Example: PCA of a matrix of fatty acids in margarines 1.0 C12 0.8 - - PowerPoint PPT Presentation

R algorithms for the calculation of markers to be used in the construction of predictive and interpolative biplot axes in routine multivariate analyses

• M. Rui Alves 1,2 and M. Beatriz Oliveira 2

(1) Escola Superior de Tecnologia e Gestão, IPVC, Viana do Castelo, Portugal

(2) REQUIMTE, Faculdade de Farmácia, Universidade do Porto, Porto, Portugal

Example: PCA of a matrix of fatty acids in margarines

Principal component 1

Principal component 2

A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2C3 C4 C5 D1 D2 D3 D4 D5 E1 E2 E3 E4 E5 F1 F3 F4 F5 G1 G2 G3 G4 G5 H1 H2 H3 H4 H5

• 10

10 20 30 40

• 40
• 30
• 20
• 10

10 20 30 Latent vector 1

Latent vector 2

• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 C18:2cc (-0.75 ; -0.40) Ttr (0.19 ; -0.28) C18:1c (0.42 ; -0.43) C16 (0.46 ; -0.01) C12 (-0.08 ; 0.72) C14 (-0.01 ; 0.19) C18 (-0.04 ; -0.10)

Journal of Chemometrics (2003), 17, 594-602 useR! 2006

Examples of matrices (fatty acids in margarines) matrix of latent values part of matrix of components

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Gower’s concepts for biplots

Predictive biplots: Interpreteing results in terms of initial variables Interpolative biplots: Positioning new units in pre-existing graphs, mainly in routine quality control

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PCA biplots: fatty acids in margarines

Predictive biplots Interpreting results Interpolative biplots Positioning new units

Journal of Chemometrics (2001), 15, 71-84 Journal of Chemometrics (2003), 17, 594-602 useR! 2006

Problems on computation

• Gower e Hand, in their book Biplots, say:

“(...) The main computational problems [of biplots] are in integrating different bits of available software and in finding good portable graphic facilities.

• To work with bipots we used:

– Genstat 5.3.1 package to develop the algorithms and carry out the analyses – Statistica for Windows to draw all the graphs based on the converted ASCII outputs produced by Genstat

• We started to work with R in an attempt to provide a final, complete

and more covenient solution

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Projection of markers

Projection of Markers

[ μ p− 1

x p s p

− 1][e p t V ρ − 1 t

][e p

t V ρ − 1 t

V ρ

− 1e p] − 1

[ μ p− 1

x p s p

− 1][e p t V ρ] useR! 2006

Building biplots (PCA of fatty acids in sunflower oils)

Componente principal 1

Componente principal 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7 Componente principal 1

Componente principal 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 51 52 53 54

56 58 60 62 64

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7

• 7
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• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7 C 18:2cc componente 1

componente 2

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10 variáveis muito longas 5.0 5.5 6.0 6.5 7.0 0.05 0.10 0.15 0.20 24.5 26.5 28.5 30.5 56 58 60 62 64 0.28 0.30 0.32 0.34 0.36 C 16 C 16:1 C 18:1 C 18:2 C 20

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Strategies for the automation of the process

Two strategies were devised: The first, more obvious, is to create an object containing the markers, the axis, the scale values and variable’s name.

• Project the object.
• Live it in the graph if it fits well
• Delete it otherwise (too long or too short vectors)
• This procedure would require interactivity facilities

The second, more mathematical:

• Find a way for the evaluation of variables’ predictive powers
• Leave in the graph variables displaying high predictive power
• Delete them otherwise
• Draw the graphs (only with automatically selected variables)

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Algorithm for the selection based on predictive powers un  pred =read in the graph=0,5 un inicial =initial value=0,4 erro

u n=0,5− 0,4=0,1

erro

 x p=1

N ∑n=1

N

un pred− unincial s p Define a tolerance value if error

x p¿ tolerance⇒accept x p

if errorx p¿ tolerance⇒reject x p

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Pr ediction=X ⊗V 2dim ⊗V [k ,]

2 dimt

UnitsStdE=abs  X [ ,k ]− Pr ediction MeanStdE=N − 1×1t ⊗UnitSdtE

Evaluation of predictive power and decision

if MeanStdETolerance  project var iable else pr int ital deleted 

[ μ p− 1

x p s p

− 1][e p t V ρ − 1 t

][e p

t V ρ − 1 t

V ρ

− 1e p] − 1

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for (i in 1:(Q-1)) { for (j in (i+1):Q) { print("component"); print(i); print("component"); print(j) # latent variables for a pair of components only V2Dim[,1] <- RedV[,i] V2Dim[,2] <- RedV[,j] MStdE <- list() for (k in 1:P) { # evaluation of variable' s predictive power print("variavel") ; print(k) VarDir <- matrix(data=(V2Dim[k,]),nrow=1,ncol=2) Pred <- XStd %*% V2Dim %*% t(VarDir) UnitStdE <- abs(XStd[,k] - Pred) VarE <- (t(ColN1s) %*% UnitStdE)/N MStdE[[k]] <- VarE print(MStdE[[k]]) if (MStdE[[k]] < Tolerance) { Zeros <- matrix(0,c(P,1)) Zeros[k,] <- 1 Adj1 <- t(Zeros) %*% V2Dim %*% t(V2Dim) %*% Zeros Adj2 <- (ColM1s %*% (1/Adj1) %*% t(Col2_1s)) EPred <- Adj2 * (ScMat[[k]] %*% V2Dim) print(EPred) plot(EPred[,1],EPred[,2]) } else print("deleted") } } }

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[1] "component" [1] 1 [1] "component" [1] 2 [1] "variavel" [1] 1 [,1] [1,] 0.6583496 [1] "deleted" [1] "variavel" [1] 2 [,1] [1,] 0.5805472 [1] "deleted" [1] "variavel" [1] 3 [,1] [1,] 0.2413512 [,1] [,2] [1,] 2.6066496 4.1854190 [2,] 1.4223328 2.2837971 [3,] 0.2380160 0.3821751 [4,] -0.9463008 -1.5194468 [5,] -2.1306176 -3.4210688

Example of results provided by the algorithm

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What needs to be done and acknowledgments

• Reminding Gower and Hand: “(...) The main computational

problems [dos biplots] are in integrating different bits of available software and in finding good portable graphic facilities.

• The algorithms are done, although probably not in the best or more

efficient way, but they provide correct results

• Produce biplots by making the graphs of components and merging

the graphs of individual variables as objects

• Multivariate analyses can therefore be made fully automatic,

including the interpretation processes

• Paul Murrell and Robert Gittins provided valuable information on

graphs and strategies to finalize the work, but unfortunately we did not find the time necessary to do it

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Thank you

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