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Wikipedia

Spatial analysis

Spatial analysis is any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques using different analytic approaches, especially spatial statistics. It may be applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, or to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is geospatial analysis, the technique applied to structures at the human scale, most notably in the analysis of geographic data. It may also be applied to genomics, as in transcriptomics data.

Map by Dr. John Snow of London, showing clusters of cholera cases in the 1854 Broad Street cholera outbreak. This was one of the first uses of map-based spatial analysis.

Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research. The most fundamental of these is the problem of defining the spatial location of the entities being studied. Classification of the techniques of spatial analysis is difficult because of the large number of different fields of research involved, the different fundamental approaches which can be chosen, and the many forms the data can take.

History edit

Spatial analysis began with early attempts at cartography and surveying. Land surveying goes back to at least 1,400 B.C in Egypt: the dimensions of taxable land plots were measured with measuring ropes and plumb bobs.[1] Many fields have contributed to its rise in modern form. Biology contributed through botanical studies of global plant distributions and local plant locations, ethological studies of animal movement, landscape ecological studies of vegetation blocks, ecological studies of spatial population dynamics, and the study of biogeography. Epidemiology contributed with early work on disease mapping, notably John Snow's work of mapping an outbreak of cholera, with research on mapping the spread of disease and with location studies for health care delivery. Statistics has contributed greatly through work in spatial statistics. Economics has contributed notably through spatial econometrics. Geographic information system is currently a major contributor due to the importance of geographic software in the modern analytic toolbox. Remote sensing has contributed extensively in morphometric and clustering analysis. Computer science has contributed extensively through the study of algorithms, notably in computational geometry. Mathematics continues to provide the fundamental tools for analysis and to reveal the complexity of the spatial realm, for example, with recent work on fractals and scale invariance. Scientific modelling provides a useful framework for new approaches.[citation needed]

Fundamental issues edit

Spatial analysis confronts many fundamental issues in the definition of its objects of study, in the construction of the analytic operations to be used, in the use of computers for analysis, in the limitations and particularities of the analyses which are known, and in the presentation of analytic results. Many of these issues are active subjects of modern research.[citation needed]

Common errors often arise in spatial analysis, some due to the mathematics of space, some due to the particular ways data are presented spatially, some due to the tools which are available. Census data, because it protects individual privacy by aggregating data into local units, raises a number of statistical issues. The fractal nature of coastline makes precise measurements of its length difficult if not impossible. A computer software fitting straight lines to the curve of a coastline, can easily calculate the lengths of the lines which it defines. However these straight lines may have no inherent meaning in the real world, as was shown for the coastline of Britain.[citation needed]

These problems represent a challenge in spatial analysis because of the power of maps as media of presentation. When results are presented as maps, the presentation combines spatial data which are generally accurate with analytic results which may be inaccurate, leading to an impression that analytic results are more accurate than the data would indicate.[2]

Formal Problems edit

Boundary problem edit

A boundary problem in analysis is a phenomenon in which geographical patterns are differentiated by the shape and arrangement of boundaries that are drawn for administrative or measurement purposes. The boundary problem occurs because of the loss of neighbors in analyses that depend on the values of the neighbors. While geographic phenomena are measured and analyzed within a specific unit, identical spatial data can appear either dispersed or clustered depending on the boundary placed around the data. In analysis with point data, dispersion is evaluated as dependent of the boundary. In analysis with areal data, statistics should be interpreted based upon the boundary.

Modifiable areal unit problem edit

 
An example of the modifiable areal unit problem and the distortion of rate calculations

The modifiable areal unit problem (MAUP) is a source of statistical bias that can significantly impact the results of statistical hypothesis tests. MAUP affects results when point-based measures of spatial phenomena are aggregated into spatial partitions or areal units (such as regions or districts) as in, for example, population density or illness rates.[3][4] The resulting summary values (e.g., totals, rates, proportions, densities) are influenced by both the shape and scale of the aggregation unit.[5]

For example, census data may be aggregated into county districts, census tracts, postcode areas, police precincts, or any other arbitrary spatial partition. Thus the results of data aggregation are dependent on the mapmaker's choice of which "modifiable areal unit" to use in their analysis. A census choropleth map calculating population density using state boundaries will yield radically different results than a map that calculates density based on county boundaries. Furthermore, census district boundaries are also subject to change over time,[6] meaning the MAUP must be considered when comparing past data to current data.

Modifiable temporal unit problem edit

 
Flowchart illustrating selected units of time. The graphic also shows the three celestial objects that are related to the units of time.
The Modified Temporal Unit Problem (MTUP) is a source of statistical bias that occurs in time series and spatial analysis when using temporal data that has been aggregated into temporal units.[7][8] In such cases, choosing a temporal unit (e.g., days, months, years) can affect the analysis results and lead to inconsistencies or errors in statistical hypothesis testing.[9]

Neighborhood effect averaging problem edit

The neighborhood effect averaging problem or NEAP delves into the challenges associated with understanding the influence of aggregating neighborhood-level phenomena on individuals when mobility-dependent exposures influence the phenomena.[10][11][12] The problem confounds the neighbourhood effect, which suggests that a person's neighborhood impacts their individual characteristics, such as health.[13][14] It relates to the boundary problem, in that delineated neighborhoods used for analysis may not fully account for an individuals activity space if the borders are permeable, and individual mobility crosses the boundaries. The term was first coined by Mei-Po Kwan in the peer-reviewed journal "International Journal of Environmental Research and Public Health" in 2018.[10][11]

Travelling salesman problem edit

 
Solution of a travelling salesperson problem: the black line shows the shortest possible loop that connects every red dot.

The travelling salesman problem, also known as the travelling salesperson problem (TSP), asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.

The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.

In the theory of computational complexity, the decision version of the TSP (where given a length L, the task is to decide whether the graph has a tour whose length is at most L) belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities.

The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, many heuristics and exact algorithms are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.[15]

Uncertain geographic context problem edit

The uncertain geographic context problem or UGCoP is a source of statistical bias that can significantly impact the results of spatial analysis when dealing with aggregate data.[16][17][18] The UGCoP is very closely related to the Modifiable areal unit problem (MAUP), and like the MAUP, arises from how we divide the land into areal units.[19][20] It is caused by the difficulty, or impossibility, of understanding how phenomena under investigation (such as people within a census tract) in different enumeration units interact between enumeration units, and outside of a study area over time.[16][21] It is particularly important to consider the UGCoP within the discipline of time geography, where phenomena under investigation can move between spatial enumeration units during the study period.[17] Examples of research that needs to consider the UGCoP include food access and human mobility.[22][23]
 
Schematic and example of a space-time prism using transit network data: On the right is a schematic diagram of a space-time prism, and on the left is a map of the potential path area for two different time budgets.[24]
The uncertain geographic context problem, or UGCoP, was first coined by Dr. Mei-Po Kwan in 2012.[16][17] The problem is highly related to the ecological fallacy, edge effect, and Modifiable areal unit problem (MAUP) in that, it relates to aggregate units as they apply to individuals.[20] The crux of the problem is that the boundaries we use for aggregation are arbitrary and may not represent the actual neighborhood of the individuals within them.[19][20] While a particular enumeration unit, such as a census tract, contains a person's location, they may cross its boundaries to work, go to school, and shop in completely different areas.[25][26] Thus, the geographic phenomena under investigation extends beyond the delineated boundary .[21][27][28] Different individuals, or groups may have completely different activity spaces, making an enumeration unit that is relevant for one person meaningless to another.[22][29] For example, a map that aggregates people by school districts will be more meaningful when studying a population of students than the general population.[30] Traditional spatial analysis, by necessity, treats each discrete areal unit as a self-contained neighborhood and does not consider the daily activity of crossing the boundaries.[16][17]

Weber problem edit

In geometry, the Weber problem, named after Alfred Weber, is one of the most famous problems in location theory. It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to n destination points, where different destination points are associated with different costs per unit distance.

The Weber problem generalizes the geometric median, which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the Fermat point, the geometric median of three points. For this reason it is sometimes called the Fermat–Weber problem, although the same name has also been used for the unweighted geometric median problem. The Weber problem is in turn generalized by the attraction–repulsion problem, which allows some of the costs to be negative, so that greater distance from some points is better.

Spatial characterization edit

 
Spread of bubonic plague in medieval Europe.[citation needed] The colors indicate the spatial distribution of plague outbreaks over time.

The definition of the spatial presence of an entity constrains the possible analysis which can be applied to that entity and influences the final conclusions that can be reached. While this property is fundamentally true of all analysis, it is particularly important in spatial analysis because the tools to define and study entities favor specific characterizations of the entities being studied. Statistical techniques favor the spatial definition of objects as points because there are very few statistical techniques which operate directly on line, area, or volume elements. Computer tools favor the spatial definition of objects as homogeneous and separate elements because of the limited number of database elements and computational structures available, and the ease with which these primitive structures can be created.[citation needed]

Spatial dependence edit

Spatial dependence is the spatial relationship of variable values (for themes defined over space, such as rainfall) or locations (for themes defined as objects, such as cities). Spatial dependence is measured as the existence of statistical dependence in a collection of random variables, each of which is associated with a different geographical location. Spatial dependence is of importance in applications where it is reasonable to postulate the existence of corresponding set of random variables at locations that have not been included in a sample. Thus rainfall may be measured at a set of rain gauge locations, and such measurements can be considered as outcomes of random variables, but rainfall clearly occurs at other locations and would again be random. Because rainfall exhibits properties of autocorrelation, spatial interpolation techniques can be used to estimate rainfall amounts at locations near measured locations.[31]

As with other types of statistical dependence, the presence of spatial dependence generally leads to estimates of an average value from a sample being less accurate than had the samples been independent, although if negative dependence exists a sample average can be better than in the independent case. A different problem than that of estimating an overall average is that of spatial interpolation: here the problem is to estimate the unobserved random outcomes of variables at locations intermediate to places where measurements are made, on that there is spatial dependence between the observed and unobserved random variables.[citation needed]

Tools for exploring spatial dependence include: spatial correlation, spatial covariance functions and semivariograms. Methods for spatial interpolation include Kriging, which is a type of best linear unbiased prediction. The topic of spatial dependence is of importance to geostatistics and spatial analysis.[citation needed]

Spatial auto-correlation edit

Spatial dependency is the co-variation of properties within geographic space: characteristics at proximal locations appear to be correlated, either positively or negatively.[32] Spatial dependency leads to the spatial autocorrelation problem in statistics since, like temporal autocorrelation, this violates standard statistical techniques that assume independence among observations. For example, regression analyses that do not compensate for spatial dependency can have unstable parameter estimates and yield unreliable significance tests. Spatial regression models (see below) capture these relationships and do not suffer from these weaknesses. It is also appropriate to view spatial dependency as a source of information rather than something to be corrected.[33]

Locational effects also manifest as spatial heterogeneity, or the apparent variation in a process with respect to location in geographic space. Unless a space is uniform and boundless, every location will have some degree of uniqueness relative to the other locations. This affects the spatial dependency relations and therefore the spatial process. Spatial heterogeneity means that overall parameters estimated for the entire system may not adequately describe the process at any given location.[citation needed]

Spatial association edit

Spatial association is the degree to which things are similarly arranged in space. Analysis of the distribution patterns of two phenomena is done by map overlay. If the distributions are similar, then the spatial association is strong, and vice versa.[34] In a Geographic Information System, the analysis can be done quantitatively. For example, a set of observations (as points or extracted from raster cells) at matching locations can be intersected and examined by regression analysis.

Like spatial autocorrelation, this can be a useful tool for spatial prediction. In spatial modeling, the concept of spatial association allows the use of covariates in a regression equation to predict the geographic field and thus produce a map.

The second dimension of spatial association edit

The second dimension of spatial association (SDA) reveals the association between spatial variables through extracting geographical information at locations outside samples. SDA effectively uses the missing geographical information outside sample locations in methods of the first dimension of spatial association (FDA), which explore spatial association using observations at sample locations.[35]

Scaling edit

Spatial measurement scale is a persistent issue in spatial analysis; more detail is available at the modifiable areal unit problem (MAUP) topic entry. Landscape ecologists developed a series of scale invariant metrics for aspects of ecology that are fractal in nature.[36] In more general terms, no scale independent method of analysis is widely agreed upon for spatial statistics.[citation needed]

Sampling edit

Spatial sampling involves determining a limited number of locations in geographic space for faithfully measuring phenomena that are subject to dependency and heterogeneity. [citation needed] Dependency suggests that since one location can predict the value of another location, we do not need observations in both places. But heterogeneity suggests that this relation can change across space, and therefore we cannot trust an observed degree of dependency beyond a region that may be small. Basic spatial sampling schemes include random, clustered and systematic. These basic schemes can be applied at multiple levels in a designated spatial hierarchy (e.g., urban area, city, neighborhood). It is also possible to exploit ancillary data, for example, using property values as a guide in a spatial sampling scheme to measure educational attainment and income. Spatial models such as autocorrelation statistics, regression and interpolation (see below) can also dictate sample design.[citation needed]

Common errors in spatial analysis edit

The fundamental issues in spatial analysis lead to numerous problems in analysis including bias, distortion and outright errors in the conclusions reached. These issues are often interlinked but various attempts have been made to separate out particular issues from each other.[37]

Length edit

In discussing the coastline of Britain, Benoit Mandelbrot showed that certain spatial concepts are inherently nonsensical despite presumption of their validity. Lengths in ecology depend directly on the scale at which they are measured and experienced. So while surveyors commonly measure the length of a river, this length only has meaning in the context of the relevance of the measuring technique to the question under study.[38]

Locational fallacy edit

The locational fallacy refers to error due to the particular spatial characterization chosen for the elements of study, in particular choice of placement for the spatial presence of the element.[38]

Spatial characterizations may be simplistic or even wrong. Studies of humans often reduce the spatial existence of humans to a single point, for instance their home address. This can easily lead to poor analysis, for example, when considering disease transmission which can happen at work or at school and therefore far from the home.[38]

The spatial characterization may implicitly limit the subject of study. For example, the spatial analysis of crime data has recently become popular but these studies can only describe the particular kinds of crime which can be described spatially. This leads to many maps of assault but not to any maps of embezzlement with political consequences in the conceptualization of crime and the design of policies to address the issue.[38]

Atomic fallacy edit

This describes errors due to treating elements as separate 'atoms' outside of their spatial context.[38] The fallacy is about transferring individual conclusions to spatial units.[39]

Ecological fallacy edit

The ecological fallacy describes errors due to performing analyses on aggregate data when trying to reach conclusions on the individual units.[38][40] Errors occur in part from spatial aggregation. For example, a pixel represents the average surface temperatures within an area. Ecological fallacy would be to assume that all points within the area have the same temperature.

Solutions to the fundamental issues edit

Geographic space edit

 
Manhattan distance versus Euclidean distance: The red, blue, and yellow lines have the same length (12) in both Euclidean and taxicab geometry. In Euclidean geometry, the green line has length 6×2 ≈ 8.48, and is the unique shortest path. In taxicab geometry, the green line's length is still 12, making it no shorter than any other path shown.

A mathematical space exists whenever we have a set of observations and quantitative measures of their attributes. For example, we can represent individuals' incomes or years of education within a coordinate system where the location of each individual can be specified with respect to both dimensions. The distance between individuals within this space is a quantitative measure of their differences with respect to income and education. However, in spatial analysis, we are concerned with specific types of mathematical spaces, namely, geographic space. In geographic space, the observations correspond to locations in a spatial measurement framework that capture their proximity in the real world. The locations in a spatial measurement framework often represent locations on the surface of the Earth, but this is not strictly necessary. A spatial measurement framework can also capture proximity with respect to, say, interstellar space or within a biological entity such as a liver. The fundamental tenet is Tobler's First Law of Geography: if the interrelation between entities increases with proximity in the real world, then representation in geographic space and assessment using spatial analysis techniques are appropriate.

The Euclidean distance between locations often represents their proximity, although this is only one possibility. There are an infinite number of distances in addition to Euclidean that can support quantitative analysis. For example, "Manhattan" (or "Taxicab") distances where movement is restricted to paths parallel to the axes can be more meaningful than Euclidean distances in urban settings. In addition to distances, other geographic relationships such as connectivity (e.g., the existence or degree of shared borders) and direction can also influence the relationships among entities. It is also possible to compute minimal cost paths across a cost surface; for example, this can represent proximity among locations when travel must occur across rugged terrain.

Types edit

Spatial data comes in many varieties and it is not easy to arrive at a system of classification that is simultaneously exclusive, exhaustive, imaginative, and satisfying. -- G. Upton & B. Fingelton[41]

Spatial data analysis edit

Urban and Regional Studies deal with large tables of spatial data obtained from censuses and surveys. It is necessary to simplify the huge amount of detailed information in order to extract the main trends. Multivariable analysis (or Factor analysis, FA) allows a change of variables, transforming the many variables of the census, usually correlated between themselves, into fewer independent "Factors" or "Principal Components" which are, actually, the eigenvectors of the data correlation matrix weighted by the inverse of their eigenvalues. This change of variables has two main advantages:

  1. Since information is concentrated on the first new factors, it is possible to keep only a few of them while losing only a small amount of information; mapping them produces fewer and more significant maps
  2. The factors, actually the eigenvectors, are orthogonal by construction, i.e. not correlated. In most cases, the dominant factor (with the largest eigenvalue) is the Social Component, separating rich and poor in the city. Since factors are not-correlated, other smaller processes than social status, which would have remained hidden otherwise, appear on the second, third, ... factors.

Factor analysis depends on measuring distances between observations : the choice of a significant metric is crucial. The Euclidean metric (Principal Component Analysis), the Chi-Square distance (Correspondence Analysis) or the Generalized Mahalanobis distance (Discriminant Analysis) are among the more widely used.[42] More complicated models, using communalities or rotations have been proposed.[43]

Using multivariate methods in spatial analysis began really in the 1950s (although some examples go back to the beginning of the century) and culminated in the 1970s, with the increasing power and accessibility of computers. Already in 1948, in a seminal publication, two sociologists, Wendell Bell and Eshref Shevky,[44] had shown that most city populations in the US and in the world could be represented with three independent factors : 1- the « socio-economic status » opposing rich and poor districts and distributed in sectors running along highways from the city center, 2- the « life cycle », i.e. the age structure of households, distributed in concentric circles, and 3- « race and ethnicity », identifying patches of migrants located within the city. In 1961, in a groundbreaking study, British geographers used FA to classify British towns.[45] Brian J Berry, at the University of Chicago, and his students made a wide use of the method,[46] applying it to most important cities in the world and exhibiting common social structures.[47] The use of Factor Analysis in Geography, made so easy by modern computers, has been very wide but not always very wise.[48]

Since the vectors extracted are determined by the data matrix, it is not possible to compare factors obtained from different censuses. A solution consists in fusing together several census matrices in a unique table which, then, may be analyzed. This, however, assumes that the definition of the variables has not changed over time and produces very large tables, difficult to manage. A better solution, proposed by psychometricians,[49] groups the data in a « cubic matrix », with three entries (for instance, locations, variables, time periods). A Three-Way Factor Analysis produces then three groups of factors related by a small cubic « core matrix ».[50] This method, which exhibits data evolution over time, has not been widely used in geography.[51] In Los Angeles,[52] however, it has exhibited the role, traditionally ignored, of Downtown as an organizing center for the whole city during several decades.

Spatial autocorrelation edit

Spatial autocorrelation statistics measure and analyze the degree of dependency among observations in a geographic space. Classic spatial autocorrelation statistics include Moran's  , Geary's  , Getis's   and the standard deviational ellipse. These statistics require measuring a spatial weights matrix that reflects the intensity of the geographic relationship between observations in a neighborhood, e.g., the distances between neighbors, the lengths of shared border, or whether they fall into a specified directional class such as "west". Classic spatial autocorrelation statistics compare the spatial weights to the covariance relationship at pairs of locations. Spatial autocorrelation that is more positive than expected from random indicate the clustering of similar values across geographic space, while significant negative spatial autocorrelation indicates that neighboring values are more dissimilar than expected by chance, suggesting a spatial pattern similar to a chess board.

Spatial autocorrelation statistics such as Moran's   and Geary's   are global in the sense that they estimate the overall degree of spatial autocorrelation for a dataset. The possibility of spatial heterogeneity suggests that the estimated degree of autocorrelation may vary significantly across geographic space. Local spatial autocorrelation statistics provide estimates disaggregated to the level of the spatial analysis units, allowing assessment of the dependency relationships across space.   statistics compare neighborhoods to a global average and identify local regions of strong autocorrelation. Local versions of the   and   statistics are also available.

Spatial heterogeneity edit

 
Land cover surrounding Madison, WI. Fields are colored yellow and brown, water is colored blue, and urban surfaces are colored red.
Spatial heterogeneity is a property generally ascribed to a landscape or to a population. It refers to the uneven distribution of various concentrations of each species within an area. A landscape with spatial heterogeneity has a mix of concentrations of multiple species of plants or animals (biological), or of terrain formations (geological), or environmental characteristics (e.g. rainfall, temperature, wind) filling its area. A population showing spatial heterogeneity is one where various concentrations of individuals of this species are unevenly distributed across an area; nearly synonymous with "patchily distributed."

Spatial interaction edit

Spatial interaction or "gravity models" estimate the flow of people, material or information between locations in geographic space. Factors can include origin propulsive variables such as the number of commuters in residential areas, destination attractiveness variables such as the amount of office space in employment areas, and proximity relationships between the locations measured in terms such as driving distance or travel time. In addition, the topological, or connective, relationships between areas must be identified, particularly considering the often conflicting relationship between distance and topology; for example, two spatially close neighborhoods may not display any significant interaction if they are separated by a highway. After specifying the functional forms of these relationships, the analyst can estimate model parameters using observed flow data and standard estimation techniques such as ordinary least squares or maximum likelihood. Competing destinations versions of spatial interaction models include the proximity among the destinations (or origins) in addition to the origin-destination proximity; this captures the effects of destination (origin) clustering on flows.

Spatial interpolation edit

Spatial interpolation methods estimate the variables at unobserved locations in geographic space based on the values at observed locations. Basic methods include inverse distance weighting: this attenuates the variable with decreasing proximity from the observed location. Kriging is a more sophisticated method that interpolates across space according to a spatial lag relationship that has both systematic and random components. This can accommodate a wide range of spatial relationships for the hidden values between observed locations. Kriging provides optimal estimates given the hypothesized lag relationship, and error estimates can be mapped to determine if spatial patterns exist.

Spatial regression edit

Spatial regression methods capture spatial dependency in regression analysis, avoiding statistical problems such as unstable parameters and unreliable significance tests, as well as providing information on spatial relationships among the variables involved. Depending on the specific technique, spatial dependency can enter the regression model as relationships between the independent variables and the dependent, between the dependent variables and a spatial lag of itself, or in the error terms. Geographically weighted regression (GWR) is a local version of spatial regression that generates parameters disaggregated by the spatial units of analysis.[53] This allows assessment of the spatial heterogeneity in the estimated relationships between the independent and dependent variables. The use of Bayesian hierarchical modeling[54] in conjunction with Markov chain Monte Carlo (MCMC) methods have recently shown to be effective in modeling complex relationships using Poisson-Gamma-CAR, Poisson-lognormal-SAR, or Overdispersed logit models. Statistical packages for implementing such Bayesian models using MCMC include WinBugs, CrimeStat and many packages available via R programming language.[55]

Spatial stochastic processes, such as Gaussian processes are also increasingly being deployed in spatial regression analysis. Model-based versions of GWR, known as spatially varying coefficient models have been applied to conduct Bayesian inference.[54] Spatial stochastic process can become computationally effective and scalable Gaussian process models, such as Gaussian Predictive Processes[56] and Nearest Neighbor Gaussian Processes (NNGP).[57]

Spatial neural networks edit

Spatial neural networks (SNNs) constitute a supercategory of tailored neural networks (NNs) for representing and predicting geographic phenomena. They generally improve both the statistical accuracy and reliability of the a-spatial/classic NNs whenever they handle geo-spatial datasets, and also of the other spatial (statistical) models (e.g. spatial regression models) whenever the geo-spatial datasets' variables depict non-linear relations.[58][59][60] Examples of SNNs are the OSFA spatial neural networks, SVANNs and GWNNs.

Simulation and modeling edit

Spatial interaction models are aggregate and top-down: they specify an overall governing relationship for flow between locations. This characteristic is also shared by urban models such as those based on mathematical programming, flows among economic sectors, or bid-rent theory. An alternative modeling perspective is to represent the system at the highest possible level of disaggregation and study the bottom-up emergence of complex patterns and relationships from behavior and interactions at the individual level. [citation needed]

Complex adaptive systems theory as applied to spatial analysis suggests that simple interactions among proximal entities can lead to intricate, persistent and functional spatial entities at aggregate levels. Two fundamentally spatial simulation methods are cellular automata and agent-based modeling. Cellular automata modeling imposes a fixed spatial framework such as grid cells and specifies rules that dictate the state of a cell based on the states of its neighboring cells. As time progresses, spatial patterns emerge as cells change states based on their neighbors; this alters the conditions for future time periods. For example, cells can represent locations in an urban area and their states can be different types of land use. Patterns that can emerge from the simple interactions of local land uses include office districts and urban sprawl. Agent-based modeling uses software entities (agents) that have purposeful behavior (goals) and can react, interact and modify their environment while seeking their objectives. Unlike the cells in cellular automata, simulysts can allow agents to be mobile with respect to space. For example, one could model traffic flow and dynamics using agents representing individual vehicles that try to minimize travel time between specified origins and destinations. While pursuing minimal travel times, the agents must avoid collisions with other vehicles also seeking to minimize their travel times. Cellular automata and agent-based modeling are complementary modeling strategies. They can be integrated into a common geographic automata system where some agents are fixed while others are mobile.

Calibration plays a pivotal role in both CA and ABM simulation and modelling approaches. Initial approaches to CA proposed robust calibration approaches based on stochastic, Monte Carlo methods.[61][62] ABM approaches rely on agents' decision rules (in many cases extracted from qualitative research base methods such as questionnaires).[63] Recent Machine Learning Algorithms calibrate using training sets, for instance in order to understand the qualities of the built environment.[64]

Multiple-point geostatistics (MPS) edit

Spatial analysis of a conceptual geological model is the main purpose of any MPS algorithm. The method analyzes the spatial statistics of the geological model, called the training image, and generates realizations of the phenomena that honor those input multiple-point statistics.

A recent MPS algorithm used to accomplish this task is the pattern-based method by Honarkhah.[65] In this method, a distance-based approach is employed to analyze the patterns in the training image. This allows the reproduction of the multiple-point statistics, and the complex geometrical features of the training image. Each output of the MPS algorithm is a realization that represents a random field. Together, several realizations may be used to quantify spatial uncertainty.

One of the recent methods is presented by Tahmasebi et al.[66] uses a cross-correlation function to improve the spatial pattern reproduction. They call their MPS simulation method as the CCSIM algorithm. This method is able to quantify the spatial connectivity, variability and uncertainty. Furthermore, the method is not sensitive to any type of data and is able to simulate both categorical and continuous scenarios. CCSIM algorithm is able to be used for any stationary, non-stationary and multivariate systems and it can provide high quality visual appeal model.,[67][68]

Geospatial and hydrospatial analysis edit

Geospatial and hydrospatial analysis, or just spatial analysis,[69] is an approach to applying statistical analysis and other analytic techniques to data which has a geographical or spatial aspect. Such analysis would typically employ software capable of rendering maps processing spatial data, and applying analytical methods to terrestrial or geographic datasets, including the use of geographic information systems and geomatics.[70][71][72]

Geographical information system usage edit

Geographic information systems (GIS) — a large domain that provides a variety of capabilities designed to capture, store, manipulate, analyze, manage, and present all types of geographical data — utilizes geospatial and hydrospatial analysis in a variety of contexts, operations and applications.

Basic applications edit

Geospatial and Hydrospatial analysis, using GIS, was developed for problems in the environmental and life sciences, in particular ecology, geology and epidemiology. It has extended to almost all industries including defense, intelligence, utilities, Natural Resources (i.e. Oil and Gas, Forestry ... etc.), social sciences, medicine and Public Safety (i.e. emergency management and criminology), disaster risk reduction and management (DRRM), and climate change adaptation (CCA). Spatial statistics typically result primarily from observation rather than experimentation. Hydrospatial is particularly used for the aquatic side and the members related to the water surface, column, bottom, sub-bottom and the coastal zones.

Basic operations edit

Vector-based GIS is typically related to operations such as map overlay (combining two or more maps or map layers according to predefined rules), simple buffering (identifying regions of a map within a specified distance of one or more features, such as towns, roads or rivers) and similar basic operations. This reflects (and is reflected in) the use of the term spatial analysis within the Open Geospatial Consortium (OGC) “simple feature specifications”. For raster-based GIS, widely used in the environmental sciences and remote sensing, this typically means a range of actions applied to the grid cells of one or more maps (or images) often involving filtering and/or algebraic operations (map algebra). These techniques involve processing one or more raster layers according to simple rules resulting in a new map layer, for example replacing each cell value with some combination of its neighbours’ values, or computing the sum or difference of specific attribute values for each grid cell in two matching raster datasets. Descriptive statistics, such as cell counts, means, variances, maxima, minima, cumulative values, frequencies and a number of other measures and distance computations are also often included in this generic term spatial analysis. Spatial analysis includes a large variety of statistical techniques (descriptive, exploratory, and explanatory statistics) that apply to data that vary spatially and which can vary over time. Some more advanced statistical techniques include Getis-ord Gi* or Anselin Local Moran's I which are used to determine clustering patterns of spatially referenced data.

Advanced operations edit

Geospatial and Hydrospatial analysis goes beyond 2D and 3D mapping operations and spatial statistics. It is multi-dimensional and also temporal and includes:

  • Surface analysis — in particular analysing the properties of physical surfaces, such as gradient, aspect and visibility, and analysing surface-like data “fields”;
  • Network analysis — examining the properties of natural and man-made networks in order to understand the behaviour of flows within and around such networks; and locational analysis. GIS-based network analysis may be used to address a wide range of practical problems such as route selection and facility location (core topics in the field of operations research), and problems involving flows such as those found in Hydrospatial and hydrology and transportation research. In many instances location problems relate to networks and as such are addressed with tools designed for this purpose, but in others existing networks may have little or no relevance or may be impractical to incorporate within the modeling process. Problems that are not specifically network constrained, such as new road or pipeline routing, regional warehouse location, mobile phone mast positioning or the selection of rural community health care sites, may be effectively analysed (at least initially) without reference to existing physical networks. Locational analysis "in the plane" is also applicable where suitable network datasets are not available, or are too large or expensive to be utilised, or where the location algorithm is very complex or involves the examination or simulation of a very large number of alternative configurations.
  • Geovisualization — the creation and manipulation of images, maps, diagrams, charts, 3D views and their associated tabular datasets. GIS packages increasingly provide a range of such tools, providing static or rotating views, draping images over 2.5D surface representations, providing animations and fly-throughs, dynamic linking and brushing and spatio-temporal visualisations. This latter class of tools is the least developed, reflecting in part the limited range of suitable compatible datasets and the limited set of analytical methods available, although this picture is changing rapidly. All these facilities augment the core tools utilised in spatial analysis throughout the analytical process (exploration of data, identification of patterns and relationships, construction of models, and communication of results)

Mobile geospatial and hydrospatial Computing edit

Traditionally geospatial and hydrospatial computing has been performed primarily on personal computers (PCs) or servers. Due to the increasing capabilities of mobile devices, however, geospatial computing in mobile devices is a fast-growing trend.[73] The portable nature of these devices, as well as the presence of useful sensors, such as Global Navigation Satellite System (GNSS) receivers and barometric pressure sensors, make them useful for capturing and processing geospatial and hydrospatial information in the field. In addition to the local processing of geospatial information on mobile devices, another growing trend is cloud-based geospatial computing. In this architecture, data can be collected in the field using mobile devices and then transmitted to cloud-based servers for further processing and ultimate storage. In a similar manner, geospatial and hydrospatial information can be made available to connected mobile devices via the cloud, allowing access to vast databases of geospatial and hydrospatial information anywhere where a wireless data connection is available.

Geographic information science and spatial analysis edit

 
This flow map of Napoleon's ill-fated march on Moscow is an early and celebrated example of geovisualization. It shows the army's direction as it traveled, the places the troops passed through, the size of the army as troops died from hunger and wounds, and the freezing temperatures they experienced.

Geographic information systems (GIS) and the underlying geographic information science that advances these technologies have a strong influence on spatial analysis. The increasing ability to capture and handle geographic data means that spatial analysis is occurring within increasingly data-rich environments. Geographic data capture systems include remotely sensed imagery, environmental monitoring systems such as intelligent transportation systems, and location-aware technologies such as mobile devices that can report location in near-real time. GIS provide platforms for managing these data, computing spatial relationships such as distance, connectivity and directional relationships between spatial units, and visualizing both the raw data and spatial analytic results within a cartographic context. Subtypes include:

  • Geovisualization (GVis) combines scientific visualization with digital cartography to support the exploration and analysis of geographic data and information, including the results of spatial analysis or simulation. GVis leverages the human orientation towards visual information processing in the exploration, analysis and communication of geographic data and information. In contrast with traditional cartography, GVis is typically three- or four-dimensional (the latter including time) and user-interactive.
  • Geographic knowledge discovery (GKD) is the human-centered process of applying efficient computational tools for exploring massive spatial databases. GKD includes geographic data mining, but also encompasses related activities such as data selection, data cleaning and pre-processing, and interpretation of results. GVis can also serve a central role in the GKD process. GKD is based on the premise that massive databases contain interesting (valid, novel, useful and understandable) patterns that standard analytical techniques cannot find. GKD can serve as a hypothesis-generating process for spatial analysis, producing tentative patterns and relationships that should be confirmed using spatial analytical techniques.
  • Spatial decision support systems (SDSS) take existing spatial data and use a variety of mathematical models to make projections into the future. This allows urban and regional planners to test intervention decisions prior to implementation.[74]

See also edit

General topics
Specific applications

References edit

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  • Banerjee, Sudipto; Carlin, Bradley P.; Gelfand, Alan E. (2014), Hierarchical Modeling and Analysis for Spatial Data, Second Edition, Monographs on Statistics and Applied Probability (2nd ed.), Chapman and Hall/CRC, ISBN 9781439819173
  • Benenson, I. and P. M. Torrens. (2004). Geosimulation: Automata-Based Modeling of Urban Phenomena. Wiley.
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  • Levine, N. (2010). CrimeStat: A Spatial Statistics Program for the Analysis of Crime Incident Locations. Version 3.3. Ned Levine & Associates, Houston, TX and the National Institute of Justice, Washington, DC. Ch. 1-17 + 2 update chapters
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  • Miller, H. J. and J. Han (eds.) (2001) Geographic Data Mining and Knowledge Discovery, Taylor and Francis.
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  • Fotheringham, S; Clarke, G; Abrahart, B (1997). "Geocomputation and GIS". Transactions in GIS. 2 (3): 199–200. doi:10.1111/j.1467-9671.1997.tb00010.x. S2CID 205576122.
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  • Gahegan, M (1999). "What is Geocomputation?". Transactions in GIS. 3 (3): 203–206. doi:10.1111/1467-9671.00017. S2CID 44656909.
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  • Reis, José P.; Silva, Elisabete A.; Pinho, Paulo (2016). "Spatial metrics to study urban patterns in growing and shrinking cities". Urban Geography. 37 (2): 246–271. doi:10.1080/02723638.2015.1096118. S2CID 62886095.
  • Papadimitriou, F. (2002). "Modelling indicators and indices of landscape complexity: An approach using G.I.S". Ecological Indicators. 2 (1–2): 17–25. doi:10.1016/S1470-160X(02)00052-3.
  • Fischer M., Leung Y. (2010) "GeoComputational Modelling: Techniques and Applications" Advances in Spatial Science. Springer-Verlag, Berlin.
  • Murgante B., Borruso G., Lapucci A. (2011) "Geocomputation, Sustainability and Environmental Planning" Studies in Computational Intelligence, Vol. 348. Springer-Verlag, Berlin.
  • Tahmasebi, P.; Hezarkhani, A.; Sahimi, M. (2012). "Multiple-point geostatistical modeling based on the cross-correlation functions". Computational Geosciences. 16 (3): 779–79742. doi:10.1007/s10596-012-9287-1. S2CID 62710397.
  • Geza, Tóth; Áron, Kincses; Zoltán, Nagy (2014). European Spatial Structure. LAP LAMBERT Academic Publishing. doi:10.13140/2.1.1560.2247.

External links edit

  • ICA Commission on Geospatial Analysis and Modeling
  • An educational resource about spatial statistics and geostatistics
  • A comprehensive guide to principles, techniques & software tools
  • Social and Spatial Inequalities
  • National Center for Geographic Information and Analysis (NCGIA)
  • International Cartographic Association (ICA) – the world body for mapping and GIScience professionals

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This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Spatial analysis news newspapers books scholar JSTOR January 2023 Learn how and when to remove this template message Spatial analysis is any of the formal techniques which studies entities using their topological geometric or geographic properties Spatial analysis includes a variety of techniques using different analytic approaches especially spatial statistics It may be applied in fields as diverse as astronomy with its studies of the placement of galaxies in the cosmos or to chip fabrication engineering with its use of place and route algorithms to build complex wiring structures In a more restricted sense spatial analysis is geospatial analysis the technique applied to structures at the human scale most notably in the analysis of geographic data It may also be applied to genomics as in transcriptomics data Map by Dr John Snow of London showing clusters of cholera cases in the 1854 Broad Street cholera outbreak This was one of the first uses of map based spatial analysis Complex issues arise in spatial analysis many of which are neither clearly defined nor completely resolved but form the basis for current research The most fundamental of these is the problem of defining the spatial location of the entities being studied Classification of the techniques of spatial analysis is difficult because of the large number of different fields of research involved the different fundamental approaches which can be chosen and the many forms the data can take Contents 1 History 2 Fundamental issues 2 1 Formal Problems 2 1 1 Boundary problem 2 1 2 Modifiable areal unit problem 2 1 3 Modifiable temporal unit problem 2 1 4 Neighborhood effect averaging problem 2 1 5 Travelling salesman problem 2 1 6 Uncertain geographic context problem 2 1 7 Weber problem 2 2 Spatial characterization 2 3 Spatial dependence 2 3 1 Spatial auto correlation 2 4 Spatial association 2 4 1 The second dimension of spatial association 2 5 Scaling 2 6 Sampling 2 7 Common errors in spatial analysis 2 7 1 Length 2 7 2 Locational fallacy 2 7 3 Atomic fallacy 2 7 4 Ecological fallacy 2 8 Solutions to the fundamental issues 2 8 1 Geographic space 3 Types 3 1 Spatial data analysis 3 2 Spatial autocorrelation 3 3 Spatial heterogeneity 3 4 Spatial interaction 3 5 Spatial interpolation 3 6 Spatial regression 3 7 Spatial neural networks 3 8 Simulation and modeling 3 9 Multiple point geostatistics MPS 4 Geospatial and hydrospatial analysis 4 1 Geographical information system usage 4 1 1 Basic applications 4 1 2 Basic operations 4 1 3 Advanced operations 4 1 4 Mobile geospatial and hydrospatial Computing 4 2 Geographic information science and spatial analysis 5 See also 6 References 7 Further reading 8 External linksHistory editSpatial analysis began with early attempts at cartography and surveying Land surveying goes back to at least 1 400 B C in Egypt the dimensions of taxable land plots were measured with measuring ropes and plumb bobs 1 Many fields have contributed to its rise in modern form Biology contributed through botanical studies of global plant distributions and local plant locations ethological studies of animal movement landscape ecological studies of vegetation blocks ecological studies of spatial population dynamics and the study of biogeography Epidemiology contributed with early work on disease mapping notably John Snow s work of mapping an outbreak of cholera with research on mapping the spread of disease and with location studies for health care delivery Statistics has contributed greatly through work in spatial statistics Economics has contributed notably through spatial econometrics Geographic information system is currently a major contributor due to the importance of geographic software in the modern analytic toolbox Remote sensing has contributed extensively in morphometric and clustering analysis Computer science has contributed extensively through the study of algorithms notably in computational geometry Mathematics continues to provide the fundamental tools for analysis and to reveal the complexity of the spatial realm for example with recent work on fractals and scale invariance Scientific modelling provides a useful framework for new approaches citation needed Fundamental issues editSpatial analysis confronts many fundamental issues in the definition of its objects of study in the construction of the analytic operations to be used in the use of computers for analysis in the limitations and particularities of the analyses which are known and in the presentation of analytic results Many of these issues are active subjects of modern research citation needed Common errors often arise in spatial analysis some due to the mathematics of space some due to the particular ways data are presented spatially some due to the tools which are available Census data because it protects individual privacy by aggregating data into local units raises a number of statistical issues The fractal nature of coastline makes precise measurements of its length difficult if not impossible A computer software fitting straight lines to the curve of a coastline can easily calculate the lengths of the lines which it defines However these straight lines may have no inherent meaning in the real world as was shown for the coastline of Britain citation needed These problems represent a challenge in spatial analysis because of the power of maps as media of presentation When results are presented as maps the presentation combines spatial data which are generally accurate with analytic results which may be inaccurate leading to an impression that analytic results are more accurate than the data would indicate 2 Formal Problems edit Boundary problem edit This section is an excerpt from Boundary problem spatial analysis edit A boundary problem in analysis is a phenomenon in which geographical patterns are differentiated by the shape and arrangement of boundaries that are drawn for administrative or measurement purposes The boundary problem occurs because of the loss of neighbors in analyses that depend on the values of the neighbors While geographic phenomena are measured and analyzed within a specific unit identical spatial data can appear either dispersed or clustered depending on the boundary placed around the data In analysis with point data dispersion is evaluated as dependent of the boundary In analysis with areal data statistics should be interpreted based upon the boundary Modifiable areal unit problem edit This section is an excerpt from Modifiable areal unit problem edit nbsp An example of the modifiable areal unit problem and the distortion of rate calculationsThe modifiable areal unit problem MAUP is a source of statistical bias that can significantly impact the results of statistical hypothesis tests MAUP affects results when point based measures of spatial phenomena are aggregated into spatial partitions or areal units such as regions or districts as in for example population density or illness rates 3 4 The resulting summary values e g totals rates proportions densities are influenced by both the shape and scale of the aggregation unit 5 For example census data may be aggregated into county districts census tracts postcode areas police precincts or any other arbitrary spatial partition Thus the results of data aggregation are dependent on the mapmaker s choice of which modifiable areal unit to use in their analysis A census choropleth map calculating population density using state boundaries will yield radically different results than a map that calculates density based on county boundaries Furthermore census district boundaries are also subject to change over time 6 meaning the MAUP must be considered when comparing past data to current data Modifiable temporal unit problem edit This section is an excerpt from Modifiable temporal unit problem edit nbsp Flowchart illustrating selected units of time The graphic also shows the three celestial objects that are related to the units of time The Modified Temporal Unit Problem MTUP is a source of statistical bias that occurs in time series and spatial analysis when using temporal data that has been aggregated into temporal units 7 8 In such cases choosing a temporal unit e g days months years can affect the analysis results and lead to inconsistencies or errors in statistical hypothesis testing 9 Neighborhood effect averaging problem edit This section is an excerpt from Neighborhood effect averaging problem edit The neighborhood effect averaging problem or NEAP delves into the challenges associated with understanding the influence of aggregating neighborhood level phenomena on individuals when mobility dependent exposures influence the phenomena 10 11 12 The problem confounds the neighbourhood effect which suggests that a person s neighborhood impacts their individual characteristics such as health 13 14 It relates to the boundary problem in that delineated neighborhoods used for analysis may not fully account for an individuals activity space if the borders are permeable and individual mobility crosses the boundaries The term was first coined by Mei Po Kwan in the peer reviewed journal International Journal of Environmental Research and Public Health in 2018 10 11 Travelling salesman problem edit This section is an excerpt from Travelling salesman problem edit nbsp Solution of a travelling salesperson problem the black line shows the shortest possible loop that connects every red dot The travelling salesman problem also known as the travelling salesperson problem TSP asks the following question Given a list of cities and the distances between each pair of cities what is the shortest possible route that visits each city exactly once and returns to the origin city It is an NP hard problem in combinatorial optimization important in theoretical computer science and operations research The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP In the theory of computational complexity the decision version of the TSP where given a length L the task is to decide whether the graph has a tour whose length is at most L belongs to the class of NP complete problems Thus it is possible that the worst case running time for any algorithm for the TSP increases superpolynomially but no more than exponentially with the number of cities The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization It is used as a benchmark for many optimization methods Even though the problem is computationally difficult many heuristics and exact algorithms are known so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1 15 Uncertain geographic context problem edit This section is an excerpt from Uncertain geographic context problem edit The uncertain geographic context problem or UGCoP is a source of statistical bias that can significantly impact the results of spatial analysis when dealing with aggregate data 16 17 18 The UGCoP is very closely related to the Modifiable areal unit problem MAUP and like the MAUP arises from how we divide the land into areal units 19 20 It is caused by the difficulty or impossibility of understanding how phenomena under investigation such as people within a census tract in different enumeration units interact between enumeration units and outside of a study area over time 16 21 It is particularly important to consider the UGCoP within the discipline of time geography where phenomena under investigation can move between spatial enumeration units during the study period 17 Examples of research that needs to consider the UGCoP include food access and human mobility 22 23 nbsp Schematic and example of a space time prism using transit network data On the right is a schematic diagram of a space time prism and on the left is a map of the potential path area for two different time budgets 24 The uncertain geographic context problem or UGCoP was first coined by Dr Mei Po Kwan in 2012 16 17 The problem is highly related to the ecological fallacy edge effect and Modifiable areal unit problem MAUP in that it relates to aggregate units as they apply to individuals 20 The crux of the problem is that the boundaries we use for aggregation are arbitrary and may not represent the actual neighborhood of the individuals within them 19 20 While a particular enumeration unit such as a census tract contains a person s location they may cross its boundaries to work go to school and shop in completely different areas 25 26 Thus the geographic phenomena under investigation extends beyond the delineated boundary 21 27 28 Different individuals or groups may have completely different activity spaces making an enumeration unit that is relevant for one person meaningless to another 22 29 For example a map that aggregates people by school districts will be more meaningful when studying a population of students than the general population 30 Traditional spatial analysis by necessity treats each discrete areal unit as a self contained neighborhood and does not consider the daily activity of crossing the boundaries 16 17 Weber problem edit This section is an excerpt from Weber problem edit In geometry the Weber problem named after Alfred Weber is one of the most famous problems in location theory It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to n destination points where different destination points are associated with different costs per unit distance The Weber problem generalizes the geometric median which assumes transportation costs per unit distance are the same for all destination points and the problem of computing the Fermat point the geometric median of three points For this reason it is sometimes called the Fermat Weber problem although the same name has also been used for the unweighted geometric median problem The Weber problem is in turn generalized by the attraction repulsion problem which allows some of the costs to be negative so that greater distance from some points is better Spatial characterization edit nbsp Spread of bubonic plague in medieval Europe citation needed The colors indicate the spatial distribution of plague outbreaks over time The definition of the spatial presence of an entity constrains the possible analysis which can be applied to that entity and influences the final conclusions that can be reached While this property is fundamentally true of all analysis it is particularly important in spatial analysis because the tools to define and study entities favor specific characterizations of the entities being studied Statistical techniques favor the spatial definition of objects as points because there are very few statistical techniques which operate directly on line area or volume elements Computer tools favor the spatial definition of objects as homogeneous and separate elements because of the limited number of database elements and computational structures available and the ease with which these primitive structures can be created citation needed Spatial dependence edit Spatial dependence is the spatial relationship of variable values for themes defined over space such as rainfall or locations for themes defined as objects such as cities Spatial dependence is measured as the existence of statistical dependence in a collection of random variables each of which is associated with a different geographical location Spatial dependence is of importance in applications where it is reasonable to postulate the existence of corresponding set of random variables at locations that have not been included in a sample Thus rainfall may be measured at a set of rain gauge locations and such measurements can be considered as outcomes of random variables but rainfall clearly occurs at other locations and would again be random Because rainfall exhibits properties of autocorrelation spatial interpolation techniques can be used to estimate rainfall amounts at locations near measured locations 31 As with other types of statistical dependence the presence of spatial dependence generally leads to estimates of an average value from a sample being less accurate than had the samples been independent although if negative dependence exists a sample average can be better than in the independent case A different problem than that of estimating an overall average is that of spatial interpolation here the problem is to estimate the unobserved random outcomes of variables at locations intermediate to places where measurements are made on that there is spatial dependence between the observed and unobserved random variables citation needed Tools for exploring spatial dependence include spatial correlation spatial covariance functions and semivariograms Methods for spatial interpolation include Kriging which is a type of best linear unbiased prediction The topic of spatial dependence is of importance to geostatistics and spatial analysis citation needed Spatial auto correlation edit Spatial dependency is the co variation of properties within geographic space characteristics at proximal locations appear to be correlated either positively or negatively 32 Spatial dependency leads to the spatial autocorrelation problem in statistics since like temporal autocorrelation this violates standard statistical techniques that assume independence among observations For example regression analyses that do not compensate for spatial dependency can have unstable parameter estimates and yield unreliable significance tests Spatial regression models see below capture these relationships and do not suffer from these weaknesses It is also appropriate to view spatial dependency as a source of information rather than something to be corrected 33 Locational effects also manifest as spatial heterogeneity or the apparent variation in a process with respect to location in geographic space Unless a space is uniform and boundless every location will have some degree of uniqueness relative to the other locations This affects the spatial dependency relations and therefore the spatial process Spatial heterogeneity means that overall parameters estimated for the entire system may not adequately describe the process at any given location citation needed Spatial association edit Further information Indicators of spatial association Spatial association is the degree to which things are similarly arranged in space Analysis of the distribution patterns of two phenomena is done by map overlay If the distributions are similar then the spatial association is strong and vice versa 34 In a Geographic Information System the analysis can be done quantitatively For example a set of observations as points or extracted from raster cells at matching locations can be intersected and examined by regression analysis Like spatial autocorrelation this can be a useful tool for spatial prediction In spatial modeling the concept of spatial association allows the use of covariates in a regression equation to predict the geographic field and thus produce a map The second dimension of spatial association edit The second dimension of spatial association SDA reveals the association between spatial variables through extracting geographical information at locations outside samples SDA effectively uses the missing geographical information outside sample locations in methods of the first dimension of spatial association FDA which explore spatial association using observations at sample locations 35 Scaling edit Spatial measurement scale is a persistent issue in spatial analysis more detail is available at the modifiable areal unit problem MAUP topic entry Landscape ecologists developed a series of scale invariant metrics for aspects of ecology that are fractal in nature 36 In more general terms no scale independent method of analysis is widely agreed upon for spatial statistics citation needed Sampling edit Spatial sampling involves determining a limited number of locations in geographic space for faithfully measuring phenomena that are subject to dependency and heterogeneity citation needed Dependency suggests that since one location can predict the value of another location we do not need observations in both places But heterogeneity suggests that this relation can change across space and therefore we cannot trust an observed degree of dependency beyond a region that may be small Basic spatial sampling schemes include random clustered and systematic These basic schemes can be applied at multiple levels in a designated spatial hierarchy e g urban area city neighborhood It is also possible to exploit ancillary data for example using property values as a guide in a spatial sampling scheme to measure educational attainment and income Spatial models such as autocorrelation statistics regression and interpolation see below can also dictate sample design citation needed Common errors in spatial analysis edit The fundamental issues in spatial analysis lead to numerous problems in analysis including bias distortion and outright errors in the conclusions reached These issues are often interlinked but various attempts have been made to separate out particular issues from each other 37 Length edit In discussing the coastline of Britain Benoit Mandelbrot showed that certain spatial concepts are inherently nonsensical despite presumption of their validity Lengths in ecology depend directly on the scale at which they are measured and experienced So while surveyors commonly measure the length of a river this length only has meaning in the context of the relevance of the measuring technique to the question under study 38 nbsp Britain measured using a 200 km linear measurement nbsp Britain measured using a 100 km linear measurement nbsp Britain measured using a 50 km linear measurementLocational fallacy edit The locational fallacy refers to error due to the particular spatial characterization chosen for the elements of study in particular choice of placement for the spatial presence of the element 38 Spatial characterizations may be simplistic or even wrong Studies of humans often reduce the spatial existence of humans to a single point for instance their home address This can easily lead to poor analysis for example when considering disease transmission which can happen at work or at school and therefore far from the home 38 The spatial characterization may implicitly limit the subject of study For example the spatial analysis of crime data has recently become popular but these studies can only describe the particular kinds of crime which can be described spatially This leads to many maps of assault but not to any maps of embezzlement with political consequences in the conceptualization of crime and the design of policies to address the issue 38 Atomic fallacy edit This describes errors due to treating elements as separate atoms outside of their spatial context 38 The fallacy is about transferring individual conclusions to spatial units 39 Ecological fallacy edit The ecological fallacy describes errors due to performing analyses on aggregate data when trying to reach conclusions on the individual units 38 40 Errors occur in part from spatial aggregation For example a pixel represents the average surface temperatures within an area Ecological fallacy would be to assume that all points within the area have the same temperature Solutions to the fundamental issues edit Geographic space edit nbsp Manhattan distance versus Euclidean distance The red blue and yellow lines have the same length 12 in both Euclidean and taxicab geometry In Euclidean geometry the green line has length 6 2 8 48 and is the unique shortest path In taxicab geometry the green line s length is still 12 making it no shorter than any other path shown A mathematical space exists whenever we have a set of observations and quantitative measures of their attributes For example we can represent individuals incomes or years of education within a coordinate system where the location of each individual can be specified with respect to both dimensions The distance between individuals within this space is a quantitative measure of their differences with respect to income and education However in spatial analysis we are concerned with specific types of mathematical spaces namely geographic space In geographic space the observations correspond to locations in a spatial measurement framework that capture their proximity in the real world The locations in a spatial measurement framework often represent locations on the surface of the Earth but this is not strictly necessary A spatial measurement framework can also capture proximity with respect to say interstellar space or within a biological entity such as a liver The fundamental tenet is Tobler s First Law of Geography if the interrelation between entities increases with proximity in the real world then representation in geographic space and assessment using spatial analysis techniques are appropriate The Euclidean distance between locations often represents their proximity although this is only one possibility There are an infinite number of distances in addition to Euclidean that can support quantitative analysis For example Manhattan or Taxicab distances where movement is restricted to paths parallel to the axes can be more meaningful than Euclidean distances in urban settings In addition to distances other geographic relationships such as connectivity e g the existence or degree of shared borders and direction can also influence the relationships among entities It is also possible to compute minimal cost paths across a cost surface for example this can represent proximity among locations when travel must occur across rugged terrain Types editSpatial data comes in many varieties and it is not easy to arrive at a system of classification that is simultaneously exclusive exhaustive imaginative and satisfying G Upton amp B Fingelton 41 Spatial data analysis edit Urban and Regional Studies deal with large tables of spatial data obtained from censuses and surveys It is necessary to simplify the huge amount of detailed information in order to extract the main trends Multivariable analysis or Factor analysis FA allows a change of variables transforming the many variables of the census usually correlated between themselves into fewer independent Factors or Principal Components which are actually the eigenvectors of the data correlation matrix weighted by the inverse of their eigenvalues This change of variables has two main advantages Since information is concentrated on the first new factors it is possible to keep only a few of them while losing only a small amount of information mapping them produces fewer and more significant maps The factors actually the eigenvectors are orthogonal by construction i e not correlated In most cases the dominant factor with the largest eigenvalue is the Social Component separating rich and poor in the city Since factors are not correlated other smaller processes than social status which would have remained hidden otherwise appear on the second third factors Factor analysis depends on measuring distances between observations the choice of a significant metric is crucial The Euclidean metric Principal Component Analysis the Chi Square distance Correspondence Analysis or the Generalized Mahalanobis distance Discriminant Analysis are among the more widely used 42 More complicated models using communalities or rotations have been proposed 43 Using multivariate methods in spatial analysis began really in the 1950s although some examples go back to the beginning of the century and culminated in the 1970s with the increasing power and accessibility of computers Already in 1948 in a seminal publication two sociologists Wendell Bell and Eshref Shevky 44 had shown that most city populations in the US and in the world could be represented with three independent factors 1 the socio economic status opposing rich and poor districts and distributed in sectors running along highways from the city center 2 the life cycle i e the age structure of households distributed in concentric circles and 3 race and ethnicity identifying patches of migrants located within the city In 1961 in a groundbreaking study British geographers used FA to classify British towns 45 Brian J Berry at the University of Chicago and his students made a wide use of the method 46 applying it to most important cities in the world and exhibiting common social structures 47 The use of Factor Analysis in Geography made so easy by modern computers has been very wide but not always very wise 48 Since the vectors extracted are determined by the data matrix it is not possible to compare factors obtained from different censuses A solution consists in fusing together several census matrices in a unique table which then may be analyzed This however assumes that the definition of the variables has not changed over time and produces very large tables difficult to manage A better solution proposed by psychometricians 49 groups the data in a cubic matrix with three entries for instance locations variables time periods A Three Way Factor Analysis produces then three groups of factors related by a small cubic core matrix 50 This method which exhibits data evolution over time has not been widely used in geography 51 In Los Angeles 52 however it has exhibited the role traditionally ignored of Downtown as an organizing center for the whole city during several decades Spatial autocorrelation edit Further information Tobler s first law of geography Spatial autocorrelation statistics measure and analyze the degree of dependency among observations in a geographic space Classic spatial autocorrelation statistics include Moran s I displaystyle I nbsp Geary s C displaystyle C nbsp Getis s G displaystyle G nbsp and the standard deviational ellipse These statistics require measuring a spatial weights matrix that reflects the intensity of the geographic relationship between observations in a neighborhood e g the distances between neighbors the lengths of shared border or whether they fall into a specified directional class such as west Classic spatial autocorrelation statistics compare the spatial weights to the covariance relationship at pairs of locations Spatial autocorrelation that is more positive than expected from random indicate the clustering of similar values across geographic space while significant negative spatial autocorrelation indicates that neighboring values are more dissimilar than expected by chance suggesting a spatial pattern similar to a chess board Spatial autocorrelation statistics such as Moran s I displaystyle I nbsp and Geary s C displaystyle C nbsp are global in the sense that they estimate the overall degree of spatial autocorrelation for a dataset The possibility of spatial heterogeneity suggests that the estimated degree of autocorrelation may vary significantly across geographic space Local spatial autocorrelation statistics provide estimates disaggregated to the level of the spatial analysis units allowing assessment of the dependency relationships across space G displaystyle G nbsp statistics compare neighborhoods to a global average and identify local regions of strong autocorrelation Local versions of the I displaystyle I nbsp and C displaystyle C nbsp statistics are also available Spatial heterogeneity edit This section is an excerpt from Spatial heterogeneity edit nbsp Land cover surrounding Madison WI Fields are colored yellow and brown water is colored blue and urban surfaces are colored red Spatial heterogeneity is a property generally ascribed to a landscape or to a population It refers to the uneven distribution of various concentrations of each species within an area A landscape with spatial heterogeneity has a mix of concentrations of multiple species of plants or animals biological or of terrain formations geological or environmental characteristics e g rainfall temperature wind filling its area A population showing spatial heterogeneity is one where various concentrations of individuals of this species are unevenly distributed across an area nearly synonymous with patchily distributed Spatial interaction edit Spatial interaction or gravity models estimate the flow of people material or information between locations in geographic space Factors can include origin propulsive variables such as the number of commuters in residential areas destination attractiveness variables such as the amount of office space in employment areas and proximity relationships between the locations measured in terms such as driving distance or travel time In addition the topological or connective relationships between areas must be identified particularly considering the often conflicting relationship between distance and topology for example two spatially close neighborhoods may not display any significant interaction if they are separated by a highway After specifying the functional forms of these relationships the analyst can estimate model parameters using observed flow data and standard estimation techniques such as ordinary least squares or maximum likelihood Competing destinations versions of spatial interaction models include the proximity among the destinations or origins in addition to the origin destination proximity this captures the effects of destination origin clustering on flows Spatial interpolation edit Spatial interpolation methods estimate the variables at unobserved locations in geographic space based on the values at observed locations Basic methods include inverse distance weighting this attenuates the variable with decreasing proximity from the observed location Kriging is a more sophisticated method that interpolates across space according to a spatial lag relationship that has both systematic and random components This can accommodate a wide range of spatial relationships for the hidden values between observed locations Kriging provides optimal estimates given the hypothesized lag relationship and error estimates can be mapped to determine if spatial patterns exist Spatial regression edit See also Local regression and Regression Kriging Spatial regression methods capture spatial dependency in regression analysis avoiding statistical problems such as unstable parameters and unreliable significance tests as well as providing information on spatial relationships among the variables involved Depending on the specific technique spatial dependency can enter the regression model as relationships between the independent variables and the dependent between the dependent variables and a spatial lag of itself or in the error terms Geographically weighted regression GWR is a local version of spatial regression that generates parameters disaggregated by the spatial units of analysis 53 This allows assessment of the spatial heterogeneity in the estimated relationships between the independent and dependent variables The use of Bayesian hierarchical modeling 54 in conjunction with Markov chain Monte Carlo MCMC methods have recently shown to be effective in modeling complex relationships using Poisson Gamma CAR Poisson lognormal SAR or Overdispersed logit models Statistical packages for implementing such Bayesian models using MCMC include WinBugs CrimeStat and many packages available via R programming language 55 Spatial stochastic processes such as Gaussian processes are also increasingly being deployed in spatial regression analysis Model based versions of GWR known as spatially varying coefficient models have been applied to conduct Bayesian inference 54 Spatial stochastic process can become computationally effective and scalable Gaussian process models such as Gaussian Predictive Processes 56 and Nearest Neighbor Gaussian Processes NNGP 57 Spatial neural networks edit This section is an excerpt from Spatial neural network edit Spatial neural networks SNNs constitute a supercategory of tailored neural networks NNs for representing and predicting geographic phenomena They generally improve both the statistical accuracy and reliability of the a spatial classic NNs whenever they handle geo spatial datasets and also of the other spatial statistical models e g spatial regression models whenever the geo spatial datasets variables depict non linear relations 58 59 60 Examples of SNNs are the OSFA spatial neural networks SVANNs and GWNNs Simulation and modeling edit Spatial interaction models are aggregate and top down they specify an overall governing relationship for flow between locations This characteristic is also shared by urban models such as those based on mathematical programming flows among economic sectors or bid rent theory An alternative modeling perspective is to represent the system at the highest possible level of disaggregation and study the bottom up emergence of complex patterns and relationships from behavior and interactions at the individual level citation needed Complex adaptive systems theory as applied to spatial analysis suggests that simple interactions among proximal entities can lead to intricate persistent and functional spatial entities at aggregate levels Two fundamentally spatial simulation methods are cellular automata and agent based modeling Cellular automata modeling imposes a fixed spatial framework such as grid cells and specifies rules that dictate the state of a cell based on the states of its neighboring cells As time progresses spatial patterns emerge as cells change states based on their neighbors this alters the conditions for future time periods For example cells can represent locations in an urban area and their states can be different types of land use Patterns that can emerge from the simple interactions of local land uses include office districts and urban sprawl Agent based modeling uses software entities agents that have purposeful behavior goals and can react interact and modify their environment while seeking their objectives Unlike the cells in cellular automata simulysts can allow agents to be mobile with respect to space For example one could model traffic flow and dynamics using agents representing individual vehicles that try to minimize travel time between specified origins and destinations While pursuing minimal travel times the agents must avoid collisions with other vehicles also seeking to minimize their travel times Cellular automata and agent based modeling are complementary modeling strategies They can be integrated into a common geographic automata system where some agents are fixed while others are mobile Calibration plays a pivotal role in both CA and ABM simulation and modelling approaches Initial approaches to CA proposed robust calibration approaches based on stochastic Monte Carlo methods 61 62 ABM approaches rely on agents decision rules in many cases extracted from qualitative research base methods such as questionnaires 63 Recent Machine Learning Algorithms calibrate using training sets for instance in order to understand the qualities of the built environment 64 Multiple point geostatistics MPS edit Spatial analysis of a conceptual geological model is the main purpose of any MPS algorithm The method analyzes the spatial statistics of the geological model called the training image and generates realizations of the phenomena that honor those input multiple point statistics A recent MPS algorithm used to accomplish this task is the pattern based method by Honarkhah 65 In this method a distance based approach is employed to analyze the patterns in the training image This allows the reproduction of the multiple point statistics and the complex geometrical features of the training image Each output of the MPS algorithm is a realization that represents a random field Together several realizations may be used to quantify spatial uncertainty One of the recent methods is presented by Tahmasebi et al 66 uses a cross correlation function to improve the spatial pattern reproduction They call their MPS simulation method as the CCSIM algorithm This method is able to quantify the spatial connectivity variability and uncertainty Furthermore the method is not sensitive to any type of data and is able to simulate both categorical and continuous scenarios CCSIM algorithm is able to be used for any stationary non stationary and multivariate systems and it can provide high quality visual appeal model 67 68 Geospatial and hydrospatial analysis editThis section may need to be cleaned up It has been merged from Geospatial analysis Geospatial and hydrospatial analysis or just spatial analysis 69 is an approach to applying statistical analysis and other analytic techniques to data which has a geographical or spatial aspect Such analysis would typically employ software capable of rendering maps processing spatial data and applying analytical methods to terrestrial or geographic datasets including the use of geographic information systems and geomatics 70 71 72 Geographical information system usage edit Geographic information systems GIS a large domain that provides a variety of capabilities designed to capture store manipulate analyze manage and present all types of geographical data utilizes geospatial and hydrospatial analysis in a variety of contexts operations and applications Basic applications edit Geospatial and Hydrospatial analysis using GIS was developed for problems in the environmental and life sciences in particular ecology geology and epidemiology It has extended to almost all industries including defense intelligence utilities Natural Resources i e Oil and Gas Forestry etc social sciences medicine and Public Safety i e emergency management and criminology disaster risk reduction and management DRRM and climate change adaptation CCA Spatial statistics typically result primarily from observation rather than experimentation Hydrospatial is particularly used for the aquatic side and the members related to the water surface column bottom sub bottom and the coastal zones Basic operations edit Vector based GIS is typically related to operations such as map overlay combining two or more maps or map layers according to predefined rules simple buffering identifying regions of a map within a specified distance of one or more features such as towns roads or rivers and similar basic operations This reflects and is reflected in the use of the term spatial analysis within the Open Geospatial Consortium OGC simple feature specifications For raster based GIS widely used in the environmental sciences and remote sensing this typically means a range of actions applied to the grid cells of one or more maps or images often involving filtering and or algebraic operations map algebra These techniques involve processing one or more raster layers according to simple rules resulting in a new map layer for example replacing each cell value with some combination of its neighbours values or computing the sum or difference of specific attribute values for each grid cell in two matching raster datasets Descriptive statistics such as cell counts means variances maxima minima cumulative values frequencies and a number of other measures and distance computations are also often included in this generic term spatial analysis Spatial analysis includes a large variety of statistical techniques descriptive exploratory and explanatory statistics that apply to data that vary spatially and which can vary over time Some more advanced statistical techniques include Getis ord Gi or Anselin Local Moran s I which are used to determine clustering patterns of spatially referenced data Advanced operations edit Geospatial and Hydrospatial analysis goes beyond 2D and 3D mapping operations and spatial statistics It is multi dimensional and also temporal and includes Surface analysis in particular analysing the properties of physical surfaces such as gradient aspect and visibility and analysing surface like data fields Network analysis examining the properties of natural and man made networks in order to understand the behaviour of flows within and around such networks and locational analysis GIS based network analysis may be used to address a wide range of practical problems such as route selection and facility location core topics in the field of operations research and problems involving flows such as those found in Hydrospatial and hydrology and transportation research In many instances location problems relate to networks and as such are addressed with tools designed for this purpose but in others existing networks may have little or no relevance or may be impractical to incorporate within the modeling process Problems that are not specifically network constrained such as new road or pipeline routing regional warehouse location mobile phone mast positioning or the selection of rural community health care sites may be effectively analysed at least initially without reference to existing physical networks Locational analysis in the plane is also applicable where suitable network datasets are not available or are too large or expensive to be utilised or where the location algorithm is very complex or involves the examination or simulation of a very large number of alternative configurations Geovisualization the creation and manipulation of images maps diagrams charts 3D views and their associated tabular datasets GIS packages increasingly provide a range of such tools providing static or rotating views draping images over 2 5D surface representations providing animations and fly throughs dynamic linking and brushing and spatio temporal visualisations This latter class of tools is the least developed reflecting in part the limited range of suitable compatible datasets and the limited set of analytical methods available although this picture is changing rapidly All these facilities augment the core tools utilised in spatial analysis throughout the analytical process exploration of data identification of patterns and relationships construction of models and communication of results Mobile geospatial and hydrospatial Computing edit Traditionally geospatial and hydrospatial computing has been performed primarily on personal computers PCs or servers Due to the increasing capabilities of mobile devices however geospatial computing in mobile devices is a fast growing trend 73 The portable nature of these devices as well as the presence of useful sensors such as Global Navigation Satellite System GNSS receivers and barometric pressure sensors make them useful for capturing and processing geospatial and hydrospatial information in the field In addition to the local processing of geospatial information on mobile devices another growing trend is cloud based geospatial computing In this architecture data can be collected in the field using mobile devices and then transmitted to cloud based servers for further processing and ultimate storage In a similar manner geospatial and hydrospatial information can be made available to connected mobile devices via the cloud allowing access to vast databases of geospatial and hydrospatial information anywhere where a wireless data connection is available Geographic information science and spatial analysis edit Further information Geographic information systems Spatial analysis nbsp This flow map of Napoleon s ill fated march on Moscow is an early and celebrated example of geovisualization It shows the army s direction as it traveled the places the troops passed through the size of the army as troops died from hunger and wounds and the freezing temperatures they experienced Geographic information systems GIS and the underlying geographic information science that advances these technologies have a strong influence on spatial analysis The increasing ability to capture and handle geographic data means that spatial analysis is occurring within increasingly data rich environments Geographic data capture systems include remotely sensed imagery environmental monitoring systems such as intelligent transportation systems and location aware technologies such as mobile devices that can report location in near real time GIS provide platforms for managing these data computing spatial relationships such as distance connectivity and directional relationships between spatial units and visualizing both the raw data and spatial analytic results within a cartographic context Subtypes include Geovisualization GVis combines scientific visualization with digital cartography to support the exploration and analysis of geographic data and information including the results of spatial analysis or simulation GVis leverages the human orientation towards visual information processing in the exploration analysis and communication of geographic data and information In contrast with traditional cartography GVis is typically three or four dimensional the latter including time and user interactive Geographic knowledge discovery GKD is the human centered process of applying efficient computational tools for exploring massive spatial databases GKD includes geographic data mining but also encompasses related activities such as data selection data cleaning and pre processing and interpretation of results GVis can also serve a central role in the GKD process GKD is based on the premise that massive databases contain interesting valid novel useful and understandable patterns that standard analytical techniques cannot find GKD can serve as a hypothesis generating process for spatial analysis producing tentative patterns and relationships that should be confirmed using spatial analytical techniques Spatial decision support systems SDSS take existing spatial data and use a variety of mathematical models to make projections into the future This allows urban and regional planners to test intervention decisions prior to implementation 74 See also editGeneral topicsBuffer analysis Cartography Complete spatial randomness Concepts and Techniques in Modern Geography Cost distance analysis Four traditions of geography GeoComputation Geospatial intelligence Geospatial predictive modeling Dimensionally Extended nine Intersection Model DE 9IM Geographic information science Mathematical statistics Modifiable areal unit problem Modifiable temporal unit problem Neighborhood effect averaging problem Point process Proximity analysis Spatial descriptive statistics Spatial relation Technical geography Terrain analysis Tobler s first law of geography Tobler s second law of geography List of spatial analysis software Specific applicationsBoundary problem in spatial analysis Extrapolation domain analysis Fuzzy architectural spatial analysis Geodemographic segmentation Geographic information systems Geoinformatics Geostatistics Permeability spatial and transport planning Spatial econometrics Spatial epidemiology Suitability analysis Viewshed analysisReferences edit The History of Land Surveying Accessed Dec 17 2020 https info courthousedirect com blog history of land surveying Mark Monmonier How to Lie with Maps University of Chicago Press 1996 Openshaw Stan 1983 The Modifiable Areal Unit Problem PDF ISBN 0 86094 134 5 Chen Xiang Ye Xinyue Widener Michael J Delmelle Eric Kwan Mei Po Shannon Jerry Racine Racine F Adams Aaron Liang Lu Peng Jia 27 December 2022 A systematic review of the modifiable areal unit problem MAUP in community food environmental research Urban Informatics 1 doi 10 1007 s44212 022 00021 1 S2CID 255206315 MAUP Definition Esri Support GIS Dictionary support esri com Retrieved 2017 03 09 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Geospatial Analysis Penn State Department of Geography Retrieved 27 April 2018 Quattrochi Dale A 2016 02 01 Integrating scale in remote sensing and GIS Taylor amp Francis ISBN 9781482218268 OCLC 973767077 Robinson Ws April 2009 Ecological Correlations and the Behavior of Individuals International Journal of Epidemiology 38 2 337 341 doi 10 1093 ije dyn357 PMID 19179346 Graham J Upton amp Bernard Fingelton Spatial Data Analysis by Example Volume 1 Point Pattern and Quantitative Data John Wiley amp Sons New York 1985 Harman H H 1960 Modern Factor Analysis University of Chicago Press Rummel R J 1970 Applied Factor Analysis Evanston ILL Northwestern University Press Bell W amp E Shevky 1955 Social Area Analysis Stanford University Press Moser C A amp W Scott 1961 British Towns A Statistical Study of their Social and Economic Differences Oliver amp Boyd London Berry B J amp F Horton 1971 Geographic Perspectives on Urban Systems John Wiley N Y Berry B J amp K B Smith eds 1972 City 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Hierarchical Modeling and Analysis for Spatial Data Second Edition Monographs on Statistics and Applied Probability 2nd ed Chapman and Hall CRC ISBN 9781439819173 Bivand Roger 20 January 2021 CRAN Task View Analysis of Spatial Data Retrieved 21 January 2021 Banerjee Sudipto Gelfand Alan E Finley Andrew O Sang Huiyan 2008 Gaussian predictive process models for large spatial datasets Journal of the Royal Statistical Society Series B 70 4 825 848 doi 10 1111 j 1467 9868 2008 00663 x PMC 2741335 PMID 19750209 Datta Abhirup Banerjee Sudipto Finley Andrew O Gelfand Alan E 2016 Hierarchical Nearest Neighbor Gaussian Process Models for Large Geostatistical Datasets Journal of the American Statistical Association 111 514 800 812 arXiv 1406 7343 doi 10 1080 01621459 2015 1044091 PMC 5927603 PMID 29720777 Morer I Cardillo A Diaz Guilera A Prignano L Lozano S 2020 Comparing spatial networks a one size fits all efficiency driven approach Physical Review 101 4 042301 doi 10 1103 PhysRevE 101 042301 hdl 2445 161417 PMID 32422764 S2CID 49564277 Gupta J Molnar C Xie Y Knight J Shekhar S 2021 Spatial variability aware deep neural networks SVANN a general approach ACM Transactions on Intelligent Systems and Technology 12 6 1 21 doi 10 1145 3466688 S2CID 244786699 Hagenauer J Helbich M 2022 A geographically weighted artificial neural network International Journal of Geographical Information Science 36 2 215 235 doi 10 1080 13658816 2021 1871618 S2CID 233883395 Silva E A Clarke K C 2002 Calibration of the SLEUTH urban growth model for Lisbon and Porto Portugal Computers Environment and Urban Systems 26 6 525 552 doi 10 1016 S0198 9715 01 00014 X Silva E A 2003 Complexity emergence and cellular urban models lessons learned from applying SLEUTH to two Portuguese metropolitan areas European Planning Studies 13 1 93 115 doi 10 1080 0965431042000312424 S2CID 197257 Liu and Silva 2017 Examining the dynamics of the interaction between the development of creative industries and urban spatial structure by agent based modelling A case study of Nanjing China Urban Studies 65 5 113 125 doi 10 1177 0042098016686493 S2CID 157318972 Liu Lun Silva Elisabete A Wu Chunyang Wang Hui 2017 A machine learning based method for the large scale evaluation of the qualities of the urban environment PDF Computers Environment and Urban Systems 65 113 125 doi 10 1016 j compenvurbsys 2017 06 003 Honarkhah M Caers J 2010 Stochastic Simulation of Patterns Using Distance Based Pattern Modeling Mathematical Geosciences 42 5 487 517 Bibcode 2010MaGeo 42 487H doi 10 1007 s11004 010 9276 7 S2CID 73657847 Tahmasebi P Hezarkhani A Sahimi M 2012 Multiple point geostatistical modeling based on the cross correlation functions Computational Geosciences 16 3 779 79742 doi 10 1007 s10596 012 9287 1 S2CID 62710397 Tahmasebi P Sahimi M 2015 Reconstruction of nonstationary disordered materials and media Watershed transform and cross correlation function Physical Review E 91 3 032401 Bibcode 2015PhRvE 91c2401T doi 10 1103 PhysRevE 91 032401 PMID 25871117 Tahmasebi P Sahimi M 2015 Geostatistical Simulation and Reconstruction of Porous Media by a Cross Correlation Function and Integration of Hard and Soft Data Transport in Porous Media 107 3 871 905 doi 10 1007 s11242 015 0471 3 S2CID 123432975 Graduate Program in Spatial Analysis Ryerson University Retrieved 17 December 2015 geospatial Collins English Dictionary Complete amp Unabridged 11th Edition Retrieved 5tth August 2012 from CollinsDictionary com website http www collinsdictionary com dictionary english geospatial Dictionary com s 21st Century Lexicon Copyright c 2003 2010 Dictionary com LLC http dictionary reference com browse geospatial The geospatial web blending physical and virtual spaces Archived 2011 10 02 at the Wayback Machine Arno Scharl in receiver magazine Autumn 2008 Chen Ruizhi Guinness Robert E 2014 Geospatial Computing in Mobile Devices 1st ed Norwood MA Artech House p 228 ISBN 978 1 60807 565 2 Retrieved 1 July 2014 Gonzalez Ainhoa Donnelly Alison Jones Mike Chrysoulakis Nektarios Lopes Myriam 2012 A decision support system for sustainable urban metabolism in Europe Environmental Impact Assessment Review 38 109 119 doi 10 1016 j eiar 2012 06 007 Further reading editThis further reading section may need cleanup Please read the editing guide and help improve the section June 2014 Learn how and when to remove this template message Abler R J Adams and P Gould 1971 Spatial Organization The Geographer s View of the World Englewood Cliffs NJ Prentice Hall Anselin L 1995 Local indicators of spatial association LISA Geographical Analysis 27 93 115 Awange Joseph Palancz Bela 2016 Geospatial Algebraic Computations Theory and Applications Third Edition New York Springer ISBN 978 3319254630 Banerjee Sudipto Carlin Bradley P Gelfand Alan E 2014 Hierarchical Modeling and Analysis for Spatial Data Second Edition Monographs on Statistics and Applied Probability 2nd ed Chapman and Hall CRC ISBN 9781439819173 Benenson I and P M Torrens 2004 Geosimulation Automata Based Modeling of Urban Phenomena Wiley Fotheringham A S C Brunsdon and M Charlton 2000 Quantitative Geography Perspectives on Spatial Data Analysis Sage Fotheringham A S and M E O Kelly 1989 Spatial Interaction Models Formulations and Applications Kluwer Academic Fotheringham A S Rogerson P A 1993 GIS and spatial analytical problems International Journal of Geographical Information Systems 7 3 19 doi 10 1080 02693799308901936 Goodchild M F 1987 A spatial analytical perspective on geographical information systems International Journal of Geographical Information Systems 1 4 327 44 doi 10 1080 02693798708927820 MacEachren A M and D R F Taylor eds 1994 Visualization in Modern Cartography Pergamon Levine N 2010 CrimeStat A Spatial Statistics Program for the Analysis of Crime Incident Locations Version 3 3 Ned Levine amp Associates Houston TX and the National Institute of Justice Washington DC Ch 1 17 2 update chapters Miller H J 2004 Tobler s First Law and spatial analysis Annals of the Association of American Geographers 94 2 284 289 doi 10 1111 j 1467 8306 2004 09402005 x S2CID 19172678 Miller H J and J Han eds 2001 Geographic Data Mining and Knowledge Discovery Taylor and Francis O Sullivan D and D Unwin 2002 Geographic Information Analysis Wiley Parker D C Manson S M Janssen M A Hoffmann M J Deadman P 2003 Multi agent systems for the simulation of land use and land cover change A review Annals of the Association of American Geographers 93 2 314 337 CiteSeerX 10 1 1 109 1825 doi 10 1111 1467 8306 9302004 S2CID 130096094 White R Engelen G 1997 Cellular automata as the basis of integrated dynamic regional modelling Environment and Planning B Planning and Design 24 2 235 246 doi 10 1068 b240235 S2CID 62516646 Scheldeman X amp van Zonneveld M 2010 Training Manual on Spatial Analysis of Plant Diversity and Distribution Bioversity International Fisher MM Leung Y 2001 Geocomputational Modelling techniques and applications Springer Verlag Berlin Fotheringham S Clarke G Abrahart B 1997 Geocomputation and GIS Transactions in GIS 2 3 199 200 doi 10 1111 j 1467 9671 1997 tb00010 x S2CID 205576122 Openshaw S and Abrahart RJ 2000 GeoComputation CRC Press Diappi Lidia 2004 Evolving Cities Geocomputation in Territorial Planning Ashgate England Longley PA Brooks SM McDonnell R Macmillan B 1998 Geocomputation a primer John Wiley and Sons Chichester Ehlen J Caldwell DR Harding S 2002 GeoComputation what is it Comput Environ and Urban Syst 26 4 257 265 doi 10 1016 s0198 9715 01 00047 3 Gahegan M 1999 What is Geocomputation Transactions in GIS 3 3 203 206 doi 10 1111 1467 9671 00017 S2CID 44656909 Murgante B Borruso G Lapucci A 2009 Geocomputation and Urban Planning Studies in Computational Intelligence Vol 176 Springer Verlag Berlin Reis Jose P Silva Elisabete A Pinho Paulo 2016 Spatial metrics to study urban patterns in growing and shrinking cities Urban Geography 37 2 246 271 doi 10 1080 02723638 2015 1096118 S2CID 62886095 Papadimitriou F 2002 Modelling indicators and indices of landscape complexity An approach using G I S Ecological Indicators 2 1 2 17 25 doi 10 1016 S1470 160X 02 00052 3 Fischer M Leung Y 2010 GeoComputational Modelling Techniques and Applications Advances in Spatial Science Springer Verlag Berlin Murgante B Borruso G Lapucci A 2011 Geocomputation Sustainability and Environmental Planning Studies in Computational Intelligence Vol 348 Springer Verlag Berlin Tahmasebi P Hezarkhani A Sahimi M 2012 Multiple point geostatistical modeling based on the cross correlation functions Computational Geosciences 16 3 779 79742 doi 10 1007 s10596 012 9287 1 S2CID 62710397 Geza Toth Aron Kincses Zoltan Nagy 2014 European Spatial Structure LAP LAMBERT Academic Publishing doi 10 13140 2 1 1560 2247 External links editLibrary resources about Spatial analysis Resources in your library Resources in other libraries nbsp Wikimedia Commons has media related to Spatial data analysis ICA Commission on Geospatial Analysis and Modeling An educational resource about spatial statistics and geostatistics A comprehensive guide to principles techniques amp software tools Social and Spatial Inequalities National Center for Geographic Information and Analysis NCGIA International Cartographic Association ICA the world body for mapping and GIScience professionals Retrieved from https en wikipedia org w index php title Spatial analysis amp oldid 1199866248 Spatial auto correlation, wikipedia, wiki, book, books, library,

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