With the AGU meeting coming up, I am rationalizing my confidence in the QBO and ENSO models. Earlier I was much more confident in the QBO model, as the results were so clean. But now my confidence in the ENSO model has risen to the same level as for QBO.

The confidence test is the following. I take a pair of non-overlapping and non-contiguous intervals in the ENSO SOI time-series. Then I use the solver to extract the underling components in each of the intervals and essentially compare the two. If the behavior of ENSO were either highly chaotic or red-noise Markovian, the fits would be markedly different as each would follow a different trajectory. However, if the two are composed of the same periodic factors, then the odds are that ENSO is a deterministic cycle.

Here is the lower fit with the given training interval described by the dotted green line. I split the fit into two intervals to capture the longer periods accurately.

![](http://imageshack.com/a/img924/3416/3PDDmj.png)

Here is the high interval fit, which as you can see does not overlap with the lower interval fit.

![](http://imageshack.com/a/img923/2949/Q9BUkM.png)

As I have described before, I flip the phase in the years 1980-1996 to capture the known climate shift, and use the wave-equation transform of the data instead of solving the DiffEq directly.

These are the periodic component comparisons between the lower and higher intervals

![http://imageshack.com/a/img921/3135/veqzU0.png](http://imageshack.com/a/img921/3135/veqzU0.png)

These all align very tightly, with the discrepancies indicating that it isn't some artifact of a flawed fitting process (i.e. the x=x problem).

The following shows the comparison between the annual harmonics (1/2, 1/3, etc periods with the annual period effectively filtered out in the original SOI time series) for the low and high regions and also the Mathieu modulation comparison

![http://imageshack.com/a/img924/6391/NphZ5M.png](http://imageshack.com/a/img924/6391/NphZ5M.png)

The years prior to 1910 were not used in the fit because the data appears much more noisy than the data post that date. Incidentally, no filtering was used during the model fitting process.

Here is the model fit over the interval from 1895 to 2013. The Mathieu modulation comes stronger as the correlation of the longer interval reinforces the fit/

![all](http://imageshack.com/a/img924/575/s0kJuJ.png)

Also note that only data to 2013 was used and that the extrapolated fit predicts that the 2016 SOI spike was almost as strong as the 1998 event and next to that the strongest in the last 100 years. This is a graph focused only on the last 50 years, so you can see the predictive extrapolation more closely.

![](http://imageshack.com/a/img924/9074/5VVywf.png)

These are all based on the known Earth wobble and lunar tidal periods and really confirms that ENSO is a nearly pure deterministic stationary process driven by known geophysical forcings. And like the tidal models that this ENSO model emulates, the longer the period to extract from and the more lunar periods that are included, the better the fit becomes.

It only deviates from stationary determinism in terms of the odd vs even parity of the biennial modulation. The biennial modulation flips from an even-year parity before 1980 to an odd-year parity between the years 1980-1996. I only lack confidence in how to predict these flips, much like I lack the confidence to predict volcanic events which seems to impact the QBO sporadically.

The confidence test is the following. I take a pair of non-overlapping and non-contiguous intervals in the ENSO SOI time-series. Then I use the solver to extract the underling components in each of the intervals and essentially compare the two. If the behavior of ENSO were either highly chaotic or red-noise Markovian, the fits would be markedly different as each would follow a different trajectory. However, if the two are composed of the same periodic factors, then the odds are that ENSO is a deterministic cycle.

Here is the lower fit with the given training interval described by the dotted green line. I split the fit into two intervals to capture the longer periods accurately.

![](http://imageshack.com/a/img924/3416/3PDDmj.png)

Here is the high interval fit, which as you can see does not overlap with the lower interval fit.

![](http://imageshack.com/a/img923/2949/Q9BUkM.png)

As I have described before, I flip the phase in the years 1980-1996 to capture the known climate shift, and use the wave-equation transform of the data instead of solving the DiffEq directly.

These are the periodic component comparisons between the lower and higher intervals

![http://imageshack.com/a/img921/3135/veqzU0.png](http://imageshack.com/a/img921/3135/veqzU0.png)

These all align very tightly, with the discrepancies indicating that it isn't some artifact of a flawed fitting process (i.e. the x=x problem).

The following shows the comparison between the annual harmonics (1/2, 1/3, etc periods with the annual period effectively filtered out in the original SOI time series) for the low and high regions and also the Mathieu modulation comparison

![http://imageshack.com/a/img924/6391/NphZ5M.png](http://imageshack.com/a/img924/6391/NphZ5M.png)

The years prior to 1910 were not used in the fit because the data appears much more noisy than the data post that date. Incidentally, no filtering was used during the model fitting process.

Here is the model fit over the interval from 1895 to 2013. The Mathieu modulation comes stronger as the correlation of the longer interval reinforces the fit/

![all](http://imageshack.com/a/img924/575/s0kJuJ.png)

Also note that only data to 2013 was used and that the extrapolated fit predicts that the 2016 SOI spike was almost as strong as the 1998 event and next to that the strongest in the last 100 years. This is a graph focused only on the last 50 years, so you can see the predictive extrapolation more closely.

![](http://imageshack.com/a/img924/9074/5VVywf.png)

These are all based on the known Earth wobble and lunar tidal periods and really confirms that ENSO is a nearly pure deterministic stationary process driven by known geophysical forcings. And like the tidal models that this ENSO model emulates, the longer the period to extract from and the more lunar periods that are included, the better the fit becomes.

It only deviates from stationary determinism in terms of the odd vs even parity of the biennial modulation. The biennial modulation flips from an even-year parity before 1980 to an odd-year parity between the years 1980-1996. I only lack confidence in how to predict these flips, much like I lack the confidence to predict volcanic events which seems to impact the QBO sporadically.