主成分分析 - akimitsu lab. · pdf file主成分分析 主成分分析は...

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  • []

    R

  • d2

    3

    1d

    d

    y

    y

    1

    2y (3)

    y (2)

    y (1)

  • x3

    x1

    x2

    z2

    z1

    Oz1

    z2

  • d2

    3

    1d

    d

    y

    y

    1

    2y (3)

    y (2)

    y (1)

  • x

    0

    x"

    0

    x

    0

    x'

    0

    1

    xi(p) = (x(p)i i) xi(p)

    =

    (x(p)i i)i

  • y3

    y1

    y2

    z1

    z2

    z1 = a11y1 + a12y2 + + a1ryrz2 = a21y1 + a22y2 + + a2ryr

    ... zm = am1y1 + am2y2 + + amryr

    (x1, x2, xr)

    (y1, y2, yr)

    (x(1)1 , x

    (1)2 , x(1)r

    ), ,

    (x(N)1 , x

    (N)2 , x(N)r

    )

    (y(1)1 , y

    (1)2 , y(1)r

    ), ,

    (y(N)1 , y

    (N)2 , y(N)r

    )

  • y3

    y1

    y2

    z1

    z2

    z1z2...

    zm

    =

    a11 a1r

    . . . . . .am1 amr

    y1y2......yr

    (x1, x2, xr)

    (y1, y2, yr)

    (x(1)1 , x

    (1)2 , x(1)r

    ), ,

    (x(N)1 , x

    (N)2 , x(N)r

    )

    (y(1)1 , y

    (1)2 , y(1)r

    ), ,

    (y(N)1 , y

    (N)2 , y(N)r

    )

  • d

    z1

    z2

  • y2

    y1

    z1

    y1(1),( )y2(1)

    z1= y1 y2a1 + a2

    1=a1+ a22 2

    a2

    a1

    a11

    a2

  • y2

    y1

    z1

    y1(1),( )y2(1)

    z1= y1 y2a1 + a2

    1=a1+ a22 2

    a2

    a1

    a11

    a2

  • y2

    y1

    z1

    y1(1),( )y2(1)

    z1= y1 y2a1 + a2

    1=a1+ a22 2

    a2

    a1

    a11

    a2

    y1(1)

    a1 + y 2(1)

    a2

  • y2

    y1

    z1

    y1(1),( )y2(1)

    z1= y1 y2a1 + a2

    1=a1+ a22 2

    a2

    a1

    a11

    a2

    y 1(1)

    a1

    y1(1)

    a1 + y 2(1)

    a2

    y 2(1)

    a2

  • d2

    N

    1d

    d

    y

    y

    1

    2y (N)

    y (2)

    y (1)

    (

    (d(1))2 = a21(y(1)1 )

    2 + a22(y(1)2 )

    2 + 2a1a2(y(1)1 y

    (1)2 )

    (d(p))2 = (a1y(p)1 + a2y

    (p)2 )

    2

  • d2

    N

    1d

    d

    y

    y

    1

    2y (N)

    y (2)

    y (1)

    (

    (d(1))2 = a21(y(1)1 )

    2 + a22(y(1)2 )

    2 + 2a1a2(y(1)1 y

    (1)2 )

    (d(p))2 = (a1y(p)1 + a2y

    (p)2 )

    2

    (d(2))2 = a21(y(2)1 )

    2 + a22(y(2)2 )

    2 + 2a1a2(y(2)1 y

    (2)2 )

  • d2

    N

    1d

    d

    y

    y

    1

    2y (N)

    y (2)

    y (1)

    (

    (d(1))2 = a21(y(1)1 )

    2 + a22(y(1)2 )

    2 + 2a1a2(y(1)1 y

    (1)2 )

    (d(p))2 = (a1y(p)1 + a2y

    (p)2 )

    2

    (d(2))2 = a21(y(2)1 )

    2 + a22(y(2)2 )

    2 + 2a1a2(y(2)1 y

    (2)2 )

    (d(N))2 = a21(y(N)1 )

    2 + a22(y(N)2 )

    2 + 2a1a2(y(N)1 y

    (N)2 )

  • d2

    N

    1d

    d

    y

    y

    1

    2y (N)

    y (2)

    y (1)

    (

    (d(p))2 = (a1y(p)1 + a2y

    (p)2 )

    2

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

    N N - 1

    (d(1))2 = a21(y(1)1 )

    2 + a22(y(1)2 )

    2 + 2a1a2(y(1)1 y

    (1)2 )

    (d(2))2 = a21(y(2)1 )

    2 + a22(y(2)2 )

    2 + 2a1a2(y(2)1 y

    (2)2 )

    (d(N))2 = a21(y(N)1 )

    2 + a22(y(N)2 )

    2 + 2a1a2(y(N)1 y

    (N)2 )+

  • d2

    N

    1d

    d

    y

    y

    1

    2y (N)

    y (2)

    y (1)

    (

    (d(1))2 = a21(y(1)1 )

    2 + a22(y(1)2 )

    2 + 2a1a2(y(1)1 y

    (1)2 )

    (d(p))2 = (a1y(p)1 + a2y

    (p)2 )

    2

    (d(2))2 = a21(y(2)1 )

    2 + a22(y(2)2 )

    2 + 2a1a2(y(2)1 y

    (2)2 )

    (d(N))2 = a21(y(N)1 )

    2 + a22(y(N)2 )

    2 + 2a1a2(y(N)1 y

    (N)2 )

    U(a1, a2) = a21 + 2r12a1a2 + a22

    +

    N N - 1

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

    a1a2

    s11a1 + s12a2 a1 = 0s12a1 + s22a2 a2 = 0

    a21 + a22 = 1

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

    a1a2

    s11a1 + s12a2 a1 = 0s12a1 + s22a2 a2 = 0

    a21 + a22 = 1

    (s11 s12s12 s22

    ) (a1a2

    )=

    (a1a2

    )

    a21 + a22 = 1

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

    (s11 s12s12 s22

    ) (a1a2

    )=

    (a1a2

    )

    a1a2

    s11a1 + s12a2 a1 = 0s12a1 + s22a2 a2 = 0

    a21 + a22 = 1

    a21 + a22 = 1

  • a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    U(a1, a2) = s11a21 + 2s12a1a2 + s22a22

    (s11 s12s12 s22

    ) (a1a2

    )=

    (a1a2

    )

    a1a2

    s11a1 + s12a2 a1 = 0s12a1 + s22a2 a2 = 0

    a21 + a22 = 1

    a21 + a22 = 1

  • U(a1, a2) = a21 + 2r12a1a2 + a22

    a1 + r12a2 a1 = 0r12a1 + a2 a2 = 0

    a21 + a22 = 1

    a1a2

    a21 + a22 = 1

    U(a1, a2)

    a1 a2

    G(a1, a2, ) = U(a1, a2) (a21 + a22 1)

    a21 + a22 = 1

    (1 r12

    r12 1

    ) (a1a2

    )=

    (a1a2

    )

  • U(a1, a2) = a

    21 + a

    22 + 2r12a

    1a

    2

    = a1(a1 + r12a

    2) + a

    2(r12a

    1 + a

    2)

    = (a21 + a22 ) =

    U(a1, a2) = s11a

    21 + s22a

    22 + 2s12a

    1a

    2

    = a1(s11a1 + s12a

    2) + a

    2(s12a

    1 + s22a

    2)

    = (a21 + a22 ) =

  • U(a1, a2) = a

    21 + a

    22 + 2r12a

    1a

    2

    = a1(a1 + r12a

    2) + a

    2(r12a

    1 + a

    2)

    = (a21 + a22 ) =

    U(a1, a2) = s11a

    21 + s22a

    22 + 2s12a

    1a

    2

    = a1(s11a1 + s12a

    2) + a

    2(s12a

    1 + s22a

    2)

    = (a21 + a22 ) =

    12

  • (

    A =

    s11 s12 s1rs21 s22 s2r...

    .... . .

    ...sr1 sr2 srr

    1 2 r 0

    i1 + 2 + + r

    =i

    r

    k=1

    k

    1 + + i1 + 2 + + r

    =

    i

    j=1

    j

    r

    k=1

    k

    r )

    Aai = iai ai =

    ai1ai2...

    air

  • (

    r )

    1 2 r 0

    Aai = iai ai =

    ai1ai2...

    air

    A =

    1 r12 r1rr21 1 r2r...

    .... . .

    ...rr1 rr2 1rr

    i1 + 2 + + r

    =i

    r

    k=1

    k

    1 + + i1 + 2 + + r

    =

    i

    j=1

    j

    r

    k=1

    k

  • r )

    Aai = iai ai =

    ai1ai2...

    air

    z1 = a11y1 + a12y2 + + a1ryrz2 = a21y1 + a22y2 + + a2ryr

    ... zm = am1y1 + am2y2 + + amryr

    m

    A =

    s11 s12 s1rs21 s22 s2r...

    .... . .

    ...sr1 sr2 srr

    1 + + m1 + 2 + + r

    =

    m

    j=1

    j

    r

    k=1

    k

  • r )

    Aai = iai ai =

    ai1ai2...

    air

    z1 = a11y1 + a12y2 + + a1ryrz2 = a21y1 + a22y2 + + a2ryr

    ... zm = am1y1 + am2y2 + + amryr

    m

    A =

    1 r12 r1rr21 1 r2r...

    .... . .

    ...rr1 rr2 1rr

    1 + + m1 + 2 + + r

    =

    m

    j=1

    j

    r

    k=1

    k

  • R

    d2

    3

    1d

    d

    y

    y

    1

    2y (3)

    y (2)

    y (1)

  • R

    > w1 w2 summary(w2)

    >plot(w2$x,type=n)

    >text(w2$x,rownames(w2$x))R

  • R

    name height weight

    A1 147.3 52.3

    A2 149.9 53.2

    A3 152.4 54.5

    A4 154.9 55.9

    A5 157.5 57.3

    A6 160.0 58.6

    A7 162.6 60.0

    A8 165.1 61.4

    A9 167.6 63.2

    A10 170.2 64.5

    A11 172.7 66.4

    A12 175.3 68.2

    A13 177.8 70.0

    A14 180.3 72.3

    A15 182.9 74.5

    name height weight

    A01 -1.5652476 -1.4022687

    A02 -1.3416408 -1.2732255

    A03 -1.1180340 -1.0796608

    A04 -0.8944272 -0.8860962

    A05 -0.6708204 -0.6925315

    A06 -0.4472136 -0.4989668

    A07 -0.2236068 -0.3054021

    A08 0.0000000 -0.1118374

    A09 0.2236068 0.1462489

    A10 0.4472136 0.3398136

    A11 0.6708204 0.5978998

    A12 0.8944272 0.8559861

    A13 1.1180340 1.1140723

    A14 1.3416408 1.4366802

    A15 1.5652476 1.7592880

  • R

    > w1 w2 summary(w2)

    >plot(w2$x,type=n)

    >text(w2$x,rownames(w2$x))R

  • R

    > w1 w2 summary(w2)

    >plot(w2$x,type=n)

    >text(w2$x,rownames(w2$x))R

  • d2

    3

    1d

    d

    y

    y

    1

    2y (3)

    y (2)

    y (1)

  • 1 r(

    U(a1, a2, , ar) = s11a21 + s22a22 + + srra2r

    +2(s12a1a2 + s13a

    1a

    3 + + sr1rar1ar)

    = s11a21 + s22a22 + + srra2r

    +( s12a1a2 + s13a

    1a

    3 + + s1r1ar1ar1 + s1rarar)

    +(s21a2a1 + s23a

    2a

    3 + + s2r1a2ar1 + s2ra2ar)

    +(sr1ara1 + sr2a

    ra

    2 + sr3a

    ra

    3 + + sr1rarar1 )

    = a1(s11a1 + s12a

    1 + + s1rar)

    +a2(s21a1 + s22a

    2 + + s2rar)

    +ar(sr1a

    1 + sr2a

    2 + + srrar)

    = (a21 + a22 + + a2r ) =

    r(r - 1)r