.43 1.00 1.00 0.29 0.27

σe = 0.05, δ = 0.75

MSE

X1

X2

X3

X4

X5

MLE

2.21 (0.13)

1.00 1.00 1.00 1.00 1.00

SCAD

2.19 (0.14) 5.01 × 10−3 1.00 1.00 1.00 0.88 0.86

Elastic net 11.15 (1.04) 2.76 × 10−5 0.04 1.00 1.00 0.02 0.05

MCP

2.16 (0.20) 3.29 × 10−3 0.95 1.00 1.00 0.52 0.50

σe = 0.1, δ = 0.75

MSE

X1

X2

X3

X4

X5

MLE

2.63 (0.18)

1.00 1.00 1.00 1.00 1.00

SCAD

2.63 (0.19) 3.30 × 10−3 0.85 1.00 1.00 0.82 0.81

Elastic net 10.89 (0.97) 3.16 × 10−5 0.06 1.00 1.00 0.03 0.01

MCP

2.52 (0.22) 3.26 × 10−3 0.99 1.00 1.00 0.62 0.58

Variable selection for historical functional linear model

Table 2: Results on 100 repetitions in simulation studies for

σe = 0.05, δ = 0.25

MSE

X1

X2

X3

MLE

3.03 (0.12)

1.00 1.00 1.00

SCAD

3.68 (0.20) 1.02 × 10−2 0.00 1.00 1.00

Elastic net 11.28 (0.64) 3.54 × 10−5 0.00 1.00 1.00

MCP

3.06 (0.12) 3.10 × 10−3 0.99 1.00 1.00

σe = 0.1, δ = 0.25

MSE

X1

X2

X3

MLE

3.12 (0.11)

1.00 1.00 1.00

SCAD

3.24 (0.25) 6.31 × 10−2 0.74 1.00 1.00

Elastic net 11.32 (0.65) 3.24 × 10−5 0.00 1.00 1.00

MCP

3.13 (0.13) 3.19 × 10−3 0.97 1.00 1.00

σe = 0.05, δ = 0.50

X2

X3

MSE

X1

MLE

1.97 (0.09)

1.00 1.00 1.00

SCAD

1.93 (0.12) 3.16 × 10−3 0.99 1.00 1.00

Elastic net 11.18 (0.69) 3.24 × 10−5 0.00 1.00 1.00

MCP

1.92 (0.10) 1.58 × 10−3 1.00 1.00 1.00

σe = 0.1, δ = 0.50

MSE

X1

X2

X3

MLE

2.18 (0.08)

1.00 1.00 1.00

SCAD

2.15 (0.09) 3.14 × 10−3 0.78 1.00 1.00

Elastic net 11.22 (0.72) 3.16 × 10−5 0.00 1.00 1.00

MCP

2.22 (0.18) 1.37 × 10−3 0.96 1.00 1.00

σe = 0.05, δ = 0.75

MSE

X1

X2

X3

MLE

1.99 (0.09)

1.00 1.00 1.00

SCAD

2.80 (1.83) 3.64 × 10−3 0.94 1.00 0.83

Elastic net 11.09 (0.64) 3.32 × 10−5 0.00 1.00 1.00

MCP

1.99 (0.09) 1.12 × 10−3 1.00 1.00 1.00

σe = 0.1, δ = 0.75

MSE

X1

X2

X3

MLE

2.17 (0.13)

1.00 1.00 1.00

SCAD

3.18 (2.13) 3.33 × 10−3 1.00 1.00 0.90

Elastic net 11.99 (0.52) 5.44 × 10−5 0.00 1.00 1.00

MCP

2.13 (0.09) 1.58 × 10−3 1.00 1.00 1.00

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n = 100.

X4

1.00

0.00

0.00

0.17

X5

1.00

0.00

0.00

0.09

X4

1.00

0.09

0.00

0.22

X5

1.00

0.02

0.00

0.11

X4

1.00

0.29

0.00

0.23

X5

1.00

0.31

0.00

0.26

X4

1.00

0.73

0.00

0.23

X5

1.00

0.75

0.00

0.10

X4

1.00

0.33

0.00

0.92

X5

1.00

0.31

0.00

0.93

X4

1.00

0.60

0.00

1.00

X5

1.00

0.70

0.00

1.00

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H. Matsui

Figure 3: Examples of typhoon data. Two plots in the top left (north latitude and east

longitude) are responses and the remaining data are predictors. The square brackets

indicates units of measurement, where ”nm” represents the nautical mile.

Figure 4: Functional data sets obtained by smoothing the data given in Figure 3.

Variable selection for historical functional linear model

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Figure 5: Estimated coefficient functions when the response is the north latitude (Y1 ).

Figure 6: Estimated coefficient functions when the response is the east longitude (Y2 ).

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