[1,2,3]Anouar Ben Mabrouk These authors contributed equally to this work.
[1]Department of Mathematics, Faculty of Science, University of Tabuk, King
Faisal road, Tabuk, 47512, Saudi Arabia 2]Department of Mathematics, Higher
Institute of Applied Mathematics and Computer Science, University of Kairouan,
Street of Assad Ibn Alfourat, Kairouan, 3100, Tunisia 3]Laboratory of Algebra,
Number Theory and Nonlinear Analysis, Department of Mathematics, Faculty of
Sciences, University of Monastir, Avenue of the Environment, Monastir, 5000,
Tunisia 4]Department of Quantitative Methods, Faculty of Economic Sciences and
Management, University of Monastir, Sidi Messaoud, Hiboun, 5111, Mahdia,
Tunisia 5]Laboratoire THEMA de l’ESCT Tunis, Campus Universitaire de Manouba,
Université de Carthage, Mannouba, 2010, Mannouba, Tunisia 6]Department of
Marketing, Faculty of Business Administration, University of Tabuk, King
Faisal road, Tabuk, 47512, Saudi Arabia

# Multifractal wavelet dynamic mode decomposition modeling for marketing time
series

Mohamed Elshazli A. Zidan<EMAIL_ADDRESS><EMAIL_ADDRESS>Nidhal Ben Abdallah<EMAIL_ADDRESS>Tawfeeq M. Alanazi
<EMAIL_ADDRESS>*  [  [  [  [  [

###### Abstract

Marketing is the way we ensure our sales are the best in the market, our
prices the most accessible, and our clients satisfied, thus ensuring our brand
has the widest distribution. This requires sophisticated and advanced
understanding of the whole related network. Indeed, marketing data may exist
in different forms such as qualitative and quantitative data. However, in the
literature, it is easily noted that large bibliographies may be collected
about qualitative studies, while only a few studies adopt a quantitative point
of view. This is a major drawback that results in marketing science still
focusing on design, although the market is strongly dependent on quantities
such as money and time. Indeed, marketing data may form time series such as
brand sales in specified periods, brand-related prices over specified periods,
market shares, etc. The purpose of the present work is to investigate some
marketing models based on time series for various brands. This paper aims to
combine the dynamic mode decomposition and wavelet decomposition to study
marketing series due to both prices, and volume sales in order to explore the
effect of the time scale on the persistence of brand sales in the market and
on the forecasting of such persistence, according to the characteristics of
the brand and the related market competition or competitors. Our study is
based on a sample of Saudi brands during the period 22 November 2017 to 30
December 2021.

###### keywords:

wavelets; Multifractals; dynamic mode decomposition; mathematical models;
brand sales and prices; time series.

###### pacs:

[

###### pacs:

[

JEL Classification]C63, E31, E32, E37 MSC Classification]65T60, 91B84, 91B55,
37M10

##  1 Introduction

The present decade has known many severe moments/crises, social, political,
and economic due to many dramatic movements such as the so-called Arab Spring,
the financial crisis, COVID-19 pandemic, and lastly the war between Russia and
Ukraine, which directly affected the energy market and thus the economy’s
stability in quasi all the world. These facts need sophisticated tools to
understand the situation of the market as a first player and main factor in
the country’s stability, as well as the correct policy and government.

Understanding the market is based on a good understanding and/or analysis of
the data it characterized or extracted. Generally, a good data analysis needs
scientific computation tools and high-fidelity measurements, especially in the
stock market where the data may hide dynamics, anomalies, singularities,
extreme values, …, etc. This induces immediate capabilities for prediction,
state estimation, and also good control of the whole system.

In this paper, our focus will be on developing a mathematical technique geared
toward analyzing financial time series issued from marketing brands by
combining the so-called dynamic mode decomposition (DMD) and the wavelet
decomposition (WD).

The aim of this work is to investigate a wavelet-based mathematical model for
brand sales and prices. Wavelet theory has gained great interest in the fields
of economics and finance since the last decades of the last century. Recall
that almost all financial indices usually show volatile behavior as well as
high fluctuations. As a result, their investigation using classical methods
remains insufficient. Wavelets have indeed been shown to be useful in such
investigations.

Marketing, in its simplest sense, includes the institutions, the activities,
and also the processes applied to deliver, create, and exchange goods, and to
communicate offerings and needs to customers, partners, and clients, or more
broadly to society [4]. While marketing consists of various activities, the
principles of marketing rely on four elements, i.e., product, price,
promotion, and place (distribution). Each one of these elements depends on
qualitative and quantitative data. Quantitative data show the power of
marketing in collecting, analyzing, optimizing, and executing, based on
relevant numerical data. Product choices and portfolios, price hierarchies,
numbers of outlets and stores, and, importantly, promotion budgets, especially
for branding, brand management, and brand equity are also considered. Brands
and the activity of branding are well represented in the literature, and brand
equity refers to the value added to products [13]. Firms can benefit from a
strong brand in various ways: it improves the product value and provides a
chance to extend the brand to new products, expand into new markets, and
obtain a high level of customer loyalty and tolerance. A well-known brand
acquires a price value [9]. Thus, brand sales- and equity-related data include
things such as choice data, the brand design, the brand idea, time of
purchases, the number of stores, the choice of price, and the competitors in
the market. The state of the market itself affects the persistence and the
success of the brand investment, strongly and simultaneously. Without
sophisticated mathematical models that take into account all the factors and
that permit a good understanding of the time and space evolution of the brand
distribution and its evolution in the market, the investment in the brand may
fail.

In marketing, a mathematical model may include many factors such as the number
of sales in specified periods such as weeks, months, or years, market shares
per specified period, price evolution, the history of the market and its
situation, competing brands, etc. The main characteristic of a mathematical
model is its ability to deal with many factors simultaneously, according to
time and space. It gathers the data into a time series form, permitting
ordering over time. This has a very important consequence, as it yields
information about the future state of the brand. The model uses a time series,
as the best descriptor to extract, deduce, and/or accurately describe the
properties of the empirical data.

The DMD method is a matrix decomposition technique used to discover the
singular values in a time series. However, the drawback of this method is
related to the low-rank structures extracted from it. These are associated
with temporal features as well as correlated spatial activity. To understand
the importance of DMD, researchers used the well-known principal component
analysis (PCA) and also the Fourier transform (See [16]).

The WD is a mathematical tool developed theoretically since the 1980’s era due
to petroleum extraction, where the Fourier transform failed to explain well
the extracted signal. The WD is characterized by its capability to describe,
model, and analyze many data sets issued from different fields such as
signals, images, videos, bio-signals, statistics, finance, … etc. It is also
proved to be a good approximation and estimator in theoretical functional and
statistical analysis. The principle of WD is based on a representation of the
analyzed object into a series of modes due to special functions called
wavelets issued from one source known as the wavelet mother/father by dilation
and translation permitting to reach the singularities around any point in the
whole time or space domain (See [2, 3, 10, 12, 24]).

The next section concerns a brief literature review of existing studies on
marketing time series models. Section 3 is devoted to the presentation of the
dynamic mode decomposition and the wavelet decomposition methods for the
marketing time series processing and modeling. Section 4 concerns the
empirical results, together with discussions and interpretations. Section 5
concludes the paper.

##  2 Literature Review

Generally, marketing researchers and/or modelers search for models taking into
account one or more variables solely as functions of time. Next, the model is
analyzed regarding its efficiency for interpolating (approximating) and
extrapolating (predicting) these variables in time.

Compared to other fields such as financial and economic applications, the use
of mathematical models, especially wavelet-based models, is still not well
developed in marketing. In [7], for example, the authors mention the
difficulty of time series training for traditional research models, the lack
of software adapted to marketing modeling, the lack of, and sometimes the poor
quality of, time-series data, and the lack of a substantive area for
marketing, allowing time-series-based models to be readily adopted.

The problem of time scales in marketing has been raised in several studies. In
[20], it is mentioned that marketing phenomena depend on frequencies in time
relative to decisions and interactions in the market. In classical
investigations, the most commonly used time intervals are weekly, monthly,
quarterly, and annual cycles. Each period has its special and/or specific
characteristics. Weekly cycles, for example, are for reductions in prices, and
quarterly periods are generally related to regular price adjustments and
competition.

Moreover, many phenomena such as pandemics (COVID-19), wars, climate change,
financial crises, and socio-political movements may induce severe
perturbations in the usual considerations, especially time periods, leading to
diversification in the use of time scales.

This diversification in how to choose the time scale, according to what
factors, what characteristics, and what aim, has led researchers in the
marketing field to think about including the time scale strongly in the
models. In [18], the authors applied certain types of short-, medium-, and
long-term horizons to investigate the effect of the time scale. However, their
choice was not adapted to the wavelet method.

In marketing, according to [20, 21], the time scale is mostly applied at
discrete, equally spaced intervals, with weekly periods dominating, although
this hypothesis has been criticized. In [21], the author applied a wavelet-
based method for forecasting brand sales. The problem of multicollinearity has
been investigated, resulting in correlated vectors of coefficients which have
been applied to provide the most accurate forecasting and the best dimension
reduction. In [19], for example, it is mentioned that marketing models still
present aggregations, even using weekly intervals. This allows us to conclude
that more adequate tools for time/frequency scales should be applied.

Many methods have been used in this context, such as Fourier analysis and
spectral analysis. However, these tools have raised many problems such as the
non-stationary behavior of the marketing data, the sizes of samples, etc. ([5,
6, 8, 14, 15, 23]).

Wavelet analysis was introduced into marketing models to overcome limitations
such as the non-localization aspect of spectral analysis and its concentration
in the frequency domain. This is achieved by frequency decomposition of the
statistical time series into components that are well localized in
time/frequency.

Michis mentioned a very important characteristic of wavelet theory in [22, 20,
21]. It permits investigation of the causal links between time-series cycles
in marketing according to time scale. The modelers differentiate the
marketing-driver influences in long-duration cycles of sales. Wavelets are
also good tools regarding estimation and prediction accuracy and the handling
of non-stationary time series. The application of wavelets in marketing
resides in covering/uncovering the frequency activity characteristics of
marketing models. Wavelet crystals generated from the inverse of the wavelet
transform are applied to localize the variation in economic variables over
different horizons. Recall that brand sales forecasting is important for
investors and consumers but also for modelers, due to its strong relationships
with financial resources allocation and budget planning. Therefore, a
successful sales forecast will yield a good guide to planning sufficient
production and good product distribution, according to both time and demand.
Furthermore, appropriate marketing activities will be well organized,
resulting in good performance and placement in the market [22, 20, 21].

##  3 The DMD and WD tools for time series

The WD and the DMD act by decomposing complex data into simpler modes based on
spatio-temporal coherent structures to be able to provide insights from such
data. This permits for example to best predict the behavior of the data and
the real phenomenon behind it.

Let $X_{i}=X(t_{i})$, $t=t_{1},t_{2},...,t_{N}$ be a time series due to a
dynamical system, for example, the DMD method seeks the best model

$X_{k+1}=F(X_{i},i\leq k)$ (1)

that regresses the best the data locally by minimizing an $L^{2}$-error
$\|X_{k+1}-F(X_{i},i\leq k)\|_{2}$ over the time interval $k=1,2,3,...,N-1$.

In the case of a discrete linear dynamical system $X_{k+1}=AX_{k}$, $A$ being
a fixed matrix, the solution may be expressed by means of the eigenvalues and
eigenvectors of the matrix $A$, and the initial solution as

$X_{k}=\displaystyle\sum_{i}\Phi_{i}\lambda_{i}^{k}X_{0,i},$ (2)

where the $\Phi_{i}$ are the eigenvectors, the $\lambda_{i}$ are the
eigenvalues, and $X_{0,i}$ are the coordinates of the initial solution
$X_{0}$.

Notice from equation (2) that the DMD uses in some way a functional basis to
estimate the solution. At short-time horizons, we may approximate the
eigenvalues and the eigenvectors by setting $\omega_{k}=\ln\lambda_{k}/\Delta
t$, to get an approximation of the solution

$X(t)=\displaystyle\sum_{i}\Phi_{i}\exp(\omega_{i}t)X_{0,i},$ (3)

which confirms the use of a suitable function basis (or modes), such as the
Fourier one in (3). Notice that such a basis permits to localize the dynamic
behavior of the time series $X(t)$.

Wavelet analysis or alternatively MRA starts with a fixed function $\psi\in
L^{2}(\mathbb{R})$ (square integrable, or finite variance in the discrete case
such as statistical series) known as the mother wavelet which generates next
daughters wavelets

$\psi_{s,p}(t)=\displaystyle\frac{1}{\sqrt{s}}\psi\left(\displaystyle\frac{t-p}{s}\right),$
(4)

where the parameter $s>0$ is known as the scale or the scaling parameter or
frequency, and $p$ stands for the position or translation.

In wavelet analysis and/or processing, the mother wavelet $\psi$ has
beneficial properties that serve for processing different cases, theoretical
or practical, to yield a series representation of the data, such as the
admissibility condition.

To analyze a time (statistical) series $X(t)$, we need to evaluate its wavelet
transform (called also detail coefficient) at the scale $s$ and the position
$p$ through a discrete suitable grid such as the dyadic $s=2^{j}$, and
$p=k2^{j}$, $j,k\in\mathbb{Z}$, by setting a discrete convolution

$DWT_{j,k}=\displaystyle\sum_{i}X(i)\psi_{j,k}(i),$ (5)

where we denote

$\psi_{j,k}(t)=2^{-j/2}\psi(2^{j}t-k).$ (6)

This permits to reconstruct the series $X(t)$ representing the data by means
of an (orthogonal) decomposition as

$X(t)=\displaystyle\sum_{j\in\mathbb{Z}}\displaystyle\sum_{k\in\mathbb{Z}}DWT_{j,k}\psi_{j,k}(t).$
(7)

We thus split up the series $x(t)$ into a collection of subseries by
translating the wavelet with the parameter $k$ over the entire time domain.
Moreover, by changing the scale parameter $j$, we obtain the processing of the
series $X(t)$ at different frequency bands, called also resolutions due to the
scaling of the mother wavelet $\psi$ with the parameter $j$. The combination
of translation and scaling allows for the processing of the signals at
different times and frequencies. We thus get the so-called multiresolution
analysis. For each scale (or frequency) $j$, the component

$DX_{j}=\displaystyle\sum_{k\in\mathbb{Z}}DWT_{j,k}\psi_{j,k}$ (8)

belongs to the special space $W_{j}=spann(\psi_{j,k},\,k\in\mathbb{Z})$ known
in wavelet analysis as the detail space at the resolution, or the level $j$.
It therefore reflects the details hidden in the data at this level. We will
see on the practical cases later that effectively this component represents
the dynamics, and/or the hidden behavior of the data, and that this
representation differs from level to level like a zooming process. More
precisely, we have an orthogonal decomposition

$L^{2}(\mathbb{R})=\displaystyle\bigoplus^{\perp}_{j\in\mathbb{Z}}W_{j}.$ (9)

The multiresolution analysis consists of superposing the detail subspaces of
many levels to get a whole subspace $V_{j}$ gathering all the details $l\leq
j$ resulting thus in a global shape for the series. Denote

$V_{j}=\displaystyle\bigoplus^{\perp}_{l\leq j}W_{l},$ (10)

we get here a sequence of nested subspaces $V_{j}$ of $L^{2}(\mathbb{R})$
which satisfies many beneficial properties such as the nesting, direct sum,
zooming ([2, 3]). The resolution space at the level $j$ denoted by $V_{j}$ may
be also generated analogously to $W_{j}$ by means of a fixed function $\Phi$
called the father wavelet or also the scaling function, for which we have
$V_{j}=spann(\Phi_{j,k},\,k\in\mathbb{Z})$, where the $\Phi_{j,k}$ are defined
from $\Phi$ by the same way as the $\psi_{j,k}$ from $\psi$ in (6) yielding
the approximation coefficients of the series $X(t)$ expressed as

$A_{j,k}=\displaystyle\sum_{i}X(i)\Phi_{j,k}(i).$ (11)

Applying the concept of MRA, the decomposition of the data $X(t)$ into the
wavelet series (7) will be splitted into parts relatively to the spaces
$V_{j}$ as

$X(t)=\displaystyle\sum_{k}A_{j,k}\Phi_{j,k}(t)+\displaystyle\sum_{l\geq
j+1}DX_{l}(t).$ (12)

The first part denote

$AX_{j}(t)=\displaystyle\sum_{k}A_{j,k}\Phi_{j,k}(t)$ (13)

is known as the projection of the data $X(t)$ on the approximation space
$V_{j}$, and it reflects the global shape of the data at the level $j$. In
econo-financial contexts, it represents the trend of the data at the level
$J$. As the parameter $j$ gets upper, this component approaches more to the
observed data.

The next step to investigate the type of volatility of the time series is to
test its multifractality via the multifractal wavelet spectrum. Recall that
for a time series $X(t)$, this spectrum is defined as follows. Consider the
$j$-level wavelet-based statistics defined for $p\in\mathbb{R}$ by

$\mathcal{S}_{j}(p)=\displaystyle\sum_{k}|d_{j,k}(X)|^{p},$

where $d_{j,k}(X)$ is the wavelet detail coefficient at the level $j$ and the
position $k$ of the series $X(t)$. This leads to the so-called Besov exponent
of the series $X(t)$ defined by

$b(p)=1-\displaystyle\limsup_{j\rightarrow+\infty}\displaystyle\frac{\log(\mathcal{Z}_{j}(p))}{j\log(2)}.$

The spectrum of singularities is

$d(\alpha)=\displaystyle\inf_{p}(\alpha\,p-b(p)+1).$

Whenever this spectrum is a concave function, we conclude that the data under
investigation is effectively multifractal or irregular volatile.

##  4 Data and results

To show the utility of involving wavelet time scales in the mathematical
model, we applied it to a typical case consisting of six brands from the set
of top brands in Saudi Arabia. The choice of Saudi Arabia was justified for
many reasons. Saudi Arabia is one of the biggest economies, and it is one of
the main players in petroleum activities as a result of its influence on
worldwide petroleum prices, and thus on economies. It is a member of the G7
group and OPEC. Saudi Arabia has also recently implemented the so-called
Vision 2030, which combines many projects and activities such as marketing and
which has a direct impact on both national and foreign markets, particularly
via the NEOM project. The period of study may also be a strong factor
affecting the results and their interpretation. Indeed, the period was related
to many changes such as the GCC near-embargo against Qatar, COVID-19, the
Yemen war, and all the other socio-political changes in the Arab world. The
geographical position is also important, as well as the fact that the main
holy cities for Muslims all over the world participate strongly in the
distribution of Saudi products. In addition, Saudi Arabia and the whole GCC
region is the largest workers’ community in the world. With regard to the
economy and marketing, we may recall that among the most important plans and
aims for the Vision 2030 program are the adoption of diversity in income
sources and the need to reduce the total dependence on oil, thus encouraging
industrial, commercial, agricultural, and other activities. This means that
the top national brands are of interest.

The brands to be considered here are given in Table 1. The chosen Saudi brands
are in turn justified for many reasons, mainly the fact that these are the
brands with the most worldwide distributions, and they are known in many other
countries, having extended their local origins to form a worldwide reputation,
consequently constituting a success for the kingdom’s industry. Recall that
the Kingdom of Saudi Arabia is generally known as a consumer community, which
relies on foreign imported consumer products in exchange for petroleum
exports. Encouraging national industry and national labor generally is one of
the major goals of the Vision 2030 program.

Table 1: Some top Saudi brands. Nomination | Brand | Description (Sector)
---|---|---
$B_{1}$ | Jarir bookstore | Books and electronics
$B_{2}$ | Almarai | Dairy and poultry
$B_{3}$ | STC | Telecommunications
$B_{4}$ | Al Abdullatif | Household durables
$B_{5}$ | EIC | Electrical industries company
$B_{6}$ | Al Aseel | Textiles, apparel and luxury goods

Jarir Bookstore was founded in July 1974 by Abdulrahman Nasser Al-Agil. It is
one of the largest retailers for books and electronics in the Kingdom of Saudi
Arabia, and has now expanded to many other countries, especially those in the
GCC, such as Kuwait, Qatar, and UAE. It is also one of the major components of
the Saudi TADAWUL index.

The Almarai company was originally constituted as a partnership between Prince
Sultan bin Mohammed bin Saud Al Kabeer and the Irishmen Paddy McGuckian and
Alastair McGuckian. Now, it is one of the biggest dairy companies in KSA and
also in the whole Middle East region. It was founded 40 years ago and now
specializes in dairy, poultry, juices, bakery goods, yogurt, and infant
products.

The Saudi Telecommunications Company, abbreviated STC, started 19 years ago.
It basically offers telecommunications services and products. It has also now
expanded into other GCC countries and to countries in other continents, such
as India, Turkey, South Africa, and Malaysia.

Al Abdullatif Industrial Investment Company is a national Saudi-Arabia-based
company specializing in both the distribution and manufacture of weaving
products such as blankets, rugs, and carpets, and intermediates such as nylon,
polyester, etc.

The Electrical Industries Company, abbreviated EIC, started in 2005, and since
then it has become a leading manufacturer of electrical products to satisfy
the growing demand for electrical equipment in Saudi Arabia.

The Thob Al Aseel Company now operates under the Al-Jedaie brand. Since its
foundation in 1970, it has been based in the capital Riyadh. It is subscribed
under the thobes and fabrics segments and is concerned with wholesale, ready-
made clothing and retail fabrics for development, importation, and
exportation. It products a variety of products such as underwear, thobes,
pajamas, and sleeping robes. It also produces ehrams, T-shirts, and cotton
socks. The Thob Al Aseel company is also concerned with selling children’s
fabrics, adult clothing products, and many other things.

###  4.1 Descriptive statistics of the brands portfolio

Table 2 below shows the descriptive statistics corresponding to prices and
sales for the six brands.

Table 2: Descriptive statistics for brands $B_{i}$, $1\leq i\leq 6$, prices and sales. Brand | Mean | Median | Min | Max | Std | Skewness | Kurtosis | JB $(h,p)$
---|---|---|---|---|---|---|---|---
The brand prices |
$B_{1}$ | 161.42 | 159.2 | 105.44 | 225 | 26.48 | 0.21 | 2.37 | (1,10-3)
$B_{2}$ | 52.91 | 53.3 | 36.95 | 63.7 | 4.07 |  $-$0.08 | 3.15 | (0, 0.34)
$B_{3}$ | 100.25 | 99.75 | 67.10 | 139.2 | 16.35 | 0.20 | 2.52 | (1,$10^{-3}$)
$B_{4}$ | 15.88 | 12.66 | 8.15 | 40.35 | 8.12 | 1.76 | 4.64 | (1,$10^{-3}$)
$B_{5}$ | 20 | 20.34 | 14.46 | 29.4 | 3.55 | 0.39 | 2.35 | (1,$10^{-3}$)
$B_{6}$ | 35.52 | 31.35 | 14.88 | 69.75 | 16.44 | 0.65 | 2.01 | (1,$10^{-3}$)
The brands sales |
$B_{1}$ | 153.63 | 109.31 | 7.56 | 4020 | 258.03 | 10.70 | 140.71 | (1,$10^{-3}$)
$B_{2}$ | 564.28 | 418.03 | 34.70 | 12,140 | 689.47 | 9.25 | 129.83 | (1,$10^{-3}$)
$B_{3}$ | 890.12 | 512.49 | 33.30 | 1.21310 | 3967.75 | 27.64 | 831.52 | (1,$10^{-3}$)
$B_{4}$ | 809.01 | 223.43 | 8.24 | 22,200 | 1935.15 | 6.08 | 51.00 | (1,$10^{-3}$)
$B_{5}$ | 2577.69 | 1720 | 111.39 | 37,490 | 3161.13 | 4.89 | 40.35 | (1,$10^{-3}$)
$B_{6}$ | 243.89 | 64.04 | 0.01 | 5250 | 570.35 | 4.70 | 29.97 | (1,$10^{-3}$)

The first deductions from Table 2 may arise from the Min and Max values, which
are widely different for almost all the brand prices and sales, reflecting the
existence of aberrant values or anomalies in the market. In addition, the mean
and median values are also different, even widely different, especially for
sales. This large range in the prices and sales is more adequately detected
and explained by means of time-scale modeling, as apparently it has no logical
causes.

From Table 2, we easily obtain a non-zero skewness for all the brand prices
and sales, which leads to rejection of the symmetry hypothesis. All the brand
prices and sales are spread to the right of their mean values, except in the
$B_{2}$ price distribution, which has a small skewness and a kurtosis close to
3, indicating that the quasi-normal distribution behavior is hidden for this
series of prices. A negative skewness indicates a left-spreading tail for the
$B_{1}$ prices. The kurtosis reflects non-normal behavior for all brand prices
and sales, with heavier tails than for a normal distribution.

In addition, for all the variables $B_{1}$, $B_{3}$, $B_{4}$, $B_{5}$, and
$B_{6}$, the Jarque–Bera test leads to a returned value of $h=1$ and a
returned $p$-value of the order of $10^{-3}$ at the $5\%$ significance level,
which indicates a rejection of the null hypothesis.

We also notice a large standard deviation, indicating a sparse distribution
for both prices and sales around their mean values. For some brand prices and
sales, the data are very sparsely distributed. This fact may be explained by
the use of large differences in prices and sales, and a non-uniform view of
the future on the part of both consumers and managers in this market. Consumer
demand for special types of products under the same brand leads to an increase
in both the volume of sales and prices, without taking into account the
equilibrium with other products under the same brand title.

###  4.2 Wavelet multifractal processing of the brands portfolio

To easily show the time-scale variations (fluctuations, increases, decreases)
of these variables and to further understand their behavior according to the
time scale, we reproduce the prices and sales for the brands $B_{i}$, $1\leq
i\leq 6$, graphically. This yields, among other interpretations, an easy-to-
read description of these variables according to the time scale.

Figures 1, 3, 5, 7, 9, and 11 illustrate the wavelet decomposition of the
prices for each brand at the level of decomposition $J=6$. In addition, in
Figures 2, 4, 6, 8, 10, and 12, we provide the wavelet decomposition due to
the brand sales at the same decomposition level of $J=6$. These graphs
illustrate the variables with their trends and dynamics or fluctuations. On
the $y$-axis, the variable $A_{6}$ stands for the wavelet approximation at
level 6, while the variable $D_{i}$, $1\leq i\leq 6$, stands for the detail
component at level $i$. The strong fitting between each variable and its
approximation can clearly be seen.

Figure 1: The wavelet decomposition of the Jarir brand $B_{1}$ prices at level
6. Figure 2: The wavelet decomposition of the Jarir brand $B_{1}$ sales at
level 6. Figure 3: The wavelet decomposition of the Almarai brand $B_{2}$
prices at level 6. Figure 4: The wavelet decomposition of the Almarai brand
$B_{2}$ sales at level 6. Figure 5: The wavelet decomposition of the STC brand
$B_{3}$ prices at level 4. Figure 6: The wavelet decomposition of the STC
brand $B_{3}$ sales at level 4. Figure 7: The wavelet decomposition of the Al
Abdullatif brand $B_{4}$ prices at level 6. Figure 8: The wavelet
decomposition of the Al Abdullatif brand $B_{4}$ sales at level 6. Figure 9:
The wavelet decomposition of the EIC brand $B_{5}$ prices at level 6. Figure
10: The wavelet decomposition of the EIC brand $B_{5}$ sales at level 6.
Figure 11: The wavelet decomposition of the Al Aseel brand $B_{6}$ prices at
level 6. Figure 12: The wavelet decomposition of the Al Aseel brand $B_{6}$
sales at level 6.

Figure 1, illustrating the wavelet decomposition of $B_{1}$ prices, shows a
somewhat increasing behavior, clearly illustrated by the level $6$ trend
$A_{6}$, reminiscent of some perturbations at medium-to-long scales. However,
the trend shows no cycles (periodic behavior) for the $B_{1}$ prices. The
perturbation is confirmed in the detail components $D_{1}$ to $D_{6}$.
According to $D_{1},D_{2}$ a certain pseudo-periodicity seems to take place,
with some differences in the maximum prices. This may be explained by the fact
that at short time scales (such as weeks in the classical treatments of the
time factor), prices are influenced by competitors on the one hand and the
global behavior of the market (national as well as international) on the
other. One essential factor to be considered is the COVID-19 pandemic, which
covered a great part of the period of study and which certainly resulted in
decreased sales so that, to compensate for the quantities in stock and thus to
recover losses, the company had to increase prices, as in many other cases.
This fact is easily shown in the $B_{1}$ sales graphs in Figure 2, where we
clearly observe an upward trend in sales whenever prices decreased and a
downward trend in sales whenever prices increased. In addition, both prices
and sales are volatile at long time scales, according to the detail components
$D_{3}$ to $D_{6}$. We may thus conclude for this brand that short time scales
are more comprehensive and easy to understand for consumers, investors, and
managers.

In Figure 3, there is a slight variation around the mean (median), with no
exact periodicity but instead a slight pseudo-periodic structure. This
behavior is repeated in the sales illustrated in Figure 4. However, some
anomalies appear, essentially at all time scales. This behavior can be
naturally understood from the fact that Almarai is a completely national
company, based on completely national products. Therefore, compared to
$B_{1}$, for example, its prices and volumes are not strongly affected by
importation. In addition, its products are the oldest, the best in the market
(according to the majority of consumers), and are consumed daily. All these
factors allow price stability in the market, despite a slight growth in the
volume of production. In addition, prices and sales seem to reflect the same
behavior at short and long time scales. These fluctuations are clearly
illustrated by the detail components $D_{1}$ to $D_{6}$. Nevertheless, slight
fluctuations remain around the mean for all the time scales.

Figures 5 and 6 illustrate the behavior of the brand $B_{3}$ prices and sales
for level 6 wavelet decomposition. Here also, we notice a global increase in
both series, which is clearly shown in the trend $A_{6}$. This increase is
sometimes interspersed with small fluctuations. This brand is one of the major
suppliers of communications needs in the entire Kingdom of Saudi Arabia.
During the COVID-19 quarantine period, STC sales grew (seen clearly in
$A_{6}$) due to the increase in remote and/or distance communications in
almost all domains. Nevertheless, we notice some pseudo-periodicity at short
time scales.

Figures 7 and 8 show quasi-stability at short and medium time scales, followed
by a global increase at higher horizons. This increase may also be due to high
consumer demand as an alternative during the quarantine period for COVID-19.
The pseudo-periodicity and stability appearing at short and medium horizons
effectively breaks down at long time scales for these reasons. Moreover, with
the lack of importation of similar products, consumers turned to national
brands, contributing to the increase in sales and consequently in prices. On
the other hand, a high proportion of these types of sales are imported (as raw
or elementary commodities). Therefore, with the economic recession afflicting
the market for the long COVID-19 period, merchants had to increase prices to
compensate for a part of their losses. In addition, other companies use many
of the products of this brand, such as polyester and nylon. In critical
periods such as during COVID-19, it is necessary to use national stocks. This
induces an increase in both sales and prices.

The brand $B_{5}$, illustrated by Figures 9 and 10 for prices and sales,
respectively, is characterized by a small skewness and kurtosis and quite a
small Std. This indicates some stability in both prices and sales, showing
some small fluctuations but no important extreme values. As its trend $A_{6}$
indicates, prices started to decrease at short time scales, turning to an
increase at high horizons, with a slow increase at medium levels. Sales seemed
to be stable globally at short horizons, decreasing at medium levels, and
increasing at longer time scales. Recall that EIC is a national manufacturer
of electric instruments. Such instruments need many raw materials, mainly
imported from outside the kingdom. Therefore, considering the COVID-19
situation, and with the orientation towards encouraging national and local
industry and production being one of the main goals of the Vision 2030
program, these companies have experienced expansion and development in the
volumes of sales, as well as in prices. Nevertheless, the graphical
representations of the detail components do not reflect any cycles, as might
be expected in classical studies.

Figures 11 and 12 illustrate the time-scale behavior of brand $B_{6}$ prices
and sales, according to the wavelet decomposition at level $J-6$. A global
increase in both prices and sales can clearly be seen, especially at longer
time scales. By investigating the detail components $D_{1}$ to $D_{6}$, we may
easily exclude the idea of cycles for this series, especially for low/medium
horizons. At longer time scales, the series becomes more and more volatile, as
for the preceding brands. The increase at the higher horizons is due to the
orientation of consumers to local/national products, as there was no
importation from outside. Moreover, this brand specializes mainly in national
clothing, representing a style that is not widely known outside the GCC
countries, although some companies in China target the GCC market. These
outside companies discontinued their exports during the COVID-19 period, which
therefore resulted in increased sales of national clothing products. In
addition, due to the Vision 2030 program, these local/national brands and the
producing companies are forced to implement government directions and
guidelines to develop and expand national production.

Overall, these graphs, especially their detail components, reduce the ideas of
cycles and stationarity for these marketing series and show instead a volatile
behavior, which increases with the time scale. This indicates that the market
concerned is emergent, and economic laws must be applied carefully by
investors and managers. These facts may have many reasons. One is the COVID-19
crisis, which has been positively used in some cases, especially regarding
telecommunications supplies and national products including foods (Almarai)
and clothing (Al Aseel). However, this was a disadvantageous situation for
many brands relying on imported raw materials, such as EIC. In addition, there
was an effect of labor resettlement, which resulted in a considerable amount
of non-highly-qualified labor in many sectors. This fact, although it
constitutes a principal goal in the Vision 2030 program, may have reduced
productivity.

As shown by wavelet analysis of the marketing time series associated to the
brands prices and sales, statistical data posses indeed a volatile aspect,
where fluctuations become more and more clear over time as shown by the
wavelet detail components. To further investigate the multifractal aspect of
these series, we subsequently provide the multifractality test based on the
multi-fractal wavelet spectrum as shown in Figure 13 for the brand prices, and
Figure 14 for the sales. We notice easily the high volatile dynamic behavior
of the series.

Figure 13: The wavelet multifractal spectra of some brands prices. Figure 14:
The wavelet multifractal spectra of some brands sales.

###  4.3 The DMD processing of the brands portfolio

####  4.3.1 The time series data processing

##### Brands prices raw data

The time series data presented in figure 15, exhibits several interesting
patterns. First, there are high-frequency oscillations present in the data,
characterized by rapid fluctuations with small amplitudes. These fluctuations
suggest short-term market volatility and indicate that prices are subject to
frequent and quick changes, potentially influenced by various factors such as
news events, market sentiment, or daily trading activities.

Figure 15: The plot displays the raw normalized data of the time series
representing the daily price values for the 12 brands. The data spans a period
of 1024 days. To enhance visualization, the time series have been vertically
stacked along the y-axis.

In addition to the high-frequency oscillations, there is a noticeable slow
oscillating trend observed in the data. These trend represent longer-term
price movements that persist over a significant period. It suggests that the
market dynamics underlying these brands exhibit cyclical patterns that extend
beyond the day-to-day fluctuations. The presence of such trends can be
attributed to factors like seasonal effects, supply and demand imbalances,
economic cycles, geopolitical events, or industry-specific conditions that
influence the overall market sentiment and investor behavior. Moreover, the
long-term behavior of all the brands shows a growing long-term pattern. This
observation implies that, on average, the prices of these brands have been
increasing over the 1024-day period. This upward trend indicates positive
market sentiment and suggests that the market dynamics driving these brands
have experienced overall growth and expansion during the observed time frame.

The combination of high-frequency oscillations, slow oscillating trends, and
the long-term growing pattern in the price data highlights the complex nature
of market dynamics. It signifies the presence of multiple forces at play,
including short-term volatility, medium-term cyclical patterns, and long-term
growth trends. Understanding and analyzing these patterns can provide valuable
insights into the underlying market dynamics, allowing investors and analysts
to make informed decisions, manage risk, and identify potential opportunities
in the dynamic marketplace.

##### Brands sales raw data

The analysis of the daily sales values for the 12 brands over a period of 1024
days reveals several significant patterns in the data as shown in figure 16.
Firstly, the plot of the raw data exhibits high-frequency oscillating patterns
with tiny amplitudes for all brands throughout the entire observation period.
This suggests that there are rapid fluctuations in sales occurring on a daily
basis, but the magnitude of these fluctuations is relatively small for each
brand. These high-frequency oscillations may be attributed to factors such as
daily consumer behavior, short-term market dynamics, or random noise inherent
in the sales data.

Figure 16: The plot displays the raw normalized data of the time series
representing the daily sales values for the 12 brands. The data spans a period
of 1024 days. To enhance visualization, the time series have been vertically
stacked along the y-axis.

Additionally, the plot shows a recurring pattern of spikes that consistently
appear across all the time series. However, the height of these spikes varies
over different time periods. From day 1 to day 350, the spikes for all brands
are relatively low, except for brand 1, which exhibits a notably high spike at
day 230. This suggests that there may be a specific event or factor that
significantly impacted the sales of brand 1 during that time period, leading
to the pronounced spike. From day 350 to day 650, the spikes increase in
height for all brands, indicating a period of heightened sales activity or
increased demand across the board. From day 650 to day 850, the plot displays
quite high spikes for all brands, suggesting a sustained period of strong
sales performance or market conditions favoring increased sales. Finally, from
day 850 to the last day of the observation period, the height of the spikes
decreases again, indicating a decline in sales activity or a shift in market
dynamics. Furthermore, the plot reveals a slow oscillating trend with low
frequency that persists throughout the entire observation period for all
brands. This implies the presence of long-term cyclic behavior or patterns
that repeat over extended periods. These slow oscillations could be influenced
by factors such as seasonal trends, macroeconomic fluctuations, or industry-
wide dynamics that operate on longer time scales. Understanding and analyzing
these slow oscillations can provide insights into market dynamics, strategic
planning, or identifying brands affected by cyclical trends. The long-term
behavior of the plot shows a very slow-growing trend, indicating a gradual
increase in sales over time for all brands. This suggests a positive overall
market growth or an upward trajectory in consumer demand. However, the rate of
growth is relatively slow, indicating a gradual and steady expansion rather
than rapid or exponential growth.

In summary, the analysis of the daily sales values for the 12 brands reveals
several key patterns. These include high-frequency oscillations with small
amplitudes, recurring spikes with varying heights over different time periods,
a slow oscillating trend with low frequency, and a very slow-growing long-term
trend. These patterns provide insights into the underlying market dynamics,
including daily fluctuations, short-term sales variations, event-driven
impacts, long-term cyclic behavior, and the overall growth trend.
Understanding these dynamics can inform decision-making processes, marketing
strategies, and business planning to effectively navigate the market and
capitalize on the identified patterns within the sales data.

####  4.3.2 The power spectrum

The DMD power spectrum provides valuable insights into the frequency content
and dominant modes of variation in the data. It allows us to analyze the
distribution of energy across different frequencies and understand the
importance of different oscillatory components in the system. By examining the
DMD power spectrum, we can identify the frequencies or modes that contribute
the most energy or have the strongest influence on the dynamics of brand
prices. This can help us uncover dominant oscillatory patterns, periodic
behaviors, or characteristic frequencies in the data.

The power spectrum can reveal the presence of both low-frequency and high-
frequency components in the brand price and sales dynamics. the high-frequency
modes provide insights into the short-term dynamics and fast variations of
brand prices and sales. These modes capture rapid oscillations or fluctuations
in brand prices and sales, which can reveal short-term relationships,
interactions, and market dynamics between the brands. By examining the high-
frequency modes, one can gain information about the temporal dynamics and
synchronization of price/sale changes among the brands at a fast pace. They
can reveal whether the brands tend to move together or exhibit divergent
behaviors in response to short-term market fluctuations, news events, or other
high-frequency factors. Analyzing the high-frequency modes in DMD allows for
the identification of fast-changing relationships and market dynamics between
brands. It provides insights into the speed at which brands respond to market
forces and how they interact with each other in the short run. This
information can be valuable for short-term trading strategies, risk
management, and understanding market microstructure. Moreover, the high-
frequency modes can help identify short-term anomalies, irregularities, or
sudden shifts in brand prices and sales. These modes may correspond to
transient market inefficiencies, price/sale spikes, or other fast-moving
phenomena that can be exploited or monitored for trading opportunities.

On the other hand, the low-frequency modes offer insights into the long-term
trends and overall behavior of brand prices/sales. These modes capture the
slow variations or trends in brand prices/sales, which can reveal the
underlying dynamics and relationships between the brands at longer timescales.
By examining the low-frequency modes, one can identify patterns of price/sales
changes that occur over extended periods. These trends may indicate common
movements or correlations between the prices/sales of different brands. If the
low-frequency modes exhibit similar long-term behavior across multiple brands,
it suggests a strong correlation or co-movement between their prices/sales at
slower timescales.

The low-frequency modes can provide information about the persistent dynamics
and synchronization of price/sale changes among the brands. They can reveal
whether the brands tend to move together or exhibit divergent behaviors in
response to long-term market trends or external factors. Additionally, the
low-frequency modes can help identify dominant modes of variation that drive
the overall behavior of brand prices/sales. These modes may correspond to
fundamental market factors, economic indicators, or other external influences
that have a lasting impact on the prices/sales of the brands. Analyzing the
low-frequency modes in DMD can be valuable for understanding the macroscopic
behavior and relationships between brands in terms of long-term trends, market
cycles, and the influence of underlying factors. It provides a broader
perspective on the dynamics of brand prices/sales beyond short-term
fluctuations, allowing for more strategic decision-making and market analysis.

##### Brands prices analysis

Figure 17, shows the DMD power spectrum for the prices time-series data. The
modes with the highest power are highlighted in green color. As we will see
later in this section, that, these modes display interesting characteristics
and provide valuable insights into the relationship between the brands and the
underlying market dynamics.

Figure 17: The figure shows the DMD spectrum of the time series data presented
in figure 15. The DMD spectrum reveals the market-opining price dynamics
associated with the brands under consideration.

The power spectrum analysis provides valuable insights into the underlying
driving factors and their implications for market dynamics. Table 3, shows the
frequencies and time duration of completing one cycle, for the selected eight
top-ranked modes which have the highest powers.

Table 3: The table shows the daily frequencies time duration for completing one full cycle for the selected DMD modes that exhibit the highest powers. DMD mode | Daily Frequency | Duration of one cycle (days)
---|---|---
1 | 0.0011 | 925
2 | 0.0015 | 654
3 | 0.0000 | Inf
4 | 0.2154 | 5
5 | 0.0517 | 19
6 | 0.0320 | 31
7 | 0.1436 | 7
8 | 0.0030 | 332

The presence of high power in very low-frequency modes suggests the existence
of long-term trends or cycles that significantly influence the market. These
longer-term cycles, such as the 925-day and 654-day cycles, could be driven by
factors like macroeconomic conditions, industry-specific events, or changes in
investor sentiment that occur over extended periods. Understanding and
monitoring these cycles can help investors identify long-term investment
opportunities or anticipate major market shifts.

In addition to long-term cycles, the analysis also reveals the importance of
shorter-term cycles. The presence of cycles with powers like the 5-day,
19-day, and 31-day cycles indicates the existence of recurring patterns in the
market over weekly, fortnightly, and monthly timeframes. These shorter-term
cycles may be influenced by factors such as trading patterns, market
sentiment, or the release of economic indicators at regular intervals. Traders
can leverage these insights to develop short- to medium-term trading
strategies, taking advantage of predictable market patterns and timing their
trades accordingly.

The identification of a zero-cycle with relatively high power suggests the
presence of non-periodic or irregular components impacting market dynamics.
These factors could include unexpected events, market shocks, regulatory
changes, or other unpredictable influences. Understanding the driving forces
behind this non-periodic behavior is crucial for market participants, as it
highlights the need to be prepared for sudden shifts in market conditions and
the possibility of outliers that can significantly impact investment
strategies.

The power spectrum analysis also highlights the relative powers of different
cycles, indicating the varying strengths of their influence on market
dynamics. Cycles with higher powers, like the 925-day and 654-day cycles, have
a more pronounced impact on market behavior, while cycles with lower powers,
such as the 7-day and 11-month cycles, have comparatively smaller effects.
Recognizing these differences can help investors prioritize their focus and
resources, aligning their strategies with the cycles that have the most
significant influence on the market.

The findings from the power spectrum analysis provide valuable insights into
the driving factors and implications for market dynamics. By understanding the
recurring patterns at different time scales and the factors driving them,
investors can make more informed decisions, anticipate market trends, and
adjust their strategies accordingly. It also highlights the importance of
considering both long-term and short-term cycles to gain a comprehensive
understanding of the market and capitalize on potential opportunities at
various time horizons.

##### Brands sales analysis

Figure 18, shows the DMD power spectrum extracted from the time series data
presented in figure 16. The modes with the highest power are highlighted in
green color. As we will see later in this section, that, these modes display
interesting characteristics and provide valuable insights into the
relationship between the brands and the underlying market dynamics. Table 4,
shows the frequencies and time duration of completing one cycle, for the
selected eight top-ranked modes which have the highest powers.

Figure 18: The figure shows the DMD spectrum of the time series data presented
in figure 16. The DMD spectrum reveals the market-opining sales dynamics
associated with the brands under consideration.

Upon analyzing the DMD power spectrum for the sales time-series data, several
key observations can be made regarding the frequencies associated with the
top-ranked modes. The first and second top-ranked modes having zero frequency
imply that these modes represent constant or steady components in the sales
data. They likely capture the average sales level across all brands without
any temporal variation. This could correspond to the baseline sales or the
overall market demand that remains relatively stable over time.

Table 4: The table shows the daily frequencies and time duration for completing one full cycle for the selected DMD modes that exhibit the highest powers. DMD mode | Daily Frequency | Duration of one cycle (days)
---|---|---
1 | 0.0000 | Inf
2 | 0.0000 | Inf
3 | 0.0038 | 263
4 | 0.1081 | 9
5 | 0.3511 | 3
6 | 0.3801 | 3
7 | 0.0010 | 1037
8 | 0.4502 | 2
9 | 0.1488 | 7
10 | 0.0021 | 469

The third, seventh, and tenth top-ranked modes having low frequency suggest
that they capture sales patterns characterized by long-term trends or slow
variations. These modes likely represent market dynamics that evolve over
extended periods, such as seasonal trends, economic fluctuations, or changes
in consumer behavior. Brands associated with these modes may experience
gradual shifts in sales performance, influenced by external factors that
operate on longer time scales. Understanding these modes can provide insights
into strategic decision-making, long-term forecasting, or identifying brands
affected by macro-level market dynamics. On the other hand, the fourth, fifth,
sixth, eighth, and ninth top-ranked modes having quite high frequency indicate
that they capture sales patterns with rapid fluctuations or short-term
variations. These modes are likely driven by factors that operate on shorter
time scales, such as weekly, monthly, or quarterly dynamics. Brands associated
with these modes may experience more frequent changes in sales performance due
to factors like consumer preferences, promotional activities, or periodic
events. Understanding these modes can facilitate operational decision-making,
inventory management, or identifying brands influenced by more immediate
market conditions.

The analysis of the raw data and the findings from the DMD spectrum provide
valuable insights into the underlying market dynamics and their connection to
the observed patterns. The high-frequency oscillations with small amplitudes
identified in the raw data align with the presence of modes in the DMD
spectrum that exhibit quite high frequencies. These modes likely capture the
rapid fluctuations and short-term variations observed in the sales data,
reflecting the daily consumer behavior and immediate market dynamics.
Similarly, the recurring spikes in the raw data correspond to certain modes in
the DMD spectrum that display distinctive frequencies and magnitudes. The
varying heights of these spikes over different time periods in the raw data
are indicative of the frequency characteristics associated with specific modes
in the DMD spectrum. This connection highlights the ability of the DMD
analysis to identify and extract relevant modes that represent the temporal
patterns and frequencies inherent in the sales data. Furthermore, the slow
oscillating trend with low frequency and the very slow-growing long-term trend
observed in the raw data are consistent with the identification of low-
frequency modes in the DMD spectrum. These modes capture the long-term cyclic
behavior and gradual changes in the sales data, reflecting macroeconomic
factors, seasonal trends, or industry-wide dynamics that influence the market
over extended periods. Overall, the analysis of the raw data and the insights
gained from the DMD spectrum complement each other, providing a comprehensive
understanding of the market dynamics, temporal patterns, and frequency
characteristics that drive the sales behavior of the brands.

###  4.4 Spatial and temporal modes analysis

DMD extracts coherent structures or patterns from time-series data and
represents them as modes, figure 20 and figure 22 show the extracted DMD modes
from the brand’s prices and sales time series respectively. These extracted
structures or patterns may include both short-term and long-term behaviors,
capturing the essential dynamics present within the dataset. In the context of
brand dynamics, each DMD mode represents a specific pattern or mode of motion
that describes how the brands evolve over time. These modes may include
synchronized movements, similar trends, or shared dynamics among the brands.
By analyzing and interpreting these modes, one can gain insights into the
underlying coherent structures or patterns, correlations, and relationships
that characterize the dynamics of the brands over time.

####  4.4.1 Brands prices analysis

##### The first top-ranked mode

This mode demonstrates a significant contribution from all brands, with their
contributions being relatively comparable, except for brand 2, which exhibits
a notably lower alignment with this mode. This observation suggests that the
majority of the brands share a common behavior captured by this mode, while
brand 2 deviates from this general trend. Furthermore, the frequency
associated with this mode indicates that, this mode exhibits a distinct cyclic
behavior with a period of approximately 925 days, which is approximately 2.5
years. This cyclic pattern implies that there is a recurring pattern or
process occurring in the market every 925 days, which affects the prices of
the brands included in the analysis.

Figure 19: The left panel shows the DMD spatial modes and the right panel
shows the corresponding temporal dynamic. Figure 20: The left panel shows the
DMD spatial modes, and the right panel shows the corresponding temporal
dynamic.

Understanding the nature and drivers of this cycle can provide valuable
insights into the longer-term dynamics of the market and may be indicative of
specific market events or macroeconomic factors that influence price movements
over this timeframe. The associated temporal time dynamic mode, which
represents the temporal behavior of this mode, reveals an oscillatory pattern
with a relatively high amplitude. Moreover, the frequency of this oscillation
is striking, with one complete cycle occurring over the entire 1024-day
observation period. This long-term oscillatory behavior suggests the presence
of a fundamental market dynamic or underlying factor that influences the
prices of the brands.

The combination of the high contribution from most brands, the cyclic behavior
with a 925-day period, and the long-term oscillatory pattern in the temporal
mode highlights the interconnectedness of the brands and the underlying market
dynamics. The consistent contributions from the majority of the brands suggest
a collective response to common market forces or factors that drive their
prices. On the other hand, the divergence of brand 2 from this mode may
indicate idiosyncratic factors unique to that particular brand or its specific
market conditions.

##### The second top-ranked mode

This mode demonstrates a significant contribution from brands 1, 3, 5, 9, 10,
and 11, indicating a strong alignment with this mode, and suggesting a
collective response to shared market forces or factors that drive their prices
in a similar manner. Brands 4, 6, 7, 10, and 12, on the other hand, have a
medium contribution to this mode, suggesting a moderate level of alignment,
and indicating some commonalities in their price movements. Notably, brand 2
again has the lowest contribution to this mode, indicating a weaker connection
with the behavior captured by this mode, and distinct behavior that deviates
from the overall market trend captured by this mode. Furthermore, this second
mode displays a distinct cyclic behavior with a period of approximately 654
days, which is approximately 1.8 years. This cyclic pattern suggests that
there is a recurring pattern or process occurring in the market every 654
days, which influences the prices of the brands included in the analysis.
Understanding the nature and drivers of this cycle can provide insights into
the medium-term dynamics of the market and may be indicative of specific
market events, sectoral trends, or economic indicators affecting the prices of
the brands over this timeframe. The associated temporal time dynamic mode,
representing the temporal behavior of this mode, exhibits oscillatory behavior
with a relatively high amplitude. Moreover, the frequency of this oscillation
is noteworthy, with one-and-a-half cycles occurring over the entire 1024-day
observation period. This long-term oscillatory behavior suggests the presence
of a fundamental market dynamic or underlying factor that influences the
prices of the brands captured by this mode.

The fact that the top-ranked first and second modes have very slow temporal
dynamics, indicates a persistent and influential force shaping the market
dynamics over the long term. It could reflect factors such as an increase in
inflation, also macro-economic factors, such as interest rates, economic
growth, fiscal policies, and geopolitical events, which can also influence
long-term price trends.

##### The third top-ranked mode

In this mode brands 2 and 7 exhibit the highest contributions, indicating a
strong alignment with this mode, and a significant role played by these brands
in shaping the behavior captured by this mode. Brands 1, 3, 5, 8, 9, 11, and
12 contribute to a medium extent, suggesting a moderate level of alignment. On
the other hand, brands 1, 4, 6, and 10 have the lowest contributions,
indicating a weaker connection with the behavior captured by this mode.
Interestingly, this third mode does not exhibit a distinct frequency, where
the imaginary part of the associated singular value is zero. The absence of a
frequency in this mode suggests that it does not represent any cyclic behavior
or recurring pattern within the observed data. Instead, it may capture more
subtle or complex dynamics that go beyond periodic trends.

The associated temporal time dynamic mode for this third mode shows a purely
moderate decaying behavior. The amplitude of this mode gradually decays over
approximately 400 days until it reaches almost zero. This decaying behavior
implies that there are underlying factors or forces in the market that
gradually dampen the influence or impact of the factors captured by this mode.
It suggests that the contribution of the underlying factors driving this mode
to the overall market dynamics diminishes over time.

##### The fourth, fifth, and seventh top-ranked modes

In all these modes brand 2 exhibits the highest contribution, indicating a
strong alignment with this mode. In contrast, all the other brands have
relatively small contributions, suggesting minimal association with the
behavior captured by this mode. The fourth mode displays a distinct cyclic
behavior with a period of 5 days. This short cyclic pattern implies that there
is a recurring pattern or process occurring in the market every 5 days,
specifically related to brand 2. Understanding the nature and drivers of this
5-day cycle can provide insights into the intraday dynamics of the market and
may be indicative of specific trading patterns, market microstructure effects,
or idiosyncratic factors impacting brand 2. The fifth mode displays a notable
cyclic behavior with a period of 19 days. This medium-term cyclic pattern
suggests that there is a recurring pattern or process occurring in the market
approximately every 19 days, specifically related to brand 2. Understanding
the nature and drivers of this 19-day cycle can provide insights into the
dynamics of the market over this time frame and may be indicative of specific
market events, seasonal trends, or other factors impacting brand 2. The
seventh mode displays a distinct cyclic behavior with a period of 7 days. This
short-term cyclic pattern suggests that there is a recurring pattern or
process occurring in the market approximately every 7 days, specifically
related to brand 2. Understanding the nature and drivers of this 7-day cycle
can provide insights into the weekly dynamics of the market and may be
indicative of specific market events, trading patterns, or other factors
impacting brand 2.

The associated temporal time dynamic mode for all these modes exhibits highly
decaying oscillatory behavior. The amplitude of the oscillations decays quite
rapidly, reaching almost zero over a short period of time. This decaying
behavior suggests that the influence of the dynamics captured by these modes
diminishes rapidly over time.

Brand 2 plays a significant role in shaping the behavior represented by all
these modes, while the other brands have minimal influence. The recurring
5-day, 19-day, and weekly cycles specific to brand 2 suggest that there are
intraday, medium-term, and weekly dynamics or trading patterns related to this
brand that occur on a regular basis. The rapid decay in amplitude demonstrated
by the temporal evolution indicates that these dynamics have a short-term
impact and quickly diminish over time. Understanding the dynamics captured by
this fifth top mode can provide valuable insights for investors and analysts,
particularly those interested in medium-term trading strategies or monitoring
brand 2 closely.

##### The sixth top-ranked mode

In this mode, brand 2 exhibits the highest contribution, indicating a strong
alignment with this mode. Additionally, Brand 9 has a moderate contribution,
while all the other brands have relatively small contributions, suggesting
limited association with the behavior captured by this mode. This mode
displays a distinct cyclic behavior with a period of 31 days. This medium-term
cyclic pattern suggests that there is a recurring pattern or process occurring
in the market approximately every 30 days, specifically related to brand 2 and
to a lesser extent, brand 9. Understanding the nature and drivers of this
31-day cycle can provide insights into the dynamics of the market over this
time frame and may be indicative of specific market events, seasonal trends,
or other factors impacting Brand 2 and Brand 9.

The associated temporal time dynamic mode for this mode exhibits decaying
oscillatory behavior, transitioning between negative and positive values. The
amplitude of this oscillation decays over time, gradually reaching almost zero
after several cycles, in about 200 days. This decaying behavior indicates that
the influence of the dynamics captured by this mode diminishes over time. The
combination of brand 2’s highest contribution, Brand 9’s moderate
contribution, the 31-day cyclic behavior, and the decaying oscillatory pattern
in the temporal mode highlight the connection between brand 2, brand 9, and
the underlying market dynamics. Brand 2 plays a significant role in shaping
the behavior represented by this mode, while Brand 9 also contributes to a
moderate extent. The recurring 31-day cycle specific to these brands suggests
that there are medium-term dynamics or trends related to these brands that
occur periodically. The decay in amplitude indicates that these dynamics have
a medium-term impact and gradually diminish over time.

Understanding the dynamics captured by this sixth top mode can provide
valuable insights for investors and analysts, particularly those interested in
medium-term trading strategies or monitoring brand 2 and brand 9 closely.

##### The eight top-ranked mode

In this mode brands 2, Brand 5, and Brand 11 exhibit the highest
contributions, indicating a strong alignment with this mode. Conversely, all
the other brands have relatively small contributions, suggesting a limited
association with the behavior captured by this mode. This eighth mode displays
a distinct cyclic behavior with a period of 11 months. This long-term cyclic
pattern suggests that there is a recurring pattern or process occurring in the
market approximately every 11 months, specifically related to brand 2, brand
5, and brand 11. Understanding the nature and drivers of this cycle can
provide insights into the annual dynamics of the market and may be indicative
of seasonal trends, macroeconomic factors, or other long-term influences
impacting these brands. The associated temporal time dynamic for this mode
exhibits long-term decaying oscillatory behavior, transitioning between
negative and positive values. The oscillations complete three cycles in 1024
days, corresponding to the full period of observations. Furthermore, the
amplitude of this oscillation is reduced to half its initial value after the
full period of observations, indicating a gradual decay of influence over
time. This long-term decay in amplitude indicates that these annual dynamic
trends have a lasting impact, while gradually diminishing over the full
observation period.

This comparative analysis highlights the diverse dynamics and relationships
between brands in different modes. It indicates that certain brands
consistently contribute more significantly to specific modes, while others
have relatively smaller or negligible contributions. Understanding these
variations in brand contribution and alignment can provide valuable insights
into the underlying market dynamics and help identify key players or factors
influencing different modes. Gaining insights from the dominant patterns
captured by the highest-ranked modes can be highly valuable for investors and
analysts. It allows them to identify prevalent market trends, cyclic patterns
spanning short-term, medium-term, and long-term durations, and the underlying
factors influencing brand prices during those periods. This knowledge proves
particularly useful for those interested in devising investment strategies
across different timeframes.

By understanding and recognizing these patterns, market participants can make
more informed decisions regarding their short-term, medium-term, and long-term
investment strategies. They can effectively manage risks and potentially
uncover trading opportunities associated with the observed dynamics.
Additionally, studying these modes enables market participants to pinpoint the
specific factors or events driving the cyclic behavior exhibited by these
prominent modes. This awareness can help them seize short-term, medium-term,
and long-term investment opportunities tied to the observed market dynamics.

####  4.4.2 Brands sales analysis

##### The first top-ranked mode

The high magnitudes of brands 2, 4, and 10 in this mode imply that these
specific brands play a significant role in driving the overall sales patterns
captured by the mode. Their strong presence suggests that these brands are
likely contributing to the majority of sales within the analyzed time series.
This could be attributed to various factors such as brand popularity,
effective marketing strategies, product differentiation, or consumer
preferences favoring these particular brands. The high magnitudes highlight
the prominence of these brands in the market and their potential influence on
the overall sales dynamics.

Figure 21: The left panels are the DMD spatial modes for the selected top-
ranked modes, and in the right panel shows the corresponding temporal dynamic
of these modes. Figure 22: The left panels are the DMD spatial modes for the
selected top-ranked modes, and in the right panel shows the corresponding
temporal dynamic of these modes. Figure 23: The left panel shows the DMD
spatial modes, and the right panel shows the corresponding temporal dynamic.

On the other hand, the low magnitudes of the remaining brands in this mode
indicate that their contributions to the overall sales patterns captured by
the mode are relatively minor. These brands might have lower market shares or
weaker sales performance compared to brands 2, 4, and 10. The lower magnitudes
suggest that the sales of these brands have a relatively smaller impact on the
mode’s temporal dynamics. Understanding the differences in magnitude among the
brands within this mode can provide insights into market segmentation, brand
positioning, or competitive dynamics within the industry. The zero frequency
of this mode suggests a stationary behavior, indicating that the underlying
market dynamics driving the sales patterns in this mode do not exhibit
oscillatory or cyclical characteristics. Instead, the mode represents a more
stable or consistent aspect of the market dynamics. This could be associated
with factors such as long-term brand loyalty, steady demand for specific
products, or stable market conditions. The absence of frequency indicates that
the sales patterns captured by this mode are driven by non-time-varying
factors.

The fast pure decay observed in the temporal dynamic of this mode indicates
that the influence or impact of the underlying market dynamics diminishes
rapidly over time. This suggests that the factors driving the sales patterns
in this mode have a short duration or a limited effect on the brands involved.
It could imply that short-term promotional campaigns, temporary market
conditions, or specific events are the driving forces behind the sales
dynamics captured by this mode. The fast decay highlights the transient nature
of the market dynamics associated with this mode, suggesting a need for agile
and responsive strategies to capitalize on the identified sales patterns.

##### The second top-ranked mode

The high magnitude of brand 8 in this mode suggests that it plays a
significant role in driving the sales patterns captured by the mode. This
particular brand likely contributes substantially to the overall sales within
the analyzed time series. The high magnitude underscores the prominence of
brand 8 in the market and its potential influence on the mode’s sales
dynamics. Factors such as brand reputation, marketing efforts, product
quality, or customer loyalty could contribute to the strong performance of
brand 8 within this mode. Understanding the significance of brand 8 provides
insights into its market position, competitive advantage, or unique selling
proposition. The medium magnitude of brand 2 indicates its moderate
contribution to the sales patterns captured by this mode. While not as
influential as brand 8, brand 2 still plays a notable role in driving the
overall sales dynamics within the mode. This suggests that brand 2 has a
relatively stable market presence and a consistent level of consumer demand.
Factors such as product portfolio, customer satisfaction, or market
positioning could contribute to the moderate performance of brand 2 within
this mode. Understanding the medium magnitude of brand 2 allows for insights
into its market share, customer base, or competitive positioning. The low
magnitudes of the remaining brands in this mode indicate their relatively
minor contributions to the overall sales patterns captured by the mode. These
brands might have lower market shares or weaker sales performance compared to
brand 8 and brand 2. The lower magnitudes suggest that the sales of these
brands have a limited impact on the mode’s temporal dynamics. Understanding
the differences in magnitude among the brands within this mode can provide
insights into market segmentation, brand performance, or competitive dynamics
within the industry.

Similar to the first top-ranked mode, the zero frequency of the second top-
ranked mode indicates a stationary behavior, signifying that the underlying
market dynamics driving the sales patterns in this mode do not exhibit
oscillatory or cyclical characteristics. Instead, the mode represents a stable
or consistent aspect of the market dynamics. This could be associated with
factors such as long-term customer preferences, steady demand for specific
products, or relatively stable market conditions. The absence of frequency
implies that the sales patterns captured by this mode are driven by non-time-
varying factors. The fast pure decay observed in the temporal dynamic of this
mode suggests that the influence or impact of the underlying market dynamics
diminishes rapidly over time. This indicates that the factors driving the
sales patterns in this mode have a short duration or a limited effect. It
suggests the presence of short-term market trends, transient consumer
preferences, or specific events influencing the sales dynamics captured by
this mode. The fast decay highlights the need for agile and responsive
strategies to capitalize on the identified sales patterns within this mode.

##### The third top-ranked mode

This mode exhibits distinct characteristics that offer insights into the
underlying market dynamics and the connection with specific brands. In this
mode, the magnitude of brand 8 is high, while the magnitudes of all other
brands are relatively low. Furthermore, this mode demonstrates an
approximately 9-month cyclic behavior. The associated temporal dynamic of the
mode displays a slow decay with a low frequency. The high magnitude of brand 8
in this mode indicates its significant role in driving the sales patterns
captured by the mode. Brand 8 likely holds a dominant position in the market,
contributing substantially to the overall sales within the analyzed time
series. Factors such as brand reputation, customer loyalty, effective
marketing strategies, or product differentiation could contribute to the
strong performance of brand 8 within this mode. Understanding the prominence
of brand 8 provides insights into its market share, competitive advantage, or
customer perception. The low magnitudes of the other brands within this mode
suggest their relatively minor contributions to the overall sales patterns
captured. These brands might have lower market shares or weaker sales
performance compared to brand 8. The lower magnitudes indicate that the sales
of these brands have a limited impact on the mode’s temporal dynamics.
Understanding the differences in magnitude among the brands within this mode
can provide insights into market segmentation, brand performance, or
competitive dynamics within the industry.

The approximately 9-month cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics. This suggests that there are
underlying factors or influences that operate on a roughly 9-month cycle,
leading to periodic fluctuations in sales. These factors could include
seasonal variations, consumer behavior patterns, or industry-specific trends
that repeat in a cyclic manner. Understanding the cyclic behavior allows for
targeted marketing campaigns, inventory management, or product development
strategies to align with the identified sales patterns within this mode.

The slow decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
diminishes gradually over time. This suggests that the factors driving the
sales patterns within this mode have a more prolonged effect compared to the
previously discussed modes. The slow decay suggests a sustained influence that
lingers even after the initial impact. This could be attributed to factors
such as long-term consumer preferences, market trends that evolve gradually,
or product life cycles. The low frequency of the mode further supports the
notion of a slow and sustained influence on the sales dynamics.

##### The fourth top-ranked mode

This mode reveals specific characteristics that provide insights into the
underlying market dynamics and the connection with specific brands. In this
mode, the magnitudes of brands 2, 8, and 10 are high, while the magnitudes of
all other brands are relatively low. Additionally, this mode exhibits an
approximately 9-day cyclic behavior. The associated temporal dynamic of the
mode displays a fast decay with a high frequency. The high magnitudes of
brands 2, 8, and 10 within this mode highlight their significant roles in
driving the sales patterns captured. These brands likely hold prominent
positions in the market and contribute substantially to the overall sales
within the analyzed time series. Factors such as brand reputation, product
popularity, effective marketing strategies, or customer loyalty could
contribute to the strong performance of brands 2, 8, and 10 within this mode.
Understanding the high magnitudes of these brands provides insights into their
market shares, competitive advantages, or customer preferences. The low
magnitudes of the other brands within this mode suggest their relatively minor
contributions to the overall sales patterns captured. These brands might have
lower market shares or weaker sales performance compared to brands 2, 8, and
10. The lower magnitudes indicate that the sales of these brands have a
limited impact on the mode’s temporal dynamics. Understanding the differences
in magnitude among the brands within this mode can provide insights into
market segmentation, brand performance, or competitive dynamics within the
industry.

The approximately 9-day cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics. This suggests that there are
underlying factors or influences that operate on a roughly 9-day cycle,
leading to periodic fluctuations in sales. These factors could include weekly
or bi-weekly consumer behavior patterns, promotional activities, or specific
events that impact sales at regular intervals. Understanding the cyclic
behavior allows for better timing of marketing campaigns, inventory
management, or resource allocation to align with the identified sales patterns
within this mode. The fast decay observed in the associated temporal dynamic
of this mode indicates that the influence or impact of the underlying market
dynamics diminishes rapidly over time. This suggests that the factors driving
the sales patterns within this mode have a short duration or a limited effect.
The fast decay implies a transient influence that fades quickly after the
initial impact. This could be attributed to factors such as short-term
promotions, short-lived consumer trends, or time-sensitive market conditions.
The high frequency of the mode further supports the notion of a fast-paced and
rapidly changing influence on the sales dynamics.

##### The fifth top-ranked mode

This mode exhibits distinct characteristics that provide valuable insights
into the underlying market dynamics and the connection with specific brands.
In this mode, the magnitude of brand 8 is high, while the magnitudes of brands
1, 2, 9, and 10 are medium. The magnitudes of all other brands within this
mode are relatively low. Moreover, this mode demonstrates an approximately
3-day cyclic behavior. The associated temporal dynamic of the mode displays a
fast decay with a high frequency. The high magnitude of brand 8 within this
mode indicates its significant role in driving the sales patterns captured.
Brand 8 likely holds a dominant position in the market, contributing
substantially to the overall sales within the analyzed time series. Factors
such as brand reputation, customer loyalty, effective marketing strategies, or
product differentiation could contribute to the strong performance of brand 8
within this mode. Understanding the prominence of brand 8 provides insights
into its market share, competitive advantage, or customer perception. The
medium magnitudes of brands 1, 2, 9, and 10 suggest their moderate
contributions to the overall sales patterns captured by this mode. These
brands likely have relatively stable market positions and consistent levels of
consumer demand. Factors such as product quality, customer satisfaction,
competitive pricing, or marketing efforts could contribute to the moderate
performance of these brands within this mode. Understanding the medium
magnitudes of these brands allows for insights into their market shares,
customer bases, or competitive positioning. The low magnitudes of the other
brands within this mode indicate their relatively minor contributions to the
overall sales patterns captured. These brands might have lower market shares
or weaker sales performance compared to brand 8 and the aforementioned medium-
magnitude brands. The lower magnitudes suggest that the sales of these brands
have a limited impact on the mode’s temporal dynamics. Understanding the
differences in magnitude among the brands within this mode can provide
insights into market segmentation, brand performance, or competitive dynamics
within the industry.

The approximately 3-day cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics. This suggests that there are
underlying factors or influences that operate on a roughly 3-day cycle,
leading to periodic fluctuations in sales. These factors could include
specific weekly or bi-weekly consumer behavior patterns, short-term
promotions, or other temporal patterns that impact sales at regular intervals.
Understanding the cyclic behavior allows for targeted marketing campaigns,
inventory management, or resource allocation to align with the identified
sales patterns within this mode.

The fast decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
diminishes rapidly over time. This suggests that the factors driving the sales
patterns within this mode have a short duration or a limited effect. The fast
decay implies a transient influence that fades quickly after the initial
impact. This could be attributed to factors such as short-term consumer
trends, time-sensitive market conditions, or specific events affecting sales
within a short timeframe. The high frequency of the mode further supports the
notion of a fast-paced and rapidly changing influence on the sales dynamics.

##### The sixth top-ranked mode

This mode reveals distinct characteristics that provide valuable insights into
the underlying market dynamics and the connection with specific brands. In
this mode, the magnitudes of brand 2, brand 8, and brand 10 are high, while
the magnitudes of all other brands are relatively low. Additionally, this mode
exhibits an approximately 3-day cyclic behavior. The associated temporal
dynamic of the mode displays a fast decay with a high frequency. The high
magnitudes of brand 2, brand 8, and brand 10 within this mode highlight their
significant roles in driving the sales patterns captured. These brands likely
hold prominent positions in the market and contribute substantially to the
overall sales within the analyzed time series. Factors such as brand
reputation, product popularity, effective marketing strategies, or customer
loyalty could contribute to the strong performance of these brands within this
mode. Understanding the high magnitudes of these brands provides insights into
their market shares, competitive advantages, or customer preferences. The low
magnitudes of the other brands within this mode suggest their relatively minor
contributions to the overall sales patterns captured. These brands might have
lower market shares or weaker sales performance compared to brand 2, brand 8,
and brand 10. The lower magnitudes indicate that the sales of these brands
have a limited impact on the mode’s temporal dynamics. Understanding the
differences in magnitude among the brands within this mode can provide
insights into market segmentation, brand performance, or competitive dynamics
within the industry.

The approximately 3-day cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics. This suggests that there are
underlying factors or influences that operate on a roughly 3-day cycle,
leading to periodic fluctuations in sales. These factors could include
specific weekly or bi-weekly consumer behavior patterns, short-term
promotions, or other temporal patterns that impact sales at regular intervals.
Understanding the cyclic behavior allows for targeted marketing campaigns,
inventory management, or resource allocation to align with the identified
sales patterns within this mode.

The fast decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
diminishes rapidly over time. This suggests that the factors driving the sales
patterns within this mode have a short duration or a limited effect. The fast
decay implies a transient influence that fades quickly after the initial
impact. This could be attributed to factors such as short-term consumer
trends, time-sensitive market conditions, or specific events affecting sales
within a short timeframe. The high frequency of the mode further supports the
notion of a fast-paced and rapidly changing influence on the sales dynamics.

##### The seventh top-ranked mode

This mode reveals distinct characteristics that provide valuable insights into
the underlying market dynamics and the connection with specific brands. In
this mode, the magnitudes of brand 1, brand 2, and brand 3 are low, while the
magnitudes of all other brands are high. Additionally, this mode exhibits an
approximately 34.5-month cyclic behavior. The associated temporal dynamic of
the mode displays a slow decay with a very low frequency, completing only one
cycle over the entire observation period of 1024 days. The low magnitudes of
brand 1, brand 2, and brand 3 within this mode suggest their relatively minor
contributions to the overall sales patterns captured. These brands might have
lower market shares or weaker sales performance compared to the other brands
within this mode. The low magnitudes indicate that the sales of these brands
have a limited impact on the mode’s temporal dynamics. Understanding the
differences in magnitude among the brands within this mode can provide
insights into market segmentation, brand performance, or competitive dynamics
within the industry. On the other hand, the high magnitudes of the other
brands within this mode highlight their significant roles in driving the sales
patterns captured. These brands likely hold prominent positions in the market
and contribute substantially to the overall sales within the analyzed time
series. Factors such as brand reputation, product popularity, effective
marketing strategies, or customer loyalty could contribute to the strong
performance of these brands within this mode. Understanding the high
magnitudes of these brands provides insights into their market shares,
competitive advantages, or customer preferences.

The approximately 34.5-month cyclic behavior observed in this mode implies a
long-term recurring pattern in the sales dynamics. This suggests that there
are underlying factors or influences that operate on a roughly 34.5-month
cycle, leading to periodic fluctuations in sales. These factors could include
macroeconomic trends, seasonal variations, industry cycles, or other long-term
patterns that impact sales. Understanding the cyclic behavior allows for
strategic planning, forecasting, or identifying opportunities and challenges
that arise within this specific time frame.

The slow decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
persists over an extended period. This suggests that the factors driving the
sales patterns within this mode have a long-lasting effect, potentially
spanning multiple years. The slow decay implies a sustained influence that
gradually changes over time. This could be attributed to factors such as long-
term market trends, changes in consumer preferences, or industry-wide shifts
that unfold gradually. The very low frequency of the mode, completing only one
cycle over the entire observation period, further emphasizes the long-term
nature of the underlying market dynamics.

##### The eighth top-ranked mode

This mode reveals distinct characteristics that provide valuable insights into
the underlying market dynamics and the connection with specific brands. In
this mode, the magnitudes of brand 2 and brand 10 are high, while the
magnitudes of all other brands are relatively low. Additionally, this mode
exhibits an approximately 2-day cyclic behavior. The associated temporal
dynamic of the mode displays a fast decay with a high frequency. The high
magnitudes of brand 2 and brand 10 within this mode highlight their
significant roles in driving the sales patterns captured. These brands likely
hold prominent positions in the market and contribute substantially to the
overall sales within the analyzed time series. Factors such as brand
reputation, product popularity, effective marketing strategies, or customer
loyalty could contribute to the strong performance of these brands within this
mode. Understanding the high magnitudes of these brands provides insights into
their market shares, competitive advantages, or customer preferences. The low
magnitudes of the other brands within this mode suggest their relatively minor
contributions to the overall sales patterns captured. These brands might have
lower market shares or weaker sales performance compared to brand 2 and brand
10. The lower magnitudes indicate that the sales of these brands have a
limited impact on the mode’s temporal dynamics. Understanding the differences
in magnitude among the brands within this mode can provide insights into
market segmentation, brand performance, or competitive dynamics within the
industry.

The approximately 2-day cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics within a short time period. This
suggests that there are underlying factors or influences that operate on a
roughly 2-day cycle, leading to periodic fluctuations in sales. These factors
could include daily consumer behavior patterns, weekday vs. weekend sales
variations, or other short-term temporal patterns that impact sales at regular
intervals. Understanding the cyclic behavior allows for targeted marketing
campaigns, inventory management, or resource allocation to align with the
identified sales patterns within this mode.

The fast decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
diminishes rapidly over time. This suggests that the factors driving the sales
patterns within this mode have a short duration or a limited effect. The fast
decay implies a transient influence that fades quickly after the initial
impact. This could be attributed to factors such as daily or short-term
consumer trends, time-sensitive market conditions, or specific events
affecting sales within a short timeframe. The high frequency of the mode
further supports the notion of a fast-paced and rapidly changing influence on
the sales dynamics.

##### The ninth top-ranked mode

This mode reveals distinct characteristics that provide valuable insights into
the underlying market dynamics and the connection with specific brands. In
this mode, the magnitudes of brand 5, brand 6, brand 9, brand 10, brand 11,
and brand 12 are high, while the magnitudes of all other brands are relatively
low. Additionally, this mode exhibits an approximately 7-day cyclic behavior.
The associated temporal dynamic of the mode displays a slow decay with a high
frequency. The high magnitudes of brand 5, brand 6, brand 9, brand 10, brand
11, and brand 12 within this mode highlight their significant roles in driving
the sales patterns captured. These brands likely hold prominent positions in
the market and contribute substantially to the overall sales within the
analyzed time series. Factors such as brand reputation, product popularity,
effective marketing strategies, or customer loyalty could contribute to the
strong performance of these brands within this mode. Understanding the high
magnitudes of these brands provides insights into their market shares,
competitive advantages, or customer preferences. The low magnitudes of the
other brands within this mode suggest their relatively minor contributions to
the overall sales patterns captured. These brands might have lower market
shares or weaker sales performance compared to brand 5, brand 6, brand 9,
brand 10, brand 11, and brand 12. The lower magnitudes indicate that the sales
of these brands have a limited impact on the mode’s temporal dynamics.
Understanding the differences in magnitude among the brands within this mode
can provide insights into market segmentation, brand performance, or
competitive dynamics within the industry.

The approximately 7-day cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics with a weekly cycle. This suggests
that there are underlying factors or influences that operate on a roughly
7-day cycle, leading to periodic fluctuations in sales. These factors could
include weekly consumer behavior patterns, day-of-the-week effects, or other
weekly temporal patterns that impact sales at regular intervals. Understanding
the cyclic behavior allows for targeted marketing campaigns, inventory
management, or resource allocation to align with the identified sales patterns
within this mode.

The slow decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
persists over time. This suggests that the factors driving the sales patterns
within this mode have a lasting effect beyond the initial impact. The slow
decay implies a sustained influence that gradually changes over time. This
could be attributed to factors such as weekly consumer habits, long-term
trends in demand, or consistent purchasing patterns that unfold over the
course of a week. The high frequency of the mode further supports the notion
of a weekly cycle and the regularity of the influence on the sales dynamics.

##### The tenth top-ranked mode

This mode reveals distinct characteristics that provide valuable insights into
the underlying market dynamics and the connection with specific brands. In
this mode, the magnitude of brand 9 is high, while the magnitudes of brands 4,
5, 6, 7, 8, 10, 11, and 12 are medium. On the other hand, the magnitudes of
brands 1, 2, and 3 are low. Additionally, this mode exhibits an approximately
16-month cyclic behavior. The associated temporal dynamic of the mode displays
a slow decay with quite a low frequency, completing only two cycles over the
entire observation period of 1024 days. The high magnitude of brand 9 within
this mode highlights its significant role in driving the sales patterns
captured. This brand likely holds a prominent position in the market and
contributes substantially to the overall sales within the analyzed time
series. Factors such as brand reputation, product uniqueness, effective
marketing strategies, or customer loyalty could contribute to the strong
performance of brand 9 within this mode. Understanding the high magnitude of
this brand provides insights into its market share, competitive advantage, or
customer preferences. The medium magnitudes of brands 4, 5, 6, 7, 8, 10, 11,
and 12 within this mode suggest their relatively moderate contributions to the
overall sales patterns captured. These brands might have a sizable market
share or a significant customer base but are not as dominant as brand 9. The
medium magnitudes indicate that the sales of these brands have a noticeable
impact on the mode’s temporal dynamics but do not overshadow the influence of
brand 9. Understanding the relative magnitudes among these brands within this
mode can provide insights into market segmentation, brand performance, or the
competitive dynamics within the industry. The low magnitudes of brands 1, 2,
and 3 within this mode indicate their relatively minor contributions to the
overall sales patterns captured. These brands might have lower market shares
or weaker sales performance compared to the other brands within the mode. The
low magnitudes suggest that the sales of these brands have limited influence
on the mode’s temporal dynamics. Understanding the differences in magnitude
among these brands within this mode can provide insights into market
segmentation, brand performance, or the competitive landscape within the
industry.

The approximately 16-month cyclic behavior observed in this mode implies a
recurring pattern in the sales dynamics with a long-term cycle. This suggests
that there are underlying factors or influences that operate on a roughly
16-month cycle, leading to periodic fluctuations in sales. These factors could
include seasonal trends, economic cycles, or other long-term temporal patterns
that impact sales at regular intervals. Understanding the cyclic behavior
allows for long-term forecasting, strategic planning, or targeted marketing
efforts aligned with the identified sales patterns within this mode.

The slow decay observed in the associated temporal dynamic of this mode
indicates that the influence or impact of the underlying market dynamics
persists over time. This suggests that the factors driving the sales patterns
within this mode have a lasting effect beyond the initial impact. The slow
decay implies a sustained influence that gradually changes over time. This
could be attributed to factors such as long-term consumer trends,
macroeconomic conditions, or product life cycles that unfold over the course
of several months. The quite low frequency of the mode, completing only two
cycles over the whole observation period, further emphasizes the long-term
nature of the influence on the sales dynamics.

###  4.5 Magnitude and phase analysis

The elements-wise magnitude and phase-wise information of the DMD mode provide
valuable insights into the contributions, alignment, and synchronization of
different brands in response to the underlying factors driving the mode.

The elements-wise magnitude of the DMD mode represents the relative magnitudes
of contribution or alignment of the different brands to the mode. Higher
magnitudes indicate a stronger influence or contribution of a particular brand
to the mode, whereas lower magnitudes suggest a relatively weaker
contribution. This information allows us to identify which brands are more
closely aligned or have a greater impact on the dynamics captured by the DMD
mode. By examining the elements-wise magnitude, we can understand the extent
to which each brand’s price variations contribute to the overall behavior
represented by the DMD mode. Brands with higher magnitudes are more
influential in shaping the mode, indicating that their price movements have a
greater impact on the observed dynamics. On the other hand, the elements
phase-wise of the DMD mode capture the synchronization of the corresponding
brands in their response to the underlying factors driving the mode. The phase
information represents the relative timing or alignment of the oscillatory
patterns among the brands.

Analyzing the elements phase-wise allows us to understand if the brands
exhibit synchronized or coordinated movements in response to the underlying
factors. If the phase values of the brands are similar or closely aligned, it
suggests that they are synchronized and tend to move together in response to
the driving factors. Conversely, if the phase values are more dispersed or
varied, it indicates that the brands may exhibit divergent behaviors or have
different responses to the underlying factors. The phase information provides
insights into the temporal coordination and relationships between the brands
captured by the DMD mode. It allows us to identify whether the brands tend to
move in unison or display varying degrees of synchronization in their response
to the underlying dynamics.

By considering both the elements-wise magnitude and phase-wise information, we
can gain a comprehensive understanding of the contributions, alignment, and
synchronization of different brands in relation to the underlying factors
driving the DMD mode. This knowledge can be valuable for identifying key
influencers, understanding interdependencies between brands, and predicting
their collective behavior in response to market forces or other driving
factors.

Figure 24: The elements-wise magnitude and the corresponding elements-wise
phase of the top-ranked brands price DMD modes, are combined and plotted as
data points Figure 25: The elements-wise magnitude and the corresponding
elements-wise phase of the top-ranked brands sales DMD modes, are combined and
plotted as data points

Figure 24 shows the phase-magnitude plot for the eight top-ranked modes,
associated with the price time-series data, and figure 25 shows the phase-
magnitude plot for the ten top-ranked modes associated with the sales time
series data. Across these modes, we observe diverse patterns in the phase and
magnitudes of the 12 brands. In some modes, we see brands that stand alone in
specific regions, indicating their unique market behavior and potential
influence on the overall mode. These brands may be driven by distinct market
dynamics, industry-specific factors, or company-specific strategies that set
them apart from others. On the other hand, we also notice brands that are
spread out in certain regions within the phase-magnitude scatter plots. This
suggests that these brands have similarities in their market behavior and may
be influenced by common underlying market dynamics or driving factors. The
alignment of these brands within the modes indicates a collective contribution
to the formation of the respective patterns.

####  4.5.1 Brands prices analysis

The analysis of the highest power top-ranked mode reveals interesting patterns
in terms of the element-wise phase and magnitude distribution among the
different brands. The mode exhibits a cyclic behavior with a period of 925
days, completing one full cycle over the entire time frame of 1024
observations. Upon examining the phase and magnitude plot, we observe distinct
clusters of brands in different regions. Brands 1, 5, 6, 7, 8, 10, 11, and 12
are tightly clustered in the high magnitude low phase region. This indicates
that these brands have a strong alignment and contribute significantly to the
dynamics captured by the mode. Their high magnitudes suggest that their price
variations have a substantial influence on the overall behavior described by
the mode. The low phase values indicate that these brands tend to be
synchronized and exhibit similar responses to the underlying factors driving
the mode. In contrast, brands 3, 4, and 9 are located in the high magnitude
high phase region. This suggests that these brands also contribute
significantly to the mode, but their responses to the underlying factors are
characterized by a phase shift compared to the tightly clustered brands. The
high magnitudes of these brands highlight their impact on the mode, while the
high phase values indicate that they exhibit a delayed or shifted response
compared to the tightly clustered brands.

Brand 2 stands alone in the low magnitude low phase region. Its low magnitude
suggests a relatively weaker contribution to the mode compared to the other
brands. The low phase value indicates that this brand exhibits a similar phase
alignment to the tightly clustered brands, implying a degree of
synchronization. However, its lower magnitude suggests that its price
variations have a relatively smaller influence on the overall dynamics
captured by the mode. The alignment of these brands with the mode and the
underlying market dynamics can provide insights into their interconnectedness
and response to common driving factors. The tightly clustered brands in the
high magnitude low phase region suggest a strong correlation and synchronized
behavior, indicating that they are influenced by shared market forces,
economic factors, or other common drivers. Their collective response to these
factors contributes significantly to the cyclic behavior captured by the mode.
To fully understand the underlying market dynamics and driving factors, a
deeper analysis of each brand, industry trends, economic conditions, and other
relevant factors would be necessary. However, based on the provided phase-
magnitude scatter plots, we can infer that the alignment of brands within the
modes and their varying phase and magnitude values indicate a complex
interplay of factors shaping their market behavior.

####  4.5.2 Brands sales analysis

The analysis of the first highest power top-ranked mode, which exhibits a zero
frequency and a fast pure decay in its associated temporal dynamics, reveals
interesting insights into the alignment of brands with this mode and the
underlying market dynamics. In terms of phase and magnitude, brand 2 stands
out as it is located in the high magnitude and almost zero phase region. This
suggests that brand 2 is strongly aligned with the dynamics captured by this
mode. Its high magnitude indicates that brand 2 contributes significantly to
the overall behavior described by this mode, while the almost zero phase
suggests that its sales patterns are in sync with the dominant oscillatory
behavior of this mode. On the other hand, brands 3, 5, and 11 are located in
the low magnitude and almost zero phase region. This implies that these brands
have relatively weaker contributions to the overall behavior described by the
mode and their sales patterns exhibit minimal phase shift compared to the
oscillations captured by the mode. These brands may have sales patterns that
are less influenced by the dominant oscillatory behavior and may be driven by
other factors or dynamics.

The remaining brands (1, 4, 6, 7, 8, 9, 10, and 12) are located in the same
high phase region but differ in magnitudes. This indicates that these brands
exhibit similar phase shifts in their sales patterns corresponding to the
dominant oscillatory behavior of this mode. However, the variation in
magnitudes suggests that they contribute differently to the overall behavior
described by the mode. Brands 4 and 10 have high magnitudes, indicating that
they contribute significantly to the overall behavior captured by the mode. On
the other hand, brands 1, 6, 7, 8, 9, and 12 have low magnitudes, suggesting
relatively weaker contributions. The connection between these findings and the
underlying market dynamics suggests that this mode captures a dominant
oscillatory pattern that is shared by several brands. The alignment of brand 2
with high magnitude and almost zero phase indicates that it closely follows
the oscillatory behavior described by this mode. This suggests that brand 2
may be strongly influenced by the factors driving this dominant oscillatory
pattern in the market.

Brands 3, 5, and 11, with low magnitudes and almost zero phase, exhibit sales
patterns that are less influenced by the dominant oscillatory behavior
captured by this mode. It is possible that these brands operate in niche
markets or are driven by different factors that lead to their distinct
behavior. The brands located in the same high-phase region but with different
magnitudes indicate that they share a similar phase shift in their sales
patterns, aligning with the dominant oscillatory behavior. However, the
variation in magnitudes suggests differences in the magnitude of influence or
contribution to this behavior. This may imply variations in market share,
consumer preferences, or competitive positioning among these brands.

###  4.6 Further deductions

By considering the relationships between the raw data patterns, the
characteristics of the spatial and temporal modes, and the underlying market
dynamics, businesses can gain valuable insights into brands positioning,
market segmentation, and the overall market dynamics that drive sales
behavior. Also, this analysis helps to identify the presence of steady
components, long-term trends, and the impact of external factors on the sales
behavior of the brands.

####  4.6.1 Brands prices

The analysis of the daily price values of the 12 brands over a 1024-day period
reveals that all brands exhibit a noticeable long-term slow-growing trend,
except for brand 2, which displays a slow decaying trend. This distinction in
the trend patterns indicates that brand 2 operates in a different market
dynamic compared to the other brands. The first top-ranked mode in the
analysis captures this long-term growing trend observed in the data. It is
characterized by a cyclic behavior with a period of approximately 925 days,
equivalent to approximately 2.5 years. This suggests the presence of a
recurring pattern or influence that drives the long-term growth exhibited by
the majority of brands. The associated temporal dynamic of this mode showcases
a slow oscillation pattern, completing only one cycle over the entire 1024-day
observation period. This slow oscillation further emphasizes the long-term
nature of the trend captured by this mode, highlighting that the growth or
decay observed in brand prices unfolds gradually over time. Interestingly, the
first top-ranked mode has a high contribution from all brands, except for
brand 2, which has a relatively small contribution. This indicates that the
long-term growing trend is a common feature shared by most brands, except for
brand 2. The distinct behavior of brand 2 suggests that it operates under
different market dynamics or is influenced by unique factors that result in a
slow decaying trend instead of the common growth observed in the other brands.
The connection between these findings and the underlying market dynamics
suggests that the first top-ranked mode captures the long-term growing trend
present in the data. The high contributions from all brands, except brand 2,
indicate that this trend is a shared characteristic driven by market-wide
factors, such as macroeconomic conditions, industry trends, or consumer
behaviors. The contrasting behavior of brand 2, with its slow decaying trend
and small contribution to the first mode, suggests that it operates under
different market dynamics or factors that set it apart from the common long-
term growth observed in the other brands. This distinction implies that brand
2 may be influenced by unique factors, such as specific market conditions,
competitive positioning, or consumer preferences, which contribute to its slow
decay instead of growth.

Upon analyzing the raw data, it is evident that brand 2 exhibits significant
and distinct high-frequency oscillations. These oscillations are not commonly
observed in the other brands, indicating that brand 2 experiences unique
market dynamics or factors that drive its price fluctuations. These
observations are further supported by the fourth, fifth, and seventh top-
ranked modes in the analysis. These modes exhibit cyclic behavior with periods
of approximately 5, 19, and 7 days, respectively. The presence of these
distinct periods suggests the existence of regular patterns or influences on
the price fluctuations of brand 2. These cyclic behaviors may be attributed to
factors such as seasonal demand, economic indicators, or industry-specific
events that impact the pricing dynamics of brand 2. The associated temporal
dynamics of these modes exhibit fast decaying oscillation patterns, indicating
that the price fluctuations captured by these modes are short-lived. This
implies that the market dynamics driving these oscillations may be transient
or subject to rapid changes. It suggests that brand 2’s pricing behavior is
responsive to short-term market forces and may require agile strategies to
adapt to the dynamic nature of the market.

Furthermore, these modes show a high contribution from brand 2, suggesting
that brand 2 plays a dominant role in driving the cyclic behaviors observed in
the price fluctuations. This indicates that brand 2’s pricing decisions,
market positioning, or consumer demand patterns have a significant influence
on the overall market dynamics captured by these modes. It implies that brand
2 holds a unique position in the market, potentially serving as a trendsetter
or representing a distinct segment with specific pricing dynamics. The
connection between these findings and the underlying market dynamics implies
that brand 2 operates within an environment characterized by distinct high-
frequency oscillations in its pricing behavior. The cyclic patterns observed
in the fourth, fifth, and seventh modes suggest the presence of external
factors or market forces that drive these price fluctuations. These dynamics
may be influenced by factors such as consumer behavior, market competition, or
industry-specific trends.

Also, the analysis of the raw data reveals that brands 2, 8, and 11 exhibit a
relatively similar slow oscillating pattern, setting them apart from the other
brands. This observation is reflected in the eighth mode, which captures this
cyclic behavior characterized by a period of 11 months. The associated
temporal time dynamic of this mode displays a long-term slow decaying
oscillatory behavior. Over the 1024-day observation period, the oscillations
complete three cycles, indicating a gradual and extended pattern. This
suggests that brands 2, 8, and 11 experience similar market dynamics or are
influenced by common factors that result in this long-term oscillating
behavior. Furthermore, the eighth mode demonstrates a high contribution from
brands 2, 8, and 11, while the remaining brands have low contributions. This
high contribution from the three brands suggests that they play a significant
role in driving the cyclic behavior captured by this mode. Their pricing
patterns and market dynamics align closely with the oscillatory nature of this
mode. The connection between these findings and the underlying market dynamics
implies that brands 2, 8, and 11 share common characteristics or are
influenced by similar factors that lead to their apparent similar slow
oscillating patterns. These factors could include industry-specific trends,
consumer behaviors, or external market forces affecting these brands uniquely.

The high contribution from these brands in the eighth mode suggests that they
are key drivers of the cyclic behavior observed. Their pricing decisions,
market positioning, or product offerings likely align with the oscillatory
nature of this mode, indicating that they are attuned to the market dynamics
captured by this mode. In contrast, the low contributions from the other
brands indicate that they have different pricing patterns or market dynamics
compared to brands 2, 8, and 11. They may be influenced by distinct factors or
operate in separate market segments, leading to their divergent behavior from
the cyclic patterns captured by the eighth mode.

The analysis of the raw data reveals a noticeable common slow oscillating
trend that is prominently observed in brands 1, 3, 5, 9, and 11 while being
less evident in brands 4, 6, 7, 10, and 12. Additionally, brands 2 and 8
exhibit distinct slow oscillating patterns, setting them apart from the other
brands. These observations are captured by the second top-ranked mode, which
characterizes the cyclic behavior with a period of approximately 654 days,
equivalent to approximately 1.8 years. This indicates the presence of a
recurring pattern or influence that drives the long-term oscillations observed
in the prices of the aforementioned brands.

The associated temporal dynamic of this mode showcases a slow oscillation
pattern, completing only one and a half cycles over the entire 1024-day
observation period. This slow oscillation further emphasizes the long-term
nature of the trend captured by this mode, indicating that the growth or decay
observed in the prices of the relevant brands unfolds gradually over time.
Furthermore, the second top-ranked mode exhibits high contributions from all
brands, except for brands 2 and 8, which have negligible contributions. This
suggests that the common slow oscillating trend is shared by brands 1, 3, 5,
9, and 11, while brands 2 and 8 exhibit different pricing dynamics that do not
align with this mode. The connection between these findings and the underlying
market dynamics implies that brands 1, 3, 5, 9, and 11 are subject to common
factors or market dynamics that result in the observed slow oscillating
patterns. These factors may include industry-specific trends, consumer
behaviors, or macroeconomic influences that impact these brands similarly. On
the other hand, brands 2 and 8 display distinctive slow oscillating patterns,
suggesting that they operate under different market dynamics or are influenced
by unique factors that set them apart from the common trend observed in the
other brands. Understanding these market dynamics and their connection to
specific brands is essential for businesses to make informed decisions.
Recognizing the shared slow oscillating trend in brands 1, 3, 5, 9, and 11
allows businesses to identify common market influences and tailor their
strategies accordingly. This knowledge can guide pricing decisions, marketing
campaigns, and resource allocation to align with the cyclic market dynamics
captured by the second mode. Additionally, acknowledging the distinct
behaviors of brands 2 and 8 provides insights into their unique market
positions and dynamics. Businesses can develop targeted approaches to address
the specific factors driving their pricing patterns and market trends,
optimizing their strategies to leverage the different market dynamics observed
in these brands.

####  4.6.2 Brands sales

Interestingly, in the analysis of the top-ten ranked spatial modes, a distinct
mode emerges with high contributions from almost all brands, except for brands
1, 2, and 3, which have low contributions to this mode, also the temporal
dynamics of this mode show a low-frequency pattern. This particular mode is
suspected to capture the pattern of the spikes observed in the raw data. The
low contributions from brands 1, 2, and 3 to this mode indicate that their
sales behavior deviates from the common spike patterns observed in the other
brands. This distinction suggests that these three brands may be influenced by
unique market dynamics or factors that set them apart from the other brands
regarding spike patterns. Analyzing the characteristics of the seventh-highest
mode further supports the connection between brands and underlying market
dynamics. The connection between these findings and the underlying market
dynamics suggests that the spike patterns in the raw data can be attributed to
specific factors or events that influence most of the brands, except for
brands 1, 2, and 3. Understanding these dynamics and their connection to
brands is crucial for market analysis and decision-making. Recognizing the
distinct spike patterns and the brands’ deviations from common patterns allows
businesses to identify unique market factors, consumer preferences, or
competitive strategies that impact their sales behavior.

Besides, in the analysis of the raw data, it becomes evident that brand 3
exhibits a unique pattern that sets it apart from the other brands. This
distinct pattern is characterized by a different sales behavior or market
dynamics compared to the rest of the brands. This observation is further
supported by the finding that brand 3 makes relatively low contributions to
the top-ranked spatial modes in the DMD analysis. This suggests that the sales
patterns of brand 3 deviate from the common trends captured by these modes,
indicating that its dynamics are distinct and less aligned with the overall
market behavior. Similarly, brand 1 shows a pattern that is less common among
the other brands in the raw data. This implies that brand 1 operates in a
different manner or experiences unique market dynamics compared to the other
brands. This distinction is reflected in the spatial modes of the top ten
ranked modes, as brand 1 exhibits moderate contributions to these modes. This
indicates that while brand 1 shares some similarities with the other brands,
it also possesses certain unique characteristics or sales patterns that set it
apart from the rest of the group. The connection between these observations
and the underlying market dynamics suggests that each brand operates within
its own market niche or experiences specific factors that drive its sales
behavior. Brand 3’s distinct pattern and low contributions to the top-ranked
spatial modes suggest that it may be influenced by unique market dynamics,
targeted consumer segments, or differentiated marketing strategies. Similarly,
brand 1’s less common pattern and moderate contributions to the spatial modes
indicate that it possesses both shared market characteristics and individual
brand-specific dynamics. Understanding these brand-specific dynamics is
crucial for market analysis and decision-making. Recognizing the unique
patterns and contributions of each brand allows for a more comprehensive
understanding of the market landscape, enabling businesses to tailor
strategies, allocate resources effectively, and make informed decisions that
align with the specific dynamics of each brand.

###  4.7 Further comments and interpretations

Economically speaking, the results are of great importance. From a theoretical
point of view, they confirm the presence of the time-scale factor or behavior
in marketing time series and thus can lead scientists in the field to take
this factor into consideration in future models. From a practical point of
view, or considering economic and financial reality, the results clearly
indicate the movements of the markets studied. We notice stability at short
time scales, with a slight growth for many brands. Then, the volumes and the
prices become disturbed for medium horizons, sometimes with large
fluctuations. Finally, for long horizons (high levels), we notice a remarkable
increase. These facts can be justified and explained economically. Indeed, the
study period is strongly related to several sociological and political
phenomena that had a great influence on the economic state of the market. At
the beginning of the period, the local market was affected by the turmoil in
several Arab countries, which negatively affected sales, especially since a
large part of the workforce originally belonged to these countries. The Qatar
embargo and the wars that broke out near the kingdom (forcing it to be
involved in some of them) had a great impact on marketing. Thus, the local
market lost several tributaries and extensions.

The beginning of the period of study also coincided with the discussion about
the added tax on all sales, which negatively affected the purchasing power of
citizens and residents and led to a kind of stagnation in the market.
Furthermore, some severe laws relating to residents, such as taxes on family
members, also led to the permanent deportation of a large number of residents
and their families in their countries of origin. This factor had the greatest
impact on sales and commercial movements.

As a result of this stagnation, the authorities resorted to ways to compensate
for the added value through grants to employees and support for national
production, which paralleled the idea of reducing the dependence on oil as a
sole income, instead encouraging national industries, of which commercial