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Intrinsic�Auto�Regressi�e�Models Spa�ial da�a anal��i� in S�an S�e Ma���e� Da�a Scie��i�� a� The R�ckefelle� F���da�i�� /
H�! � · S�e Ma���e�, Manage� and Da�a Scien�i�� a� The Rockefelle� Fo�nda�ion · P�e�io��l�, � �a� a Da�a Scien�i�� a� B���Feed and a S�a�i��ical gene�ici�� a� The Fein��ein �n��i���e fo� Medical Re�ea�ch · Hold a G�ad�a�e Di�loma in S�a�i��ic� and S�ocha��ic P�oce��e� f�om �he Uni�e��i�� of Melbo��ne, A����alia. 3/43 /
S���c���e �f �hi� �a�� � � 1. M��i�a�i�g ��e��i��: S�ike i� ca�e� �f de�g�e i� Jali�c�. 2. Wh� �e ca�'� ha�e �ice �hi�g�! 3. Wha� a�e C��di�i��al A���-Reg�e��i�e ��del� (CAR)? 4. Wha� i� �he i���i�ic �a�� �f �he ����i��ic A���-Reg�e��i�e ��del� (�CAR)? 5. ���le�e��a�i�� �f �CAR i� S�a� 4/43 /
� � Mo�i�a�ing q�es�ion: Spike in cases of deng�e in Jalisco. 5/43 /
De�g�e ca�e� i� Ja�i�c� 6/43 /
7/43 /
Jali�c� ��l�g�n Jalisco Polygon (https://datos.jalisco.gob.mx/dataset/mapa- general-de-jalisco-limite-estatal) 8/43 /
Deng�e data M�DE Jali�c� (h����://�e�lan.a��.jali�c�.g�b.m�/mide/�anelCi�dadan�/�ablaDa���? ni�elTablaDa���=3&�e�i�dicidadTablaDa���=an�al&indicad��TablaDa���=772&acci�nReg 9/43 /
� � Wh� �e can�� ha�e nice �hings� 10/�3 /
Why we can't have nice things! � � � � � S�a�ial a���c���ela�i�n 11/�3 /
S�a�ial a���c���ela�i�n A���c���ela�i�� is a measurement of similarit� between close observations of the same phenomenon. E�ample �ith temporal a�tocorrelation: �f �ou measure �our weight, two observations close in time are ver� similar than distant ones. S�a�ial a���c���ela�i�� is more nuanced because, unlike time, spatial variables are at least two-dimensional. Spatial a�tocorrelation : Describe the e�tent to which two observations from neighboring regions e�hibit higher correlation than distant ones. 12/43 /
A���c���ela�i�� i� ��a�ial da�a · �n regression analysis, one of the standard assumptions is that errors are uncorrelated. · Correlated errors suggest we have additional information in the data that has not been accounted for in the model as it is. · �n the case of spatial data, adjacent residuals tend to be similar and therefore a�tocorrelated . Mai� ���ble�: if autocorrelation is not exploited in your model, your explanatoy variables coe�cients will display an unusual explanatory power, which might be the consequence of of just �tting spatial noise. 13/43 /
��i�ial ��e��i�� ab��� de�g�e 14/43 /
Sim�le model 1�/43 /
Le�'� add c��a�ia�e� A����i�g �ha� everything else d�e� ��� a�ec� water capacity �hi� ��de� �h���d be dece��. 16/43 /
When ever�thing else contains spatial correlation We a�e ���i�g �hi� B�� i� �eali��, �e ha�e �hi�: O�r coe�cient estimates �ill be �rong! 17/43 /
Moran's � (autocorrelation statistic) � · Analogous to the the standard correlation concept. · Numerator measuring deviatiom from the mean for adjacent units. · Denominator standardi�es the quantity to re�ect the variability of the quantity of interest. 18/43 /
Moran's � (Jalisco data) 1�/43 /
Moran's � �es� (Jalisco da�a) 20/�3 /
Moran's � �es� (Jalisco da�a) � M����-C���� ���������� �� M���� I ����: ��������$������������ �������: �������� ������ �� ����������� + 1: 601 ��������� = 0.21246, �������� ���� = 597, �-����� = 0.006656 ����������� ����������: ������� 21/�3 /
B�� S�e, i� �hi� �eall� a p�oblem in o�he� �e�ea�ch a�ea�? Kell�, M��gan, The S�andard Errors of Persis�ence (J�ne 3, 2019) (h����://�a�e��.���n.c�m/��l3/�a�e��.cfm? ab���ac�_id=3398303) 22/43 /
� � Wha� are Condi�ional A��o-Regressi�e models (CAR)? 23/43 /
Condi�ional A��o-Reg�e��i�e model� (CAR) � � · CAR model� are a cla�� of �pa�ial model� ��ed �o e��ima�e �pa�ial a��ocorrela�ion. · The�e model� are �idel� ��ed in Ecolog�, Economic� and Epidemiolog�. · CAR �a� �r�� de�eloped b� J�lian Be�ag in hi� no� cla��ic 1974 paper Spa�ial �n�e�ac�ion and �he S�a�i��ical Anal��i� of La��ice S���em� . 24/43 /
CAR ��ec�f�ca����� � · Single aggregated measure per spatial unit, it can be continuous, binar� or discrete count. Example: Number of car accidents at the count� level. · Finite set of non-overlapping spatial units. · For spatial units, the relationship is de�ned in terms of adjacenc�. 25/43 /
CAR m�del � Let N be the total number of spatial units from a region. A neighbor relationship is de�ned as where . This relationship is s�mmetric (i.e if ) but not re�e�ive (i.e. a region cannot be neighbor of itself). 26/43 /
Adjacenc�! There are two matrices describing di�erent measures of adjacenc� in this model. 1) Adjacenc� matri� , de�ning � neighbor relationship. � 27/43 /
Adjacenc�! There are two matrices describing di�erent measures of adjacenc� in this model. 1) Diagonal matrix , de�ning number � of adjacent units. � 28/43 /
CAR m�del S�a�ial in�e�ac�ion be��een a�eal �ni�� i� modelled c��di�i��a��� a� a no�mall� di���ib��ed �andom �a�iable, �e��e�en�ed b� �he -leng�h �ec�o� (i.e. ). The�efo�e, �he condi�ional di���ib��ion of EACH i� de�ned a� follo��, �he�e i� �he �eigh�ed �al�e� of �he neighbo��. F�om Bane�jee, Ca�lin, and Gelfand, 2004, �ec. 3.2, i� follo�� �ha� �he join� di���ib��ion 29/43 /
CAR m�del Recap! · : between 0 and 1, it represents the strength of the spatial association, with 0 meaning spatial independence. · D is our diagonal matrix. · W is the adjacenc� matrix. 30/43 /
� � Wha� is �he in�risic par� of �he In�rinsic A��o-Regressi�e models (ICAR)? 31/43 /
The in��in�ic c�ndi�i�nal a����eg�e��i�e (�CAR) The di�e�ence be��een CAR and �CAR i� �ha� �he �a�ame�e� i� �e� �o 1. · = 1 · D i� o�� diagonal ma��i�. · W i� �he adjacenc� ma��i�. Ho�e�e�, �e��ing c�ea�e� a challenge beca��e become� a �ing�la� ma��i� (i.e. non-in�e��ible). Thankf�ll�, incl�ding �he con���ain� �ol�e� �hi� challenge. 32/43 /
Pai��i�e de�i�a�ion �CAR c�����e�� �� ��e� de��ed a� f������, a�d af�e� ���e a�geb�a, ��e ��g ���bab����� de����� bec��e�: 33/43 /
Stan � S�an i� an open-�o�rce probabili��ic programming lang�age. ��'� �ri��en in C++ and and genrall� �peaking, i� i� ��ed �o �pecif� Ba�e�ian ��a�i��ical model�. S�an e��ima�e parame�er� b� calc�la�ing �he l�g ���babili�� de��i�� . (Tr� m�l�ipl�ing a large n�mber of ob�er�a�ion� �i�h �in� n�mber�, �o� �ill q�ickl� r�n in�o n�merical error�.) 34/43 /
S�a� m�del ����c���e // T�� ����� ���� �� � ������ '�' �� ������ 'N'. ���� � ���<�����=0> N; �������N� �; � // T�� ���������� �������� �� ��� �����. ���������� � ���� ��; ����<�����=0> �����; � // T�� ����� ����� '�' �� �������� ����������� ���� ���� '��' // ��� �������� ��������� '�����'. ����� � � � ������(��, �����); � 35/43 /
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