Detecting Spatial Outliers: the Role of the Local
Influence Function
Giuseppe Arbia, Vincenzo Nardelli
Abstract
In the analysis of large spatial datasets, identifying and treating spatial
outliers is essential for accurately interpreting geographical phenomena.
While spatial correlation measures, particularly Local Indicators of Spa-
tial Association (LISA), are widely used to detect spatial patterns, the
presence of abnormal observations frequently distorts the landscape and
conceals critical spatial relationships. These outliers can significantly im-
pact analysis due to the inherent spatial dependencies present in the data.
Traditional influence function (IF) methodologies, commonly used in sta-
tistical analysis to measure the impact of individual observations, are not
directly applicable in the spatial context because the influence of an ob-
servation is determined not only by its own value but also by its spatial
location, its connections with neighboring regions, and the values of those
neighboring observations. In this paper, we introduce a local version of
the influence function (LIF) that accounts for these spatial dependen-
cies. Through the analysis of both simulated and real-world datasets, we
demonstrate how the LIF provides a more nuanced and accurate detec-
tion of spatial outliers compared to traditional LISA measures and local
impact assessments, improving our understanding of spatial patterns.
Some key words: Robust statistics, Empirical influence function, Local in-
dicator of spatial association, local Moran, spatial correlation.
1
Introduction
In the world of big spatial data, it is more and more necessary to develop tools
that can help summarizing the many geographical features that may be observed
in spatial datasets. In this context, the existing global measures of spatial corre-
lation (such as, e. g., the Moran (1950) coefficient), very rarely can characterize
in an exhaustive way the many different facets of spatial distributions and, to
overcome this limitation, local measures have been introduce. Amongst them,
an important category is represented by the class of Local Indicators of Spatial
Association (LISA), introduced by Luc Anselin (Anselin, 1995). Indeed, when
analyzing real data, the LISA maps are easy-to-interpret visualization tools that
help identifying local phenomena of clustering (such as those associated with
significant HH and LL local Moran indicators) and spatial outliers (such as those
1
arXiv:2410.18261v1  [stat.ME]  23 Oct 2024

associated with HL and HL local Moran indicators). In this paper, we approach
the study of spatial outliers from a different point of view, and we develop a new
method for spatial outlier detection that is alternative (and complementary) to
the classical LISA indicators. In particular, we will approach the study of spatial
outliers starting from the classical literature on robust statistics (Hampel, 1974;
Rousseeuw and Leroy, 1897; Huber, 1981; Hampel et al., 1986; Maronna et al.,
2006). In this area an important role is played by concepts like the ”breakdown
point”, the ”sensitivity curve” and the ”influence function” (IF) that can help
quantifying how much a statistical measure is unduly affected by the presence
of abnormal observations. However, as it is shown in Nardelli and Arbia (2024),
when data are distributed in space, the traditional definition of influence func-
tion cannot be employed because, due to the intrinsic dependence characterizing
spatial data, the influence of an outlier is not independent of its location and it
is strongly affected by the spatial context where it is observed. In this paper,
we first introduce a local version of the spatial influence function (proposed in
Nardelli and Arbia (2024)), we then characterize such a function parametrically
to facilitate the interpretation and the associated visualization, and we test its
effectiveness in describing geographical features through the analysis of simu-
lated and real data. The proposed local influence function (LIF) differs from
LISA indicators in several key aspects. While LISA measures focus primarily on
identifying local spatial autocorrelation, the LIF complements this by empha-
sizing the role of individual outliers within the spatial context, accounting not
only for their value but also for their influence on neighboring areas. LISA ef-
fectively identifies clusters (high-high and low-low) or spatial outliers (high-low
and low-high) based on local similarity or dissimilarity in values, but it does not
directly measure the extent to which outliers distort the overall spatial pattern.
The LIF enhances this analysis by directly measuring the influence of extreme
observations, considering both their magnitude and spatial dependencies. To-
gether, LISA and the LIF provide a more comprehensive approach to identifying
outliers and assessing their impact on the surrounding spatial structure.
The rest of the paper develops as follows.
Section 2 will be devoted to
introducing some important definitions and to develop a local version of the
spatial influence functions proposed in Nardelli and Arbia (2024). Section 3 has
the aim to show the empirical relevance of the proposed new tool through a
series of Monte Carlo experiments, while Section 4 illustrates it examining two
real datasets. Finally, Sections 5 draws some tentative conclusions and examines
some possible future developments.
2
The spatial influence function and its local
version
In this section we wish to introduce a set of tools that are necessary to assess
the robustness of spatial correlation measures. To start with, suppose that we
have at our disposal, say, n observations of a random variable Z such that Z =
2

(z1, z2, . . . , zn) centered around their mean and, without loss of generality, with
unitary variance). The observations n are distributed on a (possibly irregoular)
lattice. The celebrated Moran spatial correlation coefficient (Moran, 1950) can
then be defined as follows:
MC =
Pn
i=1 (zi) L [zi]
Pn
i=1 (zi)2
= ZT L(Z)
ZT Z
(1)
with L (zi) = Pn
i=1 wijzj the spatial lag, wij



= 0 if i = j
> 0 if j ∈N(i)
= 0 otherwise
(wij ∈W) .
and with N(i) representing the set of locations connected with location i. Fol-
lowing a consolidated tradition, in the reminder of the paper we will further
assume that the W matrix is row-standardized so that Pn
j=1 wij = 1 for each i.
When assessing the robustness of Moran’s coefficient, in principle, we could
follow the traditional approach introduced by Hampel (1974) and look at the
associated empirical influence function. In general, the finite sample version
of Hampel’s influence function Hampel (1974) can be defined as I+ (θ, zo) =
(n + 1)

ˆθ+ −ˆθ

with ˆθ as an estimator of a generic parameter θ, based on n
observations, and ˆθ+an estimator of the same form of ˆθ based on the same n
observations, but also on one additional observation, say zo. However, when
used within a spatial context, such a definition cannot be employed because
a spatial sample, in general, is constituted by a collection of, say n, areas (or
points) which are given once and for all (e. g. the region in one country) so that
we cannot imagine an empirically relevant situation where we can introduce
extra observations (e. g. an extra region) in the dataset. In this case Nardelli
and Arbia (2024) proposed to use the analogous measure:
Ic (θ, z1) = n

bθc −bθ

(2)
where and ˆθc is an estimator of the same form of ˆθ, but where one of the units
(say unit 1 without loss of generality) is contaminated with abnormal values.
In a spatial context in general, the quantity Ic (θ, z1) depends not only on the
amount of contamination z1, but also on the location where the contamination
is observed, on its connection with the neighboring locations and on the values
observed in the neighbouring locations.
In fact, given the nature of spatial
dependence between observations, if the contaminated a generic location i is
strongly connected with other locations (so that i can be considered a dominant
unit according to the definition of Pesaran and Yang (2020)), the influence of
z1 will be stronger because, in this case, its disturbing effects propagate also to
the neighboring units corrupting more substantially the parameter estimation.
Let us now refer more specifically to the influence function of the MC co-
efficient introduced in Equation (1) and let us consider, without losing gener-
ality, the further hypothesis that the contaminated unit before contamination
assumed a value equal to zero. Under these hypotheses (Nardelli and Arbia,
3

2024) proved that the Moran coefficient after contamination can be expressed
as:
d
MCc =
1
n
hPn
i=2
Pn
j=2 wijzizj + 2z1
Pn
i=2 wi1zi −n−1z2
1
i
1
n

n + n−1
n z12
(2)
so that, as a consequence, the influence function of MC can be expressed
as:
Ic (MC, z1) =
2nz1
n−1
n z12 + n
n
X
i=2
wi1Zi −
z2
1
n−1
n z12 + n(MC + 1)
(3)
Equation (4) shows that the influence function of the Moran coefficient de-
pends on the contaminated value z1, as it is obvious, but also on the average
of the values observed in the neighbourhood of the contaminated location (the
term Pn
i=2 wi1zi ) and on the level of spatial correlation before the contamina-
tion (the term MC).
Figure 1 shows the behaviour of the Moran’s influence function as an in-
creasing function of both z1 and Pn
i=2 wi1zi at 7 different levels of the original
MC coefficient ranging from strong negative ( MC = −0.7 ) to strong positive
( MC = 0.07 ) spatial correlation. The seven maps show that the level of MC
only marginally affects the influence function. Furthermore, Figure 2 display
the behaviour of the MC influence as a function of MC and z1 setting and
Pn
i=2 wi1zi = 0 just for the sake of visualizing the relationship.
Figure 1: Moran’s influence function expressed as function of z1 and Pn
i=2 wi1zi
for 7 different levels of the original MC coefficient ranging from strong negative
(MC=-0.7; top left corner) to strong positive (MC = 0.07bottomrightcorner)
;
An important consequence of Equation (4) is that the Moran’s influence
function cannot be represented univocally for a given spatial dataset. On the
4

−0.4
0.0
0.4
−10
−5
0
5
10
z1
MC
IF
−0.8
−0.6
−0.4
−0.2
Constant sum_wz = 0
Figure 2: Figure 2: Moran’s influence function expressed as function of z1 and
MC assuming and Pn
i=2 wi1zi = 0
contrary, in principle, in each of the different location of a map we can observe
different levels of the IF related to the spatial location of the contaminated
unit. To parametrize this space-varying effect, we propose to characterize each
influence function with the integral of its absolute value in the range [−2σ, +2σ]:
LIF (MC, z1) =
Z 2σ
−2σ
|Ic (MC, z1)| dz1
=
Z 2σ
−2σ

2nz1
n−1
n z2
1 + n
n
X
i=2
wi1zi −
z2
1
n−1
n z2
1 + n(MC + 1)
 dz1
= 2
"
n
 
MC +
n
X
i=2
wi1zi −1
!
tan−1
2σ
n

+ 2σ
 n
X
i=2
wi1zi −1
!#
+ 2√n tan−1
2σ
n

(4)
From now on,we will refer to the expression derived in Equation (4) as to
the ”Local Influence Function”(LIF) of the MC a measure that can be employed
to identifying what are the most influential locations on a map, considering not
only the absolute values of the contaminated unit, but also where they are
observed and their spatial context. To the aim of illustrating the practical use
of the local Influence Function introduced above and its relative merits with
respect to other local measures, in the next following sections we will show the
results of a series of Monte Carlo experiments (Section 3) and the analysis of
different real data sets (Section 4).
5

3
Simulation results
In this section, through the analysis of simulated datasets, we explore the re-
lationship between the spatial distribution of a random variable Z and the as-
sociated Local Influence Function (LIF). In our Monte Carlo experiments, the
spatial data were generated on a 10-by-10 regular square lattice grid obeying
to a spatial autoregressive process. Specifically, the variable Z was generated
from a normal distribution with mean µ = 0, standard deviation σ = 1, and
an autocorrelation parameter ρ = 0.5. The model used for generating the data
follows a spatial lag model (SLM), which can be expressed formally as follows:
Z = (I −ρW)−1ε
where Z is the vector of observations, ρ is the spatial autocorrelation pa-
rameter, W is the (row-standardized) spatial weights matrix representing the
neighborhood structure (with inverse (I −ρW)−1), and ε is a vector of in-
dependent and identically distributed errors following a standardized normal
distribution:
ε ∼N(0, 1)
For the sake of illustrating the simulation process, Figure 3 reports the re-
sults of a single realization of the avariable Z on a 10-by-10 regular square lattice
grid. In particular, the left pane shows the chorplet map of the variable Z, where
darker shades represent low values, and lighter shades indicate high values, high-
lighting the spatial variation of the phenomenon across the grid. In contrast,
the choroplet map reported in the right pane of Figure 3 displays the values of
the Local Influence Function (LIF). The visual inspection of the graph makes
it clear the local contribution of each cell to the overall global spatial autocor-
relation (in this case, to Moran’s I coefficient). In particular, cells displaying a
high value of the LIF (represented in darker red), indicate regions where local
spatial relationships are most influential on the global autocorrelation. These
areas correspond to local clusters of similar values (either high or low), where
the spatial dependence is strongest.
Thus, the relationship between the spatial distribution of Z and the LIF is
that cells with high LIF values identify regions where the spatial pattern of Z is
most pronounced. These cells highlight local clusters of extreme values, which
contribute significantly to the overall spatial autocorrelation as measured by the
global Moran’s I.
The complete results of the simulated maps in 1,000 replications of the simu-
lation are reported in Figure 4, which shows the Local Influence Function (LIF)
of Moran’s I as it is observed in the two cells where the influence reaches, respec-
tively, its minimum and its maximum values. In the graph reported in Figure 4,
the horizontal axis represents the values of the variable z1, while the vertical axis
represents the Local Influence Function (IF) corresponding to each value of the
contaminationz1. In particular, the two curves illustrate the behavior of the LIF
as it is observed in cell 82 and in cell 96 (cells are numbered progressively from
6

z
−2
−1
0
1
2
LIF
0.03
0.05
0.07
Figure 3: (a) A single realization of the simulated map and (b) Local influence
function of the Moran’s I
the top-left corner to the bottom right corner). Cell 82 exhibits maximum local
influence on spatial autocorrelation, with the Local Influence Function (LIF)
increasing steeply as values deviate from 0. This indicates that extreme values
observed in this location significantly impact on Moran’s I. In contrast, the LIF
observed in Cell 96 demonstrates minimal local influence, with a relatively flat
curve indicating a weaker contribution of the values observed in this cell to the
global spatial autocorrelation measure.
The complementary nature of LIF and traditional LISA measures is evident
in these results. While LISA effectively detects local spatial patterns such as
clusters or outliers based on similarity or dissimilarity with neighboring cells, the
LIF adds value by directly quantifying the extent to which individual outliers
influence the overall spatial correlation. The LIF captures the specific impact of
extreme values, taking into account both their magnitude and spatial context.
Together, LISA and LIF provide a more comprehensive understanding of spatial
patterns, with LISA identifying clusters and LIF offering deeper insights into
the broader influence of outliers on the spatial structure.
4
Real Data Analysis
4.1
House Prices in Columbus
Having shown in the previous section the behaviour of the proposed new tool
of the Local Influence Function using aseries of artificially generated datasets,
in this section we aim at showing the relevance of the proposed new tool in
two real-world situations. In particular, this Section 4.1 analyses the popular
dataset of house prices collected in Columbus, Ohio, while the following Section
4.2 considers some employment data collected by Eurostat, and disaggregated
at the NUTS 2 level.
7

0.00
0.02
0.04
−2
−1
0
1
2
z1
IF
Cell
82
96
Figure 4: Local influence function of the Moran’s I for the two cells where the
influence reaches its minimum and maximum.
In the first real data analysis, the key variable of interest is represented
by the median house value (the variable (HOVAL), observed in each of the 39
neighborhoods of the city of Columbus (Ohio).
The spatial distribution of housing values across Columbus is depicted in
Figure 5, which shows significant variability of house prices between the dif-
ferent neighborhoods. Indeed, the housing values range between $17.90K and
$96.40K, with higher values concentrated predominantly in the northern and
central areas, while lower values are clustered in the southern and western ar-
eas. This pattern reflects the socio-economic disparities and localized housing
market trends within the city. The observed spatial heterogeneity in housing
wealth suggests that some neighborhoods are significantly wealthier than others,
potentially due to factors such as proximity to economic centers or historical
development patterns.
The distribution of HOVAL reveals a notable bimodal pattern, as it is depicted
in the density plot reported in Figure 6(a). The distribution is characterized by
two distinct peaks: one centered around $25K and another less prominent peak
at higher values near $70K. This bimodal distribution suggests the presence
of two distinct groups of neighborhoods in Columbus, likely reflecting different
socio-economic conditions or housing markets within the city. The lower peak
around $25K indicates a concentration of neighborhoods with relatively modest
housing values, while the second peak captures neighborhoods where housing
values are significantly higher.
The separation between these two peaks may correspond to distinct geo-
graphic or economic divisions within the city. For example, neighborhoods with
lower housing values may be clustered in the southern or western parts of the
city, as seen in Figure 5, whereas neighborhoods with higher housing values are
likely concentrated in the northern and central regions. The bimodal distribu-
8

Figure 5: Spatial Distribution of Housing Values in Columbus, Ohio
tion underscores the socio-spatial disparity in housing wealth across Columbus.
Further supporting this observation, the Q-Q plot on Figure 6(b) reveals
clear signs of departures from the theoretical normal distribution in both tails.
The presence of a larger number of extreme values —both low and high— re-
inforces the idea that the distribution of house values does not follow a simple
normal pattern, but rather a bimodal one. This separation between two groups
of neighborhoods, as indicated by the density plot, highlights the polarized na-
ture of the housing market in Columbus.
Figure 7 presents the calculation of the Local Influence Function (LIF) for
Moran’s I related to the Columbus dataset and illustrates the local contribution
of each neighborhood to the overall spatial autocorrelation of housing values. In-
deed, the LIF map identifies key neighborhoods that exert a significant influence
on the global spatial pattern of house prices. Neighborhoods with higher LIF
values, highlighted in darker shades of red in Figure 7, are those that contribute
most to the clustering of housing values. Notably, a northeastern neighborhood
stands out with the highest LIF, suggesting that this area plays a dominant
role in driving the global spatial autocorrelation of house prices. This could
be attributed to its contrast in housing values compared to surrounding areas,
enhancing its local influence.
Conversely, neighborhoods with lower LIF values exhibit a more homoge-
9

Figure 6: (a) Density plot and (b) QQ Plot of Housing Values (HOVAL) in Colum-
bus, Ohio
neous distribution of house values, contributing less to the overall spatial auto-
correlation. These areas may reflect more gradual transitions in housing prices
across space, with less pronounced clusters of high or low values. The LIF anal-
ysis thus highlights the spatial heterogeneity of housing values and identifies
neighborhoods that are pivotal in shaping the overall spatial structure. This
information is crucial for urban planners and policymakers aiming at address-
ing housing inequalities and at implementing targeted policy measures in areas
with distinct spatial dynamics.
To complement the above results, Figure 8 presents the results of the Local
Moran’s I (LISA) analysis. The map in Figure 8 (a) highlights, on the left side
of the graph, the areas identified as spatial clusters based on significant Local
Moran’s I values. Specifically, it shows high-high (HH) clusters in dark red.
Additionally, the scatterplot in Figure 8 (b) illustrates the relationship between
the lagged values of the variable HOVAL and the corresponding p-values.
Due to the non-normality of the data and the presence of spatial outliers, the
Local Moran’s coefficients fail to provide meaningful information about spatial
dependence across most areas. In fact, the majority of the statistical tests are
non-significant at the 0.05 significance level, with only a few neighborhoods
exhibiting significant results corresponding to large values of the spatial lag.
Notably, these significant p-values are concentrated in a few isolated points,
suggesting that the p-values are predominantly driven by extreme values in the
spatial lag rather than revealing broader spatial patterns across the entire city.
While LISA effectively identifies clusters and outliers based on local auto-
correlation, it may not fully capture the extent to which outliers distort overall
spatial relationships. The LIF, in addition, accounts for both the magnitude
10

HOVAL
20
40
60
80
LIF
1
2
3
4
Figure 7: Local Influence Function (LIF) and Spatial Distribution of Housing
Values
of outliers and their spatial dependencies, providing a clearer understanding of
how specific outliers impact global spatial patterns.
quadrant
HH
NA
Local Moran
0.00
0.25
0.50
0.75
1.00
30
40
50
60
70
Lagged Values
P−value
Local Moran
HH
NA
Figure 8: (a) Local Moran’s I (LISA) Analysis and (b) Significance of the spatial
clusters
11

When comparing the evidences of Figure 8 to the Local Influence Function
(LIF) analysis reported in the previous Figure 7, one can observe that the
Local Moran’s I analysis identifies only a limited set of areas with significant
spatial autocorrelation.
In contrast, the LIF provides a more nuanced view
of spatial dependencies. Although an area significantly impacts on the others,
apparent in both analyses, the LIF can identify additional areas of interest that
Local Moran’s I does not capture. Thus, the LIF offers a more comprehensive
understanding of the spatial dynamics, especially in datasets characterized by
non-normal distributions and in the presence of many spatial outliers.
4.2
Cultural Employment in Europe
The second case study focuses on a set of cultural employment data derived from
the European Union Labour Force Survey (EU-LFS) led by EUROSTAT at the
NUTS 2 level. The NUTS 2 regions represent territorial units used for regional
statistics across the European Union and correspond to a regional subnational
level. It should be clarified that, according to the Eurostat nomnclature, cultural
employment includes all individuals engaged in economic activities classified as
cultural under the Statistical Classification of Economic Activities in the Euro-
pean Community (NACE Rev. 2) and the International Standard Classification
of Occupations (ISCO). This definition includes all people working in cultural
activities irrespective of their occupation.
The spatial distribution of cultural employment varies significantly across
NUTS 2 regions in Europe, as it is shown in Figure 9. Urban and economi-
cally developed regions, particularly in France, Germany and Italy, demonstrate
higher concentrations of cultural employment, while northern European regions
tend to have lower levels of employment in the cultural sector. This uneven dis-
tribution highlights disparities in the availability of jobs in the cultural sector
often associated with regional economic and infrastructure development.
The distribution of cultural employment is highly right-skewed, as it is
clearly illustrated in Figure 10, with the majority of NUTS 2 regions having
relatively low levels of employment, while a small number of regions report ex-
ceptionally high values. Such a skewed distribution is visually represented in
the density plot reported in Figure 10 (a), where a pronounced peak is observed
for regions with low employment, tapering into a long tail for regions with
higher employment levels. Furthermore, the Q-Q plot reported in Figure 10
(b), confirms the such a feature, showing that extreme high values deviate from
the expected normal distribution, thus indicating that cultural employment is
disproportionately concentrated in a few key regions.
The Local Moran’s I analysis reported in Figure 11 (a) reveals only a few
significant spatial clusters at the significance level of 0.05, with a dominance
of the high-high (HH) patterns. These clusters, where regions with high cul-
tural employment are neighbored by similar regions, are primarily influenced
by extreme employment values, as shown by the associated p-values reported
in Figure 1 (b). The Local Moran’s I generally fails to detect significant spa-
tial autocorrelation in most regions, underscoring the spatial heterogeneity of
12

Figure 9: Choroplet map of cultural employment across NUTS 2 regions in
Europe.
cultural employment across the NUTS 2 regions.
The Local Influence Function (LIF) analysis in Figure 12 provides a com-
plementary and more detailed perspective on the spatial structure of cultural
employment. While Local Moran’s I identifies only a few significant clusters, the
LIF uncovers additional regions that exert moderate influence on overall spa-
tial autocorrelation. This suggests that the LIF is more sensitive to detecting
subtler patterns of spatial dependence that Local Moran’s I may not capture,
allowing for the identification of regions that, while not forming distinct clus-
ters, still play a critical role in the spatial dynamics. By combining LISA with
the LIF, analysts gain a more comprehensive understanding of spatial patterns,
as LISA highlights clusters and the LIF adds depth by identifying the broader
influence of spatial outliers.
5
Conclusions
In this paper, we introduced the Local Influence Function (LIF) as a comple-
mentary approach to the commonly used Local Indicators of Spatial Association
(LISA) for detecting spatial outliers in large spatial datasets. Our primary focus
was on addressing the limitations posed by abnormal observations, which often
13

Figure 10: (a) Density plot and (b) Q-Q plot of cultural employment values
across NUTS2 regions.
quadrant
HH
LH
NA
Local Moran
0.00
0.25
0.50
0.75
1.00
0
25
50
75
100
Lagged Values
P−value
Local Moran
HH
LH
NA
Figure 11: (a) Local Moran’s I (LISA) Analysis and (b) Significance of the
spatial clusters for cultural employment across NUTS2 EU regions.
distort the interpretation of spatial patterns and mask critical spatial relation-
ships. The traditional definition of the Influence Function (IF) popular in the
robust statistics literature, was adapted to the spatial context, accounting for
the influence of an abnormal observation’s value together with its location and
its connections to neighboring regions.
Through both simulated and real-world datasets, we demonstrated the util-
14

0.00
0.01
0.02
0.03
0.04
0.05
−50
0
50
z0
I
LIF
1
2
3
LIF
1
2
3
Figure 12: a) Local Moran’s I (LISA) Analysis and (b) Significance of the spatial
clusters for the cultural employment across the NUTS2 EU regions.
ity of the LIF in providing a more nuanced understanding of spatial dependen-
cies. Compared to traditional LISA measures, the LIF proved more effective
in identifying spatial outliers and capturing subtler patterns of spatial depen-
dence, especially in cases where non-normal distributions and spatial outliers
are present.
Our findings highlight that the LIF is a valuable tool for spatial analysis,
offering improved sensitivity in detecting influential regions and spatial outliers.
By accounting for both the magnitude of outliers and their spatial dependencies,
the LIF provides a more nuanced understanding of how abnormal observations
affect spatial autocorrelation and regional dynamics. This has significant im-
plications for fields such as urban planning, regional policy development, and
socio-economic analysis, where spatial outliers can critically influence decision-
making processes.
Moreover, the LIF complements traditional methods like LISA by uncover-
ing subtler spatial patterns that may be overlooked when only relying on local
spatial autocorrelation measures. Together, LISA and LIF provide a compre-
hensive approach to spatial analysis, with LISA identifying clusters and LIF
measuring the broader impact of outliers.
Future research could explore the extension of the concept of the LIF to
investigate the impact of abnormal observations on spatial econometric model
estimation and testing. This would further expand the utility of LIF in fields
15

that rely on spatial modeling, allowing for a deeper understanding of how out-
liers influence both spatial relationships and econometric results.
References
Anselin, L. (1995). Local indicators of spatial association—lisa. Geographical
analysis 27(2), 93–115.
Hampel, F. R. (1974). The influence curve and its role in robust estimation.
Journal of the american statistical association 69(346), 383–393.
Hampel, F. R., E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel (1986).
Robust Statistics: The Approach Based on Influence Functions. Wiley Series
in Probability and Statistics. New York, USA: John Wiley & Sons, Inc.
Huber, P. J. (1981). Robust Statistics. Wiley Series in Probability and Statistics.
New York, USA: John Wiley & Sons, Inc.
Maronna, R. A., R. D. Martin, and V. J. Yohai (2006). Robust Statistics: Theory
and Methods. Chichester, UK: John Wiley & Sons, Ltd.
Moran,
P.
A.
(1950).
Notes
on
continuous
stochastic
phenomena.
Biometrika 37(1/2), 17–23.
Nardelli, V. and G. Arbia (2024). On robust measures of spatial correlation.
arXiv preprint arXiv:2402.07297.
Pesaran, M. H. and C. F. Yang (2020). Econometric analysis of production
networks with dominant units. Journal of Econometrics 219(2), 507–541.
Rousseeuw, P. J. and A. M. Leroy (1897). Robust regression and outlier detec-
tion. John wiley & sons.
16
