Applied Spatial Statistics
Applied Spatial Statistics STAT 5410
Utah State University
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Date Created: 10/28/15
STAT 54106410 Fall 2008 Lecture Handout 2 Statistical Estimation frequentist Recall that in statistical estimation we seek mathematical functions of the data estimators random that tell us something about population parameters for frequentists fixed and unknown of interest Desirable properties of estimators generally include o Unbiasedness On average the estimator will be equal to the thing it s estimating a Minimum Variance Among unbiased estimators we generally would like to have the highest precision possible ie lowest variability This is not always possible but generally we can nd these in simple situations In some cases we are willing to sacri ce unbiasedness for less variability Mean squared error is a measure including both variance and bias Common Types of Estimation p t Method of Moments When analytical or numerical moments of a probability distribution can be found and written as functions of the parameters we can set them equal to the corresponding sample moments ie functions of the data and solve for the parameters 0 Example Binomial Model Least Squares Define an objective function that is the squared difference between the data and some appropriate function of the parameters and then find the values for the parameters that minimizes it The parameters values that minimizes the objective function is then the least squares estimator I Ordinary Least Squares for Regression I Generalized Least Squares for Regression 3 Maximum Likelihood Optimization with respect to a probability distribution ie a probability model for the data Suppose we are sampling from a population whose pdf is a function of one parameter Given independent observations y1 y2 yquot the joint pdf is then When this pdf is viewed as a function of the parameter with the observations given it is called a likelihood function With maximum likelihood estimation we seek to nd the estimate that makes the observed data most likely to occur with the given likelihood function Thus we need to maximize the likelihood function with respect to the parameter to get the MLE Note we can also maximize the log likelihood and get the same MLE MLE Example Simple Linear Regression 4 Bayesian Estimation The only difference between the frequentist and Bayesian perspectives is that they view parameters as fixed and random respectively Since Bayesian statistics involves models with random parameters we need a way to estimate those parameters Note in general the estimation of random unknown quantities is called prediction in statistics so really Bayesians predict parameters rather than estimate them Here s how it works One bene t of using a Bayesian framework is that statistical inference can be made in terms of probability rather than frequency ie true probability intervals instead of confidence intervals One drawback is that only the simplest models yield analytically tractable solutions and calculus is always required However model construction is convenient and very complex problems can be solved using a variety of numerical and computational approaches STAT 54106410 Fall 2008 Lecture Handout 1 Introduction to Spatial Statistics Observational studies often involve data collection over multiple sites within some region of space I Spatial Data The resulting data if the locations of these sites are observed and associated with the observations Spatial Data Analysis Analysis of the data in which spatial locations are explicitly considered Spatial Statistics A particular type of spatial data analysis in which observations or locations or both are modeled as random variables and inference is often made about unobserved quantities Since nearly everything is observed in a spatial context the subject is quite large e g Observations can be univariate or multivariate Type of locational information can vary widely 0 points regions line segments 0 irregularly spaced or regular Euclidean or nonEuclidean distances considered Random or nonRandom locations and observations Gaussian or nonGaussian if random Speci c Sub elds have emerged in spatial statistics and are often categorized by data type 1 Geostatistics continuous data point observations of a continuously varying quantity over a spatial region e g o Richness of iron ore in an underground ore body 0 Annual acid rain deposition at point sites in the eastern Us 0 Level of electrical activity at point sites in the human brain 0 Features of Geostatistics 0 Locations known and not of primary interest 0 We are instead interested in phenomenon observed at those locations 0 Classic Geostatistical Problem 2 Lattice Processes area data Counts or spatial averages of a quantity over a nite number of subregions of a larger spatial region e g 0 Pixel values from remote sensing of the environment 0 Presenceabsence of a plant species in square quadrats laid out over some region 0 Number of deaths due to lung cancer in Utah counties 0 Features of Lattice Processes 0 regions may be regular or irregular e g 3 Spatial Point Processes random point observations Arrangement of a countable number of points within a region eg 0 locations of trees in a timber harvest stand 0 locations of craters on the moon 0 location of lung cancer cases in relation to location of pollution source Features of Spatial Point Processes 0 Locations are random 0 Additional measurements at locations marks may also be random 4 Spatial Prediction Predict the likely value of some process at some locations for which you do not have observations Many forms of ad hoc interpolation are also considered spatial prediction but in statistics we seek optimal prediction along side parameter estimation using formal probability models and data to make inference in the presence of uncertainty Forecasting is a special case of prediction where predictions are extrapolated in space andor time Classic Prediction Problem