University of California, Los Angeles Department of Statistics Statistics C173/C273 Introduction What is geostatistics? Geostatistics is concerned with estimation and prediction for spatially continuous phenomena, using data obtained at a limited number o
University of California, Los Angeles Department of Statistics Statistics C173/C273 Universal kriging The Ordinary Kriging (OK) that was discussed earlier is based on the constant mean model given by Z(s) = + (s) where () has mean zero and variogram 2().
University of California, Los Angeles Department of Statistics Statistics C173/C273 Cross Validation Cross validation is a technique that allows us to compare predicted values with true values. In spatial data this technique can help us to decide which va
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou
Effect of variogram parameters on kriging weights We will explore in this document how the kriging weights are affected by the variogram param
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou
Ordinary kriging using geoR and gstat In this document we will discuss kriging using the R packages geoR and gstat. We will use the numerical
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou
Ordinary kriging in terms of the covariance function The model: The model assumption is: Z(s) = + (s) where (s) is a zero mean stochastic term
University of California, Los Angeles Department of Statistics Statistics C173/C273 Ordinary kriging Kriging (Matheron 1963) owes its name to D. G. Krige a South African mining engineer and it was first applied in mining data. Kriging assumes a random fie
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Spatial statistics - prediction Spatial prediction: a. One of our goals in geostatistics is to predict values at unsampled loacations. This wi
University of California, Los Angeles Department of Statistics Statistics C173/C273 More variograms Two more varograms are presented below: a. Circular semi-variogram: (h; ) =
2 h c1 1 - cos-1 ( ) + 2h
Instructor: Nicolas Christou
1-
h2 2
, h h>
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F
University of California, Los Angeles Department of Statistics Statistics C173/C273 Co-kriging
Suppose that at each spatial location si , i = 1, . . . , n we observe k variables as follows: Z1 (s1 ) Z2 (s1 ) . . . . . . Zk (s1 ) Z1 (s2 ) Z2 (s2 ) . . Zk (
University of California, Los Angeles Department of Statistics Statistics C173/C273 Block kriging exampe
We will si s1 s2 s3 s4 s5 s6 s7 use the 7-point data from earlier lectures. Here they are: x y z(si ) 61 139 477 63 140 696 64 129 227 68 128 646 71 1
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou More on the variogram Variogram model parameters: The various parameters of the variogram model are: 1. Nugget Efffect (c0 ): If we stand by t
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Computing the variogram using the geoR package in R Spatial statistics computations can be done in R using the package geoR. Once you have ins
University of California, Los Angeles Department of Statistics Statistics C173/C273 The variogram Let Z(s) and Z(s + h) two random variables at locations s and s + h. Intrinsic stationarity is defined as follows: E(Z(s + h) - Z(s) = 0 and V ar(Z(s + h) -
University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou
Fitting a model variogram to the sample variogram The parameters of the model variogram can be estimated through ordinary or weighted least sq
University of California, Los Angeles Department of Statistics Statistics C173/C273 Spatial statistics Why spatial statistics? Noel Cressie ("Statistics for Spatial Data") writes "why, how, when" are not enough. We need to add "where". Today, spatial stat
University of California, Los Angeles Department of Statistics Statistics C173/C273 Lab
Exercise 1: These Jura data were collected by the Swiss Federal Institute of Technology at Lausanne. See Goovaerts, P. 1997, "Geostatistics for Natural Resources Evalu
University of California, Los Angeles Department of Statistics Introduction to GRASS/R spatial data analysis What is Geographic Information Systems (GIS)? - "A geographic information system (GIS) integrates hardware, software, and data for capturing, mana
University
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Department
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Statistics
C173/C273
Instructor:
Nicolas
Christou
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University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Working with gstat - An example
We will use the data from the Meuse (Dutch Maas) river. The actual data set contains many variables but here w
University of California, Los Angeles Department of Statistics Statistics C173/C273 Data with trend As we discussed earlier, if the intrinsic stationarity assumption holds, which implies E(Z(s + h) - Z(s) = 0 and V ar(Z(s + h) - Z(s) = 2(h) we can write V
Final Exam
Spring 2011
J.Y.Kim
1. Let Xi iidBern( p), that is, Xi is an independent and identical Bernoulli random variable
with probability p. Notice that the variance of X is 2 = g( p) = p(1 p).
(a) Get the MLE of the variance g( p) = p(1 p). (6pts)
(b)
Midterm Exam
Spring 2015
J.Y.Kim
1. Consider an experiment of tossing coins, C1 and C2 , with each coin having outcomes,
head (Hi ) and tail (Ti ) for i = 1, 2. Let Xi be a random variable associated with the
outcome of tossing Ci : Xi takes values 1, 0,
Final Exam
Spring 2010 ].Y.Kirn
1. (a) Let X ~ iid N (y, (72). We have X and 52, respectively, as estimators of y and (72. List all
the properties of each of these estimators, and for each of the properties show why we have
such a property. If it is not e
Midterm Exam
Spring2011
J.Y.Kim
1. Solvethe following problems. (3ptseachexcept(a)2pts)
(a) Let X = (Xr,. . . ,Xn) where X;be arandom variable. Show that X is a random vector.
(b)WritetheconditionthatPlc,;:x1<X(co)3xz,At<Y(ru)1lz,zr<Z(cu)<z2l>0in
terms of
Final Exam
Spring 2015
J.Y.Kim
1. Let X1 , ., Xn be a random sample from an exponential distribution with .
(a) Find UMVUEs for and 1/ and check whether they achieve CRLB. (hint: recall that X1 +
. + Xn ( = n, = ). (7pts)
(b) The survival function of expo
Final Exam
Spring 2013
J.Y.Kim
1. Let Xi iid (/2, 2 /2), i N. Define Yi = Xi + Xi+1 for all i, and answer the followings.
(a) Show that Yi s are identically distributed, but not independent.
(b) What is the mean and variance of Y n = n1 in=1 Yi ?
2
(c) De
Midterm Exam
Spring 2013
J.Y.Kim
1. Solve the following problems. (4pts each)
(a) Show that a -field is closed under countable union, intersection, set difference and symmestric
difference where symmestric difference is defined as AB = (A B) (B A).
(b) Le
Midterm Exam
Spring 2014
J.Y.Kim
1. Let Xi , i = 1, ., n, be iid random variables. Also, let Y j be the jth order statistic of
(X1 , ., Xn ).
(a) Derive the DF and the pdf of Y j . (8pts)
(b) Get the DF and the pdf of Y j for a Uniform(0,1) (4pts).
2. For
Final Exam
Spring 2014 ].Y.Kim
1. Let Yn be a sequence of random variables that satisfies WOOL 9) > N (O, (72) in distribu
tion. Let g be a measurable function.
@Su ose that exists and' is not zero. Prove that (4 ts)
PP 3/9 P
x/gm) - 8(9) ~+ N(0,02[g(9)]2
Studies in Statistics for Economists
Spring 2016
Taught by:
office:
e-mail:
lectures:
office hours:
Jae-Young Kim
16-607, (phone) 880-6390
[email protected]
Tuesday 2:00pm-4:50
Thursday 1:00-2:00pm & by appointment
TA: Hyejin Park
[email protected]
Eun Ji