Table 1: Summary of the regression models for the PDO (L1 ) and NPGO (L2 ) modes from
equations (1) and (2) in the main text. {Ai , Bi , Ci }: Regression coefficients. σi : Standard
deviation of the residual. κi : Condition number of the design matrix. {d(i ), d(∆i )}:
Durbin-Watson statistics for the residual and residual increments for 1 m lag. tc (i )/tc (Li ):
noise decorrelation time relative to modal decorrelation time. X (X) indicates external
factors/interactions which were (were not) part of a given model.
Model
1
2
3
4
5
Ext. factors
P1 , P2
X
X
X
X
X
L3
X
X
X
X
X
I1 , . . . , I 4
X
X
X
X
X
Mode L1
A1 (L1 )
0.9964
0.9972
X
0.6233 −0.0067
A1 (L2 )
0.0343
0.0330
X −0.0624 −0.2148
B1 (L3 )
X −0.0340
X
X
X
B1 (P1 , I1 )
X
X
0.4070
0.1732
0.4606
B1 (P2 , I2 )
X
X
0.4762
0.1954
0.5241
B1 (P1 , I3 )
X
X
0.2709
0.0907
0.2508
B1 (P2 , I3 )
X
X −0.1778 −0.0180 −0.0598
C1 (L1 , L1 , L1 )
X
X
X
X −0.0032
C1 (L1 , L2 , L2 )
X
X
X
X
0.0046
σ1
0.0735
0.0658
0.1582
0.0696
0.0800
κ1
1.1779
1.2138
2.1941
40.184
7.008
d(1 )
0.5862
0.7428
0.5886
0.6284
0.6810
d(∆1 )
2.0342
2.0190
1.3244
2.1196
1.9682
tc (1 )/tc (L1 )
0.2307
0.1954
0.3388
0.1967
0.1621
Mode L2
A2 (L1 )
−0.0282 −0.0278
X −0.0303 −0.1787
A2 (L2 )
0.9916
0.9086
X
0.9451 −0.0107
B2 (L3 )
X −0.0197
X
X
X
B2 (P1 , I1 )
X
X
0.1172
0.0001
0.2144
B2 (P2 , I1 )
X
X −0.2610 −0.0409 −0.2397
B2 (P2 , I2 )
X
X
0.1604
0.0196
0.2418
B2 (P2 , I3 )
X
X
0.4128
0.0540
0.3653
B2 (P1 , I4 )
X
X −0.4850
0.0128 −0.4414
B2 (P2 , I4 )
X
X
0.0118
0.0275 −0.0358
C2 (L1 , L1 , L2 )
X
X
X
X
0.0141
C2 (L2 , L2 , L2 )
X
X
X
X
0.0048
σ2
0.1330
0.1316
0.1590
0.1167
0.1573
κ2
1.1779
1.2138
2.3206
23.612
46.400
d(2 )
0.8099
0.8277
0.5262
0.7512
0.5213
d(∆2 )
1.9709
1.9711
2.0206
2.2984
2.0615
1
tc (2 )/tc (L2 )
0.2270
0.2270
0.2220
0.2257
0.2159
Table 2: Properties of mode I1 in the regression models for the intermittent modes
(I1 , . . . , I4 ) from equation (3) in the main text. {Ai , Bi , Ci }: Regression coefficients.
σi : Standard deviation of the residual. κi : Condition number of the design matrix.
{d(i ), d(∆i )}: Durbin-Watson statistics for the residual and residual increments for 1
m lag. tc (i )/tc (Ii ): noise decorrelation time relative to modal decorrelation time. X (X)
indicates external factors/interactions which were (were not) part of a given model.
Model
1
2
3
4
Ext. factors
P1 , P2
X
X
X
X
L1 , L2
X
X
X
X
A1 (I1 )
0.8787
X
0.2275 −0.0278
A1 (I2 )
−0.4118
X −0.0618
0.0298
A1 (I3 )
0.0348
X −0.2505 −0.3228
A1 (I4 )
0.2241
X
0.8530 −0.1051
B1 (P1 , L1 )
X
0.7031
0.0420 −0.4160
B1 (P2 , L1 )
X −0.3625 −0.1865 −0.5110
B1 (P2 , L2 )
X −0.6142
0.0091
0.9017
C1 (I1 , I1 , I1 )
X
X
X
0.0155
C1 (I1 , I2 , I2 )
X
X
X −0.0325
C1 (I2 , I2 , I2 )
X
X
X
0.0044
C1 (I1 , I3 , I3 )
X
X
X
0.0172
C1 (I1 , I4 , I4 )
X
X
X −0.0147
C1 (I4 , I4 , I4 )
X
X
X −0.0050
σ1
0.1077
0.1973
0.1056
0.2140
κ1
1.1180
1.1629
22.758
21.880
d(1 )
0.8062
0.4021
0.8444
0.5977
d(∆1 )
1.8797
1.3187
1.8916
1.4349
tc (1 )/tc (I1 )
0.1895
0.1892
0.3009
0.3181
2
Table 3: Same as Table 2, but for mode I2
Model
1
2
3
4
Ext. factors
P1 , P2
X
X
X
X
L1 , L2
X
X
X
X
A2 (I1 )
0.4168
X
0.3446 −0.0975
A2 (I2 )
0.8603
X
0.7029
0.0058
A2 (I3 )
0.2701
X
0.4159
0.2879
A2 (I4 )
-0.0800
X −0.1207 −0.2679
B2 (P1 , L1 )
X 0.6157
0.0196
0.6017
B2 (P2 , L1 )
X 0.7006
0.2062
0.8294
B2 (P2 , L2 )
X 0.3065 −0.1131 −0.0323
C2 (I1 , I1 , I1 )
X
X
X −0.0010
C2 (I1 , I1 , I2 )
X
X
X −0.0057
C2 (I2 , I2 , I2 )
X
X
X −0.0007
σ2
0.1109 0.2484
0.1059
0.1355
κ2
1.1180 1.1629
22.758
43.016
d(2 )
1.0876 0.4758
1.1286
0.8238
d(∆2 )
1.9688 1.1089
1.9597
1.7343
tc (2 )/tc (I2 )
0.4533 0.4458
0.5006
0.4217
3
Table 4:
Model
Ext. factors
P1 , P2
L1 , L2
A3 (I1 )
A3 (I2 )
A3 (I3 )
A3 (I4 )
B3 (P1 , L1 )
B3 (P2 , L1 )
B3 (P1 , L2 )
B3 (P2 , L2 )
C3 (I2 , I2 , I2 )
C3 (I1 , I1 , I3 )
C3 (I2 , I2 , I3 )
C3 (I3 , I3 , I3 )
C3 (I3 , I4 , I4 )
C3 (I4 , I4 , I4 )
σ3
κ3
d(3 )
d(∆3 )
tc (3 )/tc (I3 )
Same as Table 2, but for mode I3
1
2
3
X
X
−0.0493
−0.2835
0.8449
−0.4260
X
X
X
X
X
X
X
X
X
X
0.1345
1.1180
0.8541
1.7568
0.1985
X
X
X
X
X
X
0.2527
−0.6040
0.2921
0.7107
X
X
X
X
X
X
0.2090
1.1709
0.5706
1.3992
0.1820
4
X
X
0.0161
−0.5005
0.7992
−0.4912
0.0237
0.1872
−0.0811
0.1303
X
X
X
X
X
X
0.1302
22.846
0.8448
1.8681
0.1887
4
X
X
0.0909
0.0034
−0.0295
0.0124
0.1945
−0.6298
0.2740
0.8137
−0.0016
0.0112
−0.0005
−0.0090
−0.0078
0.0027
0.2246
21.913
0.5541
1.3739
0.1954
Table 5: Same as Table 2, but for mode
Model
1
2
3
Ext. factors
P1 , P2
X
X
X
L1 , L2
X
X
X
A4 (I1 )
−0.2194
X
0.6970
A4 (I2 )
0.0537
X
0.1936
A4 (I3 )
0.4351
X −0.0378
A4 (I4 )
0.8534
X −0.0795
B4 (P1 , L1 )
X
0.2476
0.0636
B4 (P1 , L2 )
X −0.8812 −0.2075
B4 (P2 , L2 )
X
0.2408
0.2586
C4 (I3 , I3 , I3 )
X
X
X
C4 (I1 , I1 , I4 )
X
X
X
C4 (I3 , I3 , I4 )
X
X
X
C4 (I4 , I4 , I4 )
X
X
X
σ4
0.1677
0.1981
0.1561
κ4
1.1180
1.1670
21.077
d(4 )
0.6916
0.1659
0.8271
d(∆4 )
2.0263
1.6131
1.9311
tc (4 )/tc (I4 )
0.3029
0.2260
0.2538
5
I4
4
X
X
−0.0051
−0.0304
−0.0840
0.0160
0.2693
−0.9084
0.3034
−0.0009
−0.0093
0.0017
−0.0014
0.1869
43.075
0.7822
1.6549
0.1975
Mode L1
Mode L2
1
θA = 0.02
θA = 0.02
(a)
1−P(ρij)
0.8
0.6
0.4
0.2
0
1
θB = 0.2
θB = 0.2
(b)
1−P(ρijk)
0.8
0.6
0.4
0.2
0
1
θC = 0.75
θC = 0.75
(c)
1−P(ρijkl)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
ρ/ρmax
0.8
1 0
0.2
0.4
0.6
ρ/ρmax
0.8
1
Figure 1: Cumulative distribution functions of the absolute values of the (a) double,
ρij (t) = hvi (τ )Lj (τ + t)i, (b) triple, ρijk (0, t) = hvi (τ )vj (τ )Lk (τ + t)i, and (c) quadruple, ρijkl (0, 0, t) = hvi (τ )vj (τ )vk (τ )Ll (τ + t)i, correlation coefficients evaluated for the lowfrequency modes, {L1 , L2 } = {v5 , v6 } at lead time t = 1 m. The predictor variables vi (τ )
are (a) modes P1 , P2 , L1 , L2 , L3 , I1 , . . . , I4 , (b) double products of modes P1 , P2 , I1 , . . . , I4 ,
(c) triple products of modes L1 , L2 , L3 . In each panel, the horizontal axis has been scaled
by the maximum absolute value ρmax of the correlation coefficients in the corresponding
group. The dashed vertical lines indicate the θ values used for univariate thresholding.
The admissible interactions lie to the right of those lines. Note that the coefficients in
(c) involving mode L3 were not used for regression modeling, but are included here for
reference.
6
Mode I1
Mode I2
1
θA = 0.02
θA = 0.02
(a)
1−P(ρij)
0.8
0.6
0.4
0.2
0
1
θB = 0.2
θB = 0.2
(b)
θC = 0.75
(c)
1−P(ρijk)
0.8
0.6
0.4
0.2
0
1
θC = 0.75
1−P(ρijkl)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
ρ/ρmax
0.8
1 0
0.2
0.4
0.6
ρ/ρmax
0.8
1
Figure 2: Cumulative distribution functions of the absolute values of the (a) double, ρij (t) = hvi (τ )Ij (τ + t)i, (b) triple, ρijk (0, t) = hvi (τ )vj (τ )Ik (τ + t)i, and (c)
quadruple, ρijkl (0, 0, t) = hvi (τ )vj (τ )vk (τ )Il (τ + t)i evaluated for intermittent modes
{I1 , I2 } = {v10 , v11 } at lead time t = 1 m. The predictor variables vi (τ ) are (a) modes
P1 , P2 , L1 , L2 , L3 , I1 , . . . , I4 , (b) double products of modes P1 , P2 , L1 , L2 , L3 , (c) triple products of modes I1 , . . . , I4 . In each panel, the horizontal axis has been scaled by the maximum
absolute value ρmax of the correlation coefficients in the corresponding group. The dashed
vertical lines indicate the θ values used for univariate thresholding. The admissible interactions lie to the right of those lines.
7
Mode I3
Mode I4
1
θA = 0.02
θA = 0.02
(a)
1−P(ρij)
0.8
0.6
0.4
0.2
0
1
θB = 0.2
θB = 0.2
(b)
θC = 0.75
(c)
1−P(ρijk)
0.8
0.6
0.4
0.2
0
1
θC = 0.75
1−P(ρijkl)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
ρ/ρmax
0.8
1 0
0.2
0.4
0.6
ρ/ρmax
0.8
1
Figure 3: Same as Figure 2, but for intermittent modes {I3 , I4 } = {v12 , v13 }
8
L1
L2
1E−1
Model 1
Prob. density
1E0
1E−2
1E−1
Model 2
Prob. density
1E−3
1E0
1E−2
1E−1
Model 3
Prob. density
1E−3
1E0
1E−2
1E−1
Model 4
Prob. density
1E−3
1E0
1E−2
1E−1
Model 5
Prob. density
1E−3
1E0
1E−2
1E−3
−8 −6 −4 −2 0 2 4 6
Standardized residual, εi/σi
8 −8 −6 −4 −2 0 2 4 6
Standardized residual, εi/σi
8
Figure 4: Residual histograms for models 1–5 in Table 1. The empirical histograms (black
lines) were computed by binning the n = 2687 monthly samples i (t) in the training timeseries into b = 20 uniform bins in the range [min i (t), max i (t)]. Gaussian distributions
with zero mean and unit variance are plotted in green lines for reference.
9
I1
I2
I3
I4
1E−1
Model 1
Prob. density
1E0
1E−2
1E−1
Model 2
Prob. density
1E−3
1E0
1E−2
1E−1
Model 3
Prob. density
1E−3
1E0
1E−2
1E−1
Model 4
Prob. density
1E−3
1E0
1E−2
1E−3
−8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8
Standardized residual, εi/σi
Standardized residual, εi/σi
Standardized residual, εi/σi
Standardized residual, εi/σi
Figure 5: Same as Figure 4, but for models 1–4 in Tables 2–5
10
L2
3
2
Model 1
1
0
−1
−2
−3
3
2
Model 2
1
0
−1
−2
−3
3
2
Model 3
1
0
−1
−2
−3
3
2
Model 4
1
0
−1
−2
−3
3
2
1
Model 5
Percentiles of standardized Percentiles of standardized Percentiles of standardized Percentiles of standardized Percentiles of standardized
residuals, εi/σi
residuals, εi/σi
residuals, εi/σi
residuals, εi/σi
residuals, εi/σi
L1
0
−1
−2
−3
−3
−2 −1
0
1
2
3 −3 −2 −1
0
1
2
3
Percentiles of standard normal
Percentiles of standard normal
Figure 6: Quantile-quantile plots of the standardized residuals from models 1–5 in Table 1
versus standard normal variables
11
I2
I3
I4
Model 1
Model 2
3
2
1
0
−1
−2
−3
Model 3
3
2
1
0
−1
−2
−3
3
2
1
0
−1
−2
−3
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
Percentiles of standard normal Percentiles of standard normal Percentiles of standard normal Percentiles of standard normal
Figure 7: Quantile-quantile plots of the standardized residuals from models 1–4 in Tables 2–
5 versus standard normal variables
12
Model 4
Percentiles of standardized Percentiles of standardized Percentiles of standardized Percentiles of standardized
residuals, εi/σi
residuals, εi/σi
residuals, εi/σi
residuals, εi/σi
I1
3
2
1
0
−1
−2
−3
L2
Model 1
Model 2
Model 3
Model 4
Model 5
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
L1
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0
1
2
Lag t (y)
3
4 0
1
2
Lag t (y)
3
4
Figure 8: Lagged correlation coefficients of the residuals from models 1–5 in Table 1 (blue
lines). Shown for reference are the lagged correlation coefficients of the corresponding
response variables, L1 and L2 (green lines).
13
I2
I3
I4
Model 1
0.5
0
−0.5
−1
1
Model 2
0.5
0
−0.5
−1
1
Model 3
0.5
0
−0.5
−1
1
0.5
Model 4
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
Lagged correlation,
<εi(t+τ)εi(τ)>/σ2i
I1
1
0
−0.5
−1
0
1
2
3
Lag τ (y)
4 0
1
2
3
Lag τ (y)
4 0
1
2
3
Lag τ (y)
4 0
1
2
3
Lag τ (y)
Figure 9: Lagged correlation coefficients of the residuals from models 1–4 in Tables 1–4
(blue lines). Shown for reference are the lagged correlation coefficients of the corresponding
response variables, I1 , . . . , I4 (green lines).
14
4