主成分分析与主成分回归 principal component analysis and regression

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1. Introduction. 主成分分析与主成分回归 Principal Component Analysis and Regression. 2. PCA. 3. PCR. 1. Introduction. 1.1 Chemometrics. 1.2 Necessary Knowledge. BACK. 1.1 Chemometrics. 1971 :瑞典人 S. Wold 基金项目定名时首提 1974 : S. Wold 何 B.R. Kowalski 倡议在西雅图首开学术会议 新创学术刊物 - PowerPoint PPT Presentation

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  • Principal Component Analysis and Regression1. Introduction2. PCA3. PCR

  • 1. Introduction1.1 Chemometrics1.2 Necessary Knowledge

  • 1.1 Chemometrics1.1.1. 1970S1.1.2. 1.1.3. 1.1.4. 1971S. Wold 1974S. Wold B.R. Kowalski J. Chem. Info. Comp. Sci. J. Chemometrics Chemom. Intell. Lab. Syst.Chemometrics StoichiometryBACK

  • A New trend in Analytical Chemistry ---Hyphenated Instrument ()HPLC DAD (diode-array detector)GC MS (Mass-spectrometer)

  • HPLC-DADGet more dataHPLC chromatogram of nuclueside of Cordyceps Sinensis () at one wavelength

    100

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    x 10

    6

    Retention time(s)

    Intensity

    A

    B

  • GC-MSGC chromatogram of peptic powder ()BACKTwo-way data containing both chromatography and spectra;Data matrix with more than 80 Megabytes;Data base of lots of chemical standards

  • 16PAH(Sulpeco)9[a]14BACKNext

    mAU

    Time/min

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  • Peaks 5 and 6 in the plot

  • 565516382243612942211

    b

    PC2

    PC1

    Peak 5

    Peak 6

    0.15

    0.1

    0.05

    0

    -0.05

    -0.1

    -0.15

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    -0.05

  • 1998 1998ChemLab. 1999BACK

    Wavelength/nm

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    b

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  • 1.2 Necessary Knowledge on Linear Algebra1.2.1 Vector1.2.2 1.2.3 Matrix1.2.3 RankBACK

  • na1, a2,, ank11+ k22+ + kmm=0k1+ k2 a1, a2,, ant ,,, rank 1 =( 1 2 3 4 5 6 ) 2 =( 6 5 4 3 2 1 ) 3 =( 1 1 1 1 1 1 )1+ 23 =0

  • Grade dik received by student i from professor k is: : j: factors (i,e., subjects) chem., physics, math., etc.

  • Four students three professortwo subject: Chemistry and English34

  • A(mn), A (s1, s2 ,s3) 6 A(206)Rank =Number of Eigenvalue=0

    1

    0.254223350.230372430.210785390.221105580.284593580.24872971

    0.313990120.276813510.247674940.252788070.339213850.28246164

    0.382839680.33421040.297866820.299607280.408126350.33268547

    0.456165790.396847720.35367870.354299550.485154630.39293909

    0.516010760.448673730.398985390.401407240.55069160.44597101

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    0.6237340.545541290.490525250.496278760.672756670.5523982

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    0.667809010.591765880.583151340.553624630.701561450.59786701

    0.672870640.606049060.643127920.586463930.695918080.61990738

    0.676922320.621185780.70827150.624155040.690936570.64705372

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    0.594887730.575476650.736562250.634936810.612896920.64740467

    0.543854240.539767740.716220860.620478630.570547580.6334815

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    0.104627130.132245060.172923090.20195150.170877460.23336554

    10

    3600.254223350.230372430.210785390.221105580.284593580.248729710.124219890.098927970.2588582

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    3900.456165790.396847720.35367870.354299550.485154630.392939090.120348930.205535890.48299837

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    4300.6237340.545541290.490525250.496278760.672756670.55239820.191163060.272077080.65825081

    4400.647166250.567964550.527995110.520838740.690621380.573252680.213277340.316293720.64155817

    4500.667809010.591765880.583151340.553624630.701561450.597867010.259633540.385596280.58400106

    4600.672870640.606049060.643127920.586463930.695918080.619907380.331865790.463245390.47973347

    4700.676922320.621185780.70827150.624155040.690936570.647053720.417945380.542980670.36464024

    4800.65653610.616546150.747449870.646055220.668017860.6616230.505988120.587328910.24406719

    4900.594887730.575476650.736562250.634936810.612896920.647404670.579957490.571891780.12177658

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    5200.380486010.400353910.537414070.501166340.442699430.530107980.605302810.325713160.00423861

    5300.248692990.281230450.374006750.384970660.332773690.424742220.536075590.146186830.00172138

    5400.160075190.194409850.255918030.286632540.243893620.326615330.440052510.048590180.00108337

    5500.104627130.132245060.172923090.20195150.170877460.233365540.322610860.014846320.00075006

    10

    pure

    30mg/L20mg/L20mg/L

    012

    1902.75282242.638339042.52297783

    1912.943461892.782875542.68701172

    1923.000977992.848809722.72023726

    1933.040133482.868571762.72027636

    1943.084602832.876422882.74965382

    1953.088269712.876825332.72430754

    1963.045105932.824354172.70176077

    1973.060160162.779733662.66163921

    1982.935873512.612566472.55446243

    1992.596766952.26791432.30761147

    2002.24844791.891537191.97024298

    2011.970181471.580560681.65089846

    2021.774101261.354688171.39695644

    2031.646615511.192129141.20090151

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    10980.00144577-0.000097750.00035381

    10990.001375680.00166893-0.00069904

    11000.003143310.00269604-0.00002956

    /nm

    A

    2

    0.124219890.098927970.2588582

    0.105141640.133034230.32987642

    0.106569290.170055390.40473652

    0.120348930.205535890.48299837

    0.137232780.228179930.55021715

    0.150177480.243629930.60367107

    0.170384410.254952910.64278841

    0.191163060.272077080.65825081

    0.213277340.316293720.64155817

    0.259633540.385596280.58400106

    0.331865790.463245390.47973347

    0.417945380.542980670.36464024

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    0.629292960.542492390.04325056

    0.639344690.479516980.01318741

    0.605302810.325713160.00423861

    0.536075590.146186830.00172138

    0.440052510.048590180.00108337

    0.322610860.014846320.00075006

    10

    3600.254223350.230372430.210785390.221105580.284593580.248729710.124219890.098927970.2588582

    3700.313990120.276813510.247674940.252788070.339213850.282461640.105141640.133034230.32987642

    3800.382839680.33421040.297866820.299607280.408126350.332685470.106569290.170055390.40473652

    3900.456165790.396847720.35367870.354299550.485154630.392939090.120348930.205535890.48299837

    4000.516010760.448673730.398985390.401407240.55069160.445971010.137232780.228179930.55021715

    4100.561341760.488008020.432864190.437384610.601589680.486908910.150177480.243629930.60367107

    4200.598151680.521390440.463227270.470496650.644463540.524743560.170384410.254952910.64278841

    4300.6237340.545541290.490525250.496278760.672756670.55239820.191163060.272077080.65825081

    4400.647166250.567964550.527995110.520838740.690621380.573252680.213277340.316293720.64155817

    4500.667809010.591765880.583151340.553624630.701561450.597867010.259633540.385596280.58400106

    4600.672870640.606049060.643127920.586463930.695918080.619907380.331865790.463245390.47973347

    4700.676922320.621185780.70827150.624155040.690936570.647053720.417945380.542980670.36464024

    4800.65653610.616546150.747449870.646055220.668017860.6616230.505988120.587328910.24406719

    4900.594887730.575476650.736562250.634936810.612896920.647404670.579957490.571891780.12177658

    5000.543854240.539767740.716220860.620478630.570547580.63348150.629292960.542492390.04325056

    5100.490087990.496368410.666945460.588459970.528792380.605904580.639344690.479516980.01318741

    5200.380486010.400353910.537414070.501166340.442699430.530107980.605302810.325713160.00423861

    5300.248692990.281230450.374006750.384970660.332773690.424742220.536075590.146186830.00172138

    5400.160075190.194409850.255918030.286632540.243893620.326615330.440052510.048590180.00108337

    5500.104627130.132245060.172923090.20195150.170877460.233365540.322610860.014846320.00075006

    10

    pure

    30mg/L20mg/L20mg/L

    012

    1902.75282242.638339042.52297783

    1912.943461892.782875542.68701172

    1923.000977992.848809722.72023726

    1933.040133482.868571762.72027636

    1943.084602832.876422882.74965382

    1953.088269712.876825332.72430754

    1963.045105932.824354172.70176077

    1973.060160162.779733662.66163921

    1982.935873512.612566472.55446243

    1992.596766952.26791432.30761147

    2002.24844791.891537191.97024298

    2011.970181471.580560681.65089846

    2021.774101261.354688171.39695644

    2031.646615511.192129141.20090151

    2041.572198391.078879831.05478048

    2051.536332131.002148150.94715261

    2061.525492190.948929310.86474419

    2071.53679180.912249570.80024004

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    2111.659631730.837907310.61582804

    2121.692383770.824539660.57711411

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    9090.001015190.001075740.00008583

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    9110.001018520.001131060.00004435

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    9210.001081940.001140120.00008965

    9220.001111030.001150130.00005817

    9230.001078610.001149180.00008583

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    9320.001166340.001197810.0000267

    9330.001242640.001227380.00007629

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    9600.000977990.00096798-0.00012207

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    9820.000911710.00093889-0.00014353

    9830.001053330.000956060.00005674

    9840.000982280.0009551-0.00009394

    9850.000915050.00087786-0.00020123

    9860.000860690.000947-0.00004196

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    9880.00096750.00100422-0.00003767

    9890.00088930.00091219-0.00009632

    9900.000853060.00073004-0.0002389

    9910.001023290.00087452-0.00024128

    9920.000980850.00097942-0.00007772

    9930.000881670.00080919-0.00030422

    9940.000849250.00085545-0.00031948

    9950.000901220.0007782-0.00019503

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    9980.001105790.00115156-0.00007105

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    10000.000854020.00088596-0.00026512

    10010.000816350.00089025-0.00053883

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    10040.000935550.00096703-0.00030518

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    10060.000915530.00086737-0.00038481

    10070.000774860.00115252-0.00031948

    10080.00075340.00085068-0.0002861

    10090.000922680.00087214-0.00041676

    10100.001039510.00115299-0.00014162

    10110.000878810.00079298-0.00041676

    10120.000985620.00096035-0.00014877

    10130.000957010.00077438-0.00028229

    10140.00092220.00069571-0.00041246

    10150.00099850.00078201-0.00016546

    10160.001204010.00112104-0.00044203

    10170.001140590.00102854-0.00004816

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    10200.000936990.00085306-0.00042582

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    10230.000985150.00118065-0.00026131

    10240.001159670.00124884-0.00031042

    10250.001041890.00097942-0.00016785

    10260.001080040.00121832-0.00038004

    10270.001110550.00089025-0.00020838

    10280.001060490.0010705-0.00035

    10290.00124550.00127602-0.00011826

    10300.001217370.00117731-0.00015402

    10310.001158240.00109529-0.00014639

    10320.001083370.00100946-0.00008249

    10330.001141550.000891210.0000453

    10340.000864030.00106192-0.00063705

    10350.0015440.001199250.00002432

    10360.000967980.00128698-0.00026321

    10370.00135040.001341820.00035143

    10380.001230720.001318450.00005722

    10390.001684190.001400950.00022316

    10400.001008030.00134134-0.00018549

    10410.001418110.00153971-0.00012255

    10420.001073840.00095081-0.00053072

    10430.001471040.00137520.00003481

    10440.000952720.00136089-0.00042915

    10450.001592640.0010047-0.00052023

    10460.001371380.001456260.00017595

    10470.001754760.001814840.00052023

    10480.001442430.00119448-0.00035143

    10490.001647950.00153208-0.00034904

    10500.001780030.001738070.000144

    10510.001335140.00133705-0.0000577

    10520.001418110.001495360.00023079

    10530.001552580.00141811-0.0000124

    10540.001489640.00173903-0.00008535

    10550.002229690.00201750.00038862

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    10600.001224990.00149107-0.00017214

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    10730.001854420.00186348-0.00000763

    10740.001814370.001221180.00026894

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    10760.002503870.001855850.00071955

    1077-0.000295640-0.00206327

    10780.003428940.003682140.00127077

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    10800.002120020.00052691-0.00013685

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    10820.001704220.00103712-0.00011492

    10830.000877860.00191116-0.00018215

    10840.003138070.002952580.00167418

    10850.002402310.00103760.00029945

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    10870.002671240.003040790.00064516

    10880.001233580.002258780.00030565

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    10900.001710890.002907280.00098753

    10910.002306460.002941610.00071526

    10920.000158310.00106335-0.000072

    10930.001859660.00079489-0.00015068

    10940.001707080.001667020.00093603

    10950.001625540.003211980.00108051

    10960.002790450.000958920.00039291

    10970.001981260.002018930.00034857

    10980.00144577-0.000097750.00035381

    10990.001375680.00166893-0.00069904

    11000.003143310.00269604-0.00002956

    /nm

    A

  • Eigenvalue Y=USVt S: Y U: (Scores Matrix) Vt:(Loadings Matrix) Matlab BACK

  • 2. PCA Principal Component Analysis2.1 12.2 22.3 3

  • 2.1 (PCA)BACK()PCA-species

  • 2.2 PCABACKY=USVtSRSDRE

  • BACKNIPALSY=TP S: Y U: (Scores Matrix) Vt:(Loadings Matrix) Matlab

  • -dY(mn)dRE (Real Error,)RE=RSD ()Residual Standard Deviation

  • RE d=1n-1RSD(d) YESdBACK

  • d BACK

  • 2.3 PCABACK

  • Y156PCA nc=37220.0023 0.6145 64.0 0.0017

    1

    0.254223350.230372430.210785390.221105580.284593580.24872971

    0.313990120.276813510.247674940.252788070.339213850.28246164

    0.382839680.33421040.297866820.299607280.408126350.33268547

    0.456165790.396847720.35367870.354299550.485154630.39293909

    0.516010760.448673730.398985390.401407240.55069160.44597101

    0.561341760.488008020.432864190.437384610.601589680.48690891

    0.598151680.521390440.463227270.470496650.644463540.52474356

    0.6237340.545541290.490525250.496278760.672756670.5523982

    0.647166250.567964550.527995110.520838740.690621380.57325268

    0.667809010.591765880.583151340.553624630.701561450.59786701

    0.672870640.606049060.643127920.586463930.695918080.61990738

    0.676922320.621185780.70827150.624155040.690936570.64705372

    0.65653610.616546150.747449870.646055220.668017860.661623

    0.594887730.575476650.736562250.634936810.612896920.64740467

    0.543854240.539767740.716220860.620478630.570547580.6334815

    0.490087990.496368410.666945460.588459970.528792380.60590458

    0.380486010.400353910.537414070.501166340.442699430.53010798

    0.248692990.281230450.374006750.384970660.332773690.42474222

    0.160075190.194409850.255918030.286632540.243893620.32661533

    0.104627130.132245060.172923090.20195150.170877460.23336554

    10

    3600.254223350.230372430.210785390.221105580.284593580.248729710.124219890.098927970.2588582

    3700.313990120.276813510.247674940.252788070.339213850.282461640.105141640.133034230.32987642

    3800.382839680.33421040.297866820.299607280.408126350.332685470.106569290.170055390.40473652

    3900.456165790.396847720.35367870.354299550.485154630.392939090.120348930.205535890.48299837

    4000.516010760.448673730.398985390.401407240.55069160.445971010.137232780.228179930.55021715

    4100.561341760.488008020.432864190.437384610.601589680.486908910.150177480.243629930.60367107

    4200.598151680.521390440.463227270.470496650.644463540.524743560.170384410.254952910.64278841

    4300.6237340.545541290.490525250.496278760.672756670.55239820.191163060.272077080.65825081

    4400.647166250.567964550.527995110.520838740.690621380.573252680.213277340.316293720.64155817

    4500.667809010.591765880.583151340.553624630.701561450.597867010.259633540.385596280.58400106

    4600.672870640.606049060.643127920.586463930.695918080.619907380.331865790.463245390.47973347

    4700.676922320.621185780.70827150.624155040.690936570.647053720.417945380.542980670.36464024

    4800.65653610.616546150.747449870.646055220.668017860.6616230.505988120.587328910.24406719

    4900.594887730.575476650.736562250.634936810.612896920.647404670.579957490.571891780.12177658

    5000.543854240.539767740.716220860.620478630.570547580.63348150.629292960.542492390.04325056

    5100.490087990.496368410.666945460.588459970.528792380.605904580.639344690.479516980.01318741

    5200.380486010.400353910.537414070.501166340.442699430.530107980.605302810.325713160.00423861

    5300.248692990.281230450.374006750.384970660.332773690.424742220.536075590.146186830.00172138

    5400.160075190.194409850.255918030.286632540.243893620.326615330.440052510.048590180.00108337

    5500.104627130.132245060.172923090.20195150.170877460.233365540.322610860.014846320.00075006

    10

    pure

    30mg/L20mg/L20mg/L

    012

    1902.75282242.638339042.52297783

    1912.943461892.782875542.68701172

    1923.000977992.848809722.72023726

    1933.040133482.868571762.72027636

    1943.084602832.876422882.74965382

    1953.088269712.876825332.72430754

    1963.045105932.824354172.70176077

    1973.060160162.779733662.66163921

    1982.935873512.612566472.55446243

    1992.596766952.26791432.30761147

    2002.24844791.891537191.97024298

    2011.970181471.580560681.65089846

    2021.774101261.354688171.39695644

    2031.646615511.192129141.20090151

    2041.572198391.078879831.05478048

    2051.536332131.002148150.94715261

    2061.525492190.948929310.86474419

    2071.53679180.912249570.80024004

    2081.558311940.886286260.74737978

    2091.587092880.86672020.69997501

    2101.623558040.85121870.65660763

    2111.659631730.837907310.61582804

    2121.692383770.824539660.57711411

    2131.722580910.810483930.53973389

    2141.743140220.795020580.50397348

    2151.747312070.779437540.47082376

    2161.73915720.764290810.43998337

    2171.711882590.749679090.41205978

    2181.666390420.737127780.38763618

    2191.605129240.726977830.36676645

    2201.52934980.719757560.34883928

    2211.445393090.71654320.33371735

    2221.3592310.717615130.32130814

    2231.275098320.723123070.31115341

    2241.197107320.73218870.30319881

    2251.1297040.743310450.29745626

    2261.071309090.755526070.29361725

    2271.021915910.767567630.29160357

    2280.980844020.780765060.29121542

    2290.945108890.795456890.29214287

    2300.912821290.81373930.29445839

    2310.883539680.835594650.29814816

    2320.856429580.856641770.30302763

    2330.83339930.872912880.30917072

    2340.81445360.881382940.31686449

    2350.800109390.880619530.3260355

    2360.790949820.87337160.3367362

    2370.785954480.863063810.34883547

    2380.784144880.852069850.36258554

    2390.784448150.839884280.37774611

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