【正文】 39。39。39。???Spatial Data Analysis Modelling of the autoregressive process Ordinary least squares errors AR1 Onedimensional autocorrelated ponent in field order ???????????????1000010000100001)( 22 ee IV a r ??eRiii ???? ?? ? 100 . 20 . 40 . 60 . 810 2 4 6 8 10D i s t a n c eAutocorrelationρ= ρ= ρ= Spatial Data Analysis Modelling of the autoregressive process AR1 Onedimensional autocorrelated ponent in field order iii ???? ?? ? 1????????????????????1111121222?????????????nneI???????????RSpatial Data Analysis Modelling of the autoregressive process Twodimensional separable spatially autocorrelated ponent ? ? ? ?rrcceR ??? ???? 2? ?cc ??is a firstorder autoregressive correlation matrix with an autocorrelation ρ: Spatial Data Analysis Twodimensional separable spatially autocorrelated ponent Ending up a very big matrix of nr*nc rows and nr*nc columns ??????????????????????????????????????11111111121212122??????????????????????????ccrrncccccncccnrrrrrnrrr??????????????????RSpatial Data Analysis Modelling of the autoregressive process “Nugget” effect unstructured environmental correlation: ? ? ? ?ccrrnnecrIIR ???? ? ???? 22Spatial Data Analysis Modelling of the autoregressive process RAD195 trial Dothistroma infection data (010 scores) surface plot 8 1 06 84 62 40 2Spatial Data Analysis !PART 1 Dothitr_0400 ~ mu !r Rep Plot Genotype_id !f mv 1 2 0 Prow Prow IDEN Ppos Ppos IDEN !PART 2 Dothitr_0400 ~ mu !r Rep Plot Genotype_id !f mv 1 2 0 Prow Prow AR1 Ppos Ppos AR1 !PART 3 Dothitr_0400 ~ mu !r Rep Plot Genotype_id units !f mv 1 2 0 Prow Prow AR1 Ppos Ppos AR1 Spatial Data Analysis T he r e s ult s from th e th r e e models fo r r a diata Dot hi dataLogL V r e p V plot V a dd itiv e V e r r or V s patial A Rr ow A r c ol1 1 5 4 5 .4 8 .8 4 3 .9 9 1 6 .1 1 1 .42 1 5 7 9 .5 1 8 .0 3 0 .4 2 2 2 .6 1 0 .9 0 .4 2 0 .6 63 1 5 8 2 .1 8 .0 5 0 .6 4 1 2 .9 7 .8 5 9 .7 1 0 .5 2 0 .7 2Spatial Data Analysis Model 1 gives a variogram that is flat which indicates that the residuals have little spatial structure LogL V r e p V plot V a dditiv e V e r r or V s pa tia l A Rr ow A r c ol1 1 5 4 5 .4 8 .8 4 3 .9 9 1 6 .1 1 1 .4Spatial Data Analysis Making the R matrix have an autoregressive structure (model 2) gives a considerable model improvement, with the autocorrelations are low () LogL V r e p V plot V a dd itiv e V e r r or V s patial A Rr ow A r c ol1 1 5 4 5 .4 8 .8 4 3 .9 9 1 6 .1 1 1 .42 1 5 7 9 .5 1 8 .0 3 0 .4 2 2 2 .6 1 0 .9 0 .4 2 0 .6 6Spatial Data Analysis Adding the independent error term (units) further improves the fit of the model (Model 3). The autocorrelations increased slightly (), and the additive variance returns to its previous levels LogL V r e p V plot V a dd itiv e V e r r or V s patial A Rr ow A r c ol1 1 5 4 5 .4 8 .8 4 3 .9 9 1 6 .1 1 1 .42 1 5 7 9 .5 1 8 .0 3 0 .4 2 2 2 .6 1 0 .9 0 .4 2 0 .6 63 1 5 8 2 .1 8 .0 5 0 .6 4 1 2 .9 7 .8 5 9 .7 1 0 .5 2 0 .7 2Spatial Data Analysis Spatial Data Analysis M o d el w it h o u t spat ial adju stm entS ou r c e Model term s G am m a Com po ne nt Com p/S E % CRep 6 6 4E 03 0 PP l ot 192 192 9E 03 0 PG en oty pe _i d 1379 1379 1E 02 0 PV aria nc e 1344 1171 1 4E 02 0 PM o d el w it h spat ial adju stm entS ou r c e Model term s G am m a Com po ne nt Com p/S E % CRep 6 6 3E 03 0 PP l ot 192 192 0E 02 7E 04 0 PG en oty pe _i d 1379 1379 6E 02 0 PV aria nc e 1344 1171 1 0E 02 0 PRes i du al A R= A uto R 16 0 URes i du al A R= A uto R 84 0 UM o d el w it h spat ial adju stm ent and ind epen d ent er r o rS ou r c e Model term s G am m a Com po ne nt Com p/S E % CRep 6 6 5E 03 0 PP l ot 192 192 4E 02 5E 04 0 Pun i ts 1344 1344 5E 03 0 PG en oty pe _i d 1379 1379 9E 02 0 PV aria nc e 1344 1171 1 1E 03 0 PRes i du al A R= A uto R 16 0 URes i du al A R= A uto R 84 0 UHeritability was improved 18% from to . 10. Multiple Site Analysis ? Multiple sites usually involve heterogeneous error variances and G by E interaction. In order to incorporate the heterogeneous error variances, variance structure for R should be defined. , 212121????????????????????????????????????eeeuuuyyYwith y1, y2 as vectors of individual site 1 and 2 21A? 212A? is additive variance in site 1 and is additive covariance between sites 1 and 2. AGAG 0 ??????????????????????????????????2222121212212121212121****AAAAAAAAAAAA????????? ?? iRRMultiple Site Analysis ? We use diallel as example for three sites model. 11. Multiple Trait Analysis ? Multiple traits usually involve heterogeneous error variances and geic correlation amongtraits. In order to incorporate the heterogeneous error variances and geic covariance, variance structure for error R and covariance G should be defined. , 212121????????????????????????????????????eeeuuuyyY???????????3211111tttyyyyan RIRRGAG ????? ? ? ia nd0where G0 is geic variance and covariance matrix among traits, and Ra = Ri is the amongtraits residual variance and co