Visualizing the Microscopic Structure of Bilinear Data: Two components chemical systems

Visualizing the Microscopic Structure of Bilinear Data: Two components

chemical systems

A matrix can be decomposed into the product of two significantly smaller matrices.

Factorization:

In many chemical studies, the measured or calculated properties of the system can be considered to be the linear sum of the term representing the fundamental effects in that system times appropriate weighing factors.

D = X Y + R

D X Y= + R

0 5 10 15 20 25 30 35 40 45 500

0.5Concentration Profiles

Retention Time

400 410 420 430 440 450 460 470 480 490 5000

1Spectral Profiles

Wavelength (nm)

400 410 420 430 440 450 460 470 480 490 5000

Wavelength

A simple one component system

435 440 445 450 455 4600

wavelength

0.10.15

0.20.25

0.30.35

0.40.45

0.10.15

0.20.25

0.30.35

0.40.45

Absorbance at wavelength #1Absorbance at wavelength #2

Observing the rows of data in wavelength space

0.10.15

0.20.25

Absorbance at time #1Absorbance at time #2

time #

19 20 21 22 23 24 25 26 27 28 290

Observing the columns of data in time space

D = USV = u1 s11 v1 + … + ur srr vr

Singular Value Decomposition

=D U S V

d1,:d2,:

… =u11

up1…

s11 v1 d1,:= u11 s11 v1d2,:= u21 s11 v1… …

dp,:= up1 s11 v1

For r=1Row vectors:

D = u1 s11 v1

[ d:,1 d:,2 … d:,q ] = u1 s11 [v11 v12 … v1q]

d:,1= u1 s11 v11

… …d:,2= u1 s11 v12

d:,q= u1 s11 v1q

Column vectors:

=D U S V

For r=1 D = u1 s11 v1

Rows of measured data matrix in row space:

u11s11v1

up1s11v1

u11s11u21s11…

up1s11

p points (rows of data matrix) in rows space have the following coordinates:

Columns of measured data matrix in column space:

v11s11v12s11…

v1qs11

q points (columns of data matrix) in columnss space have the following coordinates:

u1 v11 s11u1

v1q s11u1

-3 -2.5 -2 -1.5 -1 -0.5 0-1

ui1s11

Row Space

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.1

0.1Column space

v1js11

Visualizing the rows and columns of data matrix

400 410 420 430 440 450 460 470 480 490 5000

Wavelength

1 or A

0 5 10 15 20 25 30 35 40 45 500

Solutions

Pure spectrum

v1js11

Pure conc. profile

ui1s11

400 420 440 460 480 500 520 540 560 580 6000

1.4Spectral Profiles

Wavelength (nm)

0 10 20 30 40 50 60 70 80 90 1000

0.5Concentration Profiles

Retention Time

400 420 440 460 480 500 520 540 560 580 6000

Wavelength

Measured data

Two component systems

0.20.3

0.4 0.50.6

0.20.3

0.40.5

0.60.7

Absorbance at wavelength #1Absorbance at wavelength #2

470 480 490 500 510 520 5300

Wavelength

Visualizing data in three selected wavelengths

38 40 42 44 46 48 50 52 54 56 580

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.20.3

0.40.5

Absorbance at time #1Absorbance at time #2

time #

Visualizing data in three selected Times

Measured Data Matrix

Row vectors:d1,:d2,:

… =u11

s11 v1 u12

…s22 v2

d1,:= u11 s11 v1 + u12 s22 v2d2,:=… …

u21 s11 v1 + u22 s22 v2

up1 s11 v1 + up2 s22 v2

For r=2 D = u1 s11 v1 + u2 s22 v2

[ d:,1 d:,2 … d:,q ] = u1 s11 [v11 v12 … v1q]

+ u2 s22 [v21 v22 … v2q]

d:,1= s11 v11 u1 + s22 v21 u2

… …d:,2=

s11 v12 u1 + s22 v22 u2

s11 v1q u1 + s22 v2q u2

For r=2 D = u1 s11 v1 + u2 s22 v2Column vectors:

Rows of measured data matrix in row space:

u11s11

u12s22

u21s11

u22s22

up2s22

up1s11

u11s11 u12s22

…u21s11 u22s22

up1s11 up2s22

…Coordinates of rows

Columns of measured data matrix in column space:

…v2qs22

v1qs11v12s11

v22s22

v21s22

v11s11

v11s11 v12s11 . . . v1qs11 Coordinates of columns

v21s11 v22s11 . . . v2qs11

400 450 500 5500

Wavelength (nm)

400 450 500 5500

1.8Response matrix data

Wavelength (nm)

0 10 20 30 40 50 60 70 800

1.4Conc. Profiles

0 1 2 3 4 5 6-1

1Visualizing the rows of the data

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5

0.5Visualizing the columns of the data

Two component chromatographic system

0 2 4 6 8 10 12 14 16 18 200

1Concentration Profiles

400 420 440 460 480 500 520 540 560 580 6000.5

2Spectral Profiles

Wavelength (nm)

400 420 440 460 480 500 520 540 560 580 6000.4

2Simulated Spectra

Wavelength (nm)

6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4-2

5 6 7 8 9 10 11-2

2Visualizing the columns of the data

Two component Kinetic system

Two component multivariate calibration

0 5 10 15 20 25 300

Sample number

400 420 440 460 480 500 520 540 560 580 6000.02

Wavelength (nm)

400 420 440 460 480 500 520 540 560 580 6000

1.4Response matrix data

Wavelength (nm)

0 1 2 3 4 5 6 7-1

2 2.5 3 3.5 4 4.5-0.5

Position of a known profile in corresponding space:

Tv1 is the length of projection of dx on v1 vectorTv1 = dx . v1

Tv2 is the length of projection of dx on v1 vectorTv2 = dx . v2 Tv1

Tv2Coordinates of dx point:

Position of real profiles in V and U spaces

0 1 2 3 4 5 6 7-2

0 5 10 15 20 25-10

0 1 2 3 4 5 6-1.5

1.5Visualizing the rows of the data

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0 1 2 3 4 5 6 7 8-2

0 1 2 3 4 5 6 7 8 9 10 11-2

Duality in Measured Data Matrix

xijxij

Geometrical interpretation of an n x p matrix X

R= C ST = U D VT = X VT

RT= S CT = VD UT = Y UT

X = U D = RV = U YT VY= V D = RT U = V XT U

Duality based relation between column and row spaces

Non-negativity constraint and the system of inequalities:

U z ≥ 0V z ≥ 0

X = U D U= X D-1

Y = V D V= Y D-1

U z = X D-1 z ≥ 0 Hyperplanes

V z = Y D-1 z ≥ 0 Hyperplanes

U-space Y Points

V-space X Points

Duality based relation between column and row spaces

xi = [xi,1 xi,2]

Ui,1 zi,1 + Ui,2 zi,2 … Ui,N zi,N ≥ 0 The coordinates of each point in one space defines the coefficient of related hyper plane in dual space

Point x in V-space Hyper plane (D-1x) z in U-spaceFor two-component systems:

The ith point in V-space: xi

xiD-1= [Ui,1 Ui,2 … Ui,N] The ith hyperplane in U-space:

A point in V-space:

Ui,1 z1 + Ui,2 z2 ≥ 0 A half-plane in V-space:

Half-plane calculation in two-component systems:

General half-planeGeneral border line can be defined for all points that the ith element of the profile is equal to zero

0ith half-plane

ith border line

General border line

Ui,1 z1 + Ui,2 z2 ≥ 0

z2 ≥ (-Ui,2/Ui,1)z1

z2 = (-Ui,2/Ui,1)z1

0 1 2 3 4 5 6-1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

Chromatographic data

0 1 2 3 4 5 6 7

Visualizing the rows of the data

0 1 2 3 4 5 6 7-2

Chromatographic data

0 1 2 3 4 5 6 7 8 9-5

0 2 4 6 8 10 12 14 16-10

Kinetics data

0 1 2 3 4 5 6 7 8 9-5

0 2 4 6 8 10 12 14-4

Kinetics data

Calibration data

0 1 2 3 4 5 6 7 8 9-6

0 1 2 3 4 5 6 7-4

Calibration data

0 1 2 3 4 5 6 7 8 9-6

0 1 2 3 4 5 6-2

Intensity ambiguity in V space

k1ak2a

Normalization to unit length

k1ak2a

an = (1/||a||) a

Normalization to first eigenvector

k1ak2a

an = (1/(v.a))a an

51’ 2’ 3’

Normalization to unit length

a = T1 v1 + T2v2

a’ = v1 + T v2

1’ 2’ 3’

Normalization to first eigenvector

Chromatographic data- Normalized to unit length

0 0.2 0.4 0.6 0.8 1-0.5

0.5Normalization to unit length-Row Space

0 0.2 0.4 0.6 0.8 1 1.2-0.4

Normalization to unit length-Column Space

Chromatographic data- Normalized to 1th eigenvector

0 0.2 0.4 0.6 0.8 1 1.2-0.5

0.5Normalization to first eigenvector-Row Space

0 0.2 0.4 0.6 0.8 1 1.2-0.4

Normalization to first eigenvector-Column Space

Kinetics data- Normalized to unit length

0 0.2 0.4 0.6 0.8 1

Normalization to unit length-Row Space

0 0.2 0.4 0.6 0.8 1-0.4

Kinetics data- Normalized to 1th eigenvector

0 0.2 0.4 0.6 0.8 1

Normalization to first eigenvector-Row Space

0 0.2 0.4 0.6 0.8 1

Multivariate calibration data- Normalized to unit length

0 0.2 0.4 0.6 0.8 1 1.2-0.5

1Normalization to unit length-Row Space

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Multivariate calibration data- Normalized to 1th eigenvector

0 0.2 0.4 0.6 0.8 1 1.2-0.5

1Normalization to first eigenvector-Row Space

0 0.2 0.4 0.6 0.8 1 1.2

• The normalized abstract space of two component systems is one dimensional

Data points regionOne dimensional normalized space

• There are 4 critical points in normalized abstract space of two-component systems:

First inner point Second inner point

First outer point Second outer point• The 4 critical points can be calculated very easily

and so the complete resolving of two component systems is very simple

First feasible region Second feasible region

Lawton-Sylvester Plot

Microscopic Observation of Two Component systems using Lawton-Sylvester Plot

First feasible solutions Second feasible solutions

General Microscopic Structures of Two-Component Systems

Second feasible solutions

Case I)

Case II)

Case III)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

Lawton-Sylvester Plot- Multivariate calibration

Feasible Bands- Multivariate calibration

Feasible Bands- Chromatographic Data

Feasible Bands- Kinetics Data

400 420 440 460 480 500 520 540 560 580 6000.4

2Simulated Spectra

Wavelength (nm)

Kinetics Data (I)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

LS plot as a microscope for kinetics data (I)

Feasible bands for kinetics data (I)

400 420 440 460 480 500 520 540 560 580 6000

1.8Simulated Spectra

Wavelength (nm)

Kinetics Data (II)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

LS plot as a microscope for kinetics data (II)

Feasible bands for kinetics data (II)

Visualizing the Microscopic Structure of Bilinear Data: Two components chemical systems

Documents

Bilinear Mappings in Formal Cryptography

Lyapunov control of bilinear Schrodinger equations

Bilinear text regression and applications

Basic Algebra · 2018. 8. 20. · VI. MULTILINEAR ALGEBRA 248 1. Bilinear Forms and Matrices 249 2. Symmetric Bilinear Forms 253 3. Alternating Bilinear Forms 256 4. Hermitian Forms

Basic Algebra - math.mcgill.ca · Basic Algebra Advanced Algebra Basic Real Analysis, ... Bilinear Forms and Matrices 249 2. Symmetric Bilinear Forms 253 3. Alternating Bilinear Forms

Symmetric bilinear forms

Bilinear Mappings in Formal Cryptography 08.10.11

BILINEAR DECOMPOSITION FOR BLENDED EXPRESSIONS REPRESENTATION Anonymous ...€¦ · BILINEAR DECOMPOSITION FOR BLENDED EXPRESSIONS REPRESENTATION Anonymous VCIP submission Paper ID:

Encryption Schemes from Bilinear Maps

Chapter 3. Bilinear forms - Trinity College, Dublinpete/ma1212/chapter3.pdfChapter 3. Bilinear forms Lecture notes for MA1212 P. Karageorgis pete@maths.tcd.ie 1/25. Bilinear forms

Applications of Bilinear Control Theory in Nonlinear Spectroscopykmarwaha/bilinear.systems.nonlinear... · Applications of Bilinear Control Theory in Nonlinear Spectroscopy Kunal

Bilinear Form

Approximated Bilinear Modules for Temporal Modeling

SPE 7490 Cinco-Ley Bilinear

An Online Learning Algorithm for Bilinear Models · Yuanbin Wu Shiliang Sun An Online Learning Algorithm for Bilinear Models 2 / 27. Introduction: bilinear models Linear model for

Dietrich˜von˜Rosen Bilinear Regression Analysis

UNIT - 2 BILINEAR TRANSFORMATION

Intro to Bilinear Maps - Massachusetts Institute of …people.csail.mit.edu/alinush/6.857-spring-2015/papers/...Intro to Bilinear Maps Introduction Motivation Why bilinear maps? I

Linear, bilinear and quadratic formsandreea.arusoaie/Lectures... · 2019-08-04 · Symmetric bilinear forms De nition A bilinear form g : V V !™ is called symmetric if g(u,v) =

Bilinear Attention Networks - Neural Information Processing …papers.nips.cc/paper/7429-bilinear-attention-networks.pdf · 2019-02-19 · Bilinear Attention Networks Jin-Hwa Kim1⇤,