Spatial Association Definition World Geography

Spatial Association

Spatial Association, Measures of

A. Getis , in International Encyclopedia of the Social & Behavioral Sciences, 2001

Spatial association means connectedness or relationship between and among variables over space. A single variable may be spatially autocorrelated; that is, values of the variable are somehow connected or related spatially. Many variables may be associated one with another at one or more sites. If there is spatial interaction there is also spatial association. Maps can depict spatial association. A mathematical shorthand technique can be used to represent, in general, measures of spatial association. Scientists test or theorize about variables to determine whether spatial association, either observed or expected, actually can be confirmed. Statistical procedures that have been developed for identifying and measuring the existence of spatial association are outlined.

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Spatial Big Data Analytics for Cellular Communication Systems

Junbo Wang , ... Zixue Cheng , in Big Data Analytics for Sensor-Network Collected Intelligence, 2017

3.2.3 Spatial co-location discovery

Spatial co-location discovery is a process to find the subsets of features that are frequently located together in the same geographic area [17]. Spatial co-location is different in the traditional association rule problem, because no natural notion of transactions in spatial datasets are embedded in continuous geographic space [18].

A spatial co-location pattern can be formalized as follows: Assume the existence of several datasets (e.g., d₁, d₂, …, d _k ). These datasets have a spatial co-location relation if the areas with these datasets are frequently located near a neighborhood distance R _d . For example, in the case of two datasets (d _i , d _j ), we say (d _i , d _j ) has a co-location pattern if for each area/point p _a with datum d _i , area/point p _b exists with datum d _j , and the distance between p _a and p _d is less than R _d .

In the literature, co-location pattern discovery is classified into two major categories: spatial statistics-based and data mining approaches. The spatial statistics-based approaches use the measures of spatial correlation to characterize the relation between different types of spatial events (or features). Measures of spatial correlation include the cross-K function with Monte Carlo simulation, the nearest-neighborhood and the spatial regression models [19].

In the data mining approaches, the clustering-based map overlay method treats every spatial attribute as a map layer and considers spatial clusters (regions) of point data in each layer as candidates for mining associations. Association rule mining (ARM) is another typical data mining method in spatial co-location discovery. ARM was first introduced in Agrawal et al. [20] as an efficient approach for finding frequent and meaningful relations, positive associations, and stochastic plus asymmetric patterns among sets of items in a large transactional database and a spatial database [21].

The spatial ARM for co-location detection is further divided into transaction-based ARM and distance-based ARM.

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The transaction-based ARM focuses on the creation of transactions over space. The transactions over space can be defined either by a reference-feature centric model [21] (if the spatial features are user-specified) or by a data-partition model [22] using a prevalence measure that is order-sensitive instead. However, it is difficult to generalize a reference-feature centric model when user-specified features simply are not available. The transactions are often implicit in the data partition model, for example, force fitting the notion of a transaction in a continuous spatial framework leads to the loss of the implicit spatial relations across the boundary of these transactions.

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The distance-based ARM uses the concept of a proximity neighborhood. Two representative models are the k-neighboring class sets proposed in Morimoto [22] and the event-centric model proposed in Shekhar and Huang [23]. The event-centric model is more advanced than the k-neighboring class model in addressing the limitations in the case of order-sensitive measures or no available reference features. The event-centric model can also find subsets of spatial features that occur in a neighborhood around instances of given subsets of event types.

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Spatial Pattern Analysis

Arthur Getis , in Encyclopedia of Social Measurement, 2005

Local Clustering Analysis

Local spatial autocorrelation statistics are observation-specific measures of spatial association. They focus on the location of individual points and allow for the decomposition of global or general statistics, such as Moran's I, into the contribution by each individual observation. Because the statistics can be used to detect local spatial clustering around an individual location, they are particularly well suited for finding "hot spots," or areas of elevated levels of the variable, or where a single measure of global association may contribute little meaning.

Local statistics that are used to find hot spots in an additive or multiplicative situation are G_i statistics. G_i(d) and G_i *(d) were developed by Getis and Ord in 1992 and Ord and Getis in 1995. They indicate the extent to which a location is surrounded by a cluster of high or low values. The G_i(d) statistic excludes the value at i from the summation while the G_i *(d) includes it. Positive G_i(d) or G_i *(d) indicates spatial clustering of high values, whereas negative G_i(d) or G_i *(d) indicate spatial clustering of low values.

The G _i (d) statistic is written:

(3) $G_i(d)=\frac{{\sum }_{j=1,j\ne i}^N{w}_{\text{<mi mathvariant="italic" is="true">ij</mi>}}(d)x_j-{\bar{x}}_i{\sum }_{j=1,j\ne i}^N{w}_{\text{<mi mathvariant="italic" is="true">ij</mi>}}(d)}{S(i)\sqrt{\frac{\left\{(N-1){\sum }_{j=1,j\ne i}^N{w}_{\text{<mi mathvariant="italic" is="true">ij</mi>}}^2(d)-{\left[{\sum }_{j=1,j\ne i}^N{w}_{\text{<mi mathvariant="italic" is="true">ij</mi>}}(d)\right]}^2\right\}}{(N-2)}}},i\ne j$

where

${\bar{x}}_i=\frac{{\sum }_{j=1,j\ne i}^N{x}_j}{N-1}\text{and}S(i)=\sqrt{\frac{{\sum }_{j=1,j\ne i}^N{x}_j^2}{N-1}-{({\bar{x}}_i)}^2.}$

Both G_i *(d) and G_i *(d) are asymptotically normally distri-buted as d increases. Under the null hypothesis that there is no association between i and the j within d of i, the expectation is 0 and the variance is 1; thus, values of these statistics are interpreted as is the standard normal variate.

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Spatial Analysis in Geography

M.M. Fischer , in International Encyclopedia of the Social & Behavioral Sciences, 2001

2.2 Exploratory Analysis of Area Data

Exploratory analysis of area data is concerned with identifying and describing different forms of spatial variation in the data. Special attention is given to measuring spatial association between observations for one or several variables. Spatial association can be identified in a number of ways, rigorously by using an appropriate spatial autocorrelation statistic (Cliff and Ord 1981), or more informally, for example, by using a scatter-plot and plotting each value against the mean of neighboring areas (Haining 1990).

In the rigorous approach to spatial autocorrelation the overall pattern of dependence in the data is summarized in a single indicator, such as Moran's I and Geary's c. While Moran's I is based on cross-products to measure value association, Geary's c employs squared differences. Both require the choice of a spatial weights or contiguity matrix that represents the topology or spatial arrangement of the data and represents our understanding of spatial association. Getis (1991) has shown that these indicators are special cases of a general formulation (called gamma) defined by a matrix representing possible spatial associations (the spatial weights matrix) among all areal units, multiplied by a matrix representing some specified nonspatial association among the areas. The nonspatial association may be a social, economic, or other relationship. When the elements of these matrices are similar, high positive autocorrelation arises. Spatial association specified in terms of covariances leads to Moran's I, specified in terms of differences, to Geary's c.

These global measures of spatial association can be used to assess spatial interaction in the data and can be visualized easily by means of a spatial variogram, a series of spatial autocorrelation measures for different orders of contiguity. A major drawback of global statistics of spatial autocorrelation is that they are based on the assumption of spatial stationarity, which implies inter alia a constant mean (no spatial drift) and constant variance (no outliers) across space. This was useful in the analysis of small data sets characteristic of pre-GIS times but is not very meaningful in the context of thousands or even millions of spatial units that characterize current, data-rich environments.

In view of increasingly data-rich environments, a focus on local patterns of association ('hot spots') and an allowance for local instabilities in overall spatial association recently has been suggested as a more appropriate approach. Examples of techniques that reflect this perspective are the various geographical analysis machines developed by Openshaw and co-workers (see, e.g., Openshaw et al. 1990), the Moran scatter plot (Anselin 1996), and the distance-based G _i and ${\text{G}}_i^{*}$ statistics of Getis and Ord (1993). This last has gained wide acceptance. These G-indicators can be calculated for each location i in the data set as the ratio of the sum of values in neighboring locations (defined to be within a given distance or order of contiguity) to the sum over all the values. The two statistics differ with respect to the inclusion of the value observed at i in the calculation [included in ${\text{G}}_i^{*}$ , not included in G _i ]. They can be mapped easily and used in an exploratory analysis to detect the existence of pockets of local nonstationarity, to identify distances beyond which no discernible association arises, and to find the appropriate spatial scale for further analysis (e.g., see Spatial Association, Measures of ).

No doubt, ESDA provides useful means to generate insights into global and local patterns and associations in spatial data sets. The use of ESDA techniques, however, generally is restricted to expert users interacting with the data displays and statistical diagnostics to explore spatial information and to fairly simple low-dimensional data sets. In view of these limitations, there is a need for novel exploration tools sufficiently automated and powerful to cope with the data-richness-related complexity of exploratory analysis in spatial data environments (see, e.g., Openshaw and Fischer 1994).

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An Approach to Optimize Police Patrol Activities Based on the Spatial Pattern of Crime Hotspots1

Li Li , ... Wei Sun , in Service Science, Management, and Engineering:, 2012

8.2.3.2 Getis-Ord Gi*

Getis-Ord Gi*, known as hotspot analysis, is commonly used to detect spatial clusters of the magnitude of an attribute (Anselin, 1995 ). It is a multiplicative measure of overall spatial association of values that fall within a critical distance of each other. The following equation is for general Getis-Ord Gi* test ( Anselin, 1995):

(8.2) $G\left(d\right)=\frac{{\displaystyle \sum _{i=1}^K{\displaystyle \sum _{j=1}^K{w}_{i,j}\left(d\right)x_i{x}_j}}}{{\displaystyle \sum _{i=1}^K{\displaystyle \sum _{j=1}^K{x}_i{x}_j}}}$

where x_i is the value of the ith point and W_ij (d) is the weight for points i and j for distance d.

The general Getis-Ord Gi* only measures the characteristics of spatial clusters and does not test the statistical significance of the obtained measurements. Gi* values indicate whether certain values are likely to occur in one location or are equally likely to occur at any location. Positive values indicate that neighboring features are more like each other than distant features. Negative values indicate that neighboring features are unlike each other. The z-score of Getis-Ord Gi* provides an estimate of the statistical significance of the Gi* score. Depending on the assumption about the data, different functions can be used to calculate the z-score. A typical formulation of the z-score can be found in Anselin (1995).

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Misspecification in Linear Spatial Regression Models

Raymond J.G.M. Florax , Peter Nijkamp , in Encyclopedia of Social Measurement, 2005

Definition Spatial Autocorrelation

Two publications of the work of Cliff and Ord, in 1973 and 1981, induced extensive focus on the statistical properties of spatial data, in particular for spatial autocorrelation or dependence. Statistical tests for spatial association or dependence are always based on the null hypothesis of spatial independence. The general notion of independence is easily formalized as follows:

(1) $\text{Pr}(X_1<x_1,\dots ,X_n<x_n)={\displaystyle \prod _{i=1}^n\text{Pr}(X_i<x_i)},$

with i  =   1, …, n. Consequently,

(2) $\text{Pr}(X_i=a,X_j=b)=\text{Pr}(X_i=a)\cdot \text{Pr}(X_j=b).$

For independent events, such as tossing an unbiased coin, the probability of the occurrence of a series of events (e.g., throwing 6 and subsequently 5) is determined multiplicatively by the chance of the occurrence of the individual events (i.e., throwing 6 and throwing 5). In the case of dependence, this implication does not hold, because the chance of the second event occurring is somehow conditional on the occurrence of the first event.

The notion of spatial dependence is similar, but is more specific in the sense that the dependence among events is mediated through space—for instance, either through distance or adjacency. Spatial dependence is referred to using informal terms, such as spatial clustering of similar values, an organized pattern, or systematic spatial variation. Strictly speaking, spatial dependence is a characteristic of the joint probability density function. As such, it is verifiable only under simplifying conditions, such as normality. Spatial autocorrelation is simply a moment of that joint distribution. In this article, however, the common practice is followed, and spatial autocorrelation and spatial dependence are used interchangeably.

Spatial autocorrelation is easily observed in Fig. 1. In graph B, observations are randomly distributed over space, so the chances of observing a specific value are independently distributed. Graph A exhibits spatial autocorrelation or dependence, because the probability of a specific value occurring in a specific location depends on the value of neighboring locations.

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Law and Geography

N. Blomley , in International Encyclopedia of the Social & Behavioral Sciences, 2001

3.2 Geographic Dimensions of Legal Discourse

Closely linked is the study of the geographic claims and representations contained within legal discourse, it being noted that, despite its suspicion of local contingencies, law nevertheless draws upon and helps define a range of spatial patterns, divisions, and associations. These legal representations touch all aspects of legal life, including property, constitutional law, contract, crime, and inter-governmental law. Don Mitchell (1996), for example, outlines the ways in which judges and other legal actors help define the boundaries between public and private space in US cities, and regulate the activities that are deemed permissible in such spaces. This, he notes, has had particularly oppressive consequences for those people who are effectively denied private space—the homeless. Legal action and interpretation in this sense can actively create certain spatial arrangements, practices, or representations. So, for example, David Delaney (1998) analyzes legal decisions concerning the legality of restrictive covenants in the USA, designed to exclude black residents from white neighborhoods. Necessarily, this entailed various construals of urban space on the part of the judiciary. For Delaney, the effect of such apparently objective mappings was to position the judge as an active participant in the construction of ethnic spatial segregation.

The manner in which such legal constructions of space are contested are locally reworked by marginalized groups is also receiving attention; see, for example, Azuela's (1987) discussion of the ways in which squatters in Mexico City redefined concepts of property. Such legal understandings and representations of space, in other words, are not simply the provenance of formal legal actors. Blomley (1998) explores the way in which struggles over gentrification in Vancouver, Canada, entailed competing conceptions of property rights and land that, in turn, entailed specific representations of the landscape of a local area. For groups representing the poor of the area, opposed to gentrification, the buildings, parks, hotels, and streets of a local area were imbued with legal and political meanings, serving as a physical monument to past acts of oppression and displacement, but also a means by which a collective claim to a neighborhood could be figuratively staked out. In other words, to understand the conflict it was necessary to think about both the importance of contesting claims to property and at the same time, to explore the ways in which those claims were expressed in, and dependent upon local spaces, that were both material and representational.

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Background: Introduction

Rex Hartson , Pardha Pyla , in The UX Book (Second Edition), 2019

6.2.6.3 Situated, embodied, and tangible interaction

Situated interaction. While embedded, ubiquitous, and ambient technologies take something of an outside-in ecocentric perspective, as in how the environment relates to the user, situated, embodied, and tangible interactions take a somewhat user-centric inside-out perspective, as in how the user relates to the environment.

Situated awareness refers to technology that is aware of its context. As an example, this includes awareness of the presence of human users in an activity space. In a social interaction setting, this can help find specific people, help optimize deployment of a team, or help cultivate a feeling of community and belonging (Sellen, Eardley, Izadi, & Harper, 2006).

Being situated is all about a sense of "place," the place of interaction within the broader usage context. An example of situated awareness (credit not ours) is a cell phone that "knows" it is in a movie theater or that the owner is in a nonphone conversation; that is, the device or product encompasses knowledge of the rules of human social politeness.

Embodied and tangible interaction. Complementing situated awareness, embodied interaction refers to the ability to involve one's physical body in interaction with technology in a natural way, such as by gestures. Antle (2009) defines embodiment as, "how the nature of a living entity's cognition is shaped by the form of its physical manifestation in the world." As she points out, in contrast to the human as information processor view of cognition, humans are primarily active agents, not just "disembodied symbol processors." This implies bringing interaction into the human's physical world to involve the human's own physical being.

Embodied interaction, first identified by Malcolm McCullough in Digital Ground (McCullough, 2004) and further developed by Paul Dourish in Where the Action is (Dourish, 2001), is central to the changing nature of interaction. Dourish says, "how we understand the world, ourselves, and interaction comes from our location in a physical and social world of embodied factors." Embodied interaction is action situated in the world.

To make it a bit less abstract, think of a person who has just purchased something with "some assembly required." To sit with the instruction manual and just think about it pales in comparison to physically doing the assembly—holding the pieces and moving them around, trying to fit them this way and that, seeing and feeling the spatial relations and associations among the pieces, seeing the assembly take form, and feeling how each new piece fits. This is exactly the reason that physical sketching gives such a boost to invention and ideation. The involvement of the physical body, the motor movements, the visual connections, and the potentiation of hand-eye-mind collaboration lead to an embodied cognition far more effective than just sitting and thinking.

Ideation

An active, creative, exploratory, highly iterative, fast-moving, and usually collaborative, brainstorming process for forming ideas for design (Section 14.2).

Although tangible interaction seems to have a following of its own (Ishii & Ullmer, 1997), it is very closely related to embodied interaction. You could say that they complement each other. Tangible interaction involves physical actions between human users and physical objects. Industrial designers have been dealing with it for years, designing objects and products to be held, felt, and manipulated by humans. The difference now is that the object involves some kind of computation. And there is a strong emphasis on physicality, form, and tactile interaction (Baskinger & Gross, 2010).

More than ever before, tangible and embodied interaction calls for physical prototypes as three-dimensional sketches to inspire the ideation and design process.

Sketching

The rapid creation of free-hand drawings expressing preliminary design ideas, focusing on concepts rather than details. Is an essential part of ideation. A sketch is a conversation between the sketcher or designer and the artifact (Section 14.3).

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Spatial Pattern, Analysis of

B. Boots , in International Encyclopedia of the Social & Behavioral Sciences, 2001

3 Lattice Data

In lattice data, the values, (y ₁, y ₂, …, y_n ) of some variable Y are recorded for each member of a set of n areal units (a ₁, a ₂, …, a_n ) which, taken collectively, cover the study region A, i.e., a ₁∪a ₂∪…∪a_n =A. The set of data values is considered to represent one possible realization of a spatial process operating over A . Lattice data are analyzed by examining characteristics of the association between pairs of data values as some function of their spatial association. This is referred to as 'spatial autocorrelation.' Since the data sites are areas, there are many different ways in which the spatial association w_ij between any two data sites a_i , a_j may be modeled. By far the most frequently used method is to set w_ij =1 if a_i , a_j share a common boundary, and w_ij =0, if they do not.

Moran's I, the most widely used measure of spatial autocorrelation, is given by

$\begin{array}{c}I=\left[\frac{n}{{\sum }_{i-1}^n{\sum }_{j-1}^n{w}_{ij}}\right]\\ i\ne j\\ \times [\frac{{\sum }_{i-1}^n{\sum }_{j-1}^n{w}_{ij}(y_i-\overline{y})(y_j-\overline{y})}{{\sum }_{i-1}^n{(y_i-\overline{y})}^2}]\end{array}$

where: $\overline{y}=\frac{1}{n}{\sum }_{i-1}^n{y}_i$ . The first term on the right-hand side of this equation is a normalizing factor, so that the range of I is approximately −1≤I≤1. The expected value of Moran's Iunder the assumption of independence in the spatial pattern of values of Y (no spatial autocorrelation) is

$\text{E}\left(I\right)=-\frac{1}{n-1}$

and the variance is

$Var\left(I\right)=\left(\frac{n^2{S}_1-n{S}_2+3W^2}{W^2\left(n^2-1\right)}\right)-\frac{1}{{\left(n-1\right)}^2}$

where

$\begin{array}{c}W={\displaystyle \sum _{i-1}^n}{\displaystyle \sum _{j-1}^n}{w}_{ij};S_1=\frac{{\sum }_{i-1}^n{\sum }_{j-1}^n{(w_{ij}+w_{ji})}^2}{2};\\ S_2={\displaystyle \sum _{i-1}^n}{(w_i+w\mathrm{..}j)}^2;w_i.={\displaystyle \sum _{j=1}^n}{w}_{ij};w_{\mathrm{..}j}={\displaystyle \sum _{i=1}^n}{w}_{ij.}\end{array}$

An observed value of I which is larger than its expectation under the assumption of spatial independence indicates positive spatial autocorrelation (i.e., similar values of y, are found in spatial juxtaposition). In contrast, negative spatial autocorrelation occurs when neighboring values of y_i are mutually dissimilar, indicated by an observed value of Moran's I which is smaller than its expectation. For data sets of n≥50, I is approximately normally distributed so that the standard score

$z=\frac{I-E\left(I\right)}{\surd Var\left(I\right)}$

may be used to determine if the empirical pattern displays significant spatial autocorrelation.

As an example of this approach, consider Fig. 5 which shows the percentage of the 1996 population whose mother tongue was German for each census tract in the twin cities of Kitchener-Waterloo in Ontario, Canada. For this data set, n=56, E(I)=−0.018 and Var(I)=0.0057. While the observed value of I=0.115 suggests that there is a tendency for census tracts with similar proportions of German speakers to cluster in space, the standard score of z=1.77 indicates that the pattern of spatial association is not significantly different (at the 95 percent confidence level) from that which could arise by chance (i.e., no spatial autocorrelation).

Figure 5. Percentage of census tract population with German mother tongue (Kitchener-Waterloo 1996)

The measurement of spatial autocorrelation can be extended to higher-order spatial neighbors where the spatially associated areas are (k−1) intervening areas apart. A plot of the values of Ifor different spatial lags k, the spatial correlogram, is useful for detecting scale variations in the spatial pattern (Upton and Fingleton 1985).

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GIS Methods and Techniques

Shashi Shekhar , ... Zhe Jiang , in Comprehensive Geographic Information Systems, 2018

1.19.4.5.2 Statistical foundation

Spatial scan statistics (Kulldorff, 1997) are used to detect statistically significant hotspots from spatial datasets. It uses a circle to scan the space–time for candidate hotspots and perform hypothesis testing. The null hypothesis states that the activity points are distributed randomly obeying a homogeneous (i.e., same intensity) Poisson process over the geographical space. The alternative hypothesis states that the inside of the circle has higher intensity of activities than outside. A test statistic called the log likelihood ratio is computed for each candidate hotspot (or circle), and the candidate with the highest likelihood ratio can be evaluated using a significance value (i.e., p-value). Spatiotemporal scan statistics extend spatial scan statistics by considering an additional temporal dimension and change the scanning window to a cylinder.

Besides, LISAs, including Moran's I, Geary's C, or Ord Gi and Gi* functions (Anselin, 1995), are also used to detect hotspots. Contrary to spatial autocorrelation, these functions are computed within the neighborhood of a location. For example, the local Moran's I statistic is given as:

${\mathrm{I}}_{\mathrm{i}}=\frac{{\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{X}}}{{\mathrm{S}}_{\mathrm{i}}^2}\sum _{\mathrm{j}=1,\mathrm{j}\ne \mathrm{i}}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i},\mathrm{j}}\left({\mathrm{x}}_{\mathrm{j}}-\overline{\mathrm{X}}\right)$

where x _i is an attribute for object i, $\overline{X}$ is the mean of the corresponding attribute, w _i,j is the spatial weight between objects i and j, and $S_i^2=\frac{\sum _{j=1,j\ne i}^n{\left(x_j-\overline{X}\right)}^2}{n-1}$ with n equating to the total number of objects. A positive value for local Moran's I indicates that an object has neighboring objects with similarly high or low attribute values, so it may be a part of a cluster, while a negative value indicates that an object has neighboring objects with dissimilar values, so it may be an outlier. In either instance, the p-value for the object must be small enough for the cluster or outlier to be considered statistically significant.

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Spatial Association Definition World Geography

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