11institutetext:  Information Systems Group
Tallinn University of Technology
Akadeemia tee 15a, 12618 Tallinn, Estonia
11email<EMAIL_ADDRESS>
# Discretizing Numerical Attributes: An Analysis of Human Perceptions

Minakshi Kaushik  0000-0002-6658-1712 Rahul Sharma  0000-0002-9024-8768 Dirk
Draheim  0000-0003-3376-7489

###### Abstract

Machine learning (ML) has employed various discretization methods to partition
numerical attributes into intervals. However, an effective discretization
technique remains elusive in many ML applications, such as association rule
mining. Moreover, the existing discretization techniques do not reflect best
the impact of the independent numerical factor on the dependent numerical
target factor. This research aims to establish a benchmark approach for
numerical attribute partitioning. We conduct an extensive analysis of human
perceptions of partitioning a numerical attribute and compare these
perceptions with the results obtained from our two proposed measures. We also
examine the perceptions of experts in data science, statistics, and
engineering by employing numerical data visualization techniques. The analysis
of collected responses reveals that $68.7\%$ of human responses approximately
closely align with the values generated by our proposed measures. Based on
these findings, our proposed measures may be used as one of the methods for
discretizing the numerical attributes.

###### Keywords:

Machine learning data mining discretization numerical attributespartitioning

##  1 Introduction

Various types of variables are available in real-world data. However, discrete
values have explicit roles in statistics, machine learning (ML), and data
mining. Presently, there is no benchmark approach to find the optimum
partitions for discretizing complex real-world datasets. Generally, if a
factor impacts another factor, in that case, humans can easily perceive the
compartments or partitions because the human brain can easily perceive the
differences between the factors and detect the partitions. However, it is not
easy for a human or even an expert to find the appropriate compartments in
complex real-world datasets. In state-of-the-art, to find the optimum
partitions of the numerical values, various discretization techniques have
also been presented in the literature [23, 13, 22]. However, the existing
discretization techniques do not reflect best the impact of the independent
numerical factor on the dependent numerical target factor. Moreover, no
existing discretization approach uses numerical attributes as influencing and
response factors.

To find the cut-points for the cases of two-partitioning and three-
partitioning, we have proposed two measures  _Least Squared Ordinate-Directed
Impact Measure_ (LSQM) and  _Least Absolute-Difference Ordinate-Directed
Impact Measure_ (LADM) [18]. These measures provide a simple way to find
partitions of numerical attributes that reflect best the impact of one
independent numerical attribute on a dependent numerical attribute. In these
measures, we use numerical attributes as influencing and response factors to
distinguish them from the existing approaches.

In this paper, the outcome of  _LSQM_ and  _LADM_ measures are compared with
the human-perceived cut-points to assess the accuracy of the measures. We use
numerical attributes as influencing and response factors to distinguish them
from the existing approaches. A series of graphs with different data points
are used to collect the human responses. Here, data scientists, ML experts and
other non-expert persons are referred to as humans.

The idea of this research emerged from the research on partial
conditionalization [8, 9], association rule mining (ARM) [31, 29] and
numerical association rule mining (NARM) [32, 19, 20]. These papers discuss
the discretization process as an essential step for NARM. Moreover, research
on discretizing the numerical attributes is an essential step in frequent
itemset mining, especially for quantitative association rule mining (QARM)
[32].

In the same sequence, we have also presented a tool named Grand report [27]
and a framework [30] for unifying ARM, statistical reasoning, and online
analytical processing. These paper strengthens the generalization of ARM by
finding the partitions of numerical attributes that reflect best the impact of
one independent numerical attribute on a dependent numerical attribute. Our
vision is to develop an ecosystem to generalize the ML approaches by
significantly improving the ARM from different dimensions.

The paper is organized as follows. In Sect. 2, we delve into the discussion of
related work concerning discretization and its connection with human
perception. This section aims to provide a comprehensive overview of prior
research and studies that have explored the topic from different angles. In
Sect. 3, we explain the motivation for conducting this study. Sect. 4
describes the  _LSQM_ and  _LADM_ measures. Then, we describe the design of
the experiment in Sect. 5. In Sect. 6, we present the analysis and results.
The conclusion and future work are given in Sect. 7.

##  2 Related Work

Based on human perception evaluation and different discretization techniques,
we discuss the related work in the direction of discretization and clustering
techniques and human perception.

###  2.1 Discretization

Many data mining and ML algorithms are not designed to work with numeric
attributes and instead require nominal attributes as input data [13]. In order
to convert numeric attributes into nominal attributes, different
discretization methods have been employed as a pre-processing measure.
Discretization techniques divide the range of a numeric attribute into $n$
intervals, which are determined by $n-1$ cut-points. A variety of
discretization methods are available in the literature [13],[22] and [23].
Dougherty et al. [7] compared and analyzed discretization strategies along
three dimensions: global versus local, supervised versus unsupervised, and
static versus dynamic. Static approaches discretize each attribute separately,
whereas dynamic methods conduct a search through space to find
interdependencies between features. Liu et al. [23] performed a systematic
study of existing discretization methods and proposed a hierarchical framework
for discretization methods from the perspective of splitting and merging. The
unsupervised static discretization method, such as equal-width, uses the
minimum and maximum values of the continuous attribute and then divides the
range into equal-width intervals called bins. In contrast, the equal-frequency
algorithm determines an equal number of continuous values and places them in
each bin [4].

The RUDE (Relative Unsupervised Discretization) algorithm [25] performs the
discretization of numerical attributes in three steps: pre-discretizing,
structure projection, and merging split points. In 2010, Joita [15] proposed
an unsupervised method for selecting the initial cluster centers for the
clustering of a one-dimensional vector of real-valued data. This method was
based on a k-means clustering algorithm and can be used in single-attribute
discretization. In 2013, Dietrich et al. [6] also proposed a method for
obtaining cut-points that is more intuitive for human users. This smoothed
discretization approach works as a post-processing step to obtain the
intervals after using an arbitrary traditional discretization approach. The
authors also proposed two measures,  _distance-based deviation measure_ and
_instance-based deviation measure_ , for comparing the original discretization
method cut-points with modified cut-points. The discretization cut-points were
computed for each training dataset for the three general discretization
methods: equal-frequency [4] discretization, entropy-based discretization
[12], [28] and Chi2 [24] discretization. After that, the introduced smoothing
approaches were used with distance-based and instance-based modification
measures to get the desired results. There is another similar work, called the
best piecewise constant approximation [21], which deals with approximating a
single variable function. Still, it is different because we are not using
signals, and our primary focus is on data sets that use several data points
for one value of the influencing factor. Eubank used the population quantile
function as a tool to show the best piecewise constant approximation problem
[11]. Later Bergerhoff [3] suggested a method for finding optimal piecewise
constant approximations of one-dimensional signals using particle swarm
optimization.

###  2.2 Human perceptual evaluation

Discretization approaches are usually evaluated based on their mathematical
backgrounds. However, we are the first to assess the discretization measures
by considering human perceptions.

In the state of the art, many studies have used human perception to evaluate
various techniques. However, they are not completely related to
discretization. Tatu et al. [33] proposed a preliminary investigation of human
perception using visual quality criteria for multidimensional data. The
authors conducted a user study to examine the relationship between human
cluster interpretation and the measurements automatically retrieved from 2D
scatter plots. Etemadpour et al. [10] conducted a perception-based evaluation
of high-dimensional data where humans were asked to identify clusters and
analyze distances inside and across clusters. Demiralp et al. [5] used human
judgments to estimate perceptual kernels for visual encoding variables such as
shape, size, color, and combinations. The experiment used Amazon’s Mechanical
Turk platform, with twenty Turkers completing thirty MTurk jobs. In [1], a new
visual quality measure (VQM) based on perceptual data was proposed to rank
monochrome scatter plots. This experiment collected perceptual data from human
subjects, and the best clustering model was chosen to create a perceptual-
based VQM of grouping patterns. Similarly, a study by Aupetit [2] analyzed and
compared clustering algorithms through the lens of human perception in 2D
scatter plots. The primary focus of the authors was to evaluate how accurately
clustering algorithms aligned with the way humans perceive clusters. The
authors evaluated Gaussian Mixture Models, CLIQUE, DBSCAN, Agglomerative
Clustering methods, and 1437 variations of k-means on the benchmark data. Our
work is also related to considering human perceptions for evaluating our
proposed  _LSQM_ and  _LADM_ measures for discretizing numerical attributes.

##  3 Motivation

Real-world data sets contain real or numerical values frequently. However,
many data mining and ML approaches need discrete values. For years, obtaining
discrete values from numerical values has been a complex and ongoing task. The
main issue with the discretization process is obtaining the perfect intervals
with specific ranges and numbers of intervals. In state of the art, several
discretization approaches such as equi-depth, equi-width [4], MDLP [12], Chi2
[24], D2 [4], etc. have been proposed. However, determining the most effective
discretizer for each situation is still a challenging problem. The existing
methods for discretizing numerical attributes are not automated and require
expert knowledge; therefore, there is a need to develop an automated and
formal measure for finding the optimal partition of numerical attributes.

In [18], we presented an order-preserving partitioning method to find the
partitions of numerical attributes that reflect best the impact of one
independent numerical attribute on a dependent numerical attribute. In extreme
cases (such as step-functions), humans can easily visualize the perfect
partitions and even the number of compartments. However, in distinct cases,
the ideal partition range depends on the perception of data experts. In state
of the art, no investigation is available to understand the human perception
of partitioning. Moreover, the current literature provides a comparison of
discretization methods and compares their results. In this paper, we take a
different approach to compare the human perception of discretization with the
outcome of the proposed discretization method. We aim to visualize the
differences between the outcomes of the proposed methods and the human
perception of discretization.

##  4 Impact Driven discretization Method

In the  _Impact driven discretization method_ [18], we perform discretization
on the independent numerical attribute using order-preserving partitioning to
understand its impact on the numerical target attribute. The method involves
creating a total of $(k-1)$ cut-points, with $k$ being the number of
partitions recommended by the user. Below are two measures introduced in the
paper [18].

###  4.1 The LSQM Measure

The  _LSQM_ measure operates by initially computing the squared difference
between the $y$-value of each data point and the average of $y$-values within
the current partition. This measure maintains the order of the independent
variable by considering the values of data points, ensuring that the values
within one partition are consistently lower than those in the subsequent
partition. After summing up the squared differences of the several partitions,
_LSQM_ retrieves the minimum values as cut-points.

######  Definition 1 (Least Squared Ordinate-Directed Impact Measure)

Given $n\\!\geq\\!2$ real-valued data points $(<x_{i},y_{i}>)_{1\leq i\leq
n}$, we define the _least squared ordinate-directed impact measure_ for
$k$-partitions (with $k\\!-\\!1$ _cut-points_) as follows:

$\mathop{min}_{i_{0}=0<i^{\prime}_{1}<...<i^{\prime}_{k-1}<i^{\prime}_{k}=n}\sum_{\scriptsize\begin{array}[]{c}j=1\\\\[-1.42262pt]
\end{array}}^{k}\hskip 1.42262pt\sum_{i^{\prime}_{j-1}<i"\leq
i^{\prime}_{j}}(y_{i"}-\mu_{i^{\prime}_{j-1}<\phi\leq i^{\prime}_{j}})^{2}$
(1)

where the _average of data values in a partition_ $\mu_{a<\phi\leq b}$ between
indexes $a$ and $b$ ($a<b\leq n$) is defined as

$\mu_{a<\phi\leq b}=\frac{\sum\limits_{a<\phi\leq b}y_{\phi}}{b-a}$ (2)

In (1), we have that $i^{\prime}_{j}$ is the highest element in the $j$-_th_
partition, where  _highest element_ means the data point with the highest
index.

Indeed, the  _LSQM_ (Least Squares Ordinate-Directed Impact Measure) measure
may appear similar to the $k$-means clustering algorithm on the surface, as
both involve partitioning data into clusters. However, they differ
significantly in their underlying principles and applications.

$k$-means clustering is primarily an unsupervised ML technique employed for
the task of clustering data points into groups or clusters, with each data
point assigned to the cluster whose centroid is closest to it in terms of a
chosen distance metric, often the Euclidean distance. Euclidean distance
metric calculates dissimilarity between data points, which involves measuring
the geometric distance between vectors $X$ and $Y$. The primary goal of
$k$-means is to minimize the sum of squared distances between data points and
their assigned cluster centroids, and it finds applications in various
domains, including customer segmentation, image compression, and data
reduction. However, $k$-means’ effectiveness is influenced by the initial
random selection of cluster centers, which can lead to different clustering
results depending on the initialization.

In contrast,  _LSQM_ is a specialized method designed specifically for
discretizing numerical attributes. Its core objective is to partition a
numerical attribute into intervals while preserving the order of data points
within those intervals.  _LSQM_ achieves this by measuring the squared
difference between the values of data points and the average of values within
each partition, aiming to minimize the sum of squared differences. Unlike
$k$-means,  _LSQM_ is not highly dependent on the initial point chosen to
start the partitioning process, making it robust in this regard.  _LSQM_ is
primarily employed in data preprocessing tasks related to data mining,
enhancing the quality of numerical attribute discretization.

In summary, $k$-means clustering is a versatile and widely used clustering
algorithm with applications across various domains, focusing on minimizing the
squared distances between data points and cluster centroids. On the other
hand,  _LSQM_ serves a specific purpose in discretizing numerical attributes
while maintaining the order of data points, making it particularly valuable in
data preprocessing for data mining tasks.

###  4.2 The LADM Measure

For the  _LADM_ measure, we take the sum of the absolute differences of the
several partitions.

######  Definition 2 (Least Absolute-Difference Ordinate-Directed Impact
Measure)

Given $n\\!\geq\\!2$ real-valued data points $(<x_{i},y_{i}>)_{1\leq i\leq
n}$, we define the _least absolute-difference ordinate-directed impact
measure_ for $k$-partitions (with $k\\!-\\!1$ _cut-points_) as follows:

$\mathop{min}_{i_{0}=0<i^{\prime}_{1}<...<i^{\prime}_{k-1}<i^{\prime}_{k}=n}\sum_{\scriptsize\begin{array}[]{c}j=1\\\\[-1.42262pt]
\end{array}}^{k}\hskip 1.42262pt\sum_{i^{\prime}_{j-1}<i"\leq
i^{\prime}_{j}}|y_{i"}-\mu_{i^{\prime}_{j-1}<\phi\leq i^{\prime}_{j}}|$ (3)

where the _average of data values in a partition_ $\mu_{a<\phi\leq b}$ between
indexes $a$ and $b$ ($a<b\leq n$) is defined as

$\mu_{a<\phi\leq b}=\frac{\sum\limits_{a<\phi\leq b}y_{\phi}}{b-a}$ (4)

##  5 Experimental Design

To understand how humans partition numerical factors, we designed a series of
graphs and asked several experts to partition the data given in the graphs.
Initially, to produce a diverse collection of graphs with different data
points, a set of graphs was shared and discussed with our own research team.
The team consists of three early-stage researchers and one senior researcher.
These graphs include step functions, linear functions, and mixed data graphs.
Finally, twelve graphs were selected to be shared with humans (see Figs. 1, 2
and 3). These graphs are obtained from nine synthetic datasets (D1 to D9) and
three real-world datasets (D10 to D12). These synthetic datasets (D1 to D9)
have only two numerical attributes. The dataset D10 is a real-world dataset.
The data set, DC public government employees [16], contains 33,424 records of
DC public government employees and their salaries in 2011. This dataset is
sourced from the Washington Times via Freedom of Information Act (FOIA)
requests. The dataset D11 is Heart disease dataset [14], and it is sourced
from the UCI machine learning repository. This dataset has 13 attributes and
303 records. We used attribute  _{_ Age} and  _{_ Cholesterol} for drawing the
graph. The dataset D12 is a New Jersey (NJ) school teacher salaries (2016)
[26] sourced from the (NJ) Department of Education. It contains 138715 records
and 15 attributes. We have taken only an initial 23000 rows from the dataset.
We are interested in the column  _{_ experience_total} and  _{_ salary}. A
copy of all these datasets is available in the GitHub repository [17].

Figure 1: Graphs for datasets D1 to D4. Figure 2: Graphs for datasets D5 to D8. Figure 3: Graphs for datasets D9 to D12. Table 1: The comparison of human perception to identify number of partitions based on their profile. Datasets |  Number of Partitions |  Total responses from DS/ML experts $=60\%$ |  Total Responses from Non-expert People $=40\%$
---|---|---|---
% Responses |  % Responses
D1 |  2 |  93.3% |  90%
3 |  6.67% |  10%
D2 |  2 |  73.3% |  60%
3 |  26.6% |  40%
D3 |  2 |  0% |  0%
3 |  93.3% |  100%
4 |  6.66% |  0%
D4 |  0 |  33.3% |  20%
2 |  53% |  70%
3 |  13.3% |  10%
D5 |  0 |  93.3% |  90%
2 |  0% |  0%
3 |  6.6% |  10%
D6 |  2 |  13.3% |  30%
3 |  26.6% |  0%
4 |  26.6% |  40%
5 |  33.3% |  30%
D7 |  2 |  60% |  40%
3 |  20% |  40%
4 |  20% |  20%
D8 |  2 |  33.3% |  60%
3 |  66.6% |  30%
4 |  0% |  10%
D9 |  2 |  73.3% |  60%
3 |  26.6% |  30%
4 |  0% |  10%
D10 |  0 |  40% |  40%
2 |  40% |  30%
3 |  6.66% |  20%
4 |  6.66% |  0%
5 |  6.66% |  10%
D11 |  0 |  53.3% |  60%
2 |  26.6% |  30%
3 |  20% |  0%
4 |  0% |  10%
D12 |  0 |  40% |  20%
2 |  26.6% |  30%
3 |  6.66% |  30%
4 |  20% |  10%
5 |  6.66% |  10%
Table 2: The comparison of human perceived cut-points with the LSQM and LADM measures. D |  P | Human Perception |  LSQM |  LADM
---|---|---|---|---
R |  Approx. near Cut-Points |  Cut-points |  Cut-Points
D1 |  2 |  92% |  50(91.3%), 48(8.6%) |  50 |  50
|  3 |  8% |  (48,60)(50%), (20,50)(50%) |  (20, 50) |  (20, 50)
D2 |  2 |  68% |  50(88.2%), 52(11.7%) |  52 |  52
|  3 |  32% |  (50,54)(37.5%), (20,53)(25%) |  (52, 54) |  (52, 54)
D3 |  3 |  96% |  (32,52)(62%), (30,52)(16.6%) |  (32,52) |  (32,52)
|  4 |  4% |  (20,32,52)(100%) |  (32,52,55) |  (32,52,60)
D4 |  0 |  28% |  NA |  NA |  NA
|  2 |  60% |  20(86.6%), 25(13.3%) |  20 |  20
|  3 |  12% |  (20,45)(66.6%), (20,30)(33.3%) |  (12, 24) |  (12, 25)
D5 |  0 |  92% |  NA |  NA |  NA
|  2 |  0% |  0% |  20 |  19
|  3 |  8% |  (14,28)(100%) |  (13, 26) |  (13, 26)
D6 |  2 |  20% |  32(40%), 42(40%) 50(20%) |  42 |  42
|  3 |  16% |  (42,68)(50%), (32,42)(25%) |  (32, 42) |  (32, 42)
|  4 |  32% |  (32,37,42)(87.5%), (33,37,43)(12.5%) |  (32, 37, 42) |  (32, 37, 42)
|  5 |  32% |  (32,42,37,68)(87.5%), (17,32,38,42)(12.5%) |  (32, 37, 42, 56) |  (32, 37, 42, 56)
D7 |  2 |  52% |  40(84.6%), 50(7.6%), 36(7.6%) |  35 |  33
|  3 |  28% |  (32,39)(57.1%) |  (32, 39) |  (32, 39)
|  4 |  20% |  (32,39,50)(60%), (41,47,53)(40%) |  (32,39,52) |  (32,39,52)
D8 |  2 |  44% |  18(36%), 30(27%) |  40 |  40
|  3 |  52% |  (28,47)(53.8%), (18,47)(23%) |  (13, 15) |  (40,45)
|  4 |  4% |  (18,47,54)(100%) |  (11, 13, 15) |  (13,15,18)
D9 |  2 |  68% |  40(41%), 50(23.5%),47(23.5%) |  15 |  13
|  3 |  28% |  (24,36)(57%), (36,47)(28.5%) |  (14, 50) |  (8,15)
|  4 |  4% |  (24,39,47)(100%) |  (14,50,52) |  (13,15,18)
D10 |  0 |  40% |  NA |  NA |  NA
|  2 |  36% |  44(33.3%), 24(33.3%), 52(22%) |  56 |  11
|  3 |  12% |  (20,32)(66.6%), (18,45)(33.3%) |  (11,56) |  (11,56)
|  4 |  4% |  (12,29,42)(100%) |  (49,50,56) |  (11,52,56)
|  5 |  8% |  (12,24,30,40)(100%) |  (11,49,50,56) |  (11,41,50,56)
D11 |  0 |  56% |  NA |  NA |  NA
|  2 |  28% |  52(42.8%), 60(42.8%), 67(14%) |  67 |  67
|  3 |  12% |  (48,68)(66.6%), (40,68)(33.3%) |  (67,70) |  (67,70)
|  4 |  4% |  (40,48,68)(100%) |  (51, 63, 67) |  (62,67,70)
D12 |  0 |  32% |  NA |  NA |  NA
|  2 |  28% |  50(42.8%), 40(28.5%), 24(28.5%) |  17 |  15
|  3 |  16% |  (22,32)(50%), (14,34)(25%), (27,44)(25%) |  (15,51) |  (14,38)
|  4 |  16% |  (9,31,58)(50%), (20,36,48)(25%), (10,20,30)(25%) |  (15,51,52) |  (10,17,38)
|  5 |  8% |  (16,28,36,44)(50%), (7,20,28,36)(50%) |  (15,50,51,52) |  (14,37,51,52)
D: Datasets; P: number of partitions; R: percentage of responses

We designed a Google form by providing a series of graphs containing different
types of numerical data points and relevant questions to collect human
responses and their perceptions about discretization. We put some constraints
in the Google form to know whether a response is submitted by DS/ML experts or
not. By employing this procedure, we compare and comprehend the perceptions of
both DS/ML experts and non-expert responders.

The Google form was sent to fifty DS/ML experts and non-experts to estimate
the number of partitions and the ranges of these partitions to obtain the cut-
points. The following data was gathered and compiled from the experiments:
respondent identification (name), their email addresses, domain expertise
(DS/ML expert or non-expert), number of partitions identified, and ranges of
each partition.

##  6 Analysis and Result

Out of the fifty responses received via the Google form, two were incomplete;
therefore, we did not consider them for the analysis. From the rest of the
forty-eight responses, we divided the responses into two categories: expert
responses and non-expert responses.

Table 3: Similarity between human perceived, LSQM and LADM cut-points. D |  P |  LSQM Cut- Points |  LADM Cut- Points |  Human Perceived Cut-Points (Near to LSQM Cut-Points) |  Matching% |  Matching status
---|---|---|---|---|---|---
D1 |  2 |  50 |  50 |  50 (91.3%) |  91.3% |  Very High
|  3 |  (20,50) |  (20,50) |  (20,50)(50%) |  50% |  Medium
D2 |  2 |  52 |  52 |  52(11.7%) |  11.7% |  Low
|  3 |  (52,54) |  (52,54) |  (50,54)(37.5%) |  19% |  Low
D3 |  3 |  (32,52) |  (32,52) |  (32,52)(62%) |  62% |  High
|  4 |  (32,52,55) |  (32,52,60) |  (20,32,52)(100%) |  59% |  Medium
D4 |  2 |  20 |  20 |  20(80.6%) |  80.6% |  Very High
|  3 |  (12,24) |  (12,25) |  (20,30)(33.3%) |  0% |  No match
D5 |  3 |  (13,26) |  (13,26) |  (13,26)(100%) |  100% |  Very High
D6 |  2 |  42 |  42 |  42(40%) |  40% |  Medium
|  3 |  (32,42) |  (32,42) |  (32,42)(25%) |  25% |  Low
|  4 |  (32,37,42) |  (32,37,42) |  (32,37,42)(85.7%) |  85.7% |  Very High
|  5 |  (32,37,42,56) |  (32,37,42,56) |  (32,37,42,68)(85.7%) |  75% |  High
D7 |  2 |  35 |  33 |  36(7.6%) |  0% |  No match
|  3 |  (32,39) |  (32,39) |  (32,39)(57%) |  57% |  High
|  4 |  (32,39,52) |  (32,39,52) |  (32,39,52)(60%) |  60% |  High
D8 |  2 |  40 |  40 |  30(27%) |  0% |  No match
|  3 |  (13,15) |  (40,45) |  (18,47)(23%) |  0% |  No match
|  4 |  (11,13,15) |  (13,15,18) |  (18,47,54)(100%) |  0% |  No match
D9 |  2 |  15 |  13 |  40(41%) |  0% |  No match
|  3 |  (14,50) |  (8,15) |  (24,36)(57%) |  0% |  No match
|  4 |  (14,50,52) |  (10,17,33) |  (24,39,47)(100%) |  0% |  No match
D10 |  2 |  56 |  11 |  52(22%) |  0% |  No match
|  3 |  (11,56) |  (11,56) |  (18,45)(33.3%) |  0% |  No match
|  4 |  (49,50,56) |  (11,52,56) |  (12,29,42)(100%) |  0% |  No match
|  5 |  (11,49,50,56) |  (11,41,50,56) |  (12,24,30,40)(100%) |  0% |  No match
D11 |  2 |  67 |  67 |  50(42.8%) |  0% |  No match
|  3 |  (67,70) |  (67,70) |  (48,68)(66.6%) |  0% |  No match
|  4 |  (51,63,67) |  (62,67,70) |  (40,48,68)(100%) |  0% |  No match
D12 |  2 |  17 |  15 |  24(28.5%) |  0% |  No match
|  3 |  (15,51) |  (14,38) |  (14,34)(25%) |  0% |  No match
|  4 |  (15,51,52) |  (10,17,38) |  (20,36,48)(25%) |  0% |  No match
|  5 |  (15,50,51,52) |  (14,37,51,52) |  (16,28,36,44)(50%) |  0% |  No match
D: Datasets; P: number of partitions; R: percentage of responses |
Very High: 80-100%, High:60-80%, Medium:40-60%, Low:1-40%, No match: 0% |
Table 4: Analysis of unmatched datasets in regard of number of partitions ($\\#$) for LSQM, LADM and human perceived cut-points. D |  P | LSQM Method | LADM Method | Human Perception |  Remarks
---|---|---|---|---|---
LSQM cut-points |  LC |  LADM cut-points |  LC |  Human Perceived cut-points |  LC
D8 |  2 |  40 |  Yes |  40 |  Yes |  30 |  Yes |  Matter of perception.
|  3 |  (13,15) |  No |  (40,45) |  No |  (18,47) |  Yes |  LSQM, LADM to be improved.
|  4 |  (11,13,15) |  No |  (13,15,18) |  No |  (18,47,54) |  Yes |  LSQM, LADM to be improved.
D9 |  2 |  15 |  No |  13 |  No |  40 |  Yes |  LSQM, LADM need to be improved.
|  3 |  (14,50) |  Yes |  (8,15) |  No |  (24,36) |  Yes |  Matter of perception. However, LADM needs to be improved.
|  4 |  (14,50,52) |  No |  (10,17,33) |  No |  (24,39,47) |  Yes |  LSQM and LADM to be improved.
D10 |  2 |  56 |  No |  11 |  No |  52 |  No |  This dataset is an exceptional case; random cutpoints are obtained.
|  3 |  (11,56) |  No |  (11,56) |  No |  (18,45) |  Yes |
|  4 |  (49,59,56) |  No |  (11,52,56) |  No |  (12,29,42) |  Yes |
|  5 |  (11,49,50,56) |  No |  (11,41,50,56) |  No |  (12,24,30,40) |  Yes |
D11 |  2 |  67 |  Yes |  67 |  Yes |  50 |  Yes |  Matter of perception.
|  3 |  (67,70) |  No |  (67,70) |  No |  (48,68) |  Yes |  LSQM to be improved.
|  4 |  (51,63,67) |  Yes |  (62,67,70) |  No |  (40,48,68) |  Yes |  Matter of perception. However, LADM needs to be improved.
D12 |  2 |  17 |  Yes |  15 |  Yes |  24 |  Yes |  This dataset is an exceptional case; random cutpoints are obtained.
|  3 |  (15,51) |  Yes |  (14,38) |  Yes |  (14,34) |  Yes |
|  4 |  (15,51,52) |  No |  (10,17,38) |  No |  (20,36,48) |  Yes |
|  5 |  (15,50,51,52) |  No |  (14,37,51,52) |  No |  (16,28,36,44) |  Yes |
D4 |  3 |  (12,24) |  Yes |  (12,25) |  Yes |  (20,30) |  Yes |  Matter of perception.
D7 |  2 |  35 |  Yes |  33 |  Yes |  36 |  Yes |  Matter of perception.
D: Datasets; P: number of partitions; LC: Logical Correctness |

Table 1 illustrates the comparison of human perception to identify the number
of partitions between the DS/ML experts’ responses and non-expert people. We
received $60\%$ responses from DS/ML experts and $40\%$ of answers from non-
expert people. We analyzed that responses from both categories were opposite
for graph D8. Out of the total responses for D8, $33.3\%$ responses of DS/ML
experts marked two partitions and $66.6\%$ responses of experts marked three
partitions; however, $60\%$ of non-experts marked two partitions, and only
$30\%$ marked three partitions.

In graphs D3 and D5, we analyzed that no contributor (experts or non-experts)
marked two partitions. No non-expert contributors marked three partitions for
graph D6 and four partitions for graph D3, whereas $26.6\%$ of DS/ML experts
identified three partitions for D6, and $6.66\%$ experts marked four
partitions in graph D3. For graph D10, $40\%$ DS/ML experts and $40\%$ non-
experts have marked no partition.

Table 2 illustrates the comparison between the results of human perception,
the  _LSQM_ and the  _LADM_ measure. In the initial four columns, we detailed
the dataset used, the count of partitions, the response percentage, and the
approximate cut-points as observed by contributors. The last two columns
present the cut-points assessed by the measures. Table 3 describes the
similarity percentage between cut-points provided by human perceived
experiment outcome and the  _LSQM_ and the  _LADM_ measures outputs. We have
mentioned the cut-points from responses near the  _LSQM_ and the  _LADM_
provided cut-points. We determine the matching status by distributing the
matching percentage into the following categories: VH (Very High), H (High), M
(Medium), L (Low) and NM (No match). The distribution of ranges is mentioned
at the bottom of Table 3. It is clear from Table 3 that human perceived cut-
points and the cut-points identified by the proposed measures  _LSQM_ and
_LADM_ do not match for the datasets D8 to D12. In Table 4, we present an
analysis and reason for not getting similar cut-points for the datasets D8 to
D12. If we look at Fig. 1(D8), then it seems logical to have cut-points at the
data points of $40$ (LSQM, LADM cut-point) and $30$ (Human perceived cut-
point) for two partitions on the X-axis. Humans divided the scattered points
into the first partition and dense data points into the second partition. In
contrast, both measures calculated the cut-point in the middle of the dense
data points. This case can be observed as a matter of perception for human
perceived cut-points, while the cut-points marked by both the measures seem
analytically correct. For the cases of three partitions and four partitions,
human perceived cut-points $(18,47)$ and $(18,47,54)$ are good, but the cut-
points provided by the  _LSQM_ measure and the  _LADM_ measure are not
satisfactory.

Similarly, in dataset D9, human perception identified a single cut-point at
$40$ for two partitions and two cut-points at $(24,36)$ for three partitions,
which intuitively makes sense. However,_LSQM_ and _LADM_ produced cut-points
at $15$ and $13$ for two partitions, which are not analytically accurate. On
the other hand,_LSQM_ cut-points $(14,50)$ align analytically, albeit it
remains a matter of perception. Even though the three partitions provided by
_LADM_ $(8,15)$ and the four partitions by  _LSQM_ $(14,50,52)$ and  _LADM_
$(10,17,33)$ might seem illogical, human-perceived cut-points $(24,39,47)$
appear appropriate.

In dataset D11, the situation is again contingent on perception, with
discrepancies arising for both two partitions and four partitions in the case
of  _LSQM_. For four partitions,  _LADM_ suggests cut-points $(62,67,70)$,
which lack logical consistency. Meanwhile, for three partitions, both  _LSQM_
and  _LADM_ present unexpected cut-points $(67,70)$.

Datasets D4 and D7 also lack matching results for three partitions and two
partitions, respectively. In the case of D4, both  _LSQM_ and  _LADM_ suggest
cut-points $(12,24)$ and $(12,25)$, while human perception identifies
$(20,30)$ as the appropriate cut-points. This instance can be attributed to
varying perceptions.

Similarly, for D7, the proposed cut-points by  _LSQM_ and  _LADM_ for two
partitions are $35$ and $33$, respectively, which do not exactly align with
the human-perceived cut-point of $36$. However, given the scattered
distribution of data points on the graph, the difference between the proposed
measures’ cut-points and the human-perceived cut-point is negligible. In this
case, both sets of cut-points can be considered suitable, further emphasizing
the role of perception. While these cut-points do not match precisely, it does
not affect the correctness of the measures due to the lack of similarity.

For datasets D10 and D12, the responses from contributors present a unique
challenge. In the case of D10, $40\%$ of contributors indicated no partition,
while the remaining contributors marked random cut-points for two, three,
four, and five partitions. Similarly, for D12, $32\%$ of contributors opted
for no partition, while $68\%$ of contributors designated random cut-points
for various partitions. These random cut-points identified by humans are not
easily aligned with the cut-points derived from the proposed  _LSQM_ and
_LADM_ measures. Furthermore, these random cut-points lack analytical
correctness.

As a result, for datasets with such characteristics, it becomes difficult for
humans to identify the most appropriate partitions. The absence of clear
patterns or logic in the random cut-points makes it challenging to establish
meaningful partitions, emphasizing the complexity of the task in these
scenarios.

Figure 4: Percentage Responses of partitions for each dataset.

The distribution of response percentages for each partition across the
datasets is visually represented in Fig. 4. Notably, datasets D5, D10, D11,
and D12 exhibit a significant prevalence of responses indicating no partition
compared to other partition numbers. Hence, it proves that for these specific
datasets, human participants struggled to form a definitive perception. It
suggests that individuals had difficulty forming a clear and consistent
perception of these datasets, making it unclear for them to identify
appropriate cut-points.

It is worth highlighting that datasets D5 and D6 share a similar visual
appearance. However, they received different responses in terms of cut-points.
This discrepancy can be attributed to the distinct distribution of data points
within each dataset. Notably, D6 did not receive any responses suggesting no
partition, while D5 lacked responses suggesting two partitions.

Conversely, for datasets D1, D2, D4, D7, and D9, the majority of responses
predominantly indicated the presence of two partitions. This indicates a
higher degree of consensus among contributors regarding the presence of two
partitions in these datasets.

Table 3 provides an overview of the alignment between human-perceived cut-
points and those observed by the  _LSQM_ and  _LADM_ measures. The analysis
reveals that $25\%$ of responses exhibited a  _Very High_ level of similarity,
$25\%$ demonstrated a  _High_ level of similarity, $18.7\%$ displayed a
_Medium_ level of similarity, and an additional $18.7\%$ showed a  _Low_ level
of similarity. When considering the collective matching statuses, it becomes
evident that approximately $68.7\%$ of the responses closely resembled the
cut-points identified by the proposed  _LSQM_ and _LADM_ measures. This
analysis primarily pertains to the initial datasets (D1 to D7), as random cut-
points were observed in the responses for datasets D8 to D12. These random
cut-points in the latter datasets presented challenges in aligning them with
the analytically calculated cut-points generated by the proposed measures.
Further details and explanations for the dissimilarity in cut-points for
datasets D8 to D12 can be found in Table 4.

##  7 Conclusion

This paper is the first step toward understanding the human perception
regarding partitioning numerical attributes. We meticulously examine the
partitions perceived by humans and compare them with the outputs generated by
both proposed measures. Our approach involved evaluating human perception by
presenting a series of graphs containing numerical data and subsequently
comparing human-perceived cut-points for partitioning with the results
generated by the  _LSQM_ and  _LADM_ measures.

The outcomes of this study indicate a close alignment between the cut-points
produced by the proposed measures and those perceived by humans. Particularly
for the initial datasets (D1 to D7), our proposed measures yielded results
that closely approximated human perception. However, certain exceptional
cases, such as datasets D10 and D12, highlighted situations where humans faced
challenges identifying optimal partitions. The results also demonstrate that
both measures yield approximately similar outcomes. These findings represent a
promising step forward, signifying progress in the pursuit of advancing ARM by
identifying numerical attribute partitions that reflect best the impact of an
independent numerical attribute on a dependent numerical attribute. In future
research endeavors, we intend to explore inter-measures for comparing
partitions with varying numbers of $k$-partitions.

## Acknowledgements

This work has been conducted in the project “ICT programme” which was
supported by the European Union through the European Social Fund.

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Relations Between Discrete Maximal Operators in Harmonic Analysis

Sri Sakti Swarup Anupindi and Michael Alphonse
Department of Mathematics, Birla Institute of Technology and Science, Jawahar
Nagar,Hyderabad-500 078, Telangana, India
Department of Mathematics, Birla Institute of Technology and Science, Jawahar
Nagar,Hyderabad-500 078, Telangana, India
e-mail<EMAIL_ADDRESS>e-mail: alphonse@hyderabad-bits-
pilani.ac.in

###### Abstract

In this paper, we define several types of maximal operators on sequence spaces
occuring in Harmonic analysis and present various connections between them.

Keywords: Calderon-Zygmund decomposition, Hardy-Littlewood Maximal Operator,
Sharp Maximal Operator, Dyadic Maximal Operator, Good-$\lambda$ Inequality

2020 Mathematics Subject Classification: 42B35,42B20

##  1 Introduction

In this paper, we define several types of maximal operators on real valued
sequence spaces and study relationships between them. Several such operators
are defined in continuous case and studied in standard literature in harmonic
analysis. We present some of them namely, Hardy-Littlewood maximal operator
(both centered and non-centered), dyadic maximal operator and sharp maximal
opeator. A good-$\lambda$ inequality is presented which relates dyadic maximal
operator and sharp maximal operator. For details of these maximal operators on
real line, refer to [1].

The method of Calderon-Zygmund decomposition on sequence spaces plays an
important role in studying the relationship between these operators [4]. In
the discrete case, Calderon-Zygmund decomposition uses dyadic intervals. When
we study the relation between the maximal operators, we are required to double
the intervals that destroys the dyadic nature of the intervals. This challenge
is not there in the case of real line [1].

##  2 Preliminaries and Notation

Throughout this paper, $\mathcal{Z}$ denotes set of all integers and
$\mathcal{Z}_{+}$ denotes set of all positive integers. For a given interval
$I$ in $\mathcal{Z}$ , $\lvert I\rvert$ always denotes the cardinality of $I$.
For each positive integer N, consider the collection of disjoint intervals of
cardinality $2^{N}$,

$\left\\{I_{N,j}\right\\}_{j\in\mathcal{Z}}=\left\\{[(j-1)2^{N}+1,\dots,j2^{N}]\right\\}_{j\in\mathcal{Z}}$

The set of all intervals which are of the form $I_{N,j}$ where
$N\in\mathcal{Z}_{+}$ and $j\in\mathcal{Z}$ are called dyadic intervals. For
fixed $N$, we denote the set of all intervals
$\left\\{I_{N,j}\right\\}_{j\in\mathcal{Z}}$ as $\mathcal{I}_{N}$. For fixed
N, the intervals in $\mathcal{I}_{N}$ are disjoint. For a dyadic interval I,
we define

$\displaystyle 2LI$ $\displaystyle=[(j-2)2^{N}+1,\dots,j2^{N}]$ $\displaystyle
2RI$ $\displaystyle=[(j-1)2^{N}+1,\dots,(j+1)2^{N}]$ $\displaystyle 3I$
$\displaystyle=[(j-2)2^{N}+1,\dots,(j+1)2^{N}]$

Note $\lvert 3I\rvert=3.2^{N}$. Note that $2LI,2RI$ are dyadic intervals each
of length $2^{N+1}$ but $3I$ is not an dyadic interval.

The Calderon-Zygmund decomposition theorem for sequences [4] is as follows.

######  Theorem 2.1.

Let $1\leq p<\infty$ and a $\in\ell^{p}(\mathcal{Z})$. For every $t>0$, and
$0\leq\alpha<1$, there exists a sequence of disjoint dyadic intervals
$\left\\{I_{j}^{t}\right\\}$ such that

$\displaystyle(i)$ $\displaystyle\quad t<\frac{1}{\lvert
I_{j}^{t}\rvert^{1-\alpha}}\sum_{k\in I_{j}^{t}}\lvert a(k)\rvert\leq
2t,\forall j\in\mathcal{Z}$ $\displaystyle(ii)$ $\displaystyle\quad\forall
n\not\in\cup_{j}I_{j}^{t},\quad\lvert a(n\rvert\leq t$ $\displaystyle(iii)$
$\displaystyle\quad\text{If}\quad t_{1}>t_{2},\quad\textbf{then each}\quad
I_{j}^{t_{1}}\quad\textbf{is subinterval of some}\quad
I_{m}^{t_{2}},\quad\forall j,m\in\mathcal{Z}$

An operator $T$ is bounded on $\ell^{p}(\mathcal{Z})$ if $\forall
a\in\ell^{p}(\mathcal{Z})$

$\left\lVert Ta\right\rVert_{\ell^{p}(\mathcal{Z})}\leq C_{p}\left\lVert
a\right\rVert_{\ell^{p}(\mathcal{Z})}$

An operator $T$ is of weak type (1,1) on $\ell^{p}(\mathcal{Z})$ if for each
$a\in\ell_{1}(\mathcal{Z})$

$\lvert\left\\{m\in\mathcal{Z}:\lvert
Ta(m)\rvert>\lambda\right\\}\rvert\leq\frac{C}{\lambda}\left\lVert
a\right\rVert_{1}$

For $\left\\{a(n):n\in\mathcal{Z}\right\\}\in\ell^{p}(\mathcal{Z})$, norm in
$\ell^{p}(\mathcal{Z})$ (refer to [1]) is given by

$\left\lVert
a\right\rVert_{\ell^{p}(\mathcal{Z})}=\int_{0}^{\infty}p{\lambda}^{p-1}\lvert\left\\{m\in\mathcal{Z}:\lvert
a(m)\rvert>\lambda\right\\}\rvert d\lambda$

##  3  Definitions

###  3.1 Maximal Operators

Let $\left\\{a(n):n\in\mathcal{Z}\right\\}$ be a sequence. We define the
following three types of Hardy-Littlewood maximal operators as follows:

######  Definition 3.1.

If $I_{r}$ is the interval $\left\\{-r,-r+1,\dots,0,1,2,\dots,r-1,r\right\\}$,
define centered Hardy-Littlewood maximal operator

$M^{\prime}a(m)=\sup_{r>0}\frac{1}{(2r)}\sum_{n\in I_{r}}\lvert a(m-n)\rvert$

We define Hardy-Littlewood maximal operator as follows

$Ma(m)=\sup_{m\in I}\frac{1}{\lvert I\rvert}\sum_{n\in I}\lvert a(n)\rvert$

where the supremum is taken over all intervals containing $m$.

######  Definition 3.2.

We define dyadic Hardy-Littlewood maximal operator as follows:

$M_{d}a(m)=\sup_{m\in I}\frac{1}{\lvert I\rvert}\sum_{k\in I}\lvert
a(k)\rvert$

where supremum is taken over all dyadic intervals containing $m$.

Given a sequence $\left\\{a(n):n\in\mathcal{Z}\right\\}$ and an interval $I$,
let $a_{I}$ denote average of $\left\\{a(n):n\in\mathcal{Z}\right\\}$ on $I$,
i.e $a_{I}=\frac{1}{\lvert I\rvert}\sum_{m\in I}a(m)$. Define the sharp
maximal operator $M^{\\#}$ as follows

$M^{\\#}a(m)=\sup_{m\in I}\frac{1}{\lvert I\rvert}\sum_{n\in I}\lvert
a(n)-a_{I}\rvert$

where the supremum is taken over all intervals $I$ containing $m$. We say that
sequence $\left\\{a(n):n\in\mathcal{Z}\right\\}$ has bounded mean oscillation
if the sequence $M^{\\#}a$ is bounded. The space of sequences with this
property is called sequences of bounded mean oscillation and is denoted by
BMO($\mathcal{Z}$). We define a norm in BMO($\mathcal{Z}$) by $\left\lVert
a\right\rVert_{\star}=\left\lVert M^{\\#}a\right\rVert_{\infty}$. The space
BMO($\mathcal{Z}$) is studied in [2],[3].

##  4  Relation between Maximal operators

######  Theorem 4.1.

Given a sequence $\left\\{a(m):m\in\mathcal{Z}\right\\}$, the following
relation holds:

$M^{\prime}a(m)\leq Ma(m)\leq 3M^{\prime}a(m)$

###### Proof.

First inequality is obvious as $M^{\prime}a$ considers supremum over centered
intervals, while $M$ considers supremum over all intervals. For second
inequality, let $I=[m-r_{1},m-r_{1}+1,\dots,m+r_{2}-1,m+r_{2}]$ containing
$m$. Let $r=max\left\\{r_{1},r_{2}\right\\}$. Consider
$I_{1}=[m-r,m-r+1,\dots,m+r-1,m+r]$ containing $m$. Note that $\lvert
I_{1}\rvert=2r+1,\lvert I\rvert=r_{1}+r_{2}+1$. Then

$\lvert I\rvert=r_{2}+r_{1}+1\geq
r=\frac{1}{3}3r\geq\frac{1}{3}(2r+1)=\frac{1}{3}\lvert I_{1}\rvert$

This gives

$\frac{1}{\lvert I\rvert}\sum_{k\in I}\lvert a(k)\rvert\leq\frac{3}{\lvert
I_{1}\rvert}\sum_{k\in I_{1}}a(k)\leq 3M^{\prime}a(m)$

∎

######  Theorem 4.2.

If $\textbf{a}=\left\\{a(k):k\in\mathcal{Z}\right\\}$ is a sequence with
$\textbf{a}\in\ell_{1}$, then

$\lvert\left\\{m\in\mathcal{Z}:M^{\prime}a(m)>4\lambda\right\\}\rvert\leq
3\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}\rvert$

###### Proof.

Using Calderon Zygmund decomposition at height $\lambda$, we obtain a
collection of disjoint dyadic intervals
$\left\\{I_{j}:j\in\mathcal{Z}^{+}\right\\}$ such that

$\lambda<\frac{1}{\lvert I_{j}\rvert}\sum_{k\in I_{j}}\lvert a(k)\rvert\leq
2\lambda$

Then

$\cup_{j}I_{j}\subseteq\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}$

It suffices to show that

$\left\\{m\in\mathcal{Z}:M^{\prime}a(m)>4\lambda\right\\}\subset\cup_{j}3I_{j}$

Let $m\notin\cup_{j}3I_{j}$. We shall prove
$m\notin\left\\{k\in\mathcal{Z}:M^{\prime}a(k)>4\lambda\right\\}$. Let $I$ be
any interval centered at $m$. Choose $N\in\mathcal{Z}_{+}$ such that
$2^{N-1}\leq\lvert I\rvert<2^{N}$. Then $I$ intersects exactly 2 dyadic
intervals in $\mathcal{I}_{N}$ say $R_{1},R_{2}$. Assume $R_{1}$ intersects
$I$ on the left and $R_{2}$ intersects $I$ on the right. Since
$m\notin\cup_{j=1}^{\infty}3I_{j}$, $m\notin 2RI_{j},j=1\dots$ and $m\notin
2LI_{j},j=1,\dots$. But $m\in 2RR_{1}$ and $m\in 2LR_{2}$.

Therefore, both $R_{1}$ and $R_{2}$ cannot be any one of $I_{j}$.

Hence, the average of $\left\\{a(n):n\in\mathcal{Z}\right\\}$ on each
$R_{i},i=1,2$ is at most $\lambda$. Further note that $\frac{\lvert
R_{1}\rvert}{\lvert I\rvert}\leq 2,\frac{\lvert R_{2}\rvert}{\lvert
I\rvert}\leq 2$. So

$\displaystyle\frac{1}{\lvert I\rvert}\sum_{m\in I}\lvert a(m)\rvert$
$\displaystyle\leq\frac{1}{\lvert I\rvert}\biggl{(}\sum_{k\in R_{1}}\lvert
a(k)\rvert+\sum_{k\in R_{2}}\lvert a(k)\rvert\biggr{)}$
$\displaystyle=\biggl{(}\frac{1}{\lvert R_{1}\rvert}\frac{\lvert
R_{1}\rvert}{\lvert I\rvert}\sum_{k\in R_{1}}\lvert a(k)\rvert+\frac{1}{\lvert
R_{2}\rvert}\frac{\lvert R_{2}\rvert}{\lvert I\rvert}\sum_{k\in R_{2}}\lvert
a(k)\rvert\biggr{)}$ $\displaystyle\leq 2\biggl{(}\frac{1}{\lvert
R_{1}\rvert}\sum_{k\in R_{1}}\lvert a(k)\rvert+\frac{1}{\lvert
R_{2}\rvert}\sum_{k\in R_{2}}\lvert
a(k)\rvert\biggr{)}=2(\lambda+\lambda)=4\lambda$

∎

######  Corollary 4.3.

For a sequence $\left\\{a(n):n\in\mathcal{Z}\right\\}$, if
$M_{d}a\in\ell^{p}(\mathcal{Z}),1<p<\infty$, then

$\left\lVert M^{\prime}a\right\rVert_{\ell^{p}(\mathcal{Z})}\leq C\left\lVert
M_{d}a\right\rVert_{\ell^{p}(\mathcal{Z})}$

###### Proof.

$\displaystyle\left\lVert M^{\prime}a\right\rVert_{\ell^{p}(\mathcal{Z})}$
$\displaystyle=\int_{0}^{\infty}p\lambda^{p-1}\lvert\left\\{m:M^{\prime}a(m)>\lambda\right\\}\rvert
d\lambda$ $\displaystyle\leq
3(4)^{p-1}\int_{0}^{\infty}p({\frac{\lambda}{4}})^{p-1}\lvert\left\\{m:M_{d}a(m)>\frac{\lambda}{4}\right\\}\rvert
d\lambda$ $\displaystyle\leq
3(4)^{p}\int_{0}^{\infty}pu^{p-1}\lvert\left\\{m:M_{d}a(m)>u\right\\}\rvert
du$ $\displaystyle\leq 3(4)^{p}\left\lVert
M_{d}a\right\rVert_{\ell^{p}(\mathcal{Z})}$

∎

######  Remark 4.4.

From corollary[4.3], whenever $M^{\prime}$ is of weak type (1,1), then $M_{d}$
is also of weak type (1,1). From Theorem[4.2], if $M^{\prime}$ is of weak type
(1,1), then $M_{d}$ is of weak type (1,1). From Theorem[4.1], if $M$ is of
weak type (1,1), then $M^{\prime}$ is of weak type (1,1). It is well known
that $M$ is of weak type (1,1), refer to [4].

In the following lemma, we see that in the norm of BMO($\mathcal{Z}$) space,
we can replace the average $a_{I}$ of $\left\\{a(n)\right\\}$ by a constant
$b$. The proof is similar to the proof in continuous version [1]. We provide
the proof for the sake of completeness.

######  Lemma 4.5.

Consider a non-negative sequence
$\textbf{a}=\left\\{a(k):k\in\mathcal{Z}\right\\}$. Then the following are
valid.

$\displaystyle 1.$ $\displaystyle\quad\frac{1}{2}\left\lVert
a\right\rVert_{\star}\leq\sup_{m\in I}\inf_{b\in{\mathcal{R}}}\frac{1}{\lvert
I\rvert}\lvert a(m)-b\rvert\leq\left\lVert a\right\rVert_{\star}$
$\displaystyle 2.$ $\displaystyle\quad M^{\\#}(\lvert a\rvert)(i)\leq
M^{\\#}a(i),i\in\mathcal{Z}$

###### Proof.

For first inequality, note for all $b\in{\mathcal{R}}$,

$\sum_{m\in I}\lvert a(m)-a_{I}\rvert\leq\sum_{m\in I}\lvert
a(m)-b\rvert+\sum_{m\in I}\lvert b-a_{I}\rvert=A+B(say)$

Now

$\displaystyle B$ $\displaystyle=\lvert I\rvert\lvert b-a_{I}\rvert=\lvert
I\rvert\lvert b-\frac{1}{\lvert I\rvert}\sum_{k\in I}a(k)\rvert$
$\displaystyle=\lvert I\rvert\lvert\frac{1}{\lvert I\rvert}\biggl{(}\sum_{k\in
I}(b-a(k))\biggr{)}\rvert\leq\sum_{k\in I}\lvert b-a(k)\rvert$

So,

$\sum_{m\in I}\lvert a(m)-a_{I}\rvert\leq\sum_{m\in I}\lvert
a(m)-b\rvert+\sum_{m\in I}\lvert b-a_{I}\rvert\leq 2\sum_{m\in I}\lvert
a(m)-b\rvert$

Now, divide both sides by $\lvert I\rvert$, and take infimum over all $b$
followed by, supremum over all $I$. This proves

$\quad\frac{1}{2}\left\lVert a\right\rVert_{\star}\leq\sup_{m\in
I}\inf_{b\in{\mathcal{R}}}\frac{1}{\lvert I\rvert}\lvert a(m)-b\rvert$

The proof for second inequality

$\sup_{m\in I}\inf_{b\in{\mathcal{R}}}\frac{1}{\lvert I\rvert}\lvert
a(m)-b\rvert\leq\left\lVert a\right\rVert_{\star}$

is obvious.

The proof of (2) follows from the fact that $\lvert\lvert a\rvert-\lvert
b\rvert\rvert\leq\lvert a\rvert-\lvert b\rvert$ for any $a,b\in{\mathcal{R}}$.
∎

######  Lemma 4.6.

If $a\in\ell^{p_{0}}(\mathcal{Z})$ for some $p_{0}$, $1\leq p_{0}<\infty$,
then for all $\gamma>0$ and $\lambda>0$

$\lvert\left\\{n\in\mathcal{Z}:M_{d}a(n)>2\lambda,M^{\\#}a(n)\leq\gamma\lambda\right\\}\rvert\leq
2\gamma\lvert\left\\{n\in\mathcal{Z}:M_{d}a(n)>\lambda\right\\}\rvert$

###### Proof.

Perform Calderon-Zygmumd decomposition for the sequence
$\left\\{a(n):n\in\mathcal{Z}\right\\}$ at height $\lambda$, which gives
collection of intervals $\left\\{I_{j}\right\\}$ such that for each $j$,

$\lambda\leq\frac{1}{\lvert I_{j}\rvert}\sum_{k\in I_{j}}\lvert a(k)\rvert\leq
2\lambda$

Let $I$ be one of the interval in the collection $\left\\{I_{j}\right\\}$. In
Calderon-Zygmund decomposition, there exists interval $\tilde{I}$ such that
$\tilde{I}$ is either $2RI$ or $2LI$ and

$\frac{1}{\lvert\tilde{I}\rvert}\sum_{k\in\tilde{I}}\lvert
a(k)\rvert\leq\lambda$

For details, refer to section on Preliminaries and Notation.

It is easy to observe that $\forall m\in I$, $M_{d}a(m)>2\lambda$ implies
$M_{d}(a\chi_{I})(m)>2\lambda$. Let,

$\displaystyle a_{1}$ $\displaystyle=(a-a_{\tilde{I}})\chi_{I}$ $\displaystyle
a_{2}$ $\displaystyle=a_{\tilde{I}}\chi_{I}$

Then, since $M_{d}$ is sublinear

$\displaystyle a_{1}+a_{2}$
$\displaystyle=(a-a_{\tilde{I}})\chi_{I}+a_{\tilde{I}}\chi_{I}$ $\displaystyle
M_{d}(a_{1}+a_{2})$ $\displaystyle\leq
M_{d}((a-a_{\tilde{I}})\chi_{I})+M_{d}(a_{\tilde{I}}\chi_{I})$
$\displaystyle\leq M_{d}((a-a_{\tilde{I}})\chi_{I})+(a_{\tilde{I}})$

Since $M_{d}(a_{\tilde{I}}\chi_{I})(k)\leq a_{\tilde{I}}\quad\forall k$, it
follows that

$M_{d}(a_{1}+a_{2})=M_{d}(a\chi_{I})\leq
M_{d}((a-a_{\tilde{I}})\chi_{I})+(a_{\tilde{I}})$

Hence for every, $k\in I$, it follows that

$M_{d}((a-a_{\tilde{I}})\chi_{I})(k)\geq M_{d}(a\chi_{I})(k)-a_{\tilde{I}}$

So, for those $k^{\prime}s$,

$M_{d}((a-a_{\tilde{I}})\chi_{I})(k)\geq
M_{d}(a\chi_{I})(k)-a_{\tilde{I}}>\lambda$

By remark [4.4], using $weak(1,1)$ inequality for $M_{d}$

$\displaystyle\left\\{k\in\mathcal{Z}:M_{d}((a-a_{\tilde{I}})\chi_{I}))(k)>\lambda\right\\}$
$\displaystyle\leq\frac{C}{\lambda}\sum_{I}\lvert a(k)-a_{\tilde{I}}\rvert$
$\displaystyle\leq\frac{2}{{\lambda}}\lvert
I\rvert\frac{C}{\lvert\tilde{I}\rvert}\sum_{\tilde{I}}\lvert
a(k)-a_{\tilde{I}}\rvert$ $\displaystyle\leq\frac{2C}{{\lambda}}\lvert
I\rvert\inf_{m\in I}M^{\\#}a(m)$
$\displaystyle\leq\frac{2C}{{\lambda}}\gamma\lambda\lvert
I\rvert=2C\gamma\lvert I\rvert$

∎

As a consequence of good-$\lambda$ inequality, we prove the following theorem.

######  Theorem 4.7.

Let $\left\\{a(n):n\in\mathcal{Z}\right\\}$ be a nonnegative sequence in
$\ell^{p}(\mathcal{Z}),1<p<\infty$.Then

$\sum_{m\in\mathcal{Z}}\lvert M_{d}a(m)\rvert^{p}\leq
C\sum_{m\in\mathcal{Z}}\lvert M^{\\#}a(m)\rvert^{p}$

where $M_{d}$ is the dyadic maximal operator and $M^{\\#}$ is the sharp
maximal operator, whenever, the left hand side is finite.

###### Proof.

For a positive integer $N>0$, let

$I_{N}=\int_{0}^{N}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}\rvert
d\lambda$

$I_{N}$ is finite, since $a\in\ell^{p}(\mathcal{Z})$ implies
$M_{d}a\in\ell^{p}(\mathcal{Z})$

$\displaystyle I_{N}$
$\displaystyle=\int_{0}^{N}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}\rvert
d\lambda$
$\displaystyle=2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>2\lambda\right\\}\rvert
d\lambda$ $\displaystyle\leq
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>2\lambda,M^{\\#}a(m)\leq\gamma\lambda\right\\}\rvert
d\lambda+$ $\displaystyle
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>2\lambda,M^{\\#}a(m)>\gamma\lambda\right\\}\rvert
d\lambda$ $\displaystyle\leq
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}C{\gamma}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}\rvert
d\lambda+$ $\displaystyle
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M^{\\#}a(m)>\gamma\lambda\right\\}\rvert
d\lambda$ $\displaystyle\leq
2^{p}C{\gamma}\int_{0}^{N}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M_{d}a(m)>\lambda\right\\}\rvert
d\lambda\quad+$ $\displaystyle
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M^{\\#}a(m)>\gamma\lambda\right\\}\rvert
d\lambda$

It follows that

$(1-2^{p}C{\gamma})I_{N}\leq
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M^{\\#}a(m)>\gamma\lambda\right\\}\rvert
d\lambda$

Now choose $\gamma=\frac{1}{C2^{p+1}}$ such that
$(1-2^{p}C{\gamma})=\frac{1}{2}$. Then,

$\displaystyle\frac{1}{2}I_{N}\leq
2^{p}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M^{\\#}a(m)>\gamma\lambda\right\\}\rvert
d\lambda$
$\displaystyle\leq\frac{2^{p}}{\gamma^{p}}\int_{0}^{\frac{N}{2}}p\lambda^{p-1}\lvert\left\\{m\in\mathcal{Z}:M^{\\#}a(m)>\lambda\right\\}\rvert
d\lambda$

Now, take $N\to\infty$, we get

$\sum_{m\in\mathcal{Z}}M_{d}{a(m)}^{p}\leq
C\sum_{m\in\mathcal{Z}}{M^{\\#}a(m)}^{p}$

∎

##  References

  * [1] Duandikotoxea, _Fourier Analysis_ , Graduate Texts in Mathematics.
  * [2] A.Michael Alphonse and Shobha Madan , _The Commutator of the Ergodic Hilbert Transform_ , Contemporary Mathematics, Vol 189, 1995
  * [3] A.Michael Alphonse and Shobha Madan , _On Ergodic Singular Integral Operators_ , Colloquium Mathematicum, Vol LXVI, 1994.
  * [4] A.Sri Sakti Swarup and A. Michael Alphonse, _The boundedness of Fractional Hardy-Littlewood maximal operator on variable lp(Z) spaces using Calderon-Zygmund decomposition_ , arXiv:2204.04331