Y+ = (YTY)-1 YT Y+ = (YTY)-1 YT

YTY is non-singular and squaredYTY is non-singular and squared??

(Full rank)(Full rank)

Inversion is possible if:Inversion is possible if:

=Y Y+c=Y Y+cc

Y’*Y is (33) but:

rank(Y’*Y)=2 !

rank(Y) = 2 =min(#r,#c)

=> Y is full rank

rank(Y) = 2 =min(#r,#c)

=> Y is full rank

Y should have:

#rows > #col.s

Y should have:

#rows > #col.s

11

Y should not be:

Rank deficient

Y should not be:

Rank deficient

22

Column are linearly dependentColumn are linearly dependent

!!

3 compon.s, 4 samples3 compon.s, 4 samples

4 wavel.s, 4 samples4 wavel.s, 4 samples

rank(x)=min(r(c),r(s))=3

rank(x) < min(# r, #c) =4

=> x is rank deficient

pinv can be performed when x is rank deficient..

pinv

?X= I (not square and singular X)?X= I (not square and singular X)

svd & estimation of X using significant factors

svd & estimation of X using significant factors

?U**V*T=I?U**V*T=I

V**-1U*TU**V*T=IV**-1U*TU**V*T=I

pseudo inversepseudo inverse

pinv(X)= X+ = V**-1U*Tpinv(X)= X+ = V**-1U*T

U*TU*=IU*TU*=I

*-1*=I*-1*=I

V*V*T=I ?V*V*T=I ?

ksks kCkC

XX

= CC+X= CC+XX

|| -X|||| -X||X

Criterion for fitting

ks

Criterion for fitting

ks

Hard Model

Hard Model

X

Projection of X onto space of C

X = C S classicX = C S classic

1. # samp.s ≥ # compon.s 2. C : full rank

(rank(C)= #compon.s) (lin indep conc profiles)

1. # samp.s ≥ # compon.s 2. C : full rank

(rank(C)= #compon.s) (lin indep conc profiles)

ksks kCkCXX

Hard Model

Hard Model

Projection of X onto space of C

C = X Z inverseC = X Z inverse

= XX+C= XX+CCC || -C|||| -C||C

Criterion for fitting

ks

Criterion for fitting

ks 1. # samp.s ≥ # wavel.s 2. X: full rank (rank(X)= # wavel.s)

- variab. Select. - Factor based methods

1. # samp.s ≥ # wavel.s 2. X: full rank (rank(X)= # wavel.s)

- variab. Select. - Factor based methods

!!!!

X is usually near to singular…

X is usually near to singular…

# samples < # wavel.s # wavel.s > # compon.s

# samples < # wavel.s # wavel.s > # compon.s

XX+XX+

=U**V*TV**-1U*T (signif factors)

=U***-1U*T

=TT+

=U**V*TV**-1U*T (signif factors)

=U***-1U*T

=TT+

ksks kCkC

XX

Hard Model

Hard Model

Projection of C onto space of T

C = T RZ inverseC = T RZ inverse

= TT+C= TT+CCC || -C|||| -C||C

Criterion for fitting

ks

Criterion for fitting

ks 1. # samp.s ≥ # PCs 2. T: full rank (lin indep col.s)

1. # samp.s ≥ # PCs 2. T: full rank (lin indep col.s)

SVDSVD

TT

ksks kCkC

XX

Hard Model

Hard Model

Projection of T onto space of C

T = C R classicT = C R classic

1. # samp.s ≥ # compon.s 2. C : full rank (lin indep.

conc prof.s)

1. # samp.s ≥ # compon.s 2. C : full rank (lin indep.

conc prof.s)

TTSVDSVD

T

= CC+T= CC+TT|| -T|||| -T||T

Criterion for fitting

k

Criterion for fitting

k

= CC+X= CC+XX

= XX+C= XX+CC

= CC+T= CC+TT

= TT+C= TT+CC pcrC (Target Transform)

pcrC (Target Transform)

ccrX (classical curve

resolution)

ccrX (classical curve

resolution)

pcrTpcrT

ccrCccrC

T J Thurston, R G Brereton Analyst 127, 2002, 659.T J Thurston, R G Brereton Analyst 127, 2002, 659.

The considered kinetic system:The considered kinetic system:

Second order consecutiveSecond order consecutive

0 5 10 15 200

0.5

1

1.5

Time (min)

Conce

ntr

ati

on (

mic

roM

)

A+B CDA+B CD

Spectral meas. In 101 wavel.s each 30 sec (41 times)

Spectral meas. In 101 wavel.s each 30 sec (41 times)

r(C)=3# indep react.s +1

r(C)=3# indep react.s +1

420 440 460 480 500

0

10

200

0.5

1

1.5

2

Wavelength (nm)Time (min)

Abs

orba

nce

ccrC:ccrC:

X1=[X(:,50) X(:,70) X(:,90)]X1=[X(:,50) X(:,70) X(:,90)]

=X1*inv(X1‘*X1)*X1'*C=X1*inv(X1‘*X1)*X1'*CC

=X*inv(X‘*X)*X'*C=X*inv(X‘*X)*X'*CC 11

=X*pinv(X)*C=X*pinv(X)*CC

X(41x101) 41 samples r(X)=3 101 wavel.s

X(41x101) 41 samples r(X)=3 101 wavel.s

1 # samp.s ≥ # wavel.s 2 rank(X)= # wavel.s

1 # samp.s ≥ # wavel.s 2 rank(X)= # wavel.s

Information content !

Information content !

ccrX:

=C*inv(C’*C)*C’*X=C*inv(C’*C)*C’*XX C1=C(:,2:4)C1=C(:,2:4)

=C1*inv(C1’*C1)*C1’*X=C1*inv(C1’*C1)*C1’*X

=C*pinv(C)*X=C*pinv(C)*X X

X

1. # samp.s ≥ # compon.s 2. rank(C)= #compon.s

1. # samp.s ≥ # compon.s 2. rank(C)= #compon.s

C(41x4) 41 samples r(X)=3 4 compon.s

pcrT:pcrT:

=C*inv(C’*C)*C’*T=C*inv(C’*C)*C’*TT C1=C(:,2:4)C1=C(:,2:4)

=C1*inv(C1’*C1)*C1’*T=C1*inv(C1’*C1)*C1’*TT

=C*pinv(C)*T=C*pinv(C)*TT

pcrC:pcrC:

=T*T’*C=T*T’*CC

1. # samp.s ≥ # PCs 2. rank(T)= # col.s (always it is so…)

1. # samp.s ≥ # PCs 2. rank(T)= # col.s (always it is so…)

Overlap effectOverlap effect

420 440 460 480 5000

0.2

0.4

0.6

0.8

1

wavelength (nm)

Abso

rbance

0 5 10 15 200

0.5

1

1.5

Time (min)

Conce

ntr

ati

on (

mic

roM

)

420 440 460 480 500

0

10

200

0.5

1

1.5

2

Wavelength (nm)Time (min)

Abs

orba

nce

+Rand noise

+Rand noise

0.36 0.38 0.4 0.42 0.440.09

0.095

0.1

0.105

0.11ccrX

noise 0.02

k1

k2

0.3 0.35 0.4 0.45 0.5

0.08

0.09

0.1

0.11

0.12 ccrCnoise 0.02

k1

k2

0.36 0.38 0.4 0.42 0.440.09

0.095

0.1

0.105

0.11pcrT

noise 0.02

k1

k2

0.3 0.35 0.4 0.45 0.5

0.08

0.09

0.1

0.11

0.12 pcrCnoise 0.02

k1

k2

Spectral overlap (in the presence of some noise) results in some deviation in the results

from

***C

methods

Spectral overlap (in the presence of some noise) results in some deviation in the results

from

***C

methods

Results from application of ***C and ***X methods

are different …

Results from application of ***C and ***X methods

are different …

One way to obtain more similar results from ***C and ***X methods are application of

constraints

One way to obtain more similar results from ***C and ***X methods are application of

constraints

Presence of

heteroscedastic noise

+ a heterosced. noise

+ a heterosced. noise

41 reaction times &101 wavelengths

41 reaction times &101 wavelengths

050

100020

400

0.5

1

1.5

2

reaction timeVariable No.

Ab

sorb

ance

0 20 40 60 80 100-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Variable No.

Ab

so

rban

ce

0.36 0.38 0.4 0.42 0.440.094

0.096

0.098

0.1

0.102

0.104

0.106

k1

k2

Inaccurate results from ccrX !Inaccurate results from ccrX !

0 20 40 60 80 100-0.05

0

0.05

Variable No.

Ab

sorb

ance weightsweights

weighted regression…weighted regression…

||W ( -X)||||W ( -X)||X

1/SD1

1/SD2

…

1/SDn

W =W =

Accurate results from weighted ccrX !

Accurate results from weighted ccrX !

0.3995 0.4 0.40050.0995

0.1

0.1005

k1

k2

n=50n=50

FSMWFA in daset 6

-7.5

-5.5

-3.5

-1.5

0.5

2.5

300 320 340 360 380 400 420 440 460 480 500

wavelength

log

(eig

valu

e)

Recognition of the presence of heterosc. noise

Recognition of the presence of heterosc. noise

FSMWFA

0 10 20 30 400.99

0.995

1

1.005

1.01

Sample number

Sam

plin

g er

ror

coef

ficie

nt

0

5

100 10 20 30 40 50

0.99

0.995

1

1.005

1.01

1.015

Non-random sampling error

Non-random sampling error

A more serious source

of error

A more serious source

of error

Square, symmetric,

But not diagonal W matrix:

Square, symmetric,

But not diagonal W matrix:

J Chemometr 2002, 16, 378.

R. Bro, N.D. Sidiropoulos, A.K. Smilde

J Chemometr 2002, 16, 378.

R. Bro, N.D. Sidiropoulos, A.K. Smilde

Maximum likelihood fittingMaximum likelihood fitting

Presence of non-random sampling error nS=0.005

Presence of non-random sampling error nS=0.005

0.3 0.4 0.5 0.6

0.075

0.08

0.085

0.09

0.095

0.1

0.105

k1

k2

|| -X|||| -X||X ||W ( -X)||||W ( -X)||X

0.3 0.4 0.5 0.6

0.075

0.08

0.085

0.09

0.095

0.1

0.105

k1

k2

Weighted regression

Weighted regression

ccrXccrX

J Chemom 2002, 16,387. R.Bro et alJ Chemom 2002, 16,387. R.Bro et al

Presence of unknown interference

Changing interference, drift , or shiftChanging interference, drift , or shift

400 420 440 460 480 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

absorbance of pure components

wavelength (nm)

abso

rban

ce

400 420 440 460 480 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

absorbance of pure components

wavelength (nm)

abso

rban

ce

0 5 10 15 200

0.5

1

1.5

concentration profiles

reaction time(min)

con

cen

trat

ion

(m

icro

M)

0 5 10 15 200

0.5

1

1.5

concentration profiles

reaction time(min)

con

cen

trat

ion

(m

icro

M)

400 420 440 460 480 5000

0.5

1

1.5

2

2.5

Data

Wavelength (nm)

Ab

sorb

ance

rank(Data)=4rank(Data)=4

0.38 0.385 0.39 0.395 0.4 0.4050.095

0.1

0.105

0.11

0.115

0.12

0.125

0.13

k1

k2

pcrTpcrT

0.39 0.395 0.4 0.4050.097

0.098

0.099

0.1

0.101

0.102

k1

k2

pcrCpcrC

0.385 0.39 0.395 0.4 0.4050.099

0.1

0.101

0.102

0.103

k1

k2

ccrXccrX

0.39 0.395 0.4 0.4050.097

0.098

0.099

0.1

0.101

0.102

0.103

k1

k2

ccrCccrC

Presence of shift or drift (a changing interference) results in

serious deviations in

***X

Methods

(but not in ***C methods)

Presence of shift or drift (a changing interference) results in

serious deviations in

***X

Methods

(but not in ***C methods)

Why?Why?

= CC+X= CC+XX = XX+C= XX+CC= CC+T= CC+TT

In the presence of shift, drift or changing interferences:

T or X space includes 1. the concentration changes according to the model 2. variations from shift, drift or

changing interference

C space includes only the concentration changes according to the model

In the presence of shift, drift or changing interferences:

T or X space includes 1. the concentration changes according to the model 2. variations from shift, drift or

changing interference

C space includes only the concentration changes according to the model

Projection of a larger space to a smaller one

Projection of a larger space to a smaller one

Projection of a smaller space to a larger one

Projection of a smaller space to a larger one

= TT+C= TT+CC

= TT+C= TT+CC

in the presence of unknown interference, drift or shift.

Target Transform (pcrC) is the most preferred method

Constant interferenceConstant interference

400 420 440 460 480 5000

0.2

0.4

0.6

0.8

1

1.2

1.4absorbance of pure components

wavelength (nm)

abso

rban

ce

400 420 440 460 480 5000

0.2

0.4

0.6

0.8

1

1.2

1.4absorbance of pure components

wavelength (nm)

abso

rban

ce

0 5 10 15 200

0.5

1

1.5

concentration profiles

reaction time(min)

con

cen

trat

ion

(m

icro

M)

0 5 10 15 200

0.5

1

1.5

concentration profiles

reaction time(min)

con

cen

trat

ion

(m

icro

M)

400 420 440 460 480 5000

0.5

1

1.5

2

2.5

Data

Wavelength (nm)

Ab

sorb

ance

rank(Data)=3 !rank(Data)=3 !

A+B CDA+B CD

0.399 0.3995 0.4 0.4005 0.4010.099

0.0995

0.1

0.1005

0.101

k1

k2

0.399 0.3995 0.4 0.4005 0.4010.099

0.0995

0.1

0.1005

0.101

k1

k2

ccrCccrC

0.399 0.3995 0.4 0.4005 0.4010.099

0.0995

0.1

0.1005

0.101

k1

k2

ccrXccrX

0.399 0.3995 0.4 0.4005 0.4010.099

0.0995

0.1

0.1005

0.101

k1

k2

pcrTpcrT

0.399 0.3995 0.4 0.4005 0.4010.099

0.0995

0.1

0.1005

0.101

k1

k2pcrCpcrC

A constant interference does not show any significant effect the

accuracy of ***X and ***C methods.

A constant interference does not show any significant effect the

accuracy of ***X and ***C methods.

Target test fittingTarget test fitting

From:

J Chemometr. 2001, 15, 511.

P.Jandanklang, M. Maeder, A. C. whitson

From:

J Chemometr. 2001, 15, 511.

P.Jandanklang, M. Maeder, A. C. whitson

-0.1 -0.05 0 0.05 0.10

5

10

15

20

25

30

35voltammograms

E

Un

itar

y cu

rren

t

Differential pulse

Voltammetry

Differential pulse

Voltammetry

Each voltammog. depends only on its own E1/2

Each voltammog. depends only on its own E1/2

3

2

3

2

MLLM

MLLM

MLLM

33

22

]][[

][

]][[

][

]][[

][

3

2

LM

ML

LM

ML

LM

ML

ML

ML

ML

Successive complexation:

0....))1((][)(][][ 211

11

TLnTLTMn

nnTLTMn

nn

n CCCnLCCLL

)][....][][1(

][][

)][....][][1(

1][

221

221

nn

nn

n

nn

LLL

LML

LLLM

nn

n LM

ML

LM

ML

]][[

][,.....,

]][[

][1

][....][][

][....][][

nTL

nTM

MLnMLLC

MLMLMC

Analyst , 2001 , 126 , 371-377Each concn. profile includes 1,…, n

Each concn. profile includes 1,…, n

0 20 40 60 800

0.2

0.4

0.6

0.8

1

Ctotal L

Cu

rren

t m

icro

A

-0.1 -0.05 0 0.05 0.10

5

10

15

20

25

30

35voltammograms

E

Un

itar

y cu

rren

t

-0.1 -0.05 0 0.05 0.10

5

10

15

20

25

30

35

Data

E

curr

ent

X

X=CSX=CS

X=UVT=TVX=UVT=TV

= VVT s= VVT s

= UUT c = TTT c= UUT c = TTT c

s

c

voltammogrvoltammogr

concn.concn.

For estimation of concn. profiles 1,…,n

(n parameters) should be optimized

simultaneously

For estimation of concn. profiles 1,…,n

(n parameters) should be optimized

simultaneously

1,…,n are dependent parameters

1,…,n are dependent parameters

Simultaneous optimization of n dependent nonlinear

parameters:

Simultaneous optimization of n dependent nonlinear

parameters:

• Simplex method.

• Levenberg-Marquardt

•…

• Simplex method.

• Levenberg-Marquardt

•…

estimation of

(E1/2)1, …, (E1/2 )n

values for voltammograms

estimation of

(E1/2)1, …, (E1/2 )n

values for voltammograms

(E1/2)1, …, (E1/2 )n

are independent parameters

(E1/2)1, …, (E1/2 )n

are independent parameters

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M

(E1/2)ML(E1/2)ML

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M (E1/2)ML2

(E1/2)ML2(E1/2)ML

(E1/2)ML

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M

(E1/2)ML(E1/2)ML

(E1/2)ML2(E1/2)ML2

(E1/2)ML3(E1/2)ML3

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M

(E1/2)ML(E1/2)ML

(E1/2)ML2(E1/2)ML2

(E1/2)ML3(E1/2)ML3

(E1/2)L(E1/2)L

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

r = || - s|| 0s

(E1/2)M(E1/2)M

(E1/2)ML(E1/2)ML

(E1/2)ML2(E1/2)ML2

(E1/2)ML3(E1/2)ML3

(E1/2)L(E1/2)L

(E1/2)I(E1/2)I

-0.1 -0.05 0 0.05 0.1-3

-2

-1

0

1

2

E

log

10(n

orm

(r))-0.1 -0.05 0 0.05 0.1

0

5

10

15

20

25

30

35voltammograms

E

Un

itary c

urren

t

Optimum values for

n independent parameters can be estimated

by

grid search of one parameter.

Optimum values for

n independent parameters can be estimated

by

grid search of one parameter.

A difficult aspect of hard modeling is determination of correct model

Thanks.Thanks.

Thanks to:

Miss Maryam Khoshkam

and

Mr Yaser Beyad

for a number of m-files and slides.

Thanks to:

Miss Maryam Khoshkam

and

Mr Yaser Beyad

for a number of m-files and slides.