Biplots. Some ecological data to illustrate regression biplots

Transcription

1 Biplot The biplot eten the iea of a imple atterplot of two variable to the ae of man variable, with the obetive of viualizing a maimum amount of information in the ata a poible. We firt look at how a regreion moel with two eplanator variable an be epite a a ingle point, an then eten thi iea to prinipal omponent anali an orreponene anali Some eologial ata to illutrate regreion biplot SITE SPECIES COUNTS ENVIRONMENTAL VARS NO. a b e Polln Depth Temp. Semnt S C C S C G S C C S C S S C C G C G S S C S G C G S G G S G

2 (pollution Simple linear regreion peie veru pollution ( peie = (R 2 = (pollution Multiple linear regreion = Regreion moel i a (hperplane

3 * Multiple linear regreion, variable tanarize * * * =.446* +.47* * * Eplanator variable an an repone variable tanarize * Small etour: R oe to o regreion > ummar(lm(~+ Call: lm(formula = ~ + Reiual: Min 1Q Meian Q Ma Coeffiient: Etimate St. Error t value Pr(> t (Interept ** * --- Signif. oe: ***.1 **.1 * Reiual tanar error: 5.2 on 27 egree of freeom Multiple R-quare:.44, Aute R-quare:.4 F-tatiti: 1.6 on 2 an 27 DF, p-value:.1

4 From uual regreion oeffiient to tanarize one * * * ( ( ( ( ( ( b b b b b a b a D R oe to alulate tanarize regreion oeffiient > lm(~+$oeffiient (Interept > lm(~+$oeffiient[2]*(/( > lm(~+$oeffiient[]*(/(.471

5 Another view of regreion & preition Stanarize Pollution (Y* * =.446* +.47* Stanarize Depth (X* variane eplaine (R 2 : 44.2% 2 * * (pollution * Regreion biplot e a * b 26 1 (epth variane eplaine: a: 52.% b:.1% : 21.% : 44.2% e: 2.5% overall: 6.% ignifiane: a: ** b: ** : * : ** * e: * *

6 Regreion biplot: ummar A regreion moel an be repreente a a point vetor in pae (in thi ae two-imenional pae beaue two eplanator variable Reontrut the ata from proetion of ae onto variable iretion, but onl a well a meaure b R 2 If an are unorrelate, then the regreion oeffiient are ut orrelation oeffiient (in thi ae, of the peie with the eplanator * variable e b 2 a * What happen for three preitor? Eah regreion moel an be repreente a a point vetor in three-imenional pae. Reontrut the ata from proetion of ae onto variable iretion, but onl a well a meaure b R 2 ; in thi eample the inreae in eplaine variane from two-imenional to threeimenional (aing temperature a an eplanator variable i from 6.% to 7.1%, hene temperature i eplaining ver little etra variane. There will be a partiular orientation of the vetor that give maimum variane eplaine in the two-imenional proetion

7 Viualizing trivariate ontinuou ata (repeat Continuou variable X1 Purhaing power/apita (euro X2 GDP/apita (ine X inflation rate (% Countr X1 X2 X Be Belgium De Denmark Ge German Gr Greee Sp Spain Fr Frane 1..2 Ir Irelan It Ita l Lu Luembourg Ne Netherlan Po Portugal UK Unite Kingom Viualizing trivariate ontinuou ata (repeat Continuou variable X1 Purhaing power/apita (euro X2 GDP/apita (ine X inflation rate (% Countr X1 X2 X Be Belgium De Denmark Ge German Gr Greee Sp Spain Fr Frane 1..2 Ir Irelan It Ital Lu Luembourg Ne Netherlan Po Portugal UK Unite Kingom

8 Viualizing trivariate ontinuou ata (repeat Continuou variable X1 Purhaing power/apita (euro X2 GDP/apita (ine X inflation rate (% Countr X1 X2 X Be Belgium De Denmark Ge German Gr Greee Sp Spain Fr Frane 1..2 Ir Irelan It Ital Lu Luembourg Ne Netherlan Po Portugal UK Unite Kingom or X1 X2 X X X X GDP INF PP General priniple for aing biplot ae For a ample b variable matri ituate the ample in ome pae of repreentation. Thi will uuall be the lowimenional ubpae onto whih the ample point have been proete. In the peifi ae of CA it oul be the pae of the row profile, for eample, with repet to prinipal ae. Perform the regreion of eah entre variable on the (prinipal ae, an raw the variable a a vetor uing the regreion oeffiient a oorinate. In CA where the row are weighte, the regreion ha to be performe with weight. The vetor iniate the iretion of maimum lope of the regreion plane (i.e., the graient, an it ontour are perpeniular to thi vetor. Thi mean the vetor an be alibrate in unit of the (preite variable. Speial ae: ample are tanarize on unorrelate prinipal ae an variable i tanarize, then the regreion oeffiient are ut the variable ai orrelation.

9 Viualizing trivariate ontinuou ata (repeat Continuou variable X1 Purhaing power/apita (euro X2 GDP/apita (ine X inflation rate (% Countr X1 X2 X Be Belgium De Denmark Ge German Gr Greee Sp Spain Fr Frane 1..2 Ir Irelan It Ital Lu Luembourg Ne Netherlan Po Portugal UK Unite Kingom Comp Po Gr Sp Be It De UK Fr Ir Ge Ne X X Comp.1 X1 X1 X2 Lu R: biplot(prinomp(eu, or=t, ale= Ammetri map in CA are biplot The CA moel bae on the SVD of i: p i r i T D 1/ 2 r ( P r SC: D UD whih, from the row profile point of view, an be written a either of thee: p r i i k f ik S 2 F D 1/ PC: r UD α 2 G D 1/ VD α ( 1 kuikv k k k p r i i Φ D Γ D 1/ 2 r 1/ 2 U V pi f ( k ri 1/ 2 k ik f ( 1/ 2 k ik k V T lai ammetri map Gabriel biplot Greenare tanar (or ontribution biplot R a pakage: map= rowprinipal map= rowgab map= rowgreen

10 CA ammetri map of author ata 4.6 (1.7 % z 2-2 k w h g n t o e r f b a l ui m p v q.766 (4. % CA ontribution biplot of author ata (eah letter verte point i multiplie b the quare root of it ma

11 Dnami tranition to CA ontribution biplot Ammetri CA biplot of bentho.24 (26.1%.246 (1.4% eplaine inertia = 57.5%

12 Stanar CA biplot of bentho (eah peie point i multiplie b the quare root of it ma.24 (26.1%.246 (1.4% eplaine inertia = 57.5% Tranition to tanar biplot