Entropies and Frequencies

Figure 3.12 shows the distributions of the entropies and frequencies of the SOI, SST and MEI anomalies corresponding to all 589 analyzedn-blocks, each for all time series. The plots of the entropies and frequencies are characterized by a left side with blocks of zero or unity frequency and, consequently, zero entropy, and a right side with blocks of irregular frequencies and a steeply growing entropy. The plot takes this form as is ordered in the manner of an histogram in which the entropy increases. Each part of every plot is of different length: this is an expression of the proportion of the blocks which occur only once or are not present in the series (and therefore are given entropy zero) and those which appear more frequently.

It can be seen for all series that the most frequent blocks coincide in great part with those of the least entropy. This is an important result and a necessary condition for a reliable estimation of the most predictable states. Comparing the indices, the SOI series (on top) shows the largest section of populated blocks, leaving seventy of the 589 blocks empty, and containing a uniform curve of growing entropies which is steeper than that of the other two ENSO indices. To this series belongs only one block with an entropy below 0.5. In the plot in the middle, the SST anomalies series presents 217 blocks of zero-frequency, a fact which is not a consequence of finite-size effects, since its length is comparable to that of SOI. Having in addition 31 blocks with high frequency and uncertainties below 0.5, this is evidence for a predictability and order inherent to this series which is higher than that of the SOI. Due to its smaller length, the MEI anomalies series (at the bottom) shows the graph with the largest section of zero-frequency blocks. This series shows 343 empty blocks and simultaneuosly the smallest entropies of all the series in most of its 28 blocks of uncertainties below 0.5. Thus, it could be believed that the most predictable symbolic sequences of the anomalies of ENSO can be detected by

50 100 150 200 250 300 350 400 450 500 550 Conditional Entropies and Frequencies of the SOI, SST and MEI Anomalies

SOI

Figure 3.12: Conditional entropies of 589 blocks from the SOI, SST and MEI anomalies series. The histogram of the entropies (continuous curve, scale in the left y-axis) and the frequencies of the respective blocks (irregular curve, scale in the right y-axis) are depicted. The x axis denotes the numbering of the blocks, which obeys an order of increasing entropy.

the MEI quantifier, while its most complete dynamics in the symbolic phase space are found with a study of the SOI. Nevertheless, these indices present difficulties for they are extreme cases of an opposite relationship between an ex-tense distribution of the blocks in the space and the amount of blocks with low predictability. Thus, a more equilibrated variable in length and predictabil-ity like the SST should be more reliable. Having similar predictabilities as MEI and being much more deterministic than SOI, this series allows for a more significant and sharply distributed study of the lowest predictabilities of the anomalies of the Southern Oscillation, after the occurence of an arbitrary block.

Figure 3.13 shows the same plot for the persistences of ENSO, which are

considerably different. First of all, there is no visible correlation between the size of the entropies and the frequency of the blocks. In this light, it cannot be assured that most of the interesting blocks comply with enough statistics. Comparing again the indices, the SST (middle plot) is the index which populates the most possible blocks after the symbolic dynamic analysis.

There are only four symbolic events which do not occur from this series, all of them at the fifth order with a terniary partition. Hence, the SST is the variable which fluctuates in most of the symbolic phase space, while it can be seen that its entropies are distributed very steeply towards one. Contrary to the analysis of the anomalies, the SOI persistences series (on top) has more unused blocks and a more broadly distributed spectrum of frequent blocks than SST. The entropies of the series are, however, similarly very high, a fact which prevents most of the fluctuations from having a definite following block.

As ever, the MEI series (below) is the only series giving a large number of zero-frequency blocks. This fact is related to the length of the MEI series but also to a particular regularity of its anomalies, since it is shown from SST that a relatively ordered anomalies series does not imply an order in their persistences.

The steepness of the growth of the entropies is much higher for all of these series than for their anomalies, as can be illustrated with three statistics of the distributions of the entropies: for the SOI, 88% of the blocks of non-zero frequency have entropies above 0.90, 69% of them above 0.95 and 25% above 0.99. The same results for the SST are 96%, 73% and 30% respectively, as 78%, 50% and 15% for the MEI. Consequently, it is difficult to choose any of the ENSO persistence sequences for using the conditional entropies up to the fifth order in predictability problems. The general problem of finding useful predictabilities is better suited for an investigation of the anomalies series.

However, an interesting question for the study of both anomalies and persis-tences remains unanswered: which of these series gives a higher predictability

0 Conditional Entropies and Frequencies of the SOI, SST and MEI Persistences

SOI

Figure 3.13: Conditional entropies of 589 blocks from the SOI, SST and MEI persistences series. The histogram of the entropies (continuous curve, scale in the left y-axis) and the frequencies of the respective blocks (irregular curve, scale in the right y-axis) are depicted. The x axis denotes the numbering of the blocks, which obey an order of increasing entropy.

after a specific development in the Southern Oscillation, i.e. a definite n-block has been found.

The study of the frequencies of the analyzed series gives results which similarly differ for the anomalies and persistences. For the anomalies, as will be evident in the next section, it is found that the most frequent blocks represent constancies, and that these are more frequent over many lengths for some partition than over several partitions for some length. This means that, for the resolution allowed by the length of the series, the tendency for a state to be maintained is considerably preponderant over the fluctuations. This can be illustrated, for instance, with the fact that the most frequent block of a tertiary partition is found in the sixth, seventh and eight position in the histograms of

frequencies of SOI, SST and MEI, respectively, while before them only binary constancies (which are sequences of zeros and ones) are present.

For partitioning orders greater than two, the constant blocks at the ex-tremes of the partitioned space are more frequent than those in its center, and their entropies are lower in the extremes than in the center. As a fur-ther example, for the second-order blocks from tertiary partitions of the SOI anomalies, the El Niño phase is represented with 327 blocks, La Niña with 315 and the neutral state in between with 214. There are 195 transitions to an El Niño, 184 to a La Niña and 294 to the intermediate state. Remembering that the constancies of each state correspond to one block, while the amount of transitions to each of the states for this partition is given as the sum of two blocks, it should be clear that the constant blocks are the most represented even in the noisiest series.

The case of the persistences differs, since the frequency of the blocks is not related to their character of constancies or transitions, and small blocks of higher-order partitions are found with higher frequencies than larger ones of smaller partitions. The most frequent blocks of the SOI fluctuations are the elementary binary transitions 01 and 10, occuring 467 times and followed by their respective constancies 00 and 11 which occur 297 times. The same four blocks are more frequent in the SST, where the constancies appear more often (414 times) than the transitions (383 times), as occurs in the MEI, with 179 against 141 times. These results will be encountered and addressed in more detail below, in the context of their predictabilities.

For the study of individual blocks we shall introduce the following notation:

the label of the block will be followed by a p preceding the number of the partitions. Thus, for instance, the mentioned elementary blocks of dynamical order 2 with a binary partition will be denoted as ⁰00p1⁰ and ⁰11p1⁰ for the constancies, and ⁰01p1⁰,⁰10p1⁰ for the transitions.