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Explora: Environment
and Resource Statistical analysis of climate time series

Now let’s return to our sequence of one hundred throws. refers to a series with no trend in either its mean or
The experiment completed, we realize that in fact, the the standard deviation and with empirically calculated
average length of the chains of successive H (or T) is clearly correlations that are nearly zero and independent of
less than the theoretical figure of 1.94, that there are many their positions in time.
more chains of 1 and 2 than predicted by the frequencies 3. Create a third series consisting of 1s and 0s by
and much less for channels of 3 and more. The alternation assigning the value 1 to each gap if it is positive and
is therefore much faster than predicted by theory. This can 0 otherwise. Then, verify that the point frequencies of
happen, and at this point, there is no reason to question the 0s and 1s are both close to 0.5, ensuring that increases
truly binomial character of the experiment. and decreases in the studied parameter can reasonably
be considered equiprobable.
On the other hand, if we repeat this experience of
100 tosses ten times in a row, and each time we find a 4. If either of the previous two steps fails, MAL cannot
be used. However, if both conditions are satisfied, all
clearly accelerated alternation, it becomes necessary to prerequisites are met for modeling the binary series of
question the underlying cause. It is almost impossible for 1s and 0s using a Markov chain (p. 682 of Hamilton ).
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this to be a result of pure chance, and since the coin is 5. The temporal spacing (e.g., day, month, quarter, year)
perfectly balanced, the cause is most likely an interaction used in the initial time series, along with the “climatic
with an external phenomenon. The precise origin of memory” of the phenomenon, must be taken into
this interaction – whether a periodic electromagnetic account to determine the appropriate orders for the
influence or another factor – cannot be determined Markov chains. Based on these two parameters, only
without further investigation. Nonetheless, we can assert orders 0 and 1 were applicable in the four studies. 23-26
with high confidence that an external interaction is at play. The following terminology is used to describe average
This consistently shorter-than-expected average length of chain lengths: (A) Close to 1.94: Markov-0 signal;
successive H (or T) chains serves as a “signal” indicating (B) Significantly lower than 1.94: Markov-1 alternating
such an interaction. signal; (C) Significantly higher than 1.94: Markov-1

This is the core concept of MAL: if, in a given time lengthening signal. However, nothing precludes the
series with two equally probable outcomes A or B, the possibility that higher-order Markov chains may be
vast majority of chains of 100 successive outcomes more adequate, especially when the climatic memory
exhibit significantly lower average lengths of successive of the phenomenon significantly exceeds the length of
A (orB) subchains at 1.94, it is reasonable to suspect the time series.
the existence of an interaction with another entity 6. In the event of uncertainty between two possible
that explains the stronger-than-expected alternation. orders, calculating the probabilities of obtaining the
Conversely, if the average lengths are significantly higher observed chain lengths for each potential order allows
than 1.94, the hypothesis of an interaction must also be for determining which order is most appropriate for
seriously considered. modeling the series.
7. In the case of order 0, the MAL does not reveal any
If the time series in question concerns a certain climatic signal of interaction with another climatic entity,
entity with two equally probable outcomes, these average though this does not mean there is none. However, as
lengths can serve as the “signals” indicating the presence of soon as the order is ≥1, as seen in the introductory
interactions with other climatic entities. A priori, it will be example, it is a strong indication of an interaction with
necessary to identify potential interactions among entities one or more other entities.
whose numerical data exhibit concordant signals, enabling 8. In this latter case, the next step is to identify these
effective selection for further investigation. entities and verify that the signals observed in them

2.2. The main steps of MAL are consistent with the proposed explanation.
9. The MAL method concludes at this point. It is then the
Consider a certain time series concerning a climatic responsibility of climatology and modeling to validate
parameter with numerical values. The following steps the proposed explanations.
outline its processing using MAL:
1. From the initial series, generate a second series 3. Examples of MAL application in
consisting of “gaps,” defined as the difference between climatology
each term in the initial series and the one immediately
preceding it. 3.1. The first application of MAL
2. Verify whether this series of gaps can be modeled In the first study on MAL, fluctuations in the global
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as “white noise” (p. 45 of Hamilton ). “White noise” monthly average atmospheric temperature were analyzed
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Volume 2 Issue 1 (2025) 3 doi: 10.36922/eer.6109