中南大学中药现代化研究中心

Subtraction of two vectors defines the distance between the two points in n dimensional space
Numerical multiplication of vectors
ka1
k
a
=
ka
2
ka
n
The spectra of different concentrations
Inner product and projection between vectors
Outer product between two vectors producing a bilinear matrix, which is of
special importance in multivariate resolution
Thus, if we collect all the spectra to form a matrix, the rank of which should be 2.
How can we find the rank of a matrix with measurement noise?
What are noises and how do they influence the data analysis?
a1
a1b1 a1b2 a1bn
a
b
t
=
a 2
[b1,
b2,
...,
bn]
=
a
2 b1
a2b2
a
2
b
n
a
n
a n b1
anb matrix ：
A matrix A of order (nm)，its rank is the maximum number of the linearly independent row vectors (column vectors) in it , denoted by rank(A). It has the following features:
The length of a vector is also decided by magnitudes of its elements, since we have, ||a||=(a12+….+an2)1/2
different ratios between them can defines different directions in linear subspace
Geometric sense of vector subtraction
Direction and length of vector
The direction of a vector is decided by all its elements, since the different ratios between them can defines different directions in linear subspace;
The concentration profile of each chemical component: ci (i=1, 2, …, A) (first step for quantification)
Is it possible? Yes, if we have the two-way data!!
At = ( A-1)t
0 rank(A) min(n, m)
rank(AB) min[rank(A), rank(B)]
rank(A +B) rank(A)+rank(B) rank(AtA) =rank(AAt) =rank(A) If matrix A is a square matrix of order n, then when and only when det(A) is not equal to zeros， rank(A) =n
Thus, we need statistics and also algebra;
Useful technique: Principal component analysis!!
This is really a difficult problem!
Now, let’s go through this problem step by step.
Is Mathematics really useful for chemists?
Data exploding; Extract chemical
information from the data; Revolution of Information technique; Progress in Computer ability;
Data base of lots of chemical standards
The mixture spectrum of two different chemical compounds a and b according to the Lambert-Beer
law
Geometric sense of vector addition
Hyphenated Instruments, such as HPLC-DAD, GC-MS, GC-IR, HPLC-MS
Two-way data containing both chromatography and spectra;
Data matrix with more than 10 Megabytes;
Vector in analytical chemistry
All the spectra, chromatograms and etc. can be numerated into a group of numbers, which is called a vector in mathematics.
Mixture number and compound number
Rank of a matrix= the number of the chemical components in the mixture?
Unchanged are chemical compounds;
Rank of a matrix is also certain, which is not changed;
Wold and Kowalski founded the International Chemometric Society in 1974;
Analytical Chemistry published special review on “Chemometrics” every two year since 1978;
化学计量学运用数学、统计学、计算机科学、以及 其他相关学科的理论与方法，优化化学量测过程，并 从化学量测数据中最大限度地获取有用的化学信息， 可以说是一门化学量测的基础理论与方法学。
Brief History of Chemometrics
Svante Wold used it firstly for applying scientific project in 1970 in Sweden;
Inner product and outer product between the vectors
Inner or dot product between two vectors producing a number
b1
a
tb
=[a1,
a2,
...,
an]
b
2
=
ai
bi
b
n
Geometric sense of inner product between two vectors
A part of a real two-way data measured for Cortex Cinnamomi (肉桂)
Lambert-Beer Law
p
X C S E
T
mn
m p n p
mn
cisiT ei
i 1
The problem here for chemists to solve is that, with the measurement matrix at
What is the chemical meaning of rank of a matrix?
Linearly independent? Rank of a matrix=
the number of the chemical components in the mixture? Mixture number and compound number?
hand, one needs to find out :
The number of absorbing chemical components: A
The spectrum of each chemical component: si (i=1, 2, …, A) (first step of qualifications)
Two special chemometric international journals named “J.Chemom.” and “ChemLab” appeared in 1987 from both American and Europe.
Contents in chemometrics
Chemometrics consists of fundamental and
methodology of chemical measurements.