Page 43 - EER-2-1

Explora: Environment
and Resource Statistical analysis of climate time series

Often the equation is in this simple form: X(t) = X trend (t) inapplicable or inconclusive in terms of co-integration or
+ S(t) X noise (t). causality, despite the presence of MAL signals that showed
The main difficulty lies in obtaining a reliable estimate strong correspondence. In other words, MAL proves to be
for the X trend statistical trend from the known time series. more efficient and capable of detecting interrelationship
Estimation methods are well developed and include signals that remain invisible to classical techniques. These
linear curves (e.g., classic linear regression), which may signals, in turn, enable a deeper understanding of the
7,8
incorporate breaks, accelerated increases or decreases, 10,11 underlying climate mechanisms.
9
along with “bootstrap” confidence intervals. 12,13 These This article is divided into three main parts:
methods also extend to non-linear behaviors or even 1. Description of MAL: This first part presents the
non-parametric descriptions. 14-16 Mudelsee gives a fairly main idea of the MAL and what are the stages of its
6
complete overview of these methods, which are numerous implementation.
and more or less complex, aiming to extract estimates from 2. Examples of MAL utilization in climatology: Four
the data, accompanied by measures of uncertainty. recent articles on mathematic models applied in
climatology (Zeltz 23-26 ) employ this new method
Even though the method presented in this article also
starts from the raw data provided by climate time series, the for the statistical analysis of time series. As will be
discussed in subsequent sections, MAL enables
initial approach is not the same. It does not seek to obtain the identification of interactions that would likely
estimates of future trends from the data but simply looks remain undetected using traditional methods for
for possible “signals” of interactions with the time series studying numerical series, such as those presented by
of other climate entities whose signals are compatible with Mudelsee. In the four sections of this second part,
6
that of the initial series. These signals are obtained from we show the implementation of MAL and address the
the average lengths of increasing or decreasing chains of questions that arose during its initial applications. 23-26
the studied climate parameter. The method is therefore 3. Evaluation and perspective of MAL: We summarize
referred to as the method of average lengths, or simply here the strengths and weaknesses of MAL and
“MAL.” Once these signals are detected and the climate provide a detailed example illustrating its utility in
mechanisms underlying their interactions are thoroughly understanding the mechanisms at play across different
explained, they can be incorporated into a climate time steps (e.g., months, quarters). Finally, we explore
model. Each instance in which the method isolates new the potential applicability of MAL in fields beyond
interactions and elucidates their mechanisms contributes climatology.
to significant improvement in the climate model.
Other methods exist to check the compatibility of time 2. Description of MAL
series and investigate the causal relationships. Starting 2.1. Sample introduction
with the calculation of the correlation coefficient between
the two series concerned. But as we will see, it is possible Suppose there is a well-balanced coin with an equal
that two series do not have a significant correlation and probability of 0.5 for landing heads (denoted as H) or tails
yet signals MAL are in agreement. This therefore indicates (denoted as T). If this coin is tossed 100 times, we record
that MAL can detect invisible interactions by more classic each successive outcome. The probability of obtaining a
means. This is also the case with techniques that are perfectly alternating sequence such as HTHTHT.HTHT
100
nevertheless much more sophisticated than the simple is exceedingly small, specifically (½) , or less than one
calculation of the correlation coefficient. For example, the chance in a billion billion billion. This scenario is governed
use of the concept of co-integration introduced by Granger by a binomial law X~B (n = 100, P = 0.5). According
27
and Newbold in econometrics facilitates the detection of to Delmas, the expected distribution of successive
17
long-term relationships between two or more time series. heads (and similarly for tails) follows the mathematical
Several algorithms can be used to verify this, such as the expectations shown in Table 1.
Granger–Engle algorithm, the Johansen approach, the In addition, the theoretical average length of the
18
19
Stock–Watson test, or the Phillips–Ouliar test. Tests substrings is approximately 1.94.
21
20
are also available to verify causal relationships and their
significance between two digital entities represented by Table 1. Theoretical frequencies for substrings of heads (H)
time series, such as the Granger causality test. These or tails (T) of a certain length in a series of 100 coin tosses
22
methods are often well-suited for analyzing econometric Length 1 2 3 4 5 6 7 and more
time series in particular. However, in the studied climate
series, we observed that traditional methods could be Theoretical frequencies (%) 51 25 12.5 6 3.5 0.9 1.1

Volume 2 Issue 1 (2025) 2 doi: 10.36922/eer.6109