Spatial Effects and Externalities
|
|
- Bennett Stewart
- 6 years ago
- Views:
Transcription
1 Spatial Effects and Externalities Philip A. Viton November 5, Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
2 Introduction If you do certain things to your property for example, paint your house regularly this may increase the value of my property, if I am your neighbor. This is an instance of a phenomenon called an externality, or spillover effect. You do something, but I get (some of) the benefit, and this is not realized through the price system (You don t get any of the increase in my property value when I sell my house). The same thing applies if you are induced to do something by a public policy. So these externalities may be important sources of benefits in any public project, but especially in the sorts of projects undertaken by planners. The question is, can our empirical methods reveal the impacts of spatial externalities? Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
3 Externalities in Linear Models Consider our usual linear model of property values p i = β + β A i + β E i where i denotes the property in question, A i is the level of some amenity, and E i is a vector of everything else. We know that the impact of a unit change in A i on p i is given by p/ A i which for the linear model here is just β. Now consider the impact of a change in my property value p i of a change in an amenity A j on some other property, j. In this case we see that the impact, p i / A j is computed to be zero. In other words, in the models we ve considered up to now, the impact on my property value of something you do to improve your property is zero. The same applies to the other specifications (semi-log, log-log, Box-Cox) that we have considered. Thus, our models cannot account for spatial externalities. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
4 Spatial Neighbors (I) Let s see how our modelling could accommodate these effects. The first step is to clarify what it is to be a spatial neighbor. This is generally done via a spatial neighbors matrix W. The (i, j) element of this (W ij ) tells us the extent (degree) to which property i is a neighbor of property j. We take the diagonal elements W ii to be zero : property i is not a neighbor of itself. Philip A. Viton CRP 66 Spatial () Externalities November 5, 4 / 5
5 Spatial Neighbors (II) In studies where the geographical regions are fixed for example, states or counties one often takes W to be an indicator of spatial contiguity, that is, W ij = if i and j share a boundary, and zero otherwise. So the s in row i of W will pick out the neighbors of i. Philip A. Viton CRP 66 Spatial () Externalities November 5, 5 / 5
6 Spatial Neighbors : Example (I) 5 4 W = In this system, for example, area has neighbors (areas, and 5), area 4 has neighbors (areas and 5), etc. Philip A. Viton CRP 66 Spatial () Externalities November 5, 6 / 5
7 Spatial Neighbors : Example (II) It is often convenient to consider a row-standardized form of W in which each element of a row of W is divided by the sum of all the elements in that row. This form is often referred to as a spatial weights matrix. So the row-standardized spatial neighbors (= spatial weights) matrix is: W = Philip A. Viton CRP 66 Spatial () Externalities November 5, 7 / 5
8 Spatial Weights When W is row-standardized, for any spatial characteristic A, each element of the product WA represents the average of the values of A, where the average is taken over the neighbors of each geographical element (elements for which W ij = ). Philip A. Viton CRP 66 Spatial () Externalities November 5, 8 / 5
9 Spatial Weights: Example Suppose we have a variable A with values A = (A, A, A, A 4, A 5 ) for each of the areas in our spatial system. Then using the row-standardized spatial weights matrix we compute the matrix product: WA = A A A A 4 A 5 = (A + A + A 5 ) (A + A ) (A + A + A 4 ) (A + A 5 ) (A + A 4 ) So we see that each element of the product is the average value of A for that observation s spatial neighbors. Philip A. Viton CRP 66 Spatial () Externalities November 5, 9 / 5
10 Spatial Neighbors in Property-Values Models If we are analyzing a sample of property-sales data, this approach neighbors indicated by spatial contiguity will probably not help, since relatively few perhaps none of the sales will be next door (ie contiguous) to a given property sold. In this case we can set up the spatial neighbors matrix as reflecting the distance between any two observations. Since the neighbors matrix is supposed to indicate closeness, we focus on inverse distance and define: W ij = d ij where d ij is distance between properties i and j. As always, we take W ii =. We can use this as a spatial weights matrix, or we can standardize it by the sum of the distances in each row, as before. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
11 Local Spatial Effects (I) Suppose believe that p i is influenced not only by A i but also by the levels of A observed at i s immediate neighbors. Because only i s immediate neighbors matter, this is a model of local spatial effects. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
12 Local Spatial Effects (II) Then an appropriate specification could be: p i = β + β A i + β WA + β E i where, as usual, E i is everything else relevant to the determination of p i. Then if property j is a spatial neighbor of property i and W is a row-standardized spatial weights matrix, it is easy to see that p i A j = β n i where n i is the number of spatial neighbors of property i. The important point here is that now changes in A j do have an estimable impact on p i. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
13 General Spatial Effects Of course, this local-effects model has a major assumption: that only the local impacts matter. This model fails to capture another intuition: if I improve my property, this could raise your property values, if you re my neighbor. And this in turn could raise the property values of your neighbors, and so on. Here, the effects of changes at one location induce changes at (potentially) all other locations, whether immediate neighbors or not. However, as we get further away from the initial change, we expect the impacts to diminish. This is the idea of general spatial effects, and it can be modelled by a spatial autoregressive process, as we now discuss. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
14 The Spatial Autoregressive Model (I) Let p be a column vector of observations on sales prices, and let X be a matrix of property characteristics: row i gives the characteristics of property i (so row i is the concatenation of what we previously referred to as A i and E i ). Then the spatial autoregressive (SAR) model specifies: p = λwp + X β + ε where λ is a parameter, and ε is a vector of iid normal error terms. This is like the standard regression model except for the addition of the term λwp. We can now see why we took W ii = : if we didn t, this would make the determination of p i (on the left) depend on the same p i (on the right), an obvious circularity. Philip A. Viton CRP 66 Spatial () Externalities November 5, 4 / 5
15 The Spatial Autoregressive Model (II) Starting with the model: p = λwp + X β + ε we can analyze it by collecting up terms. We have: p = λwp + X β + ε (I λw )p = X β + ε where I is the identity matrix. p = (I λw ) X β + (I λw ) ε In this model, spatial spillover is indicated by a non-zero λ : if λ = the model reduces to the standard linear model. When λ =, p i depends (through W ) on all the properties in the sample. Philip A. Viton CRP 66 Spatial () Externalities November 5, 5 / 5
16 Interpretations Consider the term (I λw ) and suppose that W is row-standardized. Then it can be shown that: (I λw ) = I + λw + λ W + λ W +... where I is the identity matrix. First, terms like W can be thought of as W W, the neighbors of W, or the neighbors of the original neighbors; and similarly for higher powers. (To see this more precisely, you need a result linking graph theory and matrix algebra (multiplication), which will not be given here). Next, if W is row-standardized, then each of its elements is less than, so as the powers increase, the individual terms get less and less important. This is how areas that are spatially remote from a given observation affect the left-hand-side (dependent variable) in the spatial autoregressive model, but at a diminishing rate. Philip A. Viton CRP 66 Spatial () Externalities November 5, 6 / 5
17 The Spatial Autoregressive Model : Example (I) We can see what is going on if we consider a sample of properties, and where the value of each property depends on just elements. Then we will have p ( ), W ( ), ε ( ) and X ( ). Write B = (I λw ) Then the model becomes: p = BX β + Bε = BX β + η where η = Bε. It is also important to remember that B depends on the unknown parameter λ : B is not just a data matrix. Philip A. Viton CRP 66 Spatial () Externalities November 5, 7 / 5
18 The Spatial Autoregressive Model : Example (II) Let s write out the equations of this model: p B B B X X p = B B B X X p B B B X X Or: ( β β ) + η η η p = β (B X + B X + B X ) + β (B X + B X + B X ) + η p = β (B X + B X + B X ) + β (B X + B X + B X ) + η p = β (B X + B X + B X ) + β (B X + B X + B X ) + η Philip A. Viton CRP 66 Spatial () Externalities November 5, 8 / 5
19 The Spatial Autoregressive Model : Example (III) Let s analyze the effects in this model. First, the own effect: the impact on the price of (say) property of a change in an own-characteristic, say X, is given by: p X = β B a bit more complicated than before, because of B (which depends on λ). The impact on p of a change in the level of a characteristic of another property (say X, the first characteristic of property ) which in the standard linear model was zero, is now: which will in general be non-zero. p X = β B Philip A. Viton CRP 66 Spatial () Externalities November 5, 9 / 5
20 The Spatial Autoregressive Model : Extensions There is an interesting special case. Suppose we focus on property (observation) k. And suppose that all properties in our sample receive a -unit increase in X (the second characteristic). Each of these increases will have some impact (depending on distance, via W ) on property k s value, ie on p k. If our estimate of λ is between and, and if W (the spatial weights matrix) is row-standardized, then Kim, Phipps and Anselin (J. Environmental Economics and Management,, footnote 4) show that the total impact on property k is: p k X = β λ Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
21 The Spatial Autoregressive Model : Summary The SAR model is therefore a way of allowing for and measuring general spatial externalities (spillovers) in our modelling. It is important to note than the regression coeffi cients β do not have the simple interpretation they had in the linear model for either the own-characteristic impact or the other-property impact. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
22 Are There Spatial Effects? Before you get involved with this more complicated model setting, you might wonder: is it necessary? That is, given a dependent variable observed over space like p (the list of house selling prices), does it show spatial autocorrelation at all? If the answer is No, then there s probably no need to bother with the more complicated spatial regression models. There is a standard test for spatial autocorrelation in the literature, called Moran s Test (or Moran s I statistic). Most packages that accommodate spatial regression will allow you to compute this test statistic. Note that Moran s test depends on the particular spatial representation that you choose for your spatial neighbors or weights matrix: different matrices could give different answers. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
23 Estimation Issues In the SAR model p = (I λw ) X β + (I λw ) ε the term (I λw ) is a complicated non-linear function of λ. This means that the model is no longer linear-in-parameters, and therefore cannot be estimated by standard regression techniques. Specialized software is needed: among the packages that can estimate this model are Luc Anselin s stand-alone Geoda system (available free for most operating systems) and the spdep package (add-on) to the free R statistics system. I believe there is also a spatial add-on for the commercial Stata system. Unfortunately, as of now, the premier commercial package for discrete-choice models, LIMDEP/NLOGIT, does not cover spatial models. Philip A. Viton CRP 66 Spatial () Externalities November 5, / 5
24 Computational Issues (I) Non-spatial models can also be non-linear. But spatial models have another diffi culty. The spatial neighbors matrix W is n n, where n is the number of observations. It follows that (I λw ) is also n n, and the number of elements in either of these is n. For example, suppose you have observations on sales in your community. Then W is going to be a = 9 -element matrix. When the unit of observation is US counties of which there are about then W will have about = 9, 6, elements. All this can require significant storage. In other words, you may need to invert and keep track of some very large matrices. Philip A. Viton CRP 66 Spatial () Externalities November 5, 4 / 5
25 Computational Issues (II) However the problem can be mitigated if we know that each observation has relatively few neighbors. For example, for the US counties, the average number of contiguity-based neighbors is about 5. This means that the huge spatial neighbors matrix will be mostly s, and it should be possible to exploit this fact in order to reduce the computational and storage burden. This can in fact be done by considering W to be a so-called sparse matrix. Most spatial software goes to considerable lengths to cope with this issue: it is relatively easy to estimate models with n on the order of about, at least when the average number of neighbors is small. However, for significantly larger models this can easily occur for property-values models, where large samples are common data storage can be a serious issue. This includes the case where the spatial weights matrix is inverse-distance-based, since it is not sparse. Philip A. Viton CRP 66 Spatial () Externalities November 5, 5 / 5
Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign
GIS and Spatial Analysis Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu Outline GIS and Spatial Analysis
More informationIntroduction to Spatial Statistics and Modeling for Regional Analysis
Introduction to Spatial Statistics and Modeling for Regional Analysis Dr. Xinyue Ye, Assistant Professor Center for Regional Development (Department of Commerce EDA University Center) & School of Earth,
More informationMath 123, Week 2: Matrix Operations, Inverses
Math 23, Week 2: Matrix Operations, Inverses Section : Matrices We have introduced ourselves to the grid-like coefficient matrix when performing Gaussian elimination We now formally define general matrices
More informationGetting Started with Communications Engineering
1 Linear algebra is the algebra of linear equations: the term linear being used in the same sense as in linear functions, such as: which is the equation of a straight line. y ax c (0.1) Of course, if we
More informationRegression, part II. I. What does it all mean? A) Notice that so far all we ve done is math.
Regression, part II I. What does it all mean? A) Notice that so far all we ve done is math. 1) One can calculate the Least Squares Regression Line for anything, regardless of any assumptions. 2) But, if
More informationMeasuring The Benefits of Air Quality Improvement: A Spatial Hedonic Approach. Chong Won Kim, Tim Phipps, and Luc Anselin
Measuring The Benefits of Air Quality Improvement: A Spatial Hedonic Approach Chong Won Kim, Tim Phipps, and Luc Anselin Paper prepared for presentation at the AAEA annual meetings, Salt Lake City, August,
More information22m:033 Notes: 3.1 Introduction to Determinants
22m:033 Notes: 3. Introduction to Determinants Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman October 27, 2009 When does a 2 2 matrix have an inverse? ( ) a a If A =
More informationMath 103, Summer 2006 Determinants July 25, 2006 DETERMINANTS. 1. Some Motivation
DETERMINANTS 1. Some Motivation Today we re going to be talking about erminants. We ll see the definition in a minute, but before we get into ails I just want to give you an idea of why we care about erminants.
More informationSpatial Autocorrelation (2) Spatial Weights
Spatial Autocorrelation (2) Spatial Weights Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu Outline
More informationMath 308 Midterm Answers and Comments July 18, Part A. Short answer questions
Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to
More informationY t = log (employment t )
Advanced Macroeconomics, Christiano Econ 416 Homework #7 Due: November 21 1. Consider the linearized equilibrium conditions of the New Keynesian model, on the slide, The Equilibrium Conditions in the handout,
More informationSpatial Regression. 1. Introduction and Review. Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved
Spatial Regression 1. Introduction and Review Luc Anselin http://spatial.uchicago.edu matrix algebra basics spatial econometrics - definitions pitfalls of spatial analysis spatial autocorrelation spatial
More information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
More informationLecture 6: Hypothesis Testing
Lecture 6: Hypothesis Testing Mauricio Sarrias Universidad Católica del Norte November 6, 2017 1 Moran s I Statistic Mandatory Reading Moran s I based on Cliff and Ord (1972) Kelijan and Prucha (2001)
More informationLecture 7: Spatial Econometric Modeling of Origin-Destination flows
Lecture 7: Spatial Econometric Modeling of Origin-Destination flows James P. LeSage Department of Economics University of Toledo Toledo, Ohio 43606 e-mail: jlesage@spatial-econometrics.com June 2005 The
More informationSpatial Tools for Econometric and Exploratory Analysis
Spatial Tools for Econometric and Exploratory Analysis Michael F. Goodchild University of California, Santa Barbara Luc Anselin University of Illinois at Urbana-Champaign http://csiss.org Outline A Quick
More informationSpatial inference. Spatial inference. Accounting for spatial correlation. Multivariate normal distributions
Spatial inference I will start with a simple model, using species diversity data Strong spatial dependence, Î = 0.79 what is the mean diversity? How precise is our estimate? Sampling discussion: The 64
More informationMatrix Inverses. November 19, 2014
Matrix Inverses November 9, 204 22 The Inverse of a Matrix Now that we have discussed how to multiply two matrices, we can finally have a proper discussion of what we mean by the expression A for a matrix
More information/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Matroids and Greedy Algorithms Date: 10/31/16
60.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Matroids and Greedy Algorithms Date: 0/3/6 6. Introduction We talked a lot the last lecture about greedy algorithms. While both Prim
More informationMATH240: Linear Algebra Review for exam #1 6/10/2015 Page 1
MATH24: Linear Algebra Review for exam # 6//25 Page No review sheet can cover everything that is potentially fair game for an exam, but I tried to hit on all of the topics with these questions, as well
More informationc 1 v 1 + c 2 v 2 = 0 c 1 λ 1 v 1 + c 2 λ 1 v 2 = 0
LECTURE LECTURE 2 0. Distinct eigenvalues I haven t gotten around to stating the following important theorem: Theorem: A matrix with n distinct eigenvalues is diagonalizable. Proof (Sketch) Suppose n =
More informationMITOCW ocw f99-lec01_300k
MITOCW ocw-18.06-f99-lec01_300k Hi. This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra. And the
More informationWhat if the characteristic equation has complex roots?
MA 360 Lecture 18 - Summary of Recurrence Relations (cont. and Binomial Stuff Thursday, November 13, 01. Objectives: Examples of Recurrence relation solutions, Pascal s triangle. A quadratic equation What
More information1.6 and 5.3. Curve Fitting One of the broadest applications of linear algebra is to curve fitting, especially in determining unknown coefficients in
16 and 53 Curve Fitting One of the broadest applications of linear algebra is to curve fitting, especially in determining unknown coefficients in functions You should know that, given two points in the
More informationAN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS
AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply
More informationFactoring. there exists some 1 i < j l such that x i x j (mod p). (1) p gcd(x i x j, n).
18.310 lecture notes April 22, 2015 Factoring Lecturer: Michel Goemans We ve seen that it s possible to efficiently check whether an integer n is prime or not. What about factoring a number? If this could
More informationCreating and Managing a W Matrix
Creating and Managing a W Matrix Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Junel 22th, 2016 C. Hurtado (UIUC - Economics) Spatial Econometrics
More informationEigenvalues and eigenvectors
Roberto s Notes on Linear Algebra Chapter 0: Eigenvalues and diagonalization Section Eigenvalues and eigenvectors What you need to know already: Basic properties of linear transformations. Linear systems
More informationSelect/Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras
Select/Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras Lecture - 9 Angular Momentum in Quantum Mechanics Dimensionality of the Direct-Product
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationMITOCW ocw f99-lec09_300k
MITOCW ocw-18.06-f99-lec09_300k OK, this is linear algebra lecture nine. And this is a key lecture, this is where we get these ideas of linear independence, when a bunch of vectors are independent -- or
More informationSpatial Data Mining. Regression and Classification Techniques
Spatial Data Mining Regression and Classification Techniques 1 Spatial Regression and Classisfication Discrete class labels (left) vs. continues quantities (right) measured at locations (2D for geographic
More informationMA Lesson 25 Section 2.6
MA 1500 Lesson 5 Section.6 I The Domain of a Function Remember that the domain is the set of x s in a function, or the set of first things. For many functions, such as f ( x, x could be replaced with any
More informationMITOCW ocw f99-lec30_300k
MITOCW ocw-18.06-f99-lec30_300k OK, this is the lecture on linear transformations. Actually, linear algebra courses used to begin with this lecture, so you could say I'm beginning this course again by
More informationDot Products, Transposes, and Orthogonal Projections
Dot Products, Transposes, and Orthogonal Projections David Jekel November 13, 2015 Properties of Dot Products Recall that the dot product or standard inner product on R n is given by x y = x 1 y 1 + +
More informationThe Haar Wavelet Transform: Compression and. Reconstruction
The Haar Wavelet Transform: Compression and Damien Adams and Halsey Patterson December 14, 2006 Abstract The Haar Wavelet Transformation is a simple form of compression involved in averaging and differencing
More informationLinear Programming Redux
Linear Programming Redux Jim Bremer May 12, 2008 The purpose of these notes is to review the basics of linear programming and the simplex method in a clear, concise, and comprehensive way. The book contains
More informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationSpatial Autocorrelation
Spatial Autocorrelation Luc Anselin http://spatial.uchicago.edu spatial randomness positive and negative spatial autocorrelation spatial autocorrelation statistics spatial weights Spatial Randomness The
More informationNotes on the Matrix-Tree theorem and Cayley s tree enumerator
Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationDeterminants of 2 2 Matrices
Determinants In section 4, we discussed inverses of matrices, and in particular asked an important question: How can we tell whether or not a particular square matrix A has an inverse? We will be able
More informationDescriptive Statistics (And a little bit on rounding and significant digits)
Descriptive Statistics (And a little bit on rounding and significant digits) Now that we know what our data look like, we d like to be able to describe it numerically. In other words, how can we represent
More informationFreeing up the Classical Assumptions. () Introductory Econometrics: Topic 5 1 / 94
Freeing up the Classical Assumptions () Introductory Econometrics: Topic 5 1 / 94 The Multiple Regression Model: Freeing Up the Classical Assumptions Some or all of classical assumptions needed for derivations
More informationTESTING FOR CO-INTEGRATION
Bo Sjö 2010-12-05 TESTING FOR CO-INTEGRATION To be used in combination with Sjö (2008) Testing for Unit Roots and Cointegration A Guide. Instructions: Use the Johansen method to test for Purchasing Power
More informationIterative Methods for Solving A x = b
Iterative Methods for Solving A x = b A good (free) online source for iterative methods for solving A x = b is given in the description of a set of iterative solvers called templates found at netlib: http
More informationGAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.
More informationC if U can. Algebra. Name
C if U can Algebra Name.. How will this booklet help you to move from a D to a C grade? The topic of algebra is split into six units substitution, expressions, factorising, equations, trial and improvement
More informationExploratory Spatial Data Analysis Using GeoDA: : An Introduction
Exploratory Spatial Data Analysis Using GeoDA: : An Introduction Prepared by Professor Ravi K. Sharma, University of Pittsburgh Modified for NBDPN 2007 Conference Presentation by Professor Russell S. Kirby,
More informationMath Studio College Algebra
Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of
More informationUrban GIS for Health Metrics
Urban GIS for Health Metrics Dajun Dai Department of Geosciences, Georgia State University Atlanta, Georgia, United States Presented at International Conference on Urban Health, March 5 th, 2014 People,
More informationSpatial autocorrelation: robustness of measures and tests
Spatial autocorrelation: robustness of measures and tests Marie Ernst and Gentiane Haesbroeck University of Liege London, December 14, 2015 Spatial Data Spatial data : geographical positions non spatial
More informationsplm: econometric analysis of spatial panel data
splm: econometric analysis of spatial panel data Giovanni Millo 1 Gianfranco Piras 2 1 Research Dept., Generali S.p.A. and DiSES, Univ. of Trieste 2 REAL, UIUC user! Conference Rennes, July 8th 2009 Introduction
More informationMATH Mathematics for Agriculture II
MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3
More informationAn example to start off with
Impact Evaluation Technical Track Session IV Instrumental Variables Christel Vermeersch Human Development Human Network Development Network Middle East and North Africa Region World Bank Institute Spanish
More informationLecture 11: Extrema. Nathan Pflueger. 2 October 2013
Lecture 11: Extrema Nathan Pflueger 2 October 201 1 Introduction In this lecture we begin to consider the notion of extrema of functions on chosen intervals. This discussion will continue in the lectures
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationLecture 9: Elementary Matrices
Lecture 9: Elementary Matrices Review of Row Reduced Echelon Form Consider the matrix A and the vector b defined as follows: 1 2 1 A b 3 8 5 A common technique to solve linear equations of the form Ax
More informationIn matrix algebra notation, a linear model is written as
DM3 Calculation of health disparity Indices Using Data Mining and the SAS Bridge to ESRI Mussie Tesfamicael, University of Louisville, Louisville, KY Abstract Socioeconomic indices are strongly believed
More informationExploratory Spatial Data Analysis and GeoDa
Exploratory Spatial Data Analysis and GeoDa Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu Outline
More informationMATH 320, WEEK 7: Matrices, Matrix Operations
MATH 320, WEEK 7: Matrices, Matrix Operations 1 Matrices We have introduced ourselves to the notion of the grid-like coefficient matrix as a short-hand coefficient place-keeper for performing Gaussian
More information1 Least Squares Estimation - multiple regression.
Introduction to multiple regression. Fall 2010 1 Least Squares Estimation - multiple regression. Let y = {y 1,, y n } be a n 1 vector of dependent variable observations. Let β = {β 0, β 1 } be the 2 1
More informationAlgebraic Equations. 2.0 Introduction. Nonsingular versus Singular Sets of Equations. A set of linear algebraic equations looks like this:
Chapter 2. 2.0 Introduction Solution of Linear Algebraic Equations A set of linear algebraic equations looks like this: a 11 x 1 + a 12 x 2 + a 13 x 3 + +a 1N x N =b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 +
More informationUsing AMOEBA to Create a Spatial Weights Matrix and Identify Spatial Clusters, and a Comparison to Other Clustering Algorithms
Using AMOEBA to Create a Spatial Weights Matrix and Identify Spatial Clusters, and a Comparison to Other Clustering Algorithms Arthur Getis* and Jared Aldstadt** *San Diego State University **SDSU/UCSB
More informationName Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.
Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so
More informationEcon 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationESSAY 1 : EXPLORATIONS WITH SOME SEQUENCES
ESSAY : EXPLORATIONS WITH SOME SEQUENCES In this essay, in Part-I, I want to study the behavior of a recursively defined sequence f n =3.2 f n [ f n ] for various initial values f. I will also use a spreadsheet
More informationVisualize and interactively design weight matrices
Visualize and interactively design weight matrices Angelos Mimis *1 1 Department of Economic and Regional Development, Panteion University of Athens, Greece Tel.: +30 6936670414 October 29, 2014 Summary
More informationExploratory Spatial Data Analysis (ESDA)
Exploratory Spatial Data Analysis (ESDA) VANGHR s method of ESDA follows a typical geospatial framework of selecting variables, exploring spatial patterns, and regression analysis. The primary software
More informationWhat if the characteristic equation has a double root?
MA 360 Lecture 17 - Summary of Recurrence Relations Friday, November 30, 018. Objectives: Prove basic facts about basic recurrence relations. Last time, we looked at the relational formula for a sequence
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra, Spring 2005 Please use the following citation format: Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology:
More informationSpatial Relationships in Rural Land Markets with Emphasis on a Flexible. Weights Matrix
Spatial Relationships in Rural Land Markets with Emphasis on a Flexible Weights Matrix Patricia Soto, Lonnie Vandeveer, and Steve Henning Department of Agricultural Economics and Agribusiness Louisiana
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More informationCalculus with business applications, Lehigh U, Lecture 01 notes Summer
Calculus with business applications, Lehigh U, Lecture 01 notes Summer 2012 1 Functions 1. A company sells 100 widgets at a price of $20. Sales increase by 5 widgets for each $1 decrease in price. Write
More informationMath 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations
Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations
More informationcourses involve systems of equations in one way or another.
Another Tool in the Toolbox Solving Matrix Equations.4 Learning Goals In this lesson you will: Determine the inverse of a matrix. Use matrices to solve systems of equations. Key Terms multiplicative identity
More informationSYDE 112, LECTURE 7: Integration by Parts
SYDE 112, LECTURE 7: Integration by Parts 1 Integration By Parts Consider trying to take the integral of xe x dx. We could try to find a substitution but would quickly grow frustrated there is no substitution
More informationHow Latent Semantic Indexing Solves the Pachyderm Problem
How Latent Semantic Indexing Solves the Pachyderm Problem Michael A. Covington Institute for Artificial Intelligence The University of Georgia 2011 1 Introduction Here I present a brief mathematical demonstration
More information1 The Multiple Regression Model: Freeing Up the Classical Assumptions
1 The Multiple Regression Model: Freeing Up the Classical Assumptions Some or all of classical assumptions were crucial for many of the derivations of the previous chapters. Derivation of the OLS estimator
More informationVector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n.
Vector Spaces Definition: The usual addition and scalar multiplication of n-tuples x = (x 1,..., x n ) R n (also called vectors) are the addition and scalar multiplication operations defined component-wise:
More informationLecture 6: Backpropagation
Lecture 6: Backpropagation Roger Grosse 1 Introduction So far, we ve seen how to train shallow models, where the predictions are computed as a linear function of the inputs. We ve also observed that deeper
More informationDesigning Information Devices and Systems I Spring 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way
EECS 16A Designing Information Devices and Systems I Spring 018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra the EECS Way In this note, we will teach the basics of linear algebra and relate
More information2.6 Complexity Theory for Map-Reduce. Star Joins 2.6. COMPLEXITY THEORY FOR MAP-REDUCE 51
2.6. COMPLEXITY THEORY FOR MAP-REDUCE 51 Star Joins A common structure for data mining of commercial data is the star join. For example, a chain store like Walmart keeps a fact table whose tuples each
More informationThe Generalized Roy Model and Treatment Effects
The Generalized Roy Model and Treatment Effects Christopher Taber University of Wisconsin November 10, 2016 Introduction From Imbens and Angrist we showed that if one runs IV, we get estimates of the Local
More informationOutline. Overview of Issues. Spatial Regression. Luc Anselin
Spatial Regression Luc Anselin University of Illinois, Urbana-Champaign http://www.spacestat.com Outline Overview of Issues Spatial Regression Specifications Space-Time Models Spatial Latent Variable Models
More informationAnalyzing spatial autoregressive models using Stata
Analyzing spatial autoregressive models using Stata David M. Drukker StataCorp Summer North American Stata Users Group meeting July 24-25, 2008 Part of joint work with Ingmar Prucha and Harry Kelejian
More informationIntroduction to Karnaugh Maps
Introduction to Karnaugh Maps Review So far, you (the students) have been introduced to truth tables, and how to derive a Boolean circuit from them. We will do an example. Consider the truth table for
More informationx y = 1, 2x y + z = 2, and 3w + x + y + 2z = 0
Section. Systems of Linear Equations The equations x + 3 y =, x y + z =, and 3w + x + y + z = 0 have a common feature: each describes a geometric shape that is linear. Upon rewriting the first equation
More informationNotes on Random Variables, Expectations, Probability Densities, and Martingales
Eco 315.2 Spring 2006 C.Sims Notes on Random Variables, Expectations, Probability Densities, and Martingales Includes Exercise Due Tuesday, April 4. For many or most of you, parts of these notes will be
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationProcess Modelling. Table of Contents
Process Modelling 1 Process Modelling prepared by Wm. J. Garland, Professor, Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada More about this document Summary: The general
More informationCS168: The Modern Algorithmic Toolbox Lecture #7: Understanding Principal Component Analysis (PCA)
CS68: The Modern Algorithmic Toolbox Lecture #7: Understanding Principal Component Analysis (PCA) Tim Roughgarden & Gregory Valiant April 0, 05 Introduction. Lecture Goal Principal components analysis
More informationA primer on matrices
A primer on matrices Stephen Boyd August 4, 2007 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous
More informationLecture 3: Exploratory Spatial Data Analysis (ESDA) Prof. Eduardo A. Haddad
Lecture 3: Exploratory Spatial Data Analysis (ESDA) Prof. Eduardo A. Haddad Key message Spatial dependence First Law of Geography (Waldo Tobler): Everything is related to everything else, but near things
More informationInstructor (Brad Osgood)
TheFourierTransformAndItsApplications-Lecture26 Instructor (Brad Osgood): Relax, but no, no, no, the TV is on. It's time to hit the road. Time to rock and roll. We're going to now turn to our last topic
More informationW-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS
1 W-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS An Liu University of Groningen Henk Folmer University of Groningen Wageningen University Han Oud Radboud
More informationVolume in n Dimensions
Volume in n Dimensions MA 305 Kurt Bryan Introduction You ve seen that if we have two vectors v and w in two dimensions then the area spanned by these vectors can be computed as v w = v 1 w 2 v 2 w 1 (where
More informationBayesian Linear Regression [DRAFT - In Progress]
Bayesian Linear Regression [DRAFT - In Progress] David S. Rosenberg Abstract Here we develop some basics of Bayesian linear regression. Most of the calculations for this document come from the basic theory
More informationKnowledge Spillovers, Spatial Dependence, and Regional Economic Growth in U.S. Metropolitan Areas. Up Lim, B.A., M.C.P.
Knowledge Spillovers, Spatial Dependence, and Regional Economic Growth in U.S. Metropolitan Areas by Up Lim, B.A., M.C.P. DISSERTATION Presented to the Faculty of the Graduate School of The University
More informationAn Introduction to NeRDS (Nearly Rank Deficient Systems)
(Nearly Rank Deficient Systems) BY: PAUL W. HANSON Abstract I show that any full rank n n matrix may be decomposento the sum of a diagonal matrix and a matrix of rank m where m < n. This decomposition
More information