European Molecular Biology Laboratory (EMBL)

1
???????????? ?????? ? ???????? ?????????????
??????? ???/???

• ?????? ??????? ?????????
• ?????????? ??????????????? ????????
• European Molecular Biology Laboratory (EMBL)

???????? ? ?????? ??????????? ?????? (?????
WSC-5), 16 ??????? 2006, ??????
2
?????? ??????????? ??????????

• ?????? ??????? ?????????
• ?????????? ??????????????? ????????
• European Molecular Biology Laboratory (EMBL)

???????? ? ?????? ??????????? ?????? (?????
WSC-5), 16 ??????? 2006, ??????
3
???? ??????

• ??????????? ??????????
• Multivariate Calibration

?????? ??????????? ?????? (???????????)
Multivariate Data Analysis (Chemometrics)
4
? ??????? ? ??????? ????????????

• ?????? ???? ??????????? - ??????????
• ???????????? ?? 30 ??? ????????? ??????,
  ????????, ?????, ???????????
• ????????? ???????????? PCA, PCR, PLS, SIMCA,
  RMSEP, etc. - ?? ????????? ? ???????????
• ??????? ???????????? ????????? ??????
• ????? ?? ???????? ??!
• ???????? scores and loadings (!?)
• ????? ?????, ????? ??????? ??????? ????? ? ??????
• ? ????????? ?????? - ???????????? ????????????

5
?????????? ??? ????????????

• ? ??????? ????? ??? ??????? ???????
• ?????????? (??????? ?????????) ????????????
  ????????, ??????????? ? ????? ??????????? ?
  ????????????? ?????????????? ????????
  ??????????????? ????????????? ? (???) ???????????
  ? ?????????? ??????? ?????????
• ??????????? ??????????????? ????????, ???
  ?????? ??????? ??????????????? ???????? ???? ???
  ???????? ????? ?????????????? ??????? ?????????
  ????????...
• ?? ?????????? ??? ??????????? ??? calibration
• ??????????? ??????????? ??????
• ? ?????? ????? ?????????????? ???????????? ??????
  ??????????

6
????????? ??????????

• Regression is an approach for relating two sets
  of variables to each other Kim
  Esbensen
• Calibration is a process of constructing a
  mathematical model to relate the output of an
  instrument to properties of samples Kenne
  th Beebe
• ?????????? ?????????

7
????????????? ??????

• ???????? ?????????
• Y XB E

• ??? (PCA) ????????????? (X)
• ????????? ????????????? (X,Y)

8
???????????? ??????
??????? (X)
???????????? (Y)
9
??? ???? ????? ???????????

• ?????? ??????? ????????? ????????????? ????????,
  ?????????? ???????, ?????????????? ? ??????
• ????? ??????????? ????????? ???? ?????? ?????????
  ????????????? ???????? ????????????
• ??????
• ?????????
• ???????? ????? ???????
• ???????? ????????????
• ??????????? ??????????, ? ?. ?.
• ? ??????????? ????? ???????????? ???????? ?????
  ?????? ?????????!

10
??????? ?? ????????? ????????

• ????? ?????????? ?????????? ?1 ???????????????
  ???????
• ???????? ???????????????? ?????? ????? ????
  ????????? ??? ????? ???????
• ???????? ???????????? ??????, ????????,
  ??????????? ?????? ? ????? ?????????????????
  (??????? ??)
• ?????????? ?????? ???????? ????? ???????????
  ?????????? ??????????, ?????????? ????????????
  ????????? ??????
• ?????????? ? ??????? ???????????? ????? ????
  ???????? ?????? ?? ???????????? ??????

11
?????????? ?????????? ???? ?????????
univariate calibration
12
?????????? ?????????? ????????????????? ?????
???????????????? ?????
??????????
13
??????????? ??????????
yxbe
YXBE
14
???????????? ??????????? ??????????

• ??????????? ????????????? ????????? ???????????
  ????????????
• ??????? ? ???????? ?? ?????????? ???
  ????????????? ??????????, ? ?.?. ??????
  ????????????? ????????? (???????)
• ??????????? ??????????? ?????? ???????? ?
  ???????? ????????????
• ????????????????? ????? ? ???????? ? ???????
  ???????
• ? ?????????? ??? ????????? (PLS-R) ?????????????
  ???????? ?? ????? ????? ?? ???????? ??????????
  ??????? ???????

15
?????????? ? ????????????
16
???????????? ? ????????? ??????

• ??? ???????? ??????? ? ??????????? ??????????
• ???????????? ??? (Classical Least Squares, CLS)
  ??????? ?? ?????? ??????? ?????????
  ??????-????????-????
• A Ce X Ye
• ????????? ??? (Inverse Least Squares, ILS) ??????
  ????????? ????
• ? Ab Y Xb
• ? ????????? ?????? ?????? ILS

17
????????????? ???????? ????????? (???)
Multiple Linear Regression (MLR)
yb0 b1x1 b2x2bpxpe
??????? b (XT X)-1 XT y
18
?????????? ???

• ??? ????? ?? ?????????, ????
• ?????? ?????????????? ? X (???????)
• ???????????? ??????? ??? ???????????? ?????
  ??????????? ??????????????? (XT X)-1 XT
• ??????? ??????? ????, ?????? ? X
• ?????????? ??????, ??? ???????? (??????? ???
  ???????????? ??????)
• ???? ???????? ??????????? ????? ???????????
  ?????? X
• ?????????? ????????????? ???-???????
  ??????????????

19
?????? ???????????? ???????????????????????
????????????
n ????????? Cn, M
1 2-?????????????? 0 - 1
2 2-???????????????????? 0 - 0.5
3 3-???????????????????? 0 - 0.05
20
????????????????? ???????????? ????????? ?
???????? ??????
simdata
21
???-?????????? (Simdata)
???????? ???-?????? ??? ?3 (3-?? ??????????
????? ???) ???????????????????
22
????? ??????? ????????? (???) - ?????? ??????
??????????????

• ??? (Principle Component Analysis) -
  ?????????????? X TPT E
• ????? T (scores) ? ???????? P (loadings)
  ?????????? ???????????? ??????? ?????????
• T ???????????? ? ???????? ???????? ?????? ?? ??

• T ????? ???????????? ?????? X ??? ??????? (!)

23
????????? PCA ?? ???????
XA(430 nm) YA(550 nm) ZA(750 nm)
XA(522 nm) YA(644 nm) ZA(714 nm)
24
??? ??? ???! (PCA MLR PCR)

• ???-????? (PCA scores) T ????? ????????????
  ?????? X ??? ?????????? ???-?????? (MLR)
• MLR yXbe bXXT-1XTy ynew
  Xnewb (I)
• PCR yTbe bTTT-1TTy ynew
  Tnewb (II)
• ????? ?????????? ????????? ?? ???????
  ??????????, ??? (Principal Component Regression,
  PCR)

25
????? ??? (PCR) ?????????
PCA
MLR
26
????????????? ???-??????

• ????????????? ?????? ?????? ??? ????????
  ?????????? ????????? ??????
• ??????
• ???????
• ????? ????? X ? Y
• ??????????? ??????????? ??? (PCA) ???????? ? ???
  (PCR)
• ?????? ?????? (scores)
• ?????? ???????? (loadings)
• ?????? ?????? ? ???????? ?????? (bi-plot)
• ?????? ???????? (residuals)
• ??????????? ??????????? ???
• ?????????? ?????? ???????? X ? Y

27
?????? ???-?????? (Simdata)
28
?????? ???-?????? (simdata)
29
???????? (?????????) ??????

• ???????? (validation) ?????? ?????? ???
• ??????????? ??????????? ?????? (????? ??)
• ?????? ???????????????? ??????????? ??????
• ???????? ?????? ???????????? ? ??????? ????????
  ??????
• ???? ?? ????????? ? ???? ?? ???????? ???
  ????????? ?????? (?? ?? ??????????? ???????)
• ?????????? ????????????????
• ??? ?????-????????? (cross-validation)
• ?????? (leave-one-out, LOO)
• ?????????? (????????, Venetian blind)

30
?????????????????? ?????? ???????????? (RMSEP)

• RMSE? Root Mean Square Error of Calibration
• RMSEP Root Mean Square Error of Prediction
• ??????? ?? ?????? RMSEP ????????
  ????????? ????? ??
• RMSEP ?????? ???????? ? ???????? ????????? (!)
• RMSEP ???????????? ??? ????????? ???????

31
????? ????????? ?????? ??????? ?? ?????? RMSEP?
?????????? ??????????
?????????? ??????
32
?????? ????? ????????? ? ???

• ?????????? ????? ????? ??????? ?????????
  (principle components, PC) - ???????? ????????
  ??????????? ??????????
• ?????? ? ????????????? ?????? ?? (underfitting)
  ?? ?????????? ???? ???????? ?????????? ?? ??????
• ?????? ? ?????????? ?????? ?? (overfitting)
  ???????? ???????????? ??? (??????)
• ????? ??????????? ??????????? ???????? ????????
  ?????? (validation set)

33
????? ????????? ??? - simdata
34
????? ????????? ??? - simdata
35
?????? ????? ?? ? ??? ???????????

• ????? ??????? ????????? (??????????? ??????)
  ???????????? ? ??? (PCR) ??????? ??????????, ? ??
  ??????????? ????????? ? ??????????? ??? (PCA)
• ???????????
• ? ??? ???? RMSEP
• ??????? ???????????? ???????? ?????? (test set)
• ??????? ?? ?????? RMSEP - ???????? ?????????
  ????? ??
• ??? ???????????? ?????? ????????????? ????? ????
  ????? X-???????? (X-loadings)
• ??????? ?????? ?? ?????????!

36
?????????????? ???

• ??? (PCR) ?????? ????? ??????????? ??????????
• ????? ??????????? ???????????? ????? MLR
• ??????, ?? ?????? ????????????? ??? ??????????
• ???????????? ?? ?? ????????? ????????? Y ? ?????
  ????? X ? Y
• ????? ?? ?????? ??? ????? ??? ??????????
  ???????????? ???????
• ??, ??? ?????? PLS!

37
????????? ????????????

• ????????? PCA ????? ????????????? ?????
• X TPT E
• ?????????????? ?????????? ????????? ???????????,
  ????????? ?????? ?? ????????? ????????????
  (factor space)
• ?????? ??????? ? T ? P ?????????? ?????????
  (factors)
• ??????? ?????????? ?????? ?????? ??????????
  ????????????, ?? ?? ????????????
• ????????? ???????????? ????? ?????????????? ???
  ??????? ?????????? ??????
• ?? (PC) ?????????? ??? ???????????? ????????? X
• ??? ?????????????? ???????????? ??? ???????????

38
PLS ?????? ???????????? PCR

• ????? ???????? ?? ????????? ????????? (???) ?
  ???-????????? (???-?)
• PLS Partial Least Squares -gt
• Projection on Latent Structures
• ???-???????????? ????????? ??? ??????? ????
  ?????????? X ? Y ????????????
• ???????? ????????????? ??? ?????????
  (??????????) ? X, ??????? ??????????? ? Y
• ????????, ???????????? ?????? (X), ???????
  ???????? ?? ???????????? ??????????(??), ????????
  ? Y, ??????? ? ?????? ??????? ???
• ????? ??? ????????????? ??? ?????????????? ???????

39
???-????????? ????????????? ?????????????

• ????????? ??? ??????? X ? Y
• ??????? ?????????????? ?? ??????? ????????
  NIPALS
• gt 2 ?????? ?????? (scores) T, U ? ????????
  (loadings) P, Q ???? ??????? W ??????????
  ???????? (loading-weights)
• ???????????? ????????? ??????, ?????
  ??????????????? cov(T,U)
• ???????????? Y Tnew Bt
• Y Xnew B
• B W(PTW)-1QT

X TPT Ex Y UQT Ey
1 S. Wold, H. Martens, H. Wold, Lecture Notes
Math. 973 (1983) 286293
40
??? ????????????? ??? ???1 ? ???2

• ?????????? ??? ?????????? ????????????? ??? ???1
  (PLS1) ? ???2 (PLS2)
• ???1 ?????? ???????? ??? ???????????? ??????????
  y (????????), ????????, ??? ???????????? ??????
  ?????????? ?????
• ???? ????? ?????????? ?? ?????????? ?????????,
  ???????? ????????? ??????????? ???????
• ???2 ?????????????? ??? ?????????? ???????
  ????????????
• ????????? ????????? ??????? ??????????
  ??????????????

41
?????? ????????? ???

• ???-???????????? ???????????? ????????? NIPALS
• NIPALS Non-linear Iterative Partial Least
  Squares
• ??????? ????????? ?? ???????, ???? ?? ??????,
  ?????? ???? ???????? (??? ? SVD) ?? ??????????
• ???????????? ?????? ???????? uf -gt tf ? uf -gt tf
  ??? ?????????? ???????? ??????? f -
  ??????????????? ?????? ???2
• ???????? ???????? ?? ?????????? ????????
  ??????????
• ??????????? ? ?????????????? ??????, ??????? ?
  ????? ?????? ???2

42
NIPALS ???????? ??? ???2
0. uf ????? ?????????? ??????????? u
1. wf XfTuf/XfTuf ?????? ???????????????? ??????? ?????e???? ???????? w
2. tf XfTwf ?????? ??????? ????? t
3. qf YfT tf /YfT tf ?????? ???????????????? ??????? ???????? q
4. uf YfT qf ?????? ??????? ?????? u
5. tf.new- tf.oldlt lim? ???????? ?????????? ?? -gt go to 1.
6. pf XfTtf/tfTtf ?????? ??????? ????? p
7. bf ufTtf/tfTtf ?????? ??????????? ???????????? ????????? b
8. Xf1 Xf1 - tf Tpf Yf1 Yf1 - bftfqfT ?????? ??????? X ? Y
9. f f 1 ??????? ? ?????????? ???????
43
NIPALS ???????? ??? ???1

1. wf XfTyf/XfTyf ?????? ???????????????? ??????? ?????e???? ???????? w
2. tf XfTwf ?????? ??????? ????? t
3. qf yfT tf /tfT tf ?????? ???????? q (??????) ??????? f


4. pf XfTtf/tfTtf ?????? ??????? ????? p

5. Xf1 Xf1 - tf Tpf yf1 yf1 - qftf ?????? ??????? X ? y
6. f f 1 ??????? ? ?????????? ???????
44
NIPALS ???????? ??? ???1

1. wf XfTyf/XfTyf ?????? ???????????????? ??????? ?????e???? ???????? w
2. tf XfTwf ?????? ??????? ????? t
3. qf yfT tf /tfT tf ?????? ???????? q (??????) ??????? f
4. pf XfTtf/tfTtf ?????? ??????? ????? p
5. Xf1 Xf1 - tf Tpf yf1 yf1 - qftf ?????? ??????? X ? y
6. f f 1 ??????? ? ?????????? ???????
45
???1 ? ???2

• ???1 ?????????? ?????? ???? ?????????? y ?? ???
• ???2 ????????? ???????????? ????? ??????????
  ?????????? Y ??? ?? ?????????? ?????????
• ?? ??????? ????? ?????????? ??? ??????????
  ?????????? ???????
• ??????, ???1 ???? ?? ????????? ?????? ?? ??????
  ?? ???????????? ???????, ????????, ? ?????????
  ?????? ????????
• ?? ????? ?? ????? ??????????? ??????? ??????
  ?????? ?????????
• ???????????? ?????? ???
• ??????? ?????? ?? ????????!

46
?????? ???2-?????? (Simdata)
n ????????? Cn, M
1 2-?????????????? 0 - 1
2 2-???????????????????? 0 - 0.5
3 3-???????????????????? 0 - 0.05
47
????????????? ???-???????

• ????????????? ?????? ?????? ??? ????????
  ?????????? ????????? ??????
• ??????
• ???????
• ????????????
• ???????? ? ??? (PCR)
• X-????? ? ???????? (scores loadings)
• ???????????
• ?????? t u ????? ??????????? ????????
  (outliers)
• ??????? ???????? w w ????? ??????????
• c???????? ???? X-???????? p w ????????? Y
  ???????? ?? ???????????? X
• ?????? w q

48
????????????? ??????????2 ?????? ???
PLS2
49
????????????? ??????????1 ?????? ???2
50
????????????? ???-???????????? X ? Y (Simdata)
51
????????????? ???-????????????? (Octane)
52
???????? ????????????? ???????

• ???????? (validation) ?????? ?????????? ???
  ???????? ????
• ??????????? ???????????? ????? ?????????
• ?????? ???????? ??? ? ???
• ??????? RMSEP
• ?????? ???????????????? ??????????? ??????
• ?????? ????????????? ???????????? ?????????
  (predicted vs measured)
• RMSEP

53
???????? ????????????? ??????? simdata ???1
54
????????? ??????? Simdata
????????? ??????? ?????????? ????????????????
????? ??? (simdata)
??? MLR ??? PCR ???1-? PLS1-R ???2-? PLS2-R
C1 0.1312 0.0576 0.0575 0.0575
C2 0.0527 0.0241 0.0245 0.0245
C3 0.01579 0.00246 0.00246 0.00249

• ????? ?????? ???, ???1-?, ???2-? ????????
  ???????? ?????? ??? ?????????? ???? ?????? (???
  ??????????)
• ?????????? ??? ??????????? ????, ??? C3 -
  ????????????????????

55
????????? ??????? ??????????

• ??? (MLR) ????? ???????? ??? ??????????????????
  ??????
• ??? (PCR) ????? ??????????, ?? ?????? ????????
  ??? ?????????? ??????????
• ??? ????????? (PLS-R) ???????? ?????? ????????
  ??? ??????????? ???????????? ?????
• PLS1 ??? PLS2?
• ??? ??????? ?????? ?????????!
• ??? ?????????? ?????? ??????? RMSEP

56
???????? ????????? ? ????????????
X 100x351
r0.999
57
???????????? ??????????? ???????????? ?????
????????

• ?? ??? ???????? ????????????? ? ???????????
  ????????????? ??????!
• ??????????? ????????? ????????, ??
  ??????????????? ?????? ????????????? ??????
  ???????? ????? ?? ??????????? ?????????????
  ???????
• Deviation - ???????????? ????????,
  ??????????????? ???? ???????????? ?????? ???????
  ????????????? ??????
• ?????????? ??? ??????

58
??????????? ????????????(Simdata)
59
??????????? ???????????????1 - Simdata
C2 0 0.5 M
C1 0 1 M
C1 C2 C3
U1 0.5 0.25 0.025
U2 0.9 0 0.01
U3 0 0 0.2
U4 0.5 0 0
U5 0 0.3 0
U6 0.5 0.5 0.5
C3 0 0.05 M
60
??????? ?????????? ??????? ??????????

• ????????? ??????????? (???????) ???????
• ????????? ??????? ??????, ???? ??????????,
  ????????? ??????????????? ????????? ??????
  (pre-processing)
• ???? ?????????? ????????? ????????????/
  ??????????? (scaling/weighting)
• ???????????????? ??????, ??????? ?????????
  ??????, ??????? ? ??????? ????????? ???????
• ????????? ??????? ??????????? ??????,
  ??????????????? ??????
• ??????????????? ????????????

61
???? ????????

• ?????? 1. ???????????????? ??????????
  ???????????????? ????? ??? ?? ???????? ?
  ??-??????? ??????? (????????????? ??????).
• ????? ?????? ??????????, ????????????? ?
  ??????????? ??????, ???????????? ?? ?????????
  ??????
• ?????? 2. ??????????? ?????????? ????? ??????? ??
  ???????? ???????? ??.
• ?????????? ?? ???????? ??????, ??????????? ?
  ???????? ????????
• ?????? 3. ???????? ??????? (?????????????).
• ??????????????? ?????????? ??????????, MSC, ?????
  ??????????

62
????????????? ??????????

• Richard Kramer
• Chemometric Tchniques for Quantitative Analysis
  
• Kim H. Esbensen
• Multivariate Data Analysis - in Practice
• Kenneth R. Beebee et al.
• Chemometrics a Practical Guide
• Harald Martens, Tormod Naes
• Multivariate Calibration
• Richard G. Brereton
• Chemometrics Data Analysis for the Laboratory
  and Chemical Plant
• Edmund R. Malinowski
• Factor Analysis in Chemistry

63
?????? 1 ?????????? ????? ???

• ???? Simdata
• ???? ????????? ??????? ?????????? ? ??????????
  Unscrambler
• ??????? ?????? ?????? ?????????, ????????,
  unknown - ? ???????, ??? ????? ????????
• ????????? ?????????? ???, ???2 - ???????? ??????
• ????????? ???1 ??? ??????? ?? 3-? ???????????,
  ?????????? ??????????? ???????
• ??????? ??????? scores, loadings, T-U, predicted
  vs measured, RMSEP, Variance ??? ?1 - ?3 ?
  ?????? ??????????? ????????
• ??????????? ??????????? ???????

64
?????? 2 ??????????? ?????????? ????? ???????

• ???. 139, ???? Octane
• ???? ?????? ? ????????? ???????, ??????????? ?
  ?????????? ????????
• ??????????????? ?? ?????
• ????????? ?????????? ???1, ???????????????
• ?????????? ???????, ???????, ???????? ??????
• ????????? ?????? ?????????? ?????????, ???????
  ???????? ?????
• ????????? ???, ???????? ??????
• ??????????? ??????????? ???????

65
?????? 3 ???????? ???????

• ???. 150, ???? Wheat
• ???? ??????????????? ?????????? ?????????????
  ??????
• ?????????? ??????? ???1/2, ????????? ???????
• ??????????? ? ???????? ????????
• ?????????? MSC
• ??????????? ???????? ?????????? ??? ?????????
  ??????