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Table 3 Results of model fitting to the time series of recording rate. MA: window size, 2n, of the moving average calculation (equation 1). M: eruption magnitude. Model: selected model function. Rejected models are also shown when AIC or AICc difference between the accepted and the rejected models is less than 2 (Burnham and Anderson, 2002). Dexp: double exponential decay function. Exp: exponential decay function. Lr: relative likelihood. Note that the relative likelihood of the smaller AIC or AICc model is equal to 1. Values in the parentheses show the range of 95 percentile confidence interval estimated by the bootstrap method (Efron 1979) with 10,000 replicates. The recording rate of the datasets of M ≥ 4.0 (MA = 4, 6), M ≥ 4.5 (MA = 4) and M ≥ 5.0 (MA = 4) were not calculated because some of their consecutive events had the same ages and the denominator of equation 1 becomes zero

From: How many explosive eruptions are missing from the geologic record? Analysis of the quaternary record of large magnitude explosive eruptions in Japan

VEI

MA

Model

Lr

R1 (events/ka)

Half-life1 (×103 year)

R2 (events/ka)

Half-life2 (×103 year)

R0(=R1 + R2) (events/ka)

R1/R0

4

8

Dexp

1

20.8 (17.3–30.4)

5.56 (3.87–7.35)

1.01 (0.722–1.61)

101 (76.5–135)

21.8 (18.4 - 31.6)

0.954

4

6

Dexp

1

21.8 (18.8–33.4)

5.60 (3.84–7.29)

0.989 (0.737–1.65)

107(80.2–141)

22.7 (19.9 - 34.7)

0.957

4

4

Dexp

1

23.5 (21.9–38.2)

5.63 (3.76–7.04)

1.00 (0.797–1.71)

112 (85.6–146)

24.5 (23.0 - 39.6)

0.959

5

8

Dexp

1

2.62 (2.07–4.20)

39.2 (10.7–57.5)

0.538 (0.260–1.39)

181(122–257)

3.16 (2.72 - 5.43)

0.830

5

6

Dexp

1

2.78 (2.30–4.84)

39.7 (9.92–57.4)

0.509(0.263–1.45)

191(126–268)

3.28 (2.91 - 6.17)

0.845

5

4

Dexp

1

3.04 (2.68–5.85)

40.2 (9.48–55.8)

0.501(0.294–1.55)

202(132–272)

3.54 (3.27 - 7.34)

0.858

6

8

Exp

1

0.429 (0.343–0.636)

137 (104–184)

  

0.429 (0.343 - 0.636)

 

6

6

Exp

1

0.428 (0.351–0.659)

144 (107–192)

  

0.428 (0.351 - 0.659)

 

6

4

Exp

1

0.437 (0.379–0.710)

150 (112–200)

  

0.437 (0.379 - 0.710)

 

7

8

Exp

1

0.0582 (0.0437–0.101)

317 (205–472)

  

0.0582 (0.0437 - 0.101)

 

7

6

Exp

1

0.0599 (0.0464–0.112)

330 (212–479)

  

0.0599 (0.0464 - 0.112)

 

7

4

Exp

1

0.0627 (0.0504–0.127)

343 (227–481)

  

0.0627 (0.0504 - 0.127)

 

M

MA

Model

Lr

R1 (events/ka)

Half-life1 (×103 year)

R2 (events/ka)

Half-life2 (×103 year)

R0 (=R1 + R2) (events/ka)

R1/R0

4.0

8

Dexp

1

24.3 (16.1–35.7)

7.69(5.02–17.5)

2.17 (1.48–2.91)

163 (139–208)

26.4 (17.6 - 38.3)

0.918

4.5

8

Dexp

1

5.05 (4.46–20.7)

38.2 (2.86–51.6)

0.681 (0.424–2.36)

241 (138–310)

5.74 (5.06 - 22.9)

0.881

4.5

6

Dexp

1

5.23 (4.82–22.8)

38.7 (2.87–50.9)

0.663 (0.435–2.40)

249 (142–319)

5.90 (5.43–25.1)

0.887

5.0

8

Dexp

1

2.88 (2.45–3.88)

53.0 (38.1–71.2)

0.501 (0.307–0.866)

265 (207–341)

3.38 (3.02–4.45)

0.852

5.0

6

Dexp

1

2.99 (2.64–4.16)

53.3 (37.4–70.8)

0.488 (0.317–0.888)

274 (214–350)

3.48 (3.19–4.76)

0.860

5.5

8

Dexp

1

1.06 (0.754–1.60)

63.5 (39.9–90.4)

0.208(0.108–0.424)

300 (204–413)

1.26 (1.04–1.82)

0.836

5.5

6

Dexp

1

1.12 (0.850–1.72)

63.5 (39.3–88.5)

0.196 (0.108–0.413)

318 (221–434)

1.31 (1.12–1.96)

0.851

5.5

4

Dexp

1

1.20 (1.03–1.99)

63.2 (40.4–86.9)

0.191 (0.116–0.407)

335 (242–450)

1.40 (1.27–2.23)

0.863

6.0

8

Dexp Exp

1 0.619

0.378 (0.241–0.699)

87.3 (55.8–279)

0.183 (0.088–0.291)

303 (74.8–431)

0.560(0.405–0.853)

0.674

0.406(0.339–0.588)

223 (171–285)

0.406(0.339–0.588)

6.0

6

Dexp

1

0.428 (0.266–0.786)

86.4 (54.4–270)

0.168 (0.0862–0.298)

326 (81.0–454)

0.596 (0.455–0.938)

0.718

6.0

4

Dexp

1

0.481 (0.310–0.908)

84.9 (53.1–248)

0.161 (0.0902–0.315)

347 (103–480)

0.642 (0.534–1.07)

0.750

6.5

8

Exp

1

0.107 (0.0836–0.178)

323 (218–450)

  

0.107 (0.0836–0.178)

 

6.5

6

Exp

1

0.111 (0.0890–0.190)

335 (229–457)

  

0.111 (0.0890–0.190)

 

6.5

4

Dexp

0.400

0.139 (0.0840–0.435)

49.7(27.3–460)

0.0781 (1.15 × 10  6–0.120))

411 (35.7–546

0.217(0.108–0.509)

0.641

Exp

1

0.116 (0.0984–0.211)

347 (243–461)

0.116(0.098–0.211)

7.0

8

Exp

1

0.0613 (0.0476–0.107)

361 (225–526)

  

0.0613 (0.0476–0.107)

 

7.0

6

Exp

1

0.0634 (0.0499–0.113)

373 (246–531)

  

0.0634 (0.0499–0.113)

 

7.0

4

Exp

1

0.0651 (0.0547–0.133)

389 (256–536)

  

0.0651 (0.0547–0.133)