Parameter Estimation and Spatial Data Modeling in Statistics, Slides of Geochemistry

Various approaches to parameter estimation in statistics, focusing on multiple linear regression and hypothesis testing. It also explores the concept of spatial data modeling, which allows for second order effects and the distinction between first and second order effects. Examples of spatial models and techniques for analyzing spatial data, including point pattern techniques and spatially continuous data.

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2012/2013

Uploaded on 07/23/2013

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Parameter Estimation
This is the basic approach, but the actual estimation may
be complicated.
Parameter estimation of our multiple linear regression
involving assumptions of independence, normal
distributions and equal variance reduces to using the
method of ordinary least squares.
Relaxing the independence and equal variance, we can still
use generalised least squares.
Standard errors provide a measure of the reliability of
each parameter estimate.
Likelihood ratios can be used to compare alternative
models.
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Parameter Estimation

  • This is the basic approach, but the actual estimation may be complicated.
  • Parameter estimation of our multiple linear regression involving assumptions of independence, normal distributions and equal variance reduces to using the method of ordinary least squares.
  • Relaxing the independence and equal variance, we can still use generalised least squares.
  • Standard errors provide a measure of the reliability of each parameter estimate.
  • Likelihood ratios can be used to compare alternative models.

Hypothesis Testing

  • Hypothesis testing entails comparing the fit of two models, one of which incorporates assumptions which reflect the hypothesis, the other incorporating a less specific set of assumptions.
  • All modelling inevitably involves some assumptions about the phenomenon under study; hence hypothesis testing will always involve comparison of the fit of a hypothesised model with that of an alternative which also incorporates assumptions, albeit of a more general nature.

Spatial Data Modelling(2)

  • To allow for second order effects, spatial models may need to assume a covariance structure.
  • The second order effects may be modelled as a stationary spatial process – i.e. - Its statistical properties (mean, variance) are independent of absolute location; - Covariance depends only on relative location.
  • A process is said to be isotropic if it is stationary, and covariance depends only on distance and not direction.
  • If the mean, variance or covariance ‘drifts’ over the study area, then the process exhibits non-stationarity or heterogeneity.

Spatial Data Modelling(3)

  • Heterogeneity in the mean, combined with stationarity in second order effects, is a useful spatial modelling assumption.
  • The modelling of a spatial process often tends to proceed by first identifying any heterogeneous 'trend' in mean value and then modelling the 'residuals', or deviations from this 'trend', as a stationary process.

Point Pattern Techniques

  • Bailey and Gatrell discuss various techniques, organised by data type.
  • Point pattern techniques include:
    • Quadrat analysis
    • Kernel estimation
    • Nearest neighbour analysis
    • K-functions
  • Normally used to test null hypothesis of complete spatial randomness (i.e. homogeneous Poisson process), but can also examine heterogeneous Poisson processes.

Spatially Continous Data

  • Techniques used to explore field data.
  • Sometimes referred to as geostatistics.
    • Spatial moving averages
    • Trend surface analysis
    • Delauney triangulation / Thiesen polygons / TINs
    • Kernel estimation (for the values at sample points)
    • Variograms / covariograms / kriging
    • Principal components analysis / factor analysis
    • Procrustes analysis
    • Cluster analysis
    • Canonical correlation

Spatial Interaction Data

  • Techniques for modelling spatial interactions are most based on some variant of the gravity model.
  • This postulates that the amount of interaction between two places is a function of their sizes (measured using an appropriate metric) and is inversely related to the distance between them.

Software

  • ArcGIS. Geostatistical Analyst a step forward.
  • Idrisi. GIS Analysis | Statistics menu has a lot of options.
  • S-Plus. The S+SpatialStats addon provides a lot of options.
  • R. R is an open-source version of S-Plus. There are a number of projects currently developing tools for spatial statistics (e.g. sp, spatstat, DCluster, spgwr).
  • BUGS. Software for Bayesian statistics. There is a free version for Windows (WinBUGS). Includes a spatial sub- set called GeoBUGS.