A previous post described the use of proxy records of ENSO to fit the Southern Oscillation Index Model (SOIM). This model fit used one specific set of data that featured a disconnected record of coral measurements from the past 1000 years, see Cobb .
As the focus of this post, another set of data (the Unified ENSO Proxy set) is available as an ensemble record of various proxy measurements since 1650 -- giving an unbroken span of over 300 years to apply a SOIM fit . This ensemble features 10 different sets, which includes the Cobb coral as a subset.
To fit over this long a time span is quite a challenge as it assumes that the time series is stationary over this interval. The data has a resolution of only one year, in comparison to the monthly data previously used, so it may not have the temporal detail as the other sets, yet still worthy of investigation. (an interactive version is available here).
For the initial attempt, the same Mathieu-like differential equation structure and parameters were applied as for the previous proxy and for the modern-day SOI data. The figure below is repeated from the previous proxy post.
Parameters approximating those in Figure 1 are used in the unified ENSO Proxy fit as shown below:
This correlation coefficient is only 0.256, which is not as good as the previous model fits over shorter time spans.
To see if this fit could be improved, the strategy chosen was to start from a reduced set of forcing data. So the forcing parameters corresponding to w1 and w2 were removed, leaving only w3. A small drag factor was also included as y'[x].
This improved the correlation coefficient from 0.256 to 0.336, which is not bad considering the length of data.
As a final fit, an additional forcing sinusoid was added corresponding to 1.548 rads/year (see Figure 4). This improved the correlation coefficient marginally to above 0.36.
Why do the values of w1 and w2 of around 2.5 to 3 rad/year not appear on this extended proxy record? As these are higher frequencies, it is entirely possible that coherence can not be maintained over the 300+ year range of proxy values that the SOIM model is fit against. The sampling resolution of 1 year also works against selecting higher frequencies.
Recall that high correlation coefficients are hard to achieve on these oscillating time series. Even the correlation coefficient of Tahiti to Darwin is only -0.55 as shown in the previous post on SOIM. Any amount of noise will quickly reduce the coefficient from 1.0. However, the authors of the Unified ENSO Proxy study find high correlation coefficients ranging from 0.5 to 0.84 to the SOI and various Pacific equatorial SST data sets during the modern era. This indicates that the unified proxy reconstruction likely does what it is intended to -- represent the actual dynamics of past history without having the use of thermometers or hygrometers.
The variance in the 10 proxy reconstructions is shown below in Figure 5.
An interesting feature of the Mathieu equation fit is how it captures the variance in the ENSO signal across decades. The early years show less variance in ENSO peak-to-peak modulation than the later years, and the Mathieu equation formulation can capture this growing modulation. See  for more.
Overall the results remain encouraging and it is good to know that so many proxy reconstructions are available for ENSO along with historical dates for El Nino events, and that this data can be further mined for fitting extended time-span models.
 Cobb, Kim M, Christopher D Charles, Hai Cheng, and R Lawrence Edwards. “El Nino/Southern Oscillation and Tropical Pacific Climate during the Last Millennium.” Nature 424, no. 6946 (2003): 271–76.
 McGregor, S., A. Timmermann, and O. Timm. “A Unified Proxy for ENSO and PDO Variability since 1650.” Clim. Past 6, no. 1 (January 5, 2010): 1–17. doi:10.5194/cp-6-1-2010. PDF
 Carré, Matthieu, Julian P Sachs, Sara Purca, Andrew J Schauer, Pascale Braconnot, Rommel Angeles Falcón, Michèle Julien, and Danièle Lavallée. “Holocene History of ENSO Variance and Asymmetry in the Eastern Tropical Pacific.” Science (New York, NY), 2014.