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UNIVERSIDAD POLITECNICA SALESIANA
SEDE CUENCA CARRERA DE INGENIERIA DE SISTEMAS
Tema:
"EXTRACCIÓN DE CARACTERÍSTICAS PARA LA CLASIFICACIÓN DE IMÁGENES A TRAVÉS DE LA REDUCCIÓN DE DIMENSIONALIDAD APLICANDO ANÁLISIS DE COMPONENTES INDEPENDIENTES (ICA)"
Tesis previa a la obtención del
Título de Ingeniero de Sistemas
AUTOR:
Esteban Andrés López Pacheco
DIRECTOR:
Ing. Vladimir Robles Bykbaev
CUENCA - ECUADOR
2013
II
Breve reseña del autor e información de contacto
Esteban Andrés López Pacheco
Egresado de la Carrera de Ingeniería de Sistemas
Universidad Politécnica Salesiana
lopez.eandres@hotmail.com
III
Ing. Vladimir Robles B.
CERTIFICA
Haber dirigido y revisado prolijamente cada uno de los capítulos del informe de tesis,
realizada por el Sr. Andrés López Pacheco, y por cumplir los requisitos autorizo su
presentación.
Cuenca, Junio del 2013
Ing. Vladimir Robles B.
Director de Tesis
IV
DECLARACIÓN DE RESPONSABILIDAD
Yo, Esteban Andrés López Pacheco, portador de la cédula de ciudadanía 0105610539,
estudiante de la Ingeniería en Sistemas, certifico que los conceptos desarrollados,
análisis realizados, así como los criterios vertidos en la totalidad del presente trabajo son
de exclusiva responsabilidad del autor y autorizo a la Universidad Politécnica Salesiana el
uso de la misma con fines académicos.
Cuenca, Junio del 2013
Andrés López Pacheco
V
DEDICATORIA
Por haber estado una vez más presente en esta batalla, la más difícil en lo que lleva mi
vida, este gran triunfo que representa el terminar la universidad te lo dedico a ti. Aún
recuerdo claramente aquellos días en los que me enseñabas los más ejemplares valores
del día a día, días en los que empecé a convivir contigo, aún recuerdo aquel triste día en
el que inesperadamente te fuiste, día en el que te prometí salir adelante y afrontar
valientemente cualquier reto que el camino me tenga preparado. Desde aquel día
muchas batallas se me han presentado en mi camino, en cada una de ellas te he sentido
a mi lado ya que has causado en mi la suficiente inspiración para no dar mi brazo a
torcer. Todo este trabajo va dedicado a tu memoria abuelita Leti.
A ti abuelito Leonidas, que a pesar de todos tus años te sigues levantando firmemente
día tras día causándome una gran admiración por la fuerza con la que vives,
mostrándome que las adversidades de la vida son superables si las enfrentamos con una
gran actitud y muchas ganas de vivir.
A ustedes, mis amados padres, sin ustedes es seguro que nada de lo que ahora estoy
viviendo hubiese sido posible, su apoyo incondicional ha sido vital en los éxitos que ahora
cosecho. Padres como ustedes no los hay en todos lados, nuestra relación no siempre ha
sido la mejor, pero así mismo, la forma de mostrarles cuanto aprecio todo lo que me han
instruido es logrando superar difíciles pruebas de la vida, que al final son victorias en las
que ustedes sin falta están. Reitero, sin su amor y su apoyo, difícilmente hubiese llegado
a ser quien ahora soy.
Por último, sin eso significar menos importante, a mis hermanos, Johanna y Mateo, y
como no, a mi sobrino Isaac, fueron siempre una inspiración que me motivaba a
superarme todos los días, ya sea por seguir su ejemplo, o también ser yo quien lo ponga,
para ti Mateo, te motivo a que me superes con creces, sé que tienes un gran potencial
para lograr grandes cosas, y que así mismo representes el mejor ejemplo para nuestro
sobrino Isaac.
Andrés
VI
AGRADECIMIENTOS
Con este documento, se cierra una de las más hermosas etapas de mi vida hasta ahora,
con la mano en el corazón digo que la universidad cambio mi vida para bien, que las
personas que durante estos más de 5 años conocí han ido formando poco a poco mi
carácter y mi personalidad, aquellos docentes que año tras año me retaban a dar lo
mejor de mí y mucho más, principalmente aquellos que empezaron siendo profesores, y
los he terminado considerando amigos. Empiezo con el Ing. Vladimir Robles, a quien
conocí como docente estando aún en segundo ciclo, desde aquel momento supe que
tenerlo como docente marcaría un antes y después en mi vida académica, prueba de
aquello son los concursos que he ganado gracias a las sabias enseñanzas que fueron
impartidas en sus clases, pero más allá de ser docente, su personalidad es la que
considero ideal para un docente, ya que al final se mostró como una persona llena de
valores, totalmente amigable. Me considero totalmente afortunado de que esta tesis
haya sido dirigida por usted, aprendiendo al final mucho sobre este tema y además
sobre ortografía, gracias por ser mi docente y amigo durante casi toda mi vida
universitaria.
Además de Vladimir, docentes como la Ing. Tacuri, Ing. Ingavelez, Ing. Quinde, Ing.
Quintuña quienes en su debido momento confiaron en mí asignándome
responsabilidades que trascendían de la vida estudiantil, más que responsabilidades,
llegaron a ser oportunidades que muy agradecido las tome.
¿Qué habría sido de mi vida estudiantil sin amigos?, de seguro no sería la misma
persona que ahora soy. Siempre he considerado que en la universidad se conoce a las
personas que permanecerán para toda la vida, una de aquellas personas eres tu Fátima,
querida �primita�, realmente no tengo idea de lo mucho que te he llegado a querer,
aunque con mucha gracia recuerdo que no siempre fuimos los mejores amigos que
ahora somos, eres de las personas que deseo que estén para toda mi vida, siempre te
deseare lo mejor, me causa mucha nostalgia el recordar aquellos días en los que
coincidíamos en cada hora, aquellos largos fines de semana de tareas, y en algunas
ocasiones, amanecidas por proyectos, si que formamos un gran equipo querida prima,
�López & López�, deberíamos pensar en patentar una marca. Ojala que la vida nos reúna
de nuevo más adelante.
Una gran amiga eres tú también Priscila �picher�, tantas historias que asimismo
recuerdo desde el día en que te conocí, hiciste de mi vida universitaria bastante mejor, si
que adoro matarte de iras, hablar por horas y horas, coincidir en proyectos, apoyarnos
mutuamente en cualquier necesidad que se nos presentaba, te echo de menos pichita.
Este documento crecería demasiado si nombrara a todos aquellos amigos que aportaron
con su granito de arena para que mi vida universitaria sea la mejor, gracias Daniel,
Joshe, Pedro, Tavo, Gato, Tuto, Gordo, Punka, Eva, Marco, Edisson, Geovanny, Blanco,
Libio, Davicho, Mocho, Jenner, Lore, Dunia, Jessy, Ardilla, Kelo, Diabla, Brujita, Javier,
Diego, Jacky, Geovy, Jorge, etc.
VII
Trascendiendo las tareas universitarias, tuve la grandiosa oportunidad de ser
representante estudiantil, lo que ha dejado en mí un muy grato recuerdo, un muy cordial
agradecimiento al Eco. Luis Tobar, Vicerrector de la UPS sede Cuenca, por todas aquellas
horas de planificación y negociación de eventos. Al Padre Javier Herrán, Rector de la UPS,
por brindarme su vital apoyo en varias ideas. A diversas autoridades con las que trace
una buena amistad, gracias a Nancita, Vilmita, Sole, Paolita, Carmita, Manolin, Blas,
Andrés, Xavier, María, Ing. Díaz, Vicky, Eco. Vázquez, etc.
De parte de mi familia, a mis padres por confiar plenamente en mi al permitirme
estudiar los 5 años que duro mi carrera universitaria.
No podría olvidar el agradecer también a una persona que durante el transcurso de mi
tesis ha estado pendiente, y siempre apoyándome a que siga adelante, me refiero a ti
Anahi, han sido muy motivadoras todas aquellas largas jornadas en las que me has
acompañado, subiéndome los ánimos cuando parecía que todo estaba perdido. Gracias
amor, sin ti todo hubiese sido mucho más difícil.
A mis nuevas grandes amistades que han estado también pendientes del desarrollar de
este trabajo, pendientes por mi bienestar en mi nueva vida, gracias Tania, Lenin, Xavier y
Livania.
Mi grandiosa experiencia universitaria ha sido gracias a cada uno de ustedes. Los aprecio
demasiado.
Andrés
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−→x = [x0, x1, · · · , xn]
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i=1 x2
i
4'( $5$-+*'0 ). )$ 1$)$, $6&(,$( $* &,-,7' 1$* ).2!.$"&$ #$%&'(3
−→a = [1, 2, 3, 4, 5]
8"&'"%$)3
|−→a | =√
12 + 22 + 32 + 42 + 52
|−→a | =√55
|−→a | = 7,42
!"!"! ! 0*%$ () ,)-.'/)1
9, )!-, 1$ #$%&'($) )$ *, +!$1$ ($,*.:,( $"&($ ; #$%&'($) <!$ &.$"$" $* -.)-'
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−→a ?
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−→a = [3, 2, 1,−2] ? −→b = [2,−1, 4, 1]
@
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!"#$"%&'(
−→a +−→b = [3, 2, 1,−2] + [2,−1, 4, 1]
−→a +−→b = [(3 + 2), (2− 1), (1 + 4), (−2 + 1)]
−→a +−→b = [5, 1, 5,−1]
!"!"!#! $%&'()&(*+*(,- ./ %- /0*+&+1 )21 %- 3/*'21
−→a = [1, 2, 3, 4]
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−→a ·2 = [2, 4, 6, 8]
!"!"!4! 512.%*'2 /0*+&+1
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i=1 ai·bi
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−→a = [1, 6, 7, 4] ?−→b = [3, 2, 8, 3]
!"#$"%&'(
−→a ·−→b = [1, 6, 7, 4]·[3, 2, 8, 3]−→a ·−→b = 1·3 + 6·2 + 7·8 + 4·3
−→a ·−→b = 83
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|−→x | = 1
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−→a = [ 25 ,45 ,
25 ,
15 ]
C. 8-&(
|−→a | =√
( 25 )2 + ( 45 )
2 + ( 25 )2 + ( 15 )
2
|−→a | =√
2525
|−→a | = 1
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#2*-2%)'%# $%&'"(%# #") "('"*")+,%#9
−→a = [2, 1,−2, 4] : −→b = [3,−6, 4, 2]
:+ 4-%
−→a ·−→b = [2, 1,−2, 4]·[3,−6, 4, 2]
−→a ·−→b = [(2·3) + (1·(−6)) + (−2·4) + (4·2)]
−→a ·−→b = 0
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−→a = [ 25 ,15 ,− 2
5 ,45 ] :
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25 ,
45 ,
25 ]
:+ 4-%9
|−→a | = 1 :
∣
∣
∣
−→b∣
∣
∣= 1
:9
−→a ·−→b = 0
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A =
a0,0 a0,1 . . . a0,n
a1,0 a1,1 . . .=
=
=
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am,0 · · · · · · am,n
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A =
1 2 34 5 67 8 9
A* %&#"-8*,-%# -,&B#@
AT =
1 4 72 5 83 5 9
7# %&#"-8*,-%# -, &'0, # /#- -'0*',"%,- 8&28',+#+,-@
(AT )T = A
AT +BT = (A+B)T
(−→a ·A)T = −→a ·AT
(A·B)T = BT ·AT
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A =
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A·AT = I
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9 -(+-3*(+$ 6.' '2 .(* ,*&)-/ $)&$0$(*# -(:')2*8
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A =
a0,0 0 0 0 00 a1,1 0 0 00 0 a2,2 0 00 0 0 a3,3 00 0 0 0 a4,4
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<( '21*?$# 3$($3-+$2 3$,$ :'3&$)'2 = :*#$)'2 1)$1-$2 )'21'3&-:*,'(&'5 .(
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λ5 1)$(.(3-*+* #*,%+*8
@( '-0'(:'3&$) '2 .( :'3&$) )'1)'2'(&*+$ 3$,A(,'(&' 1$) #* #'&)* x8 B*
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D$+$2 #$2 '-0'(:*#.'2 = '-0'(:'3&$)2 2*&-2E*3'( #* 2-0.-'(&' '3.*3-F(4
A·x = λx
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3$,1#'>$28
G#*(&'*(+$ .(* +'7(-3-F( ,;2 E$),*#4
H$(2-+')*(+$ #* ,*&)-/ 3.*+)*+* "5 +'3-,$2 6.' λ '2 .( '-0'(:*#.' +' A 2-
'I-2&' .( :'3&$) x +-2&-(&$ +' ! &*# 6.' 2*&-2E*3' #* '3.*3-F( A·x = λx8 <( '2&'
3*2$5 * x 2' #$ ##*,* '-0'(:'3&$)8 "# 3$(>.(&$ λ = x 2' #$ 3$($3' 3$,$ '-0'(1*-)
+' A8
G$) '>',1#$4 J' +'2'* 2*%') 2-
x =
[
1−2
]
'2 .( '-0'(:'3&$) 3$))'21$(+-'(&' *# '-0'(:*#.' λ = 0,+' #* ,*&)-/
A =
[
6 3−2 −1
]
<,1'/*,$2 )'',1#*/*(+$ #$2 :*#$)'2 '( #* EF),.#*4
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!.
A·x = λx[
6 3−2 −1
]
·[
1−2
]
= 0
[
1−2
]
[
6·1 + 3·!"#$(−2)· %!" $·!"#$
]
=
[
00
]
[
00
]
=
[
00
]
&'(' )*'+)*,-./ 01'2 345 x 6787 972 4. :7*+'2 6,9+,.+' 67 ;</ 9, +7.7('9
4. :7*+'2 67 ;/ 7= 2794=+)6' 67 =) 7*4)*,-. 27>2797.+)2?) 4. *)9' ,(>2@*+,*'A
B =) 7*4)*,-. A·x = λx 97 =) >4767 79*2,8,2 +)(8,5. 67 =) C'2() (A−λ·I)·x =0/ 7. 6'.67 I 79 4.) ()+2,D ,67.+,6)6 67= (,9(' +)()E' 347 AA F7 6787 *4(>=,2
347 (A− λ·I) 97) .' ,.:72+,8=7A
G=)()('9 ) p(λ) = det(A−λ·I) 7= >'=,.'(,' *)2)*+72?9+,*'A B >)2+,2 67 79+7
>'=,.'(,' >'67('9 '8+7.72 ='9 7,H7.:)=479A
I. 7= 9,H4,7.+7 7J7(>=' :)('9 ) '8+7.72 ='9 7,H7.:)=479 67 =) 9,H4,7.+7 ()+2,DK
A =
[
2 −4−1 −1
]
I.+'.*79/ 7(>7D)('9 277(>=)D).6' 7. =) C-2(4=)K
p = det(A− λ·I)
p(λ) = det(
[
2 −4−1 −1
]
− λ·[
1 00 1
]
)
p(λ) = det(
[
2 −4−1 −1
]
−[
λ 00 λ
]
)
p(λ) = det(
[
(2− λ) −4−1 (−1− λ)
]
)
p(λ) = (2− λ)·(−1− λ)− (−4)·(−1)
p(λ) = (−2− 2λ+ λ+ λ2)− 4
p(λ) = (λ2 − λ− 6)
p(λ) = (λ− 3)(λ+ 2)
λ1 = 3 λ2 = −2
F7 7.*'.+2)2'. # 7,H7.:)=479/ >)2) *)6) 4.' 67 79+'9 6787('9 '8+7.72 94
*'2279>'.6,7.+7 7,H7.:7*+'2/ >)2) =' *4)= 49)('9 =) 7*4)*,-. (A − λ·I)·x = 0/7.+'.*79K
>)2) λ1 = 3K
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
(
[
2 −4−1 −1
]
− 3·[
1 00 1
]
)·[
x1
x2
]
=
[
00
]
(
[
2 −4−1 −1
]
−[
3 00 3
]
)·[
x1
x2
]
=
[
00
]
[
(2− 3) −4−1 (−1− 3)
]
·[
x1
x2
]
=
[
00
]
[
−1 −4−1 −4
]
·[
x1
x2
]
=
[
00
]
[
−x1 −4x2
−x1 −4x2
]
=
[
00
]
"#$#%&' ! #()*(+&$#' '+%+,*-#'.
−x1 − 4x2 = 0 / −x1 − 4x2 = 0
0$ ,* 1-+%#-* #()*(+2$ -##%1,*3*%&' x2 1&- t4 #$5&$(#'.
−x1 − 4t = 0
x1 = −4t
6 -##%1,*3*%&' #'5# 7*,&- #$ ,* '#8)$9* #()*(+2$.
4t− 4x2 = 0
4x2 = 4t
x2 = t
0$5&$(#'4 1*-* λ1 = 3 #$(&$5-*%&' :)# ') #+8#$7*,)# #'.
x =
[
−41
]
*;&-*4 1*-* λ1 = −2.
(
[
2 −4−1 −1
]
− (−2)·[
1 00 1
]
)·[
x1
x2
]
=
[
00
]
(
[
2 −4−1 −1
]
−[
−2 00 −2
]
)·[
x1
x2
]
=
[
00
]
[
(2 + 2) −4−1 (−1 + 2)
]
·[
x1
x2
]
=
[
00
]
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
" # $ % # & %
'()*+, #- . /)0,1 230434,1 5)+) 670+)6+ 8) 96*4) : 8) *61;4)34<= 0>543)
[
4 −4−1 1
]
·[
x1
x2
]
=
[
00
]
[
4x1 −4x2
−x1 x2
]
=
[
00
]
?6=69,1 # 63()34,=61 .
4x1 − 4x2 = 0 : −x1 + x2 = 0
@= 8) 5+496+) 63()34<= +66958)A)9,1 x2 5,+ tB 6=0,=361.
4x1 − 4t = 0
4x1 = 4t
x1 = t
C +66958)A)9,1 6106 ;)8,+ 6= 8) 16D(=*) 63()34<=.
−t+ x2 = 0
x2 = t
@=0,=361B 5)+) λ1 = 2 6=3,=0+)9,1 E(6 1( 64D6=;)8(6 61.
x =
[
11
]
!"!#! $%&'(&)* +*,)-.*,(&)*
!"!#!"! $%&'()*(+, -./(*)
F)+046=*, *6 (= 3,=3650, 16=3488,B 8) *61;4)34<= 0>543)B , 0)9G4H= 88)9)*)
*61;4)34<= 610I=*)+ J+65+616=0)*) 3,= σKB 94*6 3()=0, 16 165)+)= 8,1 *)0,1 *6
8) 96*4) *6 8,1 9419,1- F)+) ,G06=6+ 8) *61;4)34<= 0>543)B 670+)69,1 8) +)>A
3()*+)*) *6 8) ;)+4)=A)-
L) ;)+4)=A) +65+616=0)*) 5,+ σ261 8) 96*4) *6 8)1 *4M6+6=34)1 3,= 8) 96*4)
686;)*)1 )8 3()*+)*,- ',= 68 14D(46=06 6N6958, 675843)+69,1 8) ,G06=34<= *6
610,1 ;)8,+61-
O)9,1 ) P)88)+ 8) ;)+4)=A) : 8) *61;4)34<= 0>543) *6 8,1 14D(46=061 *)0,1.
F)+049,1 *6 8) ,G06=34<= *6 8) 96*4).
media = 8+1+5+7+8+6+77
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
media = 6
"#$%& $'()*)+$, -& .&%/&*0&1
σ22
(8−6)2+(1−6)2+(5−6)2+(7−6)2+(8−6)2+(6−6)2+(7−6)2
7
σ22
4+25+1+1+4+0+17
σ2 = 5,14
3 4$% 5-(/+$6 $'()*)+$, -& 7),./&8/9* (:4/8&1
σ =√5,14
σ = 2,267786
!"!#! ! $%&'()'*+'
;, %)4%),)*(&7& 8$* σxy
<* 7&($ /*()%),&*() 7) -& 8$.&%/&*0& ), =>) ),(& /*7/8& )- ,)*(/7$ 7) 8$%%)?
-&8/9* )*(%) -&, .&%/&'-), ),(>7/&7&,6 @ $')7)8) & -&, ,/A>/)*(), 4%$4/)7&7),1
B/ σxy < 0 )*($*8), -& 8$%%)-&8/$*), ), /*.)%,&C
B/ σxy > 0 )*($*8), -& 8$%%)-&8/$*), ), 7/%)8(&C
B/ σxy = 0 )*($*8), ,) ,&') =>) *$ )D/,() >*& %)-&8/9* )*(%) -&, .&%/&'-),
/*.$->8%&7&,C
!"!#!#! $%((,-'.)/*
E& 8$%%)-&8/9* %)4%),)*(& &- */.)- 7) 7)4)*7)*8/& )D/,()*() )*(%) F $ +G,
.&%/&'-), ,/(>&7&, )* >* 4-&*$ '/7/+)*,/$*&-C ;* $(%$, (H%+/*$,6 4)%+/() 7)()%?
+/*&% ,/ )- 8&+'/$ )* >*& .&%/&'-) &I)8(& & $(%& 7)*(%$ 7)- +/,+$ 4-&*$6 ,/ ),
),() )- 8&,$ ,) ,&') =>) #&@ 8$%%)-&8/9* )*(%) )--&,C
;D/,()* J A%&7$, 7) 8$%%)-&8/9*1
K/%)8(&6 )* -& =>) )- &>+)*($ )* >*& .&%/&'-) %)4%),)*(& >* /*8%)+)*($
)* -&, 7)+G, .&%/&'-),C ;,() (/4$ 7) 8$%%)-&8/9* ),(G %)4%),)*(&7$ 8$+$
>*& %)8(& 8%)8/)*()6 7) -& ,/A>/)*() +&*)%&1
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
"#$%&' () * +,&&-.'/#01 2#&-/3')
415-&6'7 /%'12, -. '%8-13, 2- %1' 5'&#'9.- 2-6-1/'2-1' -. 2-/&-8-13, 2-
.' ,3&'7 -6 &-:&-6-13'2' /,8, %1' &-/3' 2-/&-/#-13-*
"#$%&' ()(* +,&&-.'/#01 #15-&6')
;%.'* <- 2' /%'12, -. /'89#, -1 %1' 5'&#'9.- 1, &-:&-6-13' /'89#, '.$%1,
-1 -. &-63, 2- 5'&#'9.-6) <- .' &-:&-6-13' /,8, %1' 1%9- 2- :%13,6)
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
"#$%&' ()*+ ,-&&./'0#12 2%/')
34.5678 .9#7:.2 * 4#;.&.2:.7 $&'4-7 4. 0-&&./'0#12+
"%.&:.+ <. 4' 5#.2:&'7 /'7 ='&#'>/.7 .7:62 567 0.&0' 4. /' &.0:')
"#$%&' ()?+ ,-&&./'0#12 "%.&:.)
@A>#/+ ,-2:&'&#' ' /' '2:.&#-&8 .9#7:.2 0%'24- /-7 B%2:-7 .7:62 '/.C'4-7 4.
/' &.0:')
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
"#$%&' ()*+ ,-&&./'0#12 345#/)
,'5. #23#0'& 6%. .2 /'7 #89$.2.7 '2:.&#-&.7; /-7 <%2:-7 7-2 ='/-&.7 .7:'3>7?
:#0-7 &.<&.7.2:'3-7 .2 %2 7#7:.8' 3. 0--&3.2'3'7)
@/ 0-.A0#.2:. 3. 0-&&./'0#12 7. /- .B<&.7' 0-2 /' /.:&' r; C /' D1&8%/' ' %7'&7.
<'&' 0'/0%/'&/' .7 /' 7#$%#.2:.+
r =σxy
σxσy
E&-0.3.&.8-7 0-2 %2 0/'&- .F.8</-; :-8'3- 3. G(HI) J'//'& ./ 0-.A0#.2:.
3. 0-&&./'0#12 :.2#.23- 6%. /'7 2-:'7 3. ( '/%82-7 .2 ( '7#$2':%&'7 7-2 /'7
7#$%#.2:.7+
K':.89:#0'7 ( L M M * N N ! ! H O O
">7#0' L ( M M M N M N ! P O
,%'3&- ()(+ K':&#Q 3. 2-:'7; %7'3'7 <'&' %2 .F.8</-)
E'&:#8-7 &.'/#Q'23- '/$%2-7 09/0%/-7 6%. 2-7 7.&=#&92 897 '3./'2:.; 0-2
/' /.:&' x &.<&.7.2:'&.8-7 ' K':.89:#0'7; C 0-2 /' /.:&' y ' ">7#0' <'&' %2'
D'0#/#:'& /' 0-8<&.27#12+
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
xi yi xi·yi x2i y2i
" " #
$ $ % % %
# " ! & #
# # & & &
' # "( "' &
& # "# $& &
& & $& $& $&
) # "! #% &
) & #" #% $&
! ) '& &# #%
( % %( (( !
( ( (( (( ((
72 60 431 504 380
*+,-./ "0$1 2,3.45 -6 .67+83,-/7 69 :;8:+8/7 <.6=4/70
>, ?834@, A8, :/..67</9-6 , 8/7 3/3,867 -6 :,-, :/8+@9,0 BC/.,D :,8:+8,@/7
8,7 @6-4,7 ,.43@E34:,71
x = 7212 y = 60
12
x = 6 y = 5
F./:6-6@/7 , :,8:+8,. 8, :/=,.4,95,1
σxy = 43112 − 6·'
σxy = 5.92
G9:/93.,@/7 8,7 -67=4,:4/967 3H<4:,71
σx =√
50412 − 62 σy =
√
38012 − 52
σx = 2,45 σy = 2,58
I </. ?834@/D /J3696@/7 68 :/6A:46936 -6 :/..68,:4K91
r = 5,922,45·2,58
r = 0,94
L/@,9-/ 6736 :/6A:46936 -6 :/..68,:4K9D </-6@/7 -6:4. M+61
N6J4-/ , M+6 67 @+O :6.:,9/ , D 8, :/..68,:4K9 67 @+O P+6.360
N6J4-/ , M+6 67 </7434=/D 8, :/..68,:4K9 67 -4.6:3,0
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
"# $%$&'() &*+ ,$-,.#) . #/$+0-.+ #$,$+12.2$+3 $+ 2$,1-3 .( .#*(1+1+ 2$ 1&*4$#$+
+$ $5')#$ . ,)#01#/.,16#3 /#. 1&.4$# .( 7#.( $+ /#. &.0-18 9121&$#+1)#.(3 $+
2$,1-3 /#. &.0-18 ,)&'/$+0. 2$ :.()-$+ ,)&);
<14/-. =>?; @)&A#&$#0$ /#. 1&.4$# $+0* -$'-$+$#0.2. 2$ $+0. B)-&.> C# (.
74/-. :$&)+ /#. 1&.4$# $# $+,.(. 2$ 4-1+$+3 ,.2. '15$( ')+$$ /# :.()- $#0-$ D
E =FF3 ,)&) +$ G. 21,G) .#0$+3 /#. 1&.4$# $+ /#. &.0-18 2$ '15$($+>
H.-. $( $%$&'()3 /#)+ :.()-$+ I/$ +$ $#,/$#0-.# $# (. ')+1,16# x, y +)# ()+
+14/1$#0$+3 #60$+$ I/$ xi E yi +)# (.+ ,))-2$#.2.+ 2$ /# '15$(3 &1$#0-.+ I/$ fi$+ $( :.()- I/$ 21,G) '15$( 01$#$>
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
xi yi fi
! "
! "
! #
" "
$
# %
$ " #
$
&'()*+ ,$- .(/*01 2(*340)( ( 5( 6'3 4+789739/3 2'3)3 *32*3:39/(* ( '9(
07(;39,
<7231(7+: *3(501(9)+ 4=54'5+: 2*3>0+:-
xi yi fi xi·fi x2i ·fi yi·fi y2i ·fi xi·yi·fi
! " ! ! !
! " ! ! $ !
! # ! ! ? "@ !
" " $ " "
$ @ "? @ "? "?
# % "! ! "% $% #!
$ " # " $@ # # "
$ @ # $ @ "?
20 40 120 41 97 76
&'()*+ ,%- &=54'5+: *3(501()+: 39 5( 7(/*01 (9/3*0+*,
&(54'5(7+: 5(: 73)0(: (*0/7A/04(:-
x = 4020 y = 41
20
x = 2 y = 2,05
&(54'5(7+: 5(: )3:>0(40+93: /B204(:-
σ2x = 120
20 − 22 σ2y = 97
20 − 2,052
σ2x = 2 σ2
y = 0,65
σx =√2 σy =
√0,65
σx = 1,41 σy = 0,81
!"#$"!%&' "! #&(!)*!+,!-
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
σxy = 7620 − 2 ∗ 2,05
σxy = −0,3
"#$ %&'()#* #+',-,)#. ,& /#,0/(,-', 1, /#$$,&2/(3-4
r = −0,31,41·0,81
r = −0,26
5#- &# 67, /#-/&7()#. 67,4
8(,-1# -,92'(:# ,& /#,0/(,-', 1, /#$$,&2/(3-* ,.'2 ,. (-:,$.2;
<& .,$ /,$/2-# 2 =* &2 /#$$,&2/(3- ,. 1>+(&;
! ! "#$%&'(&$)%)*+ #+ ,-.&/#$ 0)+12.-/#$
! !"! #$%&'(')&
?-'$, &2. '>/-(/2. @2$2 ,& @$#/,.2)(,-'# 1, ()A9,-,. 1(9('2&,. ,-/#-'$2)#.
2 &2 1,./#)@#.(/(3- ,- :2&#$,. .(-97&2$,.* 2 &2 67, &&2)2$,)#. 8BC @#$ .7.
.(9&2. ,- (-9&>.; ?.@,/D0/2),-', ,- &2 /#)@$,.(3- 1, ()A9,-,. ,. ,- 1#-1, .,
7.2 8BC* .(,-1# ,& #+E,'(:# '$2-.)('($ 7-2 ()29,- $,17/(,-1# .7 '2)2F# @,$#
2& )(.)# '(,)@# @,$1(,-1# &2 ),-#$ /2-'(121 1, (-G#$)2/(3-;
H IJ @&2-',2 7-2 1,0-(/(3- )2. G#$)2&4 1, 7-2 )2'$(K A 1, n 0&2. @#$ m
/#&7)-2.* 2 @2$'($ 1, .7. ,(9,-:2&7,. @,$',-,/(,-',. 2 &2 )2'$(K #+',-(12 1,
AT ·A @#1,)#. #+',-,$ .7. :2&#$,. .(-97&2$,. $,@$,.,-'21#. @#$ σ /2&/7&2-1#
&2 $2DK /721$212 1, &#. )(.)#.; ?.'#. ,(9,-:2&7,. .(,)@$, .,$A- @#.('(:#. @2$2
@#1,$ #+',-,$ .7 $2DK /721$212; "#$ /212 ,(9,-:2&7, ., #+'(,-, 7- :2&#$ .(-97&2$*
2 &#. 67, #$1,-2$,)#. 1, G#$)2 1,./,-1,-',* ,. 1,/($4
σ1 ≥ σ2 ≥ ··· ≥ σm
L2 )2'$(K A @7,1, .,$ ,./$('2 1, &2 G#$)24
A = U ·Σ·V T
?- 1#-1, U ,. 7-2 )2'$(K #$'#9#-2& 1, n 0&2. M n /#&7)-2.* V ,. 7-2 )2'$(K
#$'#9#-2& 1, m 0&2. M m /#&7)-2.* M Σ ,. 7-2 )2'$(K 1, n 0&2. @#$ m /#&7)-2.
/7M2 1(29#-2& ,. G#$)212 @#$ &#. :2&#$,. .(-97&2$,. 1, A M .7. #'$#. :2&#$,. .#-
0; ?N@&(/21# '#1# ,.'#* 2 &2 ,N@$,.(3- U ·Σ·V T., &2 &&2)2 &2 1,./#)@#.(/(3- ,-
:2&#$,. .(-97&2$,. 1, A;
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!.
! ! ! "#$%&'(
!"#$%$& '( )$*+,-.,*/+/0% $% 1(',&$* */%23'(&$* )$ '( */23/$%#$ -(#&/45
A =
1 −11 −1√3 0
6-.$4(-,* -3'#/.'/+(%), '( #&(%*.3$*#( )$ '( A .,& A7 &$+,&)(-,* 83$ */
+(-"/(-,* $*#$ ,&)$%7 $' &$*3'#(), ,"#$%/), *$&9 )/:$&$%#$5
AT ·A =
[
1 1√3
−1 −1 0
]
·
1 −11 −1√3 0
AT ·A =
[
5 −2−2 2
]
;$ $*#( -(#&/4 ,"#$%/)(7 +('+3'(-,* *3* $/2$%1('3$*5
det
([
5 −2−2 2
]
− λ
[
1 00 1
])
= 0
det
([
5 −2−2 2
]
−[
λ 00 λ
])
= 0
det
([
5− λ −2−2 2− λ
])
= 0
(5− λ)(2− λ)− (−2· ) = 0
10− 5λ− 2λ+ λ2 − 4 = 0
λ2 − 7λ+ 6 = 0
(λ− 6)(λ− 1) = 0
λ1 = 1 λ2 = 6
<' #$%$& ',* $/2$%1('3$*7 .,)$-,* /%-$)/(#(-$%#$ ,"#$%$& ',* 1(',&$* */%23=
'(&$*5
σ =√λ
σ1 =√λ1 σ2 =
√λ2
σ1 = 1 σ2 =√6
<>,&( .,)$-,* :,&-(& ( '( -(#&/4 Σ5
Σ =
√6 00 10 0
?,%$-,* $% $*$ ,&)$% ( ',* 1(',&$* */%23'(&$* .,& ', .&$1/(-$%#$ $@.'/+(),A
!"#$%&' () $* +, !- ./ 0/.% ,1+ ./ .,2/+-,'+!&,.!. !
! !"! #$%&'('&)*+, -+ ./0
"#$# %& $&'()#'* &' +, -&.')()*'/ &' &+ 01,0,$)&'0# -& )$23&'&% -)3)0,+&% %&
45&-& &'(#'01,1 &+ 5%# -& 678/ &%4&(9.(,$&'0& &' +, (#$41&%)*' -& )$23&'&%:
6,;&$#% <5& 0#-, )$,3&' 45&-& %&1 1&41&%&'0,-, 4#1 5', $,01)= -& >,+#1&%/
4#1 +# <5& -,-, 5', )$,3&' &' &%(,+, -& 31)%&%/ 0&'&$#% <5& (,-, 4#%)()*'
-& +, $,01)= )'-)(, &+ ')>&+ -& 31)% -&+ (#11&%4#'-)&'0& 4)?&+/ &%0& ')>&+ &%02
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