# Google COVID-19 community mobility reports: insights from multi-criteria
decision making

Gabriela Cavalcante da Silva<EMAIL_ADDRESS>IMD, Universidade
Federal do Rio Grande do Norte (UFRN), Brazil  Sabrina Oliveira
<EMAIL_ADDRESS>Postgraduate Program in Mathematical and
Computational Modeling, CEFET-MG, Brazil  Elizabeth F. Wanner
<EMAIL_ADDRESS>Computer Engineering Dept., CEFET-MG, Brazil  Computer
Science Group, Aston University, UK  Leonardo C. T. Bezerra
<EMAIL_ADDRESS>
###### Abstract

Social distancing (SD) has been critical in the fight against the novel
coronavirus disease (COVID-19). To aid SD monitoring, many technology
companies have made available mobility data, the most prominent example being
the community mobility reports (CMR) provided by Google. Given the wide range
of research fields that have been drawing insights from CMR data, there has
been a rising concern for methodological discussion on how to use them.
Indeed, Google recently released their own guidelines, concerning the nature
of the place categories and the need for calibrating regional values. In this
work, we discuss how measures developed in the field of multi-criteria
decision making (MCDM) might benefit researchers analyzing this data.
Concretely, we discuss how Pareto dominance and performance measures adopted
in MCDM enable the mobility evaluation for (i) multiple categories for a given
time period and (ii) multiple categories over multiple time periods. We
empirically demonstrate these approaches conducting both a region- and
country-level analysis, comparing some of the most relevant outbreak examples
from different continents.

###### keywords:

COVID-19 , Social distancing , Google community mobility reports , Multi-
criteria decision making , Pareto dominance

††journal: Decision Support Systems

Declarations of interest: none

##  1 Introduction

In December 2019, the World Health Organization (WHO) Country Office in the
People’s Republic of China reported cases of pneumonia of unknown
etiology.111https://who.int/news-room/detail/29-06-2020-covidtimeline In
January 2020, WHO named SARS-CoV-2 the novel coronavirus responsible for these
cases, and the acute respiratory syndrome it caused COVID-19 (coronavirus
disease 2019). Still in January, WHO classified COVID-19 as a public health
emergency of international concern. In March, COVID-19 had cases reported from
all continents, and WHO declared it a pandemic.

The rapid spread of COVID-19 and the deaths it produced triggered a global
scientific rush. Indeed, by April over 20 surveys on COVID-19 could already be
identified [1]. However, this fast publication pace raises concerns on the
quality of the works produced. Specifically, many such works have not
undergone a peer reviewing process, casting uncertainty as to the
methodological decisions adopted by their proposers. This is even more
important in the COVID-19 context, where knowledge from complementary research
fields is required to propose multi-disciplinary solutions to fight the
pandemic.

Among the most relevant topics in this multi-disciplinary COVID-19 research
context is social distancing (SD), which WHO actively promotes as a non-
pharmaceutical intervention against COVID-19 [2]. To aid SD monitoring,
information technology (IT) companies have been publishing anonymized mobile
device location history data, among which Google [3]. In particular, the
community mobility reports (CMR) provided by Google comprise over 130
countries, some of which further detailed on a regional level. More
importantly, the data for a given country/region is a collection of time
series for six place categories. Therefore, SD analysis based on CMR data
requires theoretical approaches from the fields of time series analysis (TSA)
and multi-criteria decision making (MCDM) [4, 5, 6]. Furthermore, the
existence of six different place types renders this a many-criteria decision
making problem [7], a particularly challenging specialization of MCDM. Indeed,
Google itself released guidelines to aid this analysis [8], such its awareness
of the many different research fields it bridges.

The goal of this paper is to discuss methodological approaches that can help
analyze SD based on CMR data, with a special focus on MCDM. In principle, the
temporal nature of this data requires techniques from TSA, from which we
comment on the effects of (i) alternatives to reduce seasonality effects [9];
(ii) techniques to aggregate temporal dynamics, and; (iii) different
granularities for temporal discretization. Next, we discuss the effects of
different MCDM approaches that help provide a multi-criteria perspective to
this analysis, namely (i) Pareto dominance [4] and (ii) Pareto-related
measures that help deepen the analysis [5, 6]. Moreover, we adopt
visualization techniques that help understand the MCDM conclusions even in
this many-criteria context.

To empirically demonstrate the approaches discussed in this work, we conduct
both a region- and a country-level analyses comparing some of the most
relevant outbreak examples from several continents. Specifically, the region-
level analysis comprises locations that experienced some of the most severe
pandemic outbreaks [10], namely Lombardia (Italy), Île-de-France (France), New
York (United States), and São Paulo and Amazonas (Brazil). By its turn, the
country-level analysis comprises the Americas (Argentina and Canada), Europe
(Germany and Spain), Asia (Japan and South Korea), and Oceania (New Zealand).

Insights observed confirm the effects and importance of the methodological
resources discussed in this work. Some of these insights were already
expected, such as the effects of calibration and scaling on the comparison,
and the need for seasonality approaches to process the original raw data.
Other insights are specific to CMR assessment, such as (i) how the nature of
the different place categories interact with the experimental factors
considered and (ii) how different temporal granularities reveal contrasting
dynamics among localities. More importantly, we observe how an MCDM
perspective provides nuances that scalarized approaches would conceal, and
assist the decision-maker both visually and when configuring tolerance levels.
Concerning localities, the region-level analysis indicates that the European
regions considered were able to better promote social distancing than the
American ones, though a fine-grained temporal assessment indicates that this
was only achieved after a few weeks of mobility restriction.

Below, we summarize the main contributions of this paper:

  * •

A theoretical discussion on methodological alternatives for a multi-criteria
temporal analysis of social distancing through mobility data.

  * •

An empirical mobility reduction analysis of some of the most relevant outbreak
examples from several different continents in a region- and country-level.

The remainder of this paper is organized as follows. We initially briefly
review related work in Section 2. Next, Section 3 discusses the methodology we
adopt in this work to enable a multi-criteria social distancing analysis from
mobility data. The country- and region-level analyses are detailed in Section
4. Finally, we conclude and discuss future work in Section 5.

##  2 Related work

As previously discussed, the scientific rush stirred by the COVID-19 pandemic
has led to a significant number of works that have been made available online
not having undergone peer review. In this section, we consider only works that
have already been peer reviewed, and group them according to how they relate
to this work. Specifically, we first discuss works that relate social
distancing and COVID-19, and how mobility data can be used in this context.
Later, we review multi-criteria decision making approaches that have been
employed in the COVID-19 scenario.

##### Social distancing (SD)

WHO has appointed SD as the main non-pharmaceutical intervention against
COVID-19 [2]. Besides confirming this effectiveness, Bielecki et al. [11] have
indicated that SD may also be related to easing the symptoms patients develop.
Yet, SD adherence is subject to cultural traits [12], besides socioeconomical
factors. For instance, Asian countries are more used to facial masks, and
developing countries make less use of remote work. In this context, a
systematic quantification and analysis of the impact of SD policies on people
mobility may better inform health decision-making strategies during the
pandemic.

##### Mobility data

Early on, the relevance of SD monitoring through mobility data was highlighted
by Buckee et al. [13], who also discussed the need for anonymity through
aggregation and data privacy protocols. While local governments have often
relied on sensitive triangulation data from telecommunication companies, some
IT companies have anonymized their mobility data and published them online [8,
14, 15]. Given the fast-changing nature of this data, many works that monitor
SD using mobility data have been made available not having undergone peer
review. An exception that is closely related to the assessment we conduct in
this work used data from a Brazilian IT company [15] to evaluate SD in its
home state [16]. In addition, a few works use mobility data to retrospectively
model COVID-19 spread in different locations [17, 18].

##### Multi-criteria decision making

As previously discussed, SD policies employed to contain the dissemination of
COVID-19 must also be economically sustainable, allowing for planned social
movement. This conflictual search for balance has been modeled as
lockdown/lifting policies and studied from a multi-criteria perspective in
different works [19, 20]. Another application of multi-criteria decision
making techniques in the context of COVID-19 is model evaluation. For
instance, Singh and Avikal [21] have evaluated the importance of recommended
protocols for COVID-19 prevention, whereas Mohammed et al. [22] have evaluated
predictive diagnostic models from a multi-criteria perspective. Yet, we have
not identified works that assess mobility data from an MCDM perspective.

Figure 1: Methodology discussed in this work, comprising data preparation,
time series analysis, and multi-criteria decision making.

##  3 Methodology

As previously discussed, the mobility data we assess in this work is provided
by Google as community mobility reports (CMR). In this section, we initially
detail CMR data and how we enrich it with mobility restriction-related dates
and prepare it for analysis. Next, we discuss time series analysis concepts
and techniques, critical to this assessment as the CMR data is provided as a
collection of time series. Finally, we discuss concepts and techniques from
multi-criteria decision making (MCDM) which allow us to assess the CMR data
from a multi-criteria perspective. The whole process described in this section
is summarized in Figure 1.

###  3.1 Data acquisition, description, and enrichment

CMRs are provided by Google weekly as printed document files (PDFs), first
created on March 2020. Additionally, Google makes CMR data available as a
comma-separated values (CSV) file, which comprises data from all reports ever
produced. In total, CMR data comprises 135 countries, some of which are
further detailed on a regional level. Data is collected from users who
willingly enable their location history, and is anonymized as described in
Aktay et al. [3].

Data for each country is provided as a collection of six time series, one for
each place category created by Google, given in Table 1. Each time series
starts in February 15${}^{\text{th}}$, 2020, and currently extends until
August 16${}^{\text{th}}$, 2020. For a single timestamp and category, the
given value is computed relatively to a baseline, namely the median value, for
the corresponding day of the week, computed for the period between January
3${}^{\text{rd}}$, 2020 and February 6${}^{\text{th}}$, 2020 [8].

Table 1: Place categories and descriptions from Google CMR [8].

Category |  Places
---|---
Grocery& pharmacy |  Grocery markets, food warehouses, farmers markets, specialty shops, drug stores, and pharmacies
Parks |  Local parks, national parks, public beaches, marinas, dog parks, plazas, and public gardens
Transit stations |  Public transport hubs, e.g. subway and bus stations
Retail& recreation |  Restaurants, cafes, shopping centers, theme parks, museums, libraries, and movie theaters
Residential |  Places of residence
Workplaces |  Places of work

For the countries and regions we adopt in this analysis, we have further
enriched the data with (i) their initial mobility restriction dates and (ii)
the dates when these restrictions started to be relaxed. Specifically, we have
used school suspension dates as the initial restriction dates, as our
preliminary assessment showed this was the restriction measure that most
significantly affected mobility.

###  3.2 Data preparation

As previously discussed, Google has recently provided guidelines for the
assessment of its CMR data [8]. Among the most relevant recommendations are
(i) the need for calibrating data in a locality-wise basis; (ii) handling
noise incurred by holidays or other exceptional circumstances, and; (iii) the
difference in magnitude between categories. Here, we discuss how we handle the
first recommendation, and in the following sections we discuss the remaining
ones.

The need for calibrating data for each locality comes from the baseline period
adopted by Google when computing relative values. More precisely, the
assumption that the first five weeks of the year represent a period of average
mobility is very strong, specially in a global scenario as we investigate in
this work. Indeed, for most localities considered in this analysis, February
mobility presented significant deviations from the average mobility that the
baseline period should represent. A typical approach in machine learning when
comparing data from different sources is standardization, i.e., the data from
each source is first centered such that its mean is set to zero, and then
scaled such that its standard deviation equals one.

In this work, for each locality and category we process the whole time series
to ensure that the data previous to the first mobility restriction date
presents zero mean. Our goal is to promote comparability between different
localities, while preserving data interpretability. More precisely, our mean-
centering procedure aims to ensure that the data prior to mobility restriction
measures represent the average mobility for that locality and category.222We
remark that the magnitude of the deviations seen for February data for most
localities is such that ideally it should be discarded. Yet, this was not an
option in this investigation as we include Lombardia, for which the first
mobility restriction measure dates of February. If, instead, we had centered
the data to ensure the whole time series presented zero mean, we would have
compromised the interpretability of the series after the restriction dates. To
also preserve interpretability, we have chosen not to scale the data at this
point, but later in the MCDM analysis.

###  3.3 Time series analysis (TSA)

Given the temporal nature of the CMR data, we discuss three sets of concepts
and techniques from TSA which we employ in our assessment: (i) seasonality
effect reduction; (ii) time discretization granularity, and; (iii) temporal
dynamics aggregation. For each set, we highlight in boldface the experimental
options we adopt, summarized in Table 2.

Table 2: Experimental factors considered in this work. For brevity, we label
selected factors as highlighted in italics.

Perspective | Factor | Levels
---|---|---
Temporal | Seasonality | \- 7-day MA
|  | \- STL (trend)
| Granularity | \- 50-day period
|  | \- 10-day periods
| Aggregation | \- mean
|  | \- area under the curve (AUC)
|  | \- rank sums (RS)
Multi-criteria | Comparison | \- mean scalarized
|  | \- Pareto-dominance
|  | \- multiplicative $\epsilon$-dominance
| Tolerance ($\epsilon$) | $\\{0.01,0.05,0.1\\}$

##### Seasonality effects

Temporal raw data is embedded with different sorts of seasonality. A
straightforward approach to reduce such effects is to adopt moving averages
(MA). For instance, a seven-day MA spans over every day of the week, thus
reducing weekday seasonality. However, the more days an MA comprises, the more
it will propagate noise. As previously discussed, the relevance of noise in
CMR data is such that Google recommends preprocessing them. In contrast to MA,
more elaborate approaches such as seasonal-trend decomposition by loess (STL,
[9]) attempt to isolate the trend of the data from both seasonal effects and
noise.

##### Discretization granularity

Temporal discretization often leads to information loss regarding temporal
dynamics. This loss is directly regulated by the temporal discretization
granularity adopted, as a more fine-grained discretization still preserves
some of these dynamics (at the cost of producing yet other time series). In
this work, we analyze the data using both (i) a single time period, comprising
50 days, and (ii) multiple time periods, comprising ten consecutive days each.
In more detail, we have observed that for all localities considered, the
period between the first restriction measure and the date when it was first
relaxed comprises at least 50 days (but not much more than that). For this
reason, after we center the data and apply seasonality approaches,333In a
preliminary assessment, we have also considered restricting the time period of
our analysis prior to applying seasonality approaches. However, we have
observed that the outcomes of the techniques reflect better the original data
when the whole series are considered. we restrict our analysis to the first 50
days since the first restriction measure at each locality.444Considering days
since a given event is a common option in COVID-19 analysis, and renders raw
data further unsuited to the assessment. For instance, the 50-day period for
two localities may differ as to the number of weekends they comprise, a factor
that is critical in the context of mobility analysis. If a coarse
discretization is adopted, all 50 days comprise a single period. Instead, if a
more fine-grained discretization is chosen, the analysis is performed on five
periods comprising ten consecutive days each, producing a five-period time
series for each aggregation technique we next discuss.

##### Temporal aggregation

A straightforward approach to aggregate a time series is to compute its mean.
A more ellaborate approach is to compute the area under the curve (AUC) for
each series, provided a common reference point.555The programming library we
adopt in this work (SciPy, http://scipy.org) does not allow us to specify a
reference point for AUC computation. For this reason, before AUC aggregation
we globally shift the data so it becomes non-negative and the origin of the
y-axis can be used as reference point. Since CMR data is updated weekly and
this could interfere with global shifting, we have performed shifting on data
up to July 22nd, 2020.  Both mean and AUC can be regarded parametric
approaches, as for a given day and category they are able to capture both (i)
which localities had larger mobility reduction and (ii) the mobility reduction
differences themselves. Conversely, a non-parametric approach such as rank
sums (RS) is only able to grasp the former. Effectively, an RS helps identify
which localities were more consistently effective in mobility reduction than
others, rather than by how much.

###  3.4 Multi-criteria decision making

To assess CMR data from a multi-criteria perspective, we initally discuss how
we prepare the data, and next the metrics we can employ to compare data from
pairs or sets of localities.

##### Data preparation

Some of the most relevant recommendations from Google concerning CMR regard
the particularities of the different places categories. In this work, we
restrict our analysis to mobility in non-residential areas. Our rationale is
that the indication that residential mobility should be maximized is not as
clear as that mobility in non-residential places should be minimized. More
precisely, Google does not specify the space granularity it adopts for
residential places. As such, cultural traits such as social gatherings from
neighbors that live in a same building would have to be minimized, and we do
not know how the spatial discretization adopted by Google handles these
scenarios. To ensure comparability for all the remaining place categories,
prior to temporal aggregation we have scaled the data so that the variance
from each category during the 50-day period considered becomes unitary,
whatever the seasonality approach. Our motivation is two-fold. First, scaling
the data from the 50-day period rather than the whole series ensures
comparability between reduction rates. Second, scaling to unit variance rather
than to fit a range as traditional in MCDM literature preserves the
interpretability of the $\epsilon$-dominance analysis.

Table 3: Pareto dominance relations among localities [4].

Relation | Notation |  Interpretation
---|---|---
Strictdominance | $l_{1}\prec\prec l_{2}$ |  Mobility reduction at locality $l_{1}$ was larger than at locality $l_{2}$ for all place categories
Dominance | $l_{1}\prec l_{2}$ |  Mobility reduction at locality $l_{1}$ was not smaller than at locality $l_{2}$ for all place categories, and was larger for at least one
Weakdominance | $l_{1}\preceq l_{2}$ |  Mobility reduction at locality $l_{1}$ was not smaller than at locality $l_{2}$ for all categories
Indifference | $l_{1}\sim l_{2}$ |  Locality $l_{1}$ presents the same mobility reduction as locality $l_{2}$ for all place categories
Incomparability | $l_{1}\mid\mid l_{2}$ |  Neither $l_{1}\preceq l_{2}$ nor $l_{2}\preceq l_{1}$

##### Multi-criteria comparison

The most straightforward approach to comparing mobility from multiple
localities is to aggregate the data from the different place categories,
typically through an scalarization function. In this work, the mean of the
multiple categories is an applicable scalarization function, since we do not
pre-establish a preference over the place categories. However, aggregating the
data from the different place categories incurs in information loss. The most
commonly adopted alternatives to scalarization functions are Pareto\- or
$\epsilon$-dominance [4, 6]. The former is summarized in Table 3, and can
discriminate situations where the mobility reduction from two localities are
(i) incomparable, since each locality presents more effective mobility
reduction for at least one of the place categories being assessed, and; (ii)
comparable, when the mobility reduction for one locality is at least as
effective as the mobility reduction for the other locality, for all place
categories assessed. Complementarily, $\epsilon$-dominance is a relaxed form
of Pareto dominance, where the decision maker can configure the tolerance
level ($\epsilon$) of the analysis.

##### Dominance-compliant ranking

When assessing a set of localities, it is interesting to rank them in a
Pareto-compliant way. Among the best-established ranking approaches in MCDM
theory is nondominated sorting [5], which produces the dominance depth measure
[23]. The most relevant properties of dominance depth are that (i) localities
in a same cluster (same dominance depth) are incomparable and (ii) localities
in a cluster with larger dominance depth are dominated by some locality in a
cluster with lower dominance depth. In this work, we evaluate localities based
on their dominance depth, computed using the three different multi-criteria
comparison approaches discussed above.666Note that dominance depth coupled
with mean scalarization will most often produce singleton clusters, and is
equivalent to ranking the mean values from the different place categories.
Still, we model all comparison metrics using a common MCDM framework to have
the ranking based on mean scalarization as baseline.

##  4 Results

To empirically demonstrate the methodology discussed in the previous section,
we conduct both a region- and a country-level mobility reduction analysis for
selected localities. Concretely, we start with an analysis of regions that
experienced a severe COVID-19 outbreak, and for simplicity we limit this
analysis to the two most recent global epicentres: (i) Europe, from which we
include Lombardia (Italy) and Île-de-France (France), and; (ii) the Americas,
from which we consider the states of New York (USA) and Amazonas and São Paulo
(Brazil).777Differently from other countries, in Brazil the epicenter (São
Paulo) was not the region with the most severe outbreak (Amazonas) [10]. In
common, mobility restriction measures adopted by those countries were
originally proposed on a regional basis. By contrast, other countries enforced
nation-wide restrictions early on, which we assess with the country-level
analysis. Besides Europe (Germany and Spain) and the Americas (Argentina and
Canada), the country-level analysis also includes Asia (Japan and South Korea)
and Oceania (New Zealand).

Next, we first illustrate the effects of the first steps of the methodology
discussed in Section 3, specifically data calibration and seasonality
approaches. Later, we present our region- and country-level analyses.

###  4.1 Preliminary discussion

As discussed in the CMR guidelines [8] and also in Section 3, data from
different localities and/or categories may need to be calibrated. This is
illustrated in Figure 2, where we depict mobility data for the different place
categories in Lombardia. In the plot, the left-most vertical dashed line
indicates the date of the first mobility restriction adopted in Lombardia
(February 23rd), whereas the right-most vertical dashed line indicates the
date when restrictions were first relaxed (May 11th). The shaded region
highlights Lombardia’s 50-day period considered in this assessment. For the
dates before the first restriction, the mobility at the different categories
is in general centered around zero, indicating that the baseline period
adopted by Google (January 3rd to February 6th) can be regarded representative
of an average mobility scenario in Lombardia. However, this is not the case
for the Parks category, for which the mobility before the first restriction
measure is higher than on the baseline period.

Figure 2: Lombardia’s mobility data (y-axis) for all categories before data
calibration. Left-most vertical dashed line: first mobility restriction (Feb
23rd). Right-most vertical dashed line: restrictions were first relaxed (May
11th). Shaded region: Lombardia’s 50-day period considered in the assessment.
Figure 3: Mobility data from Parks in Lombardia after calibration, comparing
different seasonality approaches (MA and trend) with the original raw data.

| AUC/Mean/RS
---|---
| MA | Trend
Île-de-France | 1 | 1
Lombardia | 2 | 2
Amazonas | 5 | 4
São Paulo | 3 | 3
New York | 4 | 5

(a) Workplaces

| AUC/Mean | RS
---|---|---
| MA | Trend | MA | Trend
Île-de-France | 1 | 2 | 1 | 1
Lombardia | 2 | 1 | 2 | 2
Amazonas | 3 | 3 | 3 | 3
São Paulo | 4 | 4 | 4 | 3
New York | 5 | 5 | 5 | 5

(b) Grocery & Pharmacy

| AUC | Mean | RS
---|---|---|---
| MA | Trend | MA | Trend | MA | Trend
Île-de-France | 1 | 2 | 2 | 2 | 1 | 1
Lombardia | 2 | 1 | 1 | 1 | 2 | 2
Amazonas | 3 | 3 | 3 | 3 | 3 | 3
São Paulo | 4 | 4 | 4 | 4 | 4 | 4
New York | 5 | 5 | 5 | 5 | 5 | 5

(c) Parks

Table 4: Regions respectively ranked according to their mobility in (a)
Workplaces, (b) Grocery & Pharmacy, and (c) Parks, aggregated over a 50-day
period using different measures (AUC, mean and RS) and seasonality approaches
(MA and trend data). Regions are sorted to match the ranking most frequently
observed.

By contrast, Figure 3 depicts the mobility data from Parks in Lombardia after
data is calibrated (green line), and also illustrates the effect of
seasonality approaches applied post-calibration. Specifically, data processed
using seven-day moving averages is referred to as MA (brown line), and the
trend component extracted with the STL approach is referred to as trend (black
line). Concerning calibration, the mobility data from Parks in Lombardia
before the first mobility restriction now presents zero mean. As a
consequence, we observe that the whole series has been shifted down along the
y-axis, and the mobility reduction from Parks during the 50-day period
assessed is now often greater than from the remaining place categories.
Regarding seasonality approaches, data processed using moving averages is less
smooth than data processed with the STL approach. As discussed in Section 3,
STL is a technique able to partially mitigate the presence of noise, whereas
moving averages may propagate it. Indeed, in comparison to the remaining place
categories given in Figure 2, we notice that Parks is a category where
mobility follows a less-structured pattern, being more prone to exceptional
circumstances (noise).

###  4.2 Region-level analysis

After calibration and seasonality processing, we proceed to the analysis of
the selected regions. For clarity, we group the main insights observed
following the remainining steps of the methodology discussed in Section 3.

####  4.2.1 Temporal aggregation

We begin our discussion with the temporal aggregation of the 50-day series for
each region and place category. The three different aggregation approaches
(AUC, mean, and RS) were computed using MA and trend data. Table 4 shows
regions ranked for each of these combinations when considering mobility data
from Workplaces (a), Grocery & Pharmacy (b), and Parks (c). For clarity,
regions are sorted to match the ranking most frequently observed in the table,
namely: Île-de-France, Lombardia, Amazonas, São Paulo, and New York. For
brevity, results for categories Transit Stations and Retail & Recreation are
provided as supplementary material [24], as the mobility data from these
categories are highly correlated to mobility data from Workplaces [25].

Regardless of the aggregation measure or seasonality approach adopted, we can
see two distinct clusters on Table 4. The first cluster comprises Île-de-
France and Lombardia, which alternate between the first and second positions.
The second cluster comprises Amazonas, São Paulo, and New York, which
alternate among the third, fourth, and fifth positions. Differences in
rankings are least frequent for Workplaces, for which aggregation measures
agree, and most frequent for Parks, for which both aggregation measures and
seasonality approaches interact. To illustrate these effects, Figure 4 shows
the 50-day series from Parks in Lombardia (orange line) and Île-de-France
(blue line) with different seasonality approaches. As discussed in the
previous section, MA and trend present different smoothness degrees due to how
noise is handled. The different shapes of the curves affect AUC aggregation,
and results for MA diverge from results with trend data. Regarding mean
aggregation, rankings for the different seasonality approaches match in the
Parks category, though they diverge for Grocery & Pharmacy. This is also the
case for RS aggregation, which Fig. 4 also helps illustrate. In particular,
the shape differences seen for Parks do not strongly alter the relative
rankings of the regions, and so RS aggregation is not affected. We conclude
from these results that AUC aggregation is more sensitive to the choice of
seasonality approach than the remaining aggregation measures considered.

Figure 4: Mobility data from Parks comparing Île-de-France and Lombardia after
calibration, with different seasonality approaches, zoomed at the 50-day
period considered in the assessment.

####  4.2.2 Discretization granularity

Time series assessment using different levels of temporal discretization
granularity often provide complementary insights, at the cost of producing
novel time series. To assess the influence of time granularity, in this
section we compute aggregated measures for five time periods, comprising ten
consecutive days each. Concretely, given an individual place category and
region, we compute aggregated measures for each time period considering MA and
trend data. For brevity, we illustrate the resulting time series for category
Workplaces in Figure 5, which depicts series produced using AUC (top) and RS
(bottom) aggregation. In particular, we omit the series produced by mean
aggregation as they are roughly equivalent to the series produced using AUC
aggregation for this category. For each plot, the x-axis represents the 50-day
period divided into 10-consecutive-day periods, labeled according to the
number of days since the first restriction measure in the given region.

Figure 5: Multiple-time-period analysis for Workplaces using MA and trend data
and different aggregation measures. Top: AUC. Bottom: RS. Each time period is
represented by its initial day, given as a marker.

Figure 5 reveals interesting insights about the dynamics of mobility reduction
in Lombardia and Amazonas (purple line). In more detail, the single-time-
period analysis given in Table 4a shows Lombardia ranking worse only than Île-
de-France. Conversely, when multiple time periods are considered, the
reduction observed in Lombardia during the first and second time period were
among the worst of the regions considered. As of the third time period on, we
observe a significant improvement in mobility reduction in Lombardia,
eventually leading to a better rank than all the remaining regions for all
combinations of aggregation and seasonality approaches. A similar pattern is
observed for Amazonas, which often was ranked worse than all regions but New
York (green line) in Table 4a. When multiple time periods are considered, we
see that Amazonas actually starts worse than all other regions (first time
period), but improves its mobility reduction rate to the point of even
surpassing the reduction seen in São Paulo (red line).

####  4.2.3 MCDM analysis

To compare mobility from different localities taking into account all place
categories simultaneously, we employ the different multi-criteria comparison
approaches discussed in Section 3: mean scalarization and Pareto- and
$\epsilon$-dominance. Furthermore, to produce a ranking of the regions, we
couple these different comparison approaches with nondominated sorting, i.e.,
the ranking produced depicts the dominance depth of each region. For
simplicity, we start this analysis considering a single 50-day time period,
and group the most relevant insights we discuss to match the structure used in
Section 3.

Figure 6: Multi-criteria visualization of AUC-aggregated mobility trend data
comparing different regions and categories. Top: original. Bottom: scaled.

##### Data preparation

Scaling effects are illustrated in Fig. 6, where a radar chart depicts AUC
aggregation of the original (top) and scaled (bottom) trend data. In more
detail, this polar coordinate plot depicts (i) categories as angles and (ii)
mobility reduction aggregated over the 50-day period as radii. More
importantly, this multi-dimensional visualization simplifies the multi-
criteria comparison between different localities, as locality $l_{1}$
dominates locality $l_{2}$ iff the polygon given for $l_{1}$ is contained by
the polygon given for $l_{2}$. Besides the improved balance between
categories, we observe that scaling may also affect dominance depending on the
seasonality approach and/or aggregation measure considered. For instance, Fig.
6 (top) shows that only mobility reduction rates from categories Grocery &
Pharmacy and Parks were deemed greater in Lombardia than in Île-de-France,
rendering both localities Pareto-incomparable. Yet, scaling employed prior to
aggregation (Fig. 6, bottom) renders reduction rates from all categories
greater in Île-de-France than in Lombardia, and so the former dominates the
latter.

##### Multi-criteria comparison

Figure 6 (bottom) also illustrates the different multi-criteria comparison
approaches we consider. For instance, AUC-aggregated mobility reduction rates
in New York are (marginally) larger than in Amazonas for categories Retail &
Recreation, Transit stations, and Workplaces. According to Pareto-dominance,
these two localities are incomparable. Conversely, the difference from these
categories is so small that $\epsilon$-dominance considers that Amazonas
dominates New York for all tolerance levels investigated in this work.

##### Dominance-compliant ranking

Table 5 gives rankings computed using Pareto-dominance. In general, Lombardia
and Île-de-France comprise the cluster with largest mobility reduction rates.
By contrast, Amazonas and São Paulo are now a cluster that always dominates
New York. Overall, two rankings can be observed, differing as to whether Île-
de-France dominates Lombardia or not. As expected, mean and RS aggregation are
not affected by seasonality approach for this matter, whereas AUC is. In
contrast to dominance-based analysis, ranking regions using mean scalarization
produces the same total order, independently of the aggregation measures and
seasonality approaches applied: Île-de-France, Lombardia, São Paulo, Amazonas,
and New York. This order differs from the most frequent in Table 4 in that São
Paulo precedes Amazonas, but matches the order seen for Workplaces when trend
data is adopted. Besides scaling, this could also be a consequence of the
number of categories in CMR data correlated with Workplaces.

| AUC | Mean | RS
---|---|---|---
| MA | Trend | MA | Trend | MA | Trend
Lombardia | 1 | 2 | 1 | 1 | 2 | 2
Île-de-France | 1 | 1 | 1 | 1 | 1 | 1
Amazonas | 2 | 3 | 2 | 2 | 3 | 3
São Paulo | 2 | 3 | 2 | 2 | 3 | 3
New York | 3 | 4 | 3 | 3 | 4 | 4

Table 5: Regions ranked according to Pareto-dominance.

##### Tolerance configuration

When tolerance is set very low ($\epsilon=0.01$), the only difference to
Pareto-dominance rankings concerns mean aggregation of MA data, for which Île-
de-France now $\epsilon$-dominates Lombardia. Conversely, a tolerance level
set too high ($\epsilon=0.1$) renders AUC/mean rankings identical to mean-
scalarized rankings. Concerning RS, São Paulo $\epsilon$-dominates Amazonas
and New York, which are now considered $\epsilon$-incomparable. The setup
using $\epsilon=0.05$ shares characteristics from both other setups.
Specifically, AUC/mean rankings match mean-scalarized rankings, whereas RS
rankings match Pareto-dominance rankings. We conclude from these results that
RS aggregation is less sensitive to $\epsilon$-dominance.

Finally, we assess insights from the multiple-time-period analysis, in which
we investigate the effect of the temporal discretization granularity. Figure 7
depicts a parallel coordinates plot for this assessment. Each of the five time
periods is represented as a vertical axis, and each region is represented by a
polyline with vertices on the axes. The position of each vertex corresponds to
the dominance depth of the given region in the corresponding time period. For
clarity, vertices that would collide are slightly shifted along the x-axis.
Furthermore, only the analysis for scaled trend data is shown, as we next
discuss.

Figure 7: Parallel coordinates plot depicting the changes in Pareto-compliant
rankings of the different regions. Each row represents a different aggregation
measure applied to scaled trend data. Top: AUC. Bottom: RS.

##### Experimental factor effects

We observe ranking variability as a function of the seasonality approach,
aggregation measure, and time period considered. Specifically, we notice that
rankings vary less as a function of experimental factors as time progresses.
Indeed, for the first time period we observe four different rankings, whereas
for the latter three a single ranking per period is produced by all
aggregation measures and seasonality approaches. Overall, we recommend that
multiple-period analyses consider only scaled trend data as we do, given the
benefits demonstrated by these factors.

##### Temporal dynamics

Figure 7 depicts AUC and RS analyses of the scaled trend data. In particular,
mean aggregation rankings equal RS rankings, and for brevity they are not
shown. As previously discussed, disagreements in rankings are observed only on
the two initial time periods. Interestingly, Île-de-France and São Paulo
present low dominance depths on those periods regardless of the aggregation
measure considered. This contrasts with the later time-period rankings (and
also single-period rankings), in which São Paulo is generally part of a
cluster with higher dominance depth, being often incomparable w.r.t. Amazonas
and/or New York. Conversely, Lombardia is often ranked poorly in the early
time-period rankings, presenting lower dominance depth only from the third
period on. Altogether, these observations confirm the importance of bridging
sound TSA and MCDM methodology.

###  4.3 Country-level results

In the previous section, we have discussed how the different methodologies
considered have affected results. In this section, we focus on an MCDM
analysis of the different countries selected, highlighting whether insights
discussed in the region-level analysis are still observable. For simplicity,
we start our analysis considering a single time period comprising 50 days.

Figure 8: Multi-criteria visualization of AUC-aggregated scaled trend data
comparing different countries and categories.

##### Scaling and comparison

Figure 8 shows a radar chart for the AUC-aggregated mobility using scaled
trend data, which is representative of results seen for other factors. Once
again, scaling balances the importance of the categories, but this time it
does not interfere with the multi-criteria comparison. At a first glance, we
observe in Fig. 8 two groups of disjoint countries, namely (i) Spain, New
Zealand, and Argentina, and (ii) Canada, Germany, Japan, and South Korea.
Within each group, we notice that some differences are relatively small, which
impacts the later discussion on $\epsilon$-dominance.

##### Dominance-compliant ranking

Both Pareto- and $\epsilon$-dominance with $\epsilon=0.01$ converge to a same
ranking between countries, whatever the factors considered. The cluster with
lowest dominance depth comprises Spain (pink) and New Zealand (purple),
followed by a singleton comprising Argentina (blue). Figure 8 helps visualize
why dominance depth indicates that these countries comprise two clusters.
Specifically, even if New Zealand is incomparable w.r.t. to Spain and
Argentina, Spain dominates Argentina. The last two clusters comprise (i)
Canada and Germany and (ii) Japan and South Korea. Once again, a closer look
at Fig. 8 confirms that Canada (orange) dominates Japan (red) and South Korea
(brown). This way, Germany (green) is clustered with Canada, even if Germany
and Japan are incomparable. Finally, these multi-criteria nuances would have
been completely overlooked if mean scalarization had been employed, resulting
on a total order of the countries: Spain, New Zealand, Argentina, Canada,
Germany, Japan, and South Korea.

Figure 9: Parallel coordinates plot depicting the changes in Pareto-compliant
rankings of the different countries over time. Each row represents a different
aggregation measure applied to scaled trend data.

##### Tolerance configuration

As discussed above, a small tolerance value ($\epsilon=0.01)$ is unable to
affect rankings. The remaining insights observed match to some extent what had
been discussed for the region-level analysis. In more detail, AUC and mean
rankings for $\epsilon\in\\{0.05,0.1\\}$ show that Canada $\epsilon$-dominates
Germany. As a result, Canada now comprises a singleton cluster, and two new
clusters are formed: (i) Germany and Japan, and (ii) South Korea. This is also
illustrated in Figure 8, where we can see that Japan is incomparable w.r.t.
both Germany and South Korea, but the former dominates the latter. As in the
region-level analysis, RS aggregation is less sensitive to
$\epsilon$-dominance, with the only change in rankings observed concerning
$\epsilon=0.1$ and variying as a function of the seasonality approach adopted.
Specifically, rankings computed using MA data remain the same, whereas
rankings computed using trend data indicate that Japan $\epsilon$-dominates
South Korea.

We conclude this analysis changing the temporal granularity adopted. As with
the previously discussed multiple-time-period region-level analysis,
experimental factors strongly affect rankings specially for the initial time
periods. Figure 9 depicts the parallel coordinate plots using different
aggregation measures of scaled trend data. In common, rankings for all
measures agree as of the third time period, and also on the second time period
for mean (middle) and RS (bottom) aggregation. The most significant
differences are observed for the first time period, on which none of the
aggregation measures agree.

Concerning mobility dynamics, it is interesting to remark that Spain is never
ranked in the lowest depth cluster on the first time period, whatever the
aggregation measure considered. By contrast, New Zealand presented a strong
mobility reduction rate in the early periods, but was later surpassed by Spain
and even Argentina. Regarding the remaining countries, it is interesting to
observe how South Korea starts in a same- (mean) or in a lower-depth (AUC and
RS) cluster than Japan, but ends in a higher-depth cluster whichever measure
considered. Finally, we remark that none of the rankings for a given time
period match the Pareto-dominance rankings for the single-time-period
analysis, which corroborates the importance of an assessment that considers
multiple discretization granularities.

##  5 Conclusion

The novel coronavirus disease (COVID-19) has been the most serious public
health concern of the recent decades due to its quick spread across the world.
In the absence of pharmaceutical alternatives, social distancing has been a
key approach for countries trying to control the pandemic. Yet, social
distancing is economically expensive, and countries have adhered to this
measure in different rates. The community mobility reports (CMR) produced by
Google weekly are a relevant example of how technology companies have assisted
in the fight against COVID-19. Fully grasping the insights they provide
requires a multi-disciplinary approach, comprising domains as diverse as time
series analysis (TSA), data visualization, and multi-criteria decision making
(MCDM). More importantly, the speed with which the scientific community is
being pressed to produce results makes methodological soundness yet more
important.

In this work, we have discussed MCDM concepts that enable a multi-criteria
analysis of CMR data. Specifically, we have discussed how to (i) ensure
comparability between the mobility reduction rates from different place
categories, (ii) compare the mobility reduction rates from a pair of
localities through different Pareto dominance relations, and; (iii) cluster
and rank a set of localities based on their mobility reduction rates in a
Pareto-compliant way. Furthermore, we have discussed how to visually compare
localities, even in a many-criteria scenario. Finally, we have acomplished
this analysis building on sound TSA methodology, which have allowed us to (i)
mitigate the impact of seasonality and noise, (ii) aggregate temporal dynamics
using different perspectives, and; (iii) assess the data using different
temporal discretization granularities.

To empirically demonstrate the proposed approach, we have conducted both a
region- and a country-level analysis on mobility reduction rates from
localities in different continents. Some of the insights observed have been
already reported, such as the need for calibration and seasonality approaches,
as well as the drawbacks with $\epsilon$-dominance if extreme tolerance values
are adopted. Other insights concern CMR data specifically, e.g. the
disagreement between rankings as a function of the experimental factors
considered. Specifically, we have demonstrated that place categories,
aggregation measures, and seasonality approaches interact, and that a coarse
temporal discretization conceals dynamics that are critical to SD monitoring.

Though the MCDM methodology discussed here has helped identify interesting
insights, follow-up investigations can further leverage these benefits. For
instance, future work could bridge multi-criteria TSA methodology with
simulation analysis to predict the impact of SD policies. Another important
research direction is to explore more visualization techniques to improve the
MCDM process for public policymakers. Finally, sociodemographic aspects are
likely related to how localities rank, and the examples illustrated in this
work could serve as grounds for that investigation.

## References

  * Yu et al. [2020] Y. Yu, Q. Shi, P. Zheng, L. Gao, H. Li, P. Tao, B. Gu, D. Wang, H. Chen,  Assessment of the quality of systematic reviews on COVID-19: A comparative study of previous coronavirus outbreaks,  J. of Med. Virol. (2020).
  * World Health Organization [2020] World Health Organization, Advice for the public, https://who.int/emergencies/diseases/novel-coronavirus-2019/advice-for-public, 2020\.
  * Aktay et al. [2020] A. Aktay, S. Bavadekar, G. Cossoul, J. Davis, D. Desfontaines, A. Fabrikant, E. Gabrilovich, K. Gadepalli, B. Gipson, M. Guevara, et al.,  Google COVID-19 community mobility reports: Anonymization process description (version 1.0),  arXiv preprint arXiv:2004.04145 (2020).
  * Zitzler et al. [2003] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, V. G. Da Fonseca,  Performance assessment of multiobjective optimizers: An analysis and review,  IEEE Trans. on Evol. Comput. 7 (2003) 117–132.
  * Goldberg [1989] D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning, Addison Wesley Publishing, 1989.
  * Laumanns et al. [2002] M. Laumanns, L. Thiele, K. Deb, E. Zitzler,  Combining convergence and diversity in evolutionary multiobjective optimization,  Evol. Comput. 10 (2002) 263–282.
  * Li et al. [2015] B. Li, J. Li, K. Tang, X. Yao,  Many-objective evolutionary algorithms: A survey,  ACM Comput. Surv. 48 (2015) 1–35.
  * Google LLC [2020] Google LLC, Google COVID-19 community mobility reports, https://google.com/covid19/mobility, 2020\. Accessed: 17/07/2020.
  * Cleveland et al. [1990] R. B. Cleveland, W. S. Cleveland, J. E. McRae, I. Terpenning,  STL: A seasonal-trend decomposition,  J. of Off. Stat. 6 (1990) 3–73.
  * Johns Hopkins University [2020] Johns Hopkins University, COVID-19 map: Coronavirus resource center, https://coronavirus.jhu.edu/map.html, 2020\. Accessed: 17/08/2020.
  * Bielecki et al. [2020] M. Bielecki, R. Züst, D. Siegrist, D. Meyerhofer, G. A. G. Crameri, Z. G. Stanga, A. Stettbacher, T. W. Buehrer, J. W. Deuel,  Social distancing alters the clinical course of COVID-19 in young adults: A comparative cohort study,  Clin. Infect. Dis. (2020).
  * Huynh [2020] T. L. D. Huynh,  Does culture matter social distancing under the COVID-19 pandemic?,  Saf. Sci. 130 (2020) 104872.
  * Buckee et al. [2020] C. O. Buckee, S. Balsari, J. Chan, M. Crosas, F. Dominici, U. Gasser, Y. H. Grad, B. Grenfell, M. E. Halloran, M. U. G. Kraemer, M. Lipsitch, C. E. Metcalf, L. A. Meyers, T. A. Perkins, M. Santillana, S. V. Scarpino, C. Viboud, A. Wesolowski, A. Schroeder,  Aggregated mobility data could help fight COVID-19,  Science 386 (2020) 145–146.
  * Apple Inc [2020] Apple Inc, Mobility trends reports, https://apple.com/covid19/mobility, 2020\. Accessed: 30/07/2020.
  * In Loco [2020] In Loco, Social distancing index, https://inloco.com.br/covid-19, 2020\. Accessed: 30/07/2020.
  * Endo et al. [2020] P. T. Endo, I. Silva, L. Lima, L. C. T. Bezerra, R. Gomes, M. Ribeiro-Dantas, G. Alves, K. H. d. C. Monteiro, T. Lynn, V. d. S. Sampaio,  #StayHome: Monitoring and benchmarking social isolation trends in Caruaru and the Região Metropolitana do Recife during the COVID-19 pandemic,  J. of the Braz. Soc. of Trop. Med. 53 (2020).
  * Wang and Yamamoto [2020] H. Wang, N. Yamamoto,  Using a partial differential equation with Google mobility data to predict COVID-19 in Arizona,  Math. Biosci. and Eng. 17 (2020) 4891–4904.
  * Kraemer et al. [2020] M. U. G. Kraemer, C.-H. Yang, B. Gutierrez, C.-H. Wu, B. Klein, D. M. Pigott, L. du Plessis, N. R. Faria, R. Li, W. P. Hanage, J. S. Brownstein, M. Layan, A. Vespignani, H. Tian, C. Dye, O. G. Pybus, S. V. Scarpino,  The effect of human mobility and control measures on the COVID-19 epidemic in China,  Science 368 (2020) 493–497.
  * Raboisson and Lhermie [2020] D. Raboisson, G. Lhermie,  Living with COVID-19: A systemic and multi-criteria approach to enact evidence-based health policy,  Front. in Public Health 8 (2020) 1–7.
  * Sangiorgio and Parisi [2020] V. Sangiorgio, F. Parisi,  A multicriteria approach for risk assessment of COVID-19 in urban district lockdown,  Saf. Sci. 130 (2020) 104862.
  * Singh and Avikal [2020] R. Singh, S. Avikal,  Covid-19: A decision-making approach for prioritization of preventive activities,  Inter. J. of Healthc. Manag. (2020) 1–6.
  * Mohammed et al. [2020] M. A. Mohammed, K. H. Abdulkareem, A. S. Al-Waisy, S. A. Mostafa, S. Al-Fahdawi, A. M. Dinar, W. Alhakami, A. BAZ, M. N. Al-Mhiqani, H. Alhakami, N. Arbaiy, M. S. Maashi, A. A. Mutlag, B. García-Zapirain, I. D. L. T. De La Torre Díez,  Benchmarking methodology for selection of optimal COVID-19 diagnostic model based on entropy and TOPSIS methods,  IEEE Access 8 (2020) 99115–99131.
  * Bezerra [2016] L. C. T. Bezerra, A component-wise approach to multi-objective evolutionary algorithms, Ph.D. thesis, IRIDIA, ULB, Belgium, 2016.
  * da Silva et al. [2020] G. C. da Silva, S. M. Oliveira, E. F. Wanner, L. C. T. Bezerra, Supplementary material for ”Google COVID-19 community mobility reports: insights from multi-criteria decision making”, https://gabicavalcante.github.io/mcdm-covid19, 2020\.
  * Ritchie [2020] H. Ritchie, Google mobility trends: How has the pandemic changed the movement of people around the world?, https://ourworldindata.org/covid-mobility-trends, 2020\. Accessed: 23/07/2020.


∎

11institutetext:  Yaofeng Su 22institutetext: School of Mathematics, Georgia
Institute of Technology, Atlanta, USA
22email<EMAIL_ADDRESS>
# Polynomial escape rates via maximal large deviations

Yaofeng Su

###### Abstract

In this short note, we propose an approach to polynomial escape rates, which
can be applied to various open systems with intermittency. The tool of our
approach is the maximal large deviations developed in Bunimovich and Su
[2023].

##  1 introduction

In the early 80s, mathematicians suggested investigating open systems (i.e.
systems with holes or leakages) as a means of generating transient chaos
Pianigiani and Yorke [1979], retrieving information from a distribution and
deducing facts about the equivalent closed systems. More precisely, consider a
dynamical system $(X,f,\mu)$, $\mu$ is an invariant probability on $X$
preserved by the dynamics $f$, a hole is a subset $H\subsetneq X$, then the
phase space $X$ of the system is opened by $H$, leading to leakages/escapes of
trajectories through the hole $H$. Such a system is called an open system. An
escape time is defined by $\tau_{H}(x):=\inf\\{n\geq 1:f^{n}(x)\in H\\}$, its
distribution $\mu(\tau_{H}>n)$ gives the probability of the trajectories that
do not leave $X$ at time $n$. Studying the decay rate of $\mu(\tau_{H}>n)$
(called an escape rate) can deduce the facts about chaos for the closed system
$(X,f,\mu)$.

For the closed systems enjoying exponential mixing rates, e.g. some expanding
maps and Sinai billiards, exponential escape rates were proved in Bruin et al.
[2018], Haydn and Yang [2020], Freitas et al. [2015], Demers [2013], Demers et
al. [2010]. For the closed systems enjoying polynomial mixing rates, rigorous
results in this direction are limited: Liverani-Saussole-Vaienti maps were
considered and a polynomial escape rate was proved in Demers and Fernandez
[2016].

The purpose of the present paper is to propose a flexible approach to
polynomial escape rates, which can be applied to various systems with
intermittency. Our results and methods have the following new aspects.

  1. 1.

We prove that, in general, a polynomial mixing dynamical system implies a
polynomial escape rate. This is different from the general results in Haydn
and Yang [2020], Freitas et al. [2015] which proved that a polynomial mixing
dynamical system implies an exponential escape rate.

  2. 2.

When $\mu$ is an SRB measure, our Theorem 3.1 recovers one result in Demers
and Fernandez [2016]. But the requirements on our hole $H$ in Theorems 3.1 and
4.1 can be relaxed and do not need to be a small Markov hole required in
Demers and Fernandez [2016].

  3. 3.

Although our Theorems 3.1 and 4.1 only study special systems, our methods can
be adapted to various dynamical systems with intermittency, see Remarks 2 and
3.

  4. 4.

To the best of our knowledge, our Theorem 4.1 is the first result addressing
escape rates for high-dimensional hyperbolic systems with intermittency.

Structures of this paper: section 2 includes necessary definitions and
notations used throughout this paper. Section 3 addresses polynomial escape
rates for one-dimensional intermittent maps. Its method is adapted to
polynomial escape rates for high-dimensional hyperbolic systems with
intermittency in Section 4.

##  2 Definitions and notations

Throughout this paper, we consider a dynamical system $(X,f,\mu)$, $f$ is a
hyperbolic dynamics, and the probability $\mu$ preserved by $f$ is an SRB
measure on $X$, i.e., the conditional measure on unstable manifolds is
absolutely continuous w.r.t. the Lebesgue measure on the unstable manifolds. A
hole $H\subsetneq X$ satisfies $\mu(H)\in(0,1)$. An escape time is defined by
$\tau_{H}(x):=\inf\\{n\geq 1:f^{n}(x)\in H\\}$. By the Poincaré recurrence
theorem, $\tau_{H}(x)<\infty$ for almost every $x$.

Throughout this paper, the notation $cls(H)$ (resp. $int(H)$) means the
closure (resp. interior) of $H$, $a_{n}\precsim b_{n}$ (resp. $a_{n}\succsim
b_{n}$) means that there is a constant $C\geq 1$ such that $|a_{n}|\leq
C|b_{n}|$ (resp. $|a_{n}|\geq C^{-1}|b_{n}|$). $a_{n}\approx b_{n}$ means
$|a_{n}|\precsim|b_{n}|$ and $|a_{n}|\succsim|b_{n}|$.

##  3 One-dimensional intermittent maps

In this section, we present an approach to escape rates of one-dimensional
intermittent maps. Typical one-dimensional intermittent maps are Liverani-
Saussol-Vaienti maps defined as follows

$\displaystyle g_{\alpha}(x)=\begin{cases}x+2^{\alpha}x^{1+\alpha},&0\leq
x\leq\frac{1}{2}\\\ 2x-1,&\frac{1}{2}<x\leq 1\\\ \end{cases},\quad
0<\alpha<1.$ (3.1)

It was proved in Liverani et al. [1999], Young [1999] that there is an SRB
measure $\mu_{\alpha}$ preserved by $g_{\alpha}$ and the mixing rate is
$O(n^{1-1/\alpha})$.

######  Theorem 3.1 (Escape Rates I).

For the dynamical system $(X,f,\mu)$:= $([0,1],g_{\alpha},\mu_{\alpha})$, if a
(dis)connected hole $H$ satisfies $0\notin cls(H)$ and $int(H)\neq\emptyset$,
then $\mu(\tau_{H}>n)\approx n^{1-1/\alpha}$.

###### Proof.

Choose a sufficiently small $a_{m}:=g_{\alpha}|_{L}^{-m}\\{1\\}$ (here
$g_{\alpha}|_{L}$ is the left branch of $g_{\alpha}$) such that
$H\subseteq(a_{m},1]$,

1\. A first return tower. It follows from Young [1999] that the system
$(X,f,\mu)$ can be modelled by a first return Young tower
$\Delta:=\\{(x,n)\in(a_{m},1]\times\\{0,1,2,\cdots\\}:n<R(x)\\}\\}$. Here
$R(x):=\inf\\{n\geq 1:f^{n}(x)\in(a_{m},1]\\}$ defined on $(a_{m},1]$ is a
first return time, the dynamics $F:\Delta\to\Delta$ sends $(x,n)$ to $(x,n+1)$
if $n+1<R(x)$, and $(x,n)$ to $(f^{R}(x),0)$ if $n=R(x)-1$. It follows from
Young [1999] that $F$ preserves an invariant probability $\mu_{\Delta}$ on
$\Delta$ and its mixing rate is $O(n^{1-1/\alpha})$. Let $\pi:\Delta\to X$,
$\pi(x,n)=f^{n}(x)$, then $\pi_{*}\mu_{\Delta}=\mu$.
2\. Lifting. Define $\tau_{H\times\\{0\\}}:=\inf\\{n\geq 1:F^{n}\in
H\times\\{0\\}\\}$, then

$\displaystyle\mu(\tau_{H}>n)$ $\displaystyle=\mu_{\Delta}(\inf\\{n\geq
1:f^{n}\circ\pi\in H\\}>n)=\mu_{\Delta}(\inf\\{n\geq
1:F^{n}\in\pi^{-1}H\\}>n)$ $\displaystyle=\mu_{\Delta}(\inf\\{n\geq 1:F^{n}\in
H\times\\{0\\}\\}>n)=\mu_{\Delta}(\tau_{H\times\\{0\\}}>n).$

3\. Lower bounds. Note that $\bigcup_{i\geq
1}\\{(x,i):R(x)>n+i\\}\subseteq\\{\tau_{H\times\\{0\\}}>n\\}$. Then
$\mu(\tau_{H}>n)=\mu_{\Delta}(\tau_{H\times\\{0\\}}>n)\succsim\sum_{i}\mu(R>n+i)\approx
n^{1-1/\alpha}$.
4\. Several escape times. Denote the base $(a_{m},1]\times\\{0\\}$ by $B$ and
define a new escape time on $\Delta$ by
$\tau_{H,R}:=\sum_{m=0}^{\tau_{H\times\\{0\\}}}\mathbbm{1}_{B}\circ F^{m}$,
another escape time on $B$ by $\tau_{H,B}:=\tau_{H,R}|_{B}$. Since $f^{R}$ is
a uniformly expanding map, by Theorem 2.1 of Bruin et al. [2018] and the
calculation of Appendix A of Bruin et al. [2018],
$\mu_{\Delta}|_{B}(\tau_{H,B}>n\cap\\{R=i\\})\precsim\mu(R=i)e^{-C\cdot n}$
for some $C>0$ when $H$ is a small Markov hole. But this also holds for our
hole $H$ because $int(H)$ must contain a small Markov hole.
5\. Upper bounds. By maximal large deviations, i.e., Theorem 3.1 of Bunimovich
and Su [2023],

$\mu_{\Delta}\Big{\\{}\max_{N\geq
n}\Big{|}\frac{\sum_{m=0}^{N}\mathbbm{1}_{B}\circ
F^{m}}{N}-\mu_{\Delta}(B)\Big{|}\geq\frac{\mu_{\Delta}(B)}{2}\Big{\\}}\precsim
n^{1-1/\alpha}.$

Denote $\mu_{\Delta}(B)$ by K, now we can estimate an upper bound:
$\mu_{\Delta}(\tau_{H\times 0}>n)$

$\displaystyle\leq\mu_{\Delta}\Big{(}\tau_{H\times\\{0\\}}>n,\Big{|}\frac{\tau_{H,R}}{\tau_{H\times\\{0\\}}}-K\Big{|}\geq\frac{K}{2}\Big{)}+\mu_{\Delta}\Big{(}\tau_{H\times\\{0\\}}>n,\Big{|}\frac{\tau_{H,R}}{\tau_{H\times\\{0\\}}}-K\Big{|}\leq\frac{K}{2}\Big{)}$
$\displaystyle\leq\mu_{\Delta}\Big{\\{}\max_{N\geq
n}\Big{|}\frac{\sum_{m=0}^{N}\mathbbm{1}_{B}\circ
F^{m}}{N}-K\Big{|}\geq\frac{K}{2}\Big{\\}}+\mu_{\Delta}\Big{(}\tau_{H,R}>n\frac{K}{2}\Big{)}$
$\displaystyle\precsim n^{1-1/\alpha}+\sum_{i\geq
0}\mu_{\Delta}\Big{\\{}(x,i):\tau_{H,R}(x,i)>n\frac{K}{2}\Big{\\}}$
$\displaystyle\precsim n^{1-1/\alpha}+\sum_{i\geq
1}\mu_{\Delta}|_{B}\Big{\\{}\tau_{H,B}>n\frac{K}{2},R\geq i\Big{\\}}$
$\displaystyle\precsim n^{1-1/\alpha}+\sum_{i\geq 1}\mu(R\geq
i)e^{-CKn/2}\precsim n^{1-1/\alpha}.$

Hence we conclude the proof with these five steps. ∎

######  Remark 1.

From the proof of Theorem 3.1, it can be seen easily that the upper bound of
the polynomial escape rate is a general result for Young towers, the lower
bound depends on the position of the hole which avoids singularities.

######  Remark 2.

Our proof can be easily extended to various intermittent systems. For example,
a finite type $\alpha$-Farey map of exponent $\theta$ in Kesseböhmer et al.
[2012]:

$\displaystyle F_{\alpha}(x)=\begin{cases}(1-x)/a_{1},&x\in[t_{2},t_{1}]\\\
a_{n-1}(x-t_{n+1})/a_{n}+t_{n},&x\in[t_{n+1},t_{n}],n\geq 2\\\ 0,&x=0\\\
\end{cases},$ (3.2)

where $a_{n}\geq 0,t_{n}=\sum_{i\geq n}a_{n},t_{1}=1$ and
$t_{n}=O(n^{-\theta}),\theta>1$. Follow the same argument in the proof of
Theorem 3.1, if $H$ satisfies $int(H)\neq\emptyset,0\notin cls(H)$, then
$\mu(\tau_{H}>n)\approx n^{1-\theta}$.

##  4 High-dimensional hyperbolic dynamical systems with intermittency

In this section, we present a similar method to approach escape rates for
high-dimensional hyperbolic systems with intermittency. A typical example is a
solenoid with intermittency. Let $\mathcal{M}=[0,1]\times\mathbb{D}$,
$f(x,z)=(g_{\alpha}(x),\theta z+e^{2\pi ix}/2)$, where

  1. 1.

$g_{\alpha}$ is the Liverani-Saussol-Vaienti map (3.1)

  2. 2.

$\theta>0$ is so small that $\theta||Dg_{\alpha}||_{\infty}<1-\theta$.

It follows from Alves and Azevedo [2016] that the attractor is $\bigcap_{i\geq
0}f^{i}(\mathcal{M})$, an SRB probability measure $\mu_{\alpha}$ exists on the
attractor.

######  Theorem 4.1 (Escape Rates II).

For the dynamical system $(X,f,\mu):=(\bigcap_{i\geq
0}f^{i}(\mathcal{M}),f,\mu_{\alpha})$, if a (dis)connected hole $H$ satisfies
$(\\{0\\}\times\mathbb{D})\bigcap cls(H)=\emptyset$ and $int(H)\neq\emptyset$,
then $\mu(\tau_{H}>n)\approx n^{1-1/\alpha}$.

###### Proof.

Choose a sufficiently small $a_{m}:=g_{\alpha}|_{L}^{-m}\\{1\\}$ (here
$g_{\alpha}|_{L}$ is the left branch of $g_{\alpha}$) such that
$H\subseteq(a_{m},1]\times\mathbb{D}$. Let a hyperbolic product set be
$\Lambda:=([a_{m},1]\times\mathbb{D})\bigcap X$. Each stable manifold
$\gamma^{s}$ is in the form of $\\{x\\}\times\mathbb{D}$. The scheme of the
proof is similar to the proof of Theorem 3.1.

1\. A first return tower. It follows from Bunimovich and Su [2022] that the
system $(X,f,\mu)$ can be modelled by a first return hyperbolic Young tower
$\Delta:=\\{(x,n)\in\Lambda\times\\{0,1,2,\cdots\\}:n<R(x)\\}$. Here
$R(x):=\inf\\{n\geq 1:f^{n}(x)\in\Lambda\\}$ defined on $\Lambda$ is a first
return time with $\mu(R>n)\approx n^{1-1/\alpha}$, the dynamics
$F:\Delta\to\Delta$ sends $(x,n)$ to $(x,n+1)$ if $n+1<R(x)$, and $(x,n)$ to
$(f^{R}(x),0)$ if $n=R(x)-1$.

It follows from Bunimovich and Su [2022], Alves and Azevedo [2016] that
$\pi:\Delta\to X$ defined by $\pi(x,n)=f^{n}(x)$ is a bijection, a relation
$\sim$ on $\Delta$ defined by

$(x,i)\sim(y,j)\text{ if and only if }i=j,x,y\text{ are in a stable manifold
}\gamma^{s}$

is an equivalent relation, and there is a quotient Young tower
$\widetilde{\Delta}:=\Delta/\sim$, a projection
$\widetilde{\pi}_{\Delta}:\Delta\to\widetilde{\Delta}$, a quotient dynamics
$\widetilde{F}:\widetilde{\Delta}\to\widetilde{\Delta}$, a probability
$\mu_{\Delta}$ on $\Delta$ and a probability $\mu_{\widetilde{\Delta}}$ on
$\widetilde{\Delta}$ satisfying

$\displaystyle\pi_{*}\mu_{\Delta}=\mu,\quad\left(\widetilde{\pi}_{\Delta}\right)_{*}\mu_{\Delta}=\mu_{\widetilde{\Delta}},\quad\widetilde{F}_{*}\mu_{\widetilde{\Delta}}=\mu_{\widetilde{\Delta}},\quad
F_{*}\mu_{\Delta}=\mu_{\Delta},\quad\widetilde{\pi}_{\Delta}\circ
F=\widetilde{F}\circ\widetilde{\pi}_{\Delta}.$ (4.1)

2\. Lifting. $\mu(\tau_{H}>n)=\mu_{\Delta}(\inf\\{n\geq 1:f^{n}\circ\pi\in
H\\}>n)=\mu_{\Delta}(\inf\\{n\geq
1:F^{n}\in\pi^{-1}H\\}>n)=\mu_{\Delta}(\inf\\{n\geq 1:F^{n}\in
H\times\\{0\\}\\}>n),$.
3\. Lower bounds. $\mu(\tau_{H}>n)\succsim n^{1-1/\alpha}$follows from the
same arguments in step 3 of the proof in Theorem 3.1.

4\. Upper bounds. Since $int(H)\neq\emptyset$, suppose that there is a
sufficiently large $N$ and a sufficiently small interval $I\subseteq[a_{m},1]$
such that $f^{N}(I\times\mathbb{D})\subseteq H$. We choose $I$ to be a small
Markov hole of $g_{\alpha}$ for convenience. Then by (4.1) we have
$\mu(\tau_{H}>n)$

$\displaystyle\leq\mu(\inf\\{n\geq 1:f^{n}\in f^{N}(I\times\mathbb{D})\\}>n)$
$\displaystyle=\mu(\inf\\{n\geq 1:f^{n}\in I\times\mathbb{D}\\}>n)$
$\displaystyle=\mu_{\Delta}(\inf\\{n\geq
1:F^{n}\in(I\times\mathbb{D})\times\\{0\\}\\}>n)$
$\displaystyle=\mu_{\Delta}(\inf\\{n\geq
1:F^{n}\in\widetilde{\pi}_{\Delta}^{-1}(I\times\\{0\\})\\}>n)$
$\displaystyle=\mu_{\widetilde{\Delta}}(\inf\\{n\geq 1:\widetilde{F}^{n}\in
I\times\\{0\\}\\}>n).$

Now we repeat the argument of steps 4 and 5 in the proof of Theorem 3.1, by
replacing $H$, $F$ and $\Delta$ therein by $I$, $\widetilde{F}$ and
$\widetilde{\Delta}$ respectively, then we have

$\mu(\tau_{H}>n)\leq\mu_{\widetilde{\Delta}}(\inf\\{n\geq
1:\widetilde{F}^{n}\in I\times\\{0\\}\\}>n)\precsim n^{1-1/\alpha}.$

Hence we conclude the proof. ∎

######  Remark 3.

The method in this section can be easily extended to various hyperbolic
systems, e.g. replacing $g_{\alpha}$ by Farey maps (3.2) gives a new three-
dimensional hyperbolic system with intermittency, a similar argument gives
$\mu(\tau_{H}>n)\approx n^{1-\theta}$ when $(\\{0\\}\times\mathbb{D})\bigcap
cls(H)=\emptyset$ and $int(H)\neq\emptyset$.

######  Remark 4.

Bunimovich billiards are typical examples of hyperbolic systems with
intermittency. However, we will not present a decay rate of $\mu(\tau_{H}>n)$
in this paper, this is because studying a stronger result: an explicit
asymptotic expansion for the $\mu(\tau>n)$ for Bunimovich billiards reveals
many interesting facts in chaos (see Bunimovich and Vela-Arevalo [2015]). We
will address this new question for Bunimovich billiards in a separate paper.

## References

  * Alves and Azevedo [2016] J. F. Alves and D. Azevedo.  Statistical properties of diffeomorphisms with weak invariant manifolds.  _Discrete Contin. Dyn. Syst._ , 36(1):1–41, 2016.  ISSN 1078-0947.  doi: 10.3934/dcds.2016.36.1.  URL https://doi.org/10.3934/dcds.2016.36.1.
  * Bruin et al. [2018] H. Bruin, M. F. Demers, and M. Todd.  Hitting and escaping statistics: mixing, targets and holes.  _Adv. Math._ , 328:1263–1298, 2018.  ISSN 0001-8708,1090-2082.  doi: 10.1016/j.aim.2017.12.020.  URL https://doi.org/10.1016/j.aim.2017.12.020.
  * Bunimovich and Su [2022] L. A. Bunimovich and Y. Su.  Poisson approximations and convergence rates for hyperbolic dynamical systems.  _Comm. Math. Phys._ , 390(1):113–168, 2022.  ISSN 0010-3616.  doi: 10.1007/s00220-022-04309-w.  URL https://doi.org/10.1007/s00220-022-04309-w.
  * Bunimovich and Su [2023] L. A. Bunimovich and Y. Su.  Maximal large deviations and slow recurrences in weakly chaotic systems.  _Adv. Math._ , 432:Paper No. 109267, 58, 2023.  ISSN 0001-8708,1090-2082.  doi: 10.1016/j.aim.2023.109267.  URL https://doi.org/10.1016/j.aim.2023.109267.
  * Bunimovich and Vela-Arevalo [2015] L. A. Bunimovich and L. V. Vela-Arevalo.  Some new surprises in chaos.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ , 25(9):097614, 06 2015.  ISSN 1054-1500.  doi: 10.1063/1.4916330.  URL https://doi.org/10.1063/1.4916330.
  * Demers et al. [2010] M. Demers, P. Wright, and L.-S. Young.  Escape rates and physically relevant measures for billiards with small holes.  _Comm. Math. Phys._ , 294(2):353–388, 2010.  ISSN 0010-3616,1432-0916.  doi: 10.1007/s00220-009-0941-y.  URL https://doi.org/10.1007/s00220-009-0941-y.
  * Demers [2013] M. F. Demers.  Escape rates and physical measures for the infinite horizon Lorentz gas with holes.  _Dyn. Syst._ , 28(3):393–422, 2013.  ISSN 1468-9367,1468-9375.  doi: 10.1080/14689367.2013.814946.  URL https://doi.org/10.1080/14689367.2013.814946.
  * Demers and Fernandez [2016] M. F. Demers and B. Fernandez.  Escape rates and singular limiting distributions for intermittent maps with holes.  _Trans. Amer. Math. Soc._ , 368(7):4907–4932, 2016.  ISSN 0002-9947,1088-6850.  doi: 10.1090/tran/6481.  URL https://doi.org/10.1090/tran/6481.
  * Freitas et al. [2015] A. C. M. Freitas, J. M. Freitas, and M. Todd.  Speed of convergence for laws of rare events and escape rates.  _Stochastic Process. Appl._ , 125(4):1653–1687, 2015.  ISSN 0304-4149,1879-209X.  doi: 10.1016/j.spa.2014.11.011.  URL https://doi.org/10.1016/j.spa.2014.11.011.
  * Haydn and Yang [2020] N. Haydn and F. Yang.  Local escape rates for $\phi$-mixing dynamical systems.  _Ergodic Theory Dynam. Systems_ , 40(10):2854–2880, 2020.  ISSN 0143-3857,1469-4417.  doi: 10.1017/etds.2019.21.  URL https://doi.org/10.1017/etds.2019.21.
  * Kesseböhmer et al. [2012] M. Kesseböhmer, S. Munday, and B. O. Stratmann.  Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems.  _Ergodic Theory Dynam. Systems_ , 32(3):989–1017, 2012.  ISSN 0143-3857,1469-4417.  doi: 10.1017/S0143385711000186.  URL https://doi.org/10.1017/S0143385711000186.
  * Liverani et al. [1999] C. Liverani, B. Saussol, and S. Vaienti.  A probabilistic approach to intermittency.  _Ergodic Theory Dynam. Systems_ , 19(3):671–685, 1999.  ISSN 0143-3857,1469-4417.  doi: 10.1017/S0143385799133856.  URL https://doi.org/10.1017/S0143385799133856.
  * Pianigiani and Yorke [1979] G. Pianigiani and J. A. Yorke.  Expanding maps on sets which are almost invariant. Decay and chaos.  _Trans. Amer. Math. Soc._ , 252:351–366, 1979.  ISSN 0002-9947,1088-6850.  doi: 10.2307/1998093.  URL https://doi.org/10.2307/1998093.
  * Young [1999] L.-S. Young.  Recurrence times and rates of mixing.  _Israel J. Math._ , 110:153–188, 1999.  ISSN 0021-2172,1565-8511.  doi: 10.1007/BF02808180.  URL https://doi.org/10.1007/BF02808180.