I am doing multiple linear regression of several aliased tidal periods on the QBO from 1953 to present and using the same factors on multiple linear regression on ENSO from 1880 to 1980 and the agreement in the scaling of the common factors is striking.


ENSO 2nd derivative

Estimate Std. Error t value Pr(>|t|)
msS[1:1200] 0.393387 0.076489 5.143 3.17e-07 ***
msC[1:1200] 0.300668 0.076635 3.923 9.24e-05 ***
s9S[1:1200] -0.155704 0.076556 -2.034 0.042190 *
s9C[1:1200] -0.014996 0.076896 -0.195 0.845411
diS[1:1200] 0.010538 0.076490 0.138 0.890446
diC[1:1200] 0.005264 0.077649 0.068 0.945965
q4S[1:1200] 0.288473 0.076631 3.764 0.000175 ***
q4C[1:1200] 0.737927 0.076501 9.646 < 2e-16 ***
qmS[1:1200] 0.267377 0.078113 3.423 0.000641 ***
qmC[1:1200] -1.291226 0.078034 -16.547 < 2e-16 ***
q1S[1:1200] 0.802413 0.076488 10.491 < 2e-16 ***
q1C[1:1200] -0.085617 0.076619 -1.117 0.264033
cwS[1:1200] 0.419867 0.076419 5.494 4.81e-08 ***
cwC[1:1200] -0.008576 0.076778 -0.112 0.911085
sdS[1:1200] -0.476016 0.076555 -6.218 6.99e-10 ***
sdC[1:1200] 0.208593 0.076419 2.730 0.006436 **
qsS[1:1200] 0.808250 0.077621 10.413 < 2e-16 ***
qsC[1:1200] 0.220302 0.077570 2.840 0.004589 **
spS[1:1200] -0.754940 0.077362 -9.759 < 2e-16 ***
spC[1:1200] 1.306538 0.077567 16.844 < 2e-16 ***
q2S[1:1200] 0.794521 0.076619 10.370 < 2e-16 ***
q2C[1:1200] 0.487892 0.076766 6.356 2.97e-10 ***
q3S[1:1200] 0.831220 0.076436 10.875 < 2e-16 ***
q3C[1:1200] -0.547102 0.076755 -7.128 1.78e-12 ***
qbS[1:1200] -2.246223 0.077833 -28.859 < 2e-16 ***
qbC[1:1200] 0.484379 0.078057 6.205 7.55e-10 ***




QBO

Estimate Std. Error t value Pr(>|t|)
msS[877:1596] -1.26202 0.22753 -5.547 4.15e-08 ***
msC[877:1596] -1.56597 0.22743 -6.886 1.29e-11 ***
s9S[877:1596] 0.22970 0.22723 1.011 0.312445
s9C[877:1596] 0.42552 0.22646 1.879 0.060659 .
diS[877:1596] 0.19315 0.23125 0.835 0.403888
diC[877:1596] -0.03392 0.22352 -0.152 0.879440
q4S[877:1596] 0.45803 0.22857 2.004 0.045470 *
q4C[877:1596] 0.16178 0.22860 0.708 0.479365
qmS[877:1596] -0.82360 0.22496 -3.661 0.000270 ***
qmC[877:1596] -3.20132 0.22533 -14.207 < 2e-16 ***
q1S[877:1596] -0.69555 0.22555 -3.084 0.002125 **
q1C[877:1596] -0.98784 0.22509 -4.389 1.32e-05 ***
cwS[877:1596] 0.37967 0.22720 1.671 0.095156 .
cwC[877:1596] -0.11175 0.22602 -0.494 0.621151
sdS[877:1596] -0.03667 0.22830 -0.161 0.872449
sdC[877:1596] 1.46746 0.22857 6.420 2.53e-10 ***
qsS[877:1596] 0.72815 0.22558 3.228 0.001306 **
qsC[877:1596] -3.11850 0.22413 -13.914 < 2e-16 ***
spS[877:1596] 1.81587 0.22445 8.090 2.67e-15 ***
spC[877:1596] 0.86197 0.22583 3.817 0.000147 ***
q2S[877:1596] 0.07538 0.22677 0.332 0.739683
q2C[877:1596] 1.14416 0.22582 5.067 5.20e-07 ***
q3S[877:1596] -0.08424 0.22747 -0.370 0.711238
q3C[877:1596] 1.57475 0.22771 6.916 1.06e-11 ***
qbS[877:1596] 3.58416 0.22578 15.875 < 2e-16 ***
qbC[877:1596] -6.14774 0.22424 -27.416 < 2e-16 ***


This looks like many more factors than there actually are. In the multiple linear regression approach, for a particular sinusoidal factor, the Sin and Cos are considered independently so as to get the correct phase. So the character S or C is appended to the factors to distinguish between the Sin and Cos factor.

Anything with 3 asterisks (***) has a high statistical significance. If either a Sin or Cos has 3 asterisks, that is a strong contributing factor. (If both do, that only means it happens to be a linear combination)

The factors that do not add to the fit have periods greater than 4 years, and so will contribute to the actual ENSO (not the second derivative). The QBO is not as sensitive to these long-period forcing as the stratosphere has a much smaller inertia, and so responds primarily to the faster tidal factors.


Both QBO and ENSO fits have correlation coefficients greater than 0.8, which is actually difficult to achieve with non-trending pseudo-oscillating data.

![a](http://imageshack.com/a/img537/4889/82vQYf.png)


The QBO plot is truncated up to 1953 because data for QBO is not available until after this date.

![b](http://imageshack.com/a/img673/4091/OX5E2a.png)


I expect to hear some comments to the effect that "With four parameters I can fit an elephant", which is OK by me.

This may need to be changed to "With the same nine parameters I can fit an elephant and a hummingbird". Scientists have always thought that ENSO (the elephant) and QBO (the hummingbird) differed behaviorally. That is not so true as they appear to respond to a common set of tidal factors.