Siic 504 0. Aing Normliy Gry W. Ohlr School of Siic 33B For Hll 6-65-557 gry@.umn.u Mny procur um normliy. Som procur fll pr if h rn norml, whr ohr cn k lo of bu n kp going. In ihr c, i nic o know how clo o norml h r. Aing normliy i n xmpl of goon-of-fi problm. Goon-of-fi i ifficul problm. Thr r mny pproch. W will mphiz grphicl pproch n h numricl follow-on o h grph. Sr wih h univri c. Q-Q plo. Q-Q i hor for qunil-qunil. Whn ing normliy, h r lo known norml probbiliy plo n (loclly) rnki plo. Mk pir, whr i h h qunil of om horicl iribuion, n i h h qunil of h. Thn mk plo of h pir. If h com from h horicl iribuion, h poin on h plo houl fll on lin wih lop n inrcp 0. Plo will b linr if qunil r cl qunil plu conn:. +,.-/ Uully w choo lvl omhing lik or! #"%$ or &' ( *) for " " ", o h h qunil r ju h orr 035476# 80:9;4<6 6# >0@?A4 5 "=" ". For coninuou horicl iribuion wih cumuliv B, h horicl qunil r ju BDC =E. I gnrlly mk lil iffrnc which vrion of AE w u. Cm> r("",r,r,h,h,u,u) R from fil "/crom/t-8.dat" Column v REAL vcor r Column v REAL vcor r Column 3 v REAL vcor h Column 4 v REAL vcor h Column 5 v REAL vcor u Column 6 v REAL vcor u Cm> q <- invnor((run(5)-.5)/5) Cm> q <- invnor(run(5)/6) Cm> q3 <- invnor((run(5)-3/8)/(5.5)) Cm> cor(hconc(q,q,q3)) (,) 0.999 0.9999 (,) 0.999 0.9996
(3,) 0.9999 0.9996 Cm> cor(hconc(q3,rnki(q3))) (,) (,) Cm> or <- gr(r) Cm> plo(q3,r[or], xlb:"norml qunil", ylb:"or ", il:"dominn riu"). Dominn riu o r 0.9 0.8 0.7 0.6 0.5 - -.5 - -0.5 0 0.5.5 norml qunil Cm> plo(rnki(h),h, xlb:"norml qunil", ylb:"or ",il:"humru")
Humru. o r.8.6.4 - -.5 - -0.5 0 0.5.5 norml qunil Cm> yll <- hconc(r,r,h,h,u,u) Cm> yll <- or(yll) Cm> chplo(q3,yll,lin:t, xlb:"norml qunil", ylb:"or ",il:"all ") o r All 3. 3 3 34 4.8.6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 3 3 3.4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4. 3 0.8 0.6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 56 56 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 - -.5 - -0.5 0 0.5.5 norml qunil A of normliy, conir h corrlion of h norml qunil n h qunil. Lrg vlu of h corrlion r conin wih normliy; mll vlu r inconin. Cm> cor(q3,yll)[,] (,) 0.948 0.970 0.983 0.99 (,6) 0.984 0.995 3
How mll i oo mll? No impl nwr, o imul h iribuion. Cm> r <- rp(0,0000) Cm> for(i,run(0000)) x <- or(rnorm(5)) r[i] <- cor(q3,x)[];; Cm> min(r) () 0.8655 Cm> mx(r) () 0.99734 Cm> hi(r,run(.85,,.005)) 50 Hiogrm of r wih ol r 40 D n i y 30 0 0 0 0.86 0.88 0.9 0.9 0.94 0.96 0.98 r Cm> obr <- cor(q3,yll)[,-];obr (,) 0.948 0.970 0.983 0.99 (,6) 0.984 0.995 Cm> um(r<obr)/0000 (,) 0.03 0.44 0.47 0.86 0.503 (,6) 0.984 Th r h imul p-vlu for ing normliy for h ix vribl. I m no complly hppy wih mny of normliy, incluing hi on. Whn i mll, hy hv low powr. 4
- ' " Whn i big, hy cn c viion from normliy h migh b f o ignor. Thy n o b u in conjucion wih jugmn n ohr mho, uch mn of h QQ plo. Wh o w o for mulivri norml? Uul pproch i o if propri known o hol for MVN hol for h. Norml mrginl, norml coniionl, norml linr combinion, linr rgrion of ch vribl on ohr vribl, conn coniionl vrinc, chi-qur inc, c. Whn w normliy for ch vribl, w r ing mrginl normliy. Rcll h p-vlu: Cm> um(r<obr)/0000 (,) 0.03 0.44 0.47 0.86 0.503 (,6) 0.984 Shoul w rjc normliy bcu vribl on h p-vlu of.03? No ncrily. W hv mulipl ing iuion (lo cll mulipl comprion or imulnou infrnc). In hi problm w hv null hypoh H E (normliy of vribl ) n n ovrll null hypohi H which i ru if ll h H E r ru. If w ch H E lvl, hn 6 6 rjc H n o b clor o for mll, bu high corrlion mk clor o. Th Bonfrroni jumn y o rjc H if h mll iniviul p-vlu i l hn <, or quivlnly if im h mll p-vlu i l hn, whr i h numbr of. Th Bonfrroniz h. In our xmpl " ',( i no vry mll, n w woul no k h mrginl rul rong vinc gin mulivri normliy. On common rcommnion i o chck h normliy of, whr i h mrix of ignvcor for, h vrinc mrix of. Thi ro h o x of h llip n o norml ing mrginlly own ch xi. Th wor llip i in quo bov, bcu w migh no hv llipicl poin clou for nonnorml. Cm> V <- b(yll,covr:t) Cm> U <- ign(v)$vcor Cm> w <- yll%*%u; w <- or(w) Cm> obrw <- cor(q3,w)[,-];obrw (,) 0.990 0.968 0.985 0.970 0.987 (,6) 0.986 Cm> um(r<obrw)/0000 (,) 0.808 0. 0.57 0.49 0.650 (,6) 0.605 5
E C If N <, hn & If i lrg, hn 9 + E!C E houl lo b pproximly 9 9. U Q-Q plo wih E n horizonl vlu h prcn poin of 9. Th qur roo of h vlu will omim work br for mll. Cm> <- icomp(yll); () 8.866 7.459 0.848 3.59 4.374 (6) 0.83 3.569 3.08 3.54.600 ().570 9.40.897.74 5.59 (6) 7.5 6.93.560 7.74 7.70 ().057 7.345 3.9 3.607 7.4 Cm> <- or() Cm> q <- invchi((run(5)-.5)/5,6) Cm> plo(q,, xlb:"chi-qur wih 6 f qunil", ylb:"qur inc", il:"qq inc plo for bon ", how:f) Cm> lin(vcor(0,5),vcor(0,5), how:f) Cm> howplo(xmin:0,ymin:0) 9 4 QQ inc plo for bon q u r i n c 0 8 6 4 0 0 4 6 8 0 4 chi-qur wih 6 f qunil 6
Cm> plo(qˆ.5,ˆ.5, xlb:"chi wih 6 f qunil", ylb:"inc", il:"qq inc plo for bon ", how:f) Cm> lin(vcor(0,4),vcor(0,4), how:f) Cm> howplo(xmin:0,ymin:0) 3.5 3 QQ inc plo for bon i n c.5.5 0.5 0 0 0.5.5.5 3 3.5 chi wih 6 f qunil W migh wn o buil b on hi plo, bu w cn u corrlion. W rlly wn i o b on h lin wih inrcp 0 n lop. Try viion from h lin, or wigh viion from h lin. For xmpl, (-q)ˆ or (-q)ˆ/q. I upc h lrgr orr iic hv mor vrinc, o I my wn o ownwigh hm. Myb iviing by q (mor or l h xpc vlu) will work. Smpl bunch of chiqur, orr hm, n compu vrinc. Cm> D <- mrix(invchi(runi(5000),6),5) Cm> D <- or(d) Cm> D <- D Cm> b(d,vr:t) () 0.86 0.39 0.340 0.375 0.40 (6) 0.44 0.448 0.47 0.498 0.549 () 0.58 0.6 0.654 0.700 0.785 (6) 0.833 0.908.086.90.36 ().658.044.886 4.467 0.4 7
Cm> b(d,vr:t)/q () 0.5 0.86 0.54 0.46 0.38 (6) 0.3 0.7 0.3 0. 0.4 () 0.3 0. 0. 0.3 0.30 (6) 0.30 0.34 0.50 0.54 0.65 () 0.87 0. 0.7 0.370 0.673 Cm> b(d,vr:t)/qˆ () 0. 0.06 0.070 0.057 0.047 (6) 0.04 0.036 0.03 0.09 0.08 () 0.06 0.04 0.03 0.0 0.0 (6) 0.00 0.00 0.0 0.00 0.00 () 0.0 0.0 0.05 0.03 0.045 Cm> D <- D Cm> ou <- um( (D-q)ˆ/q) Cm> ou <- vcor(ou) Cm> um( (-q)ˆ/q) ().009 Cm> lngh(ou) () 000 Cm> um(ou >.009)/000 () 0.8 Thi ming roun wih q cling fcor i no h b w cn o. W im h vrinc mrix of h orr iic uring our imulion. W cn mk br uing ho vrinc. 8