• 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.

• 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.

• Spatial heterogeneity means that overall parameters estimated for the entire system may not adequately describe the process at any given location.

• [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.

• 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.

• [citation needed] Fundamental issues 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.

• [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.

• However, in spatial analysis, we are concerned with specific types of mathematical spaces, namely, geographic space.

• 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.

• Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research.

• 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.

• 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.

• 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.

• 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”.

• A better solution, proposed by psychometricians,[49] groups the data in a « cubic matrix », with three entries (for instance, locations, variables, time periods).

• 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.

• 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.

• 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.

• [citation needed] Dependency suggests that since one location can predict the value of another location, we do not need observations in both places.

• [36] In more general terms, no scale independent method of analysis is widely agreed upon for spatial statistics.

• 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.

• Local spatial autocorrelation statistics provide estimates disaggregated to the level of the spatial analysis units, allowing assessment of the dependency relationships across
  space.

• [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.

• 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.

• Studies of humans often reduce the spatial existence of humans to a single point, for instance their home address.

• [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.

• [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.

• The locations in a spatial measurement framework often represent locations on the surface of the Earth, but this is not strictly necessary.

• 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.

• [38] • Britain measured using a 200 km linear measurement • Britain measured using a 100 km linear measurement • Britain measured using a 50 km linear measurement 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.

• [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.

• [50] This method, which exhibits data evolution over time, has not been widely used in geography.

• 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.

• 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.

• 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.

• [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.

• [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.

• Spatial analysis includes a variety of techniques using different analytic approaches, especially spatial statistics.

• It is also appropriate to view spatial dependency as a source of information rather than something to be corrected.

• [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.

• This, however, assumes that the definition of the variables has not changed over time and produces very large tables, difficult to manage.

• 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.

• 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.

• Spatial interaction[edit] Spatial interaction or “gravity models” estimate the flow of people, material or information between locations in geographic space.

• 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.

• [citation needed] These problems represent a challenge in spatial analysis because of the power of maps as media of presentation.

• [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.

• [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.

• 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.

• Analysis of the distribution patterns of two phenomena is done by map overlay.

• 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.

• A Three-Way Factor Analysis produces then three groups of factors related by a small cubic « core matrix ».

• The possibility of spatial heterogeneity suggests that the estimated degree of autocorrelation may vary significantly across geographic space.

• — 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.

• 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.

• A spatial measurement framework can also capture proximity with respect to, say, interstellar space or within a biological entity such as a liver.

• Types 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.

• 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.

• [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.

• 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.

• [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][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.

• 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.

• [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).

• In geographic space, the observations correspond to locations in a spatial measurement framework that capture their proximity in the real world.

 

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