Post on 11-Nov-2018
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Mestrado Bolonha em Tecnologias Biomédicas
Introdução à Engenharia Biomédica
IST – FMUL,
2014-2015
Spaces
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Linear Spaces
Definition A linear space of vectors, S, defined over a scalar set R is a set of objects, called vectors, for which an addition operation between vectors is defined as well as a multiplication operation by scalars, with the following properties: 1. Addition Closeness. For any 2. There is a null element, , such that for any 3. Inverse. For every element there is an element such that 4. The addition is associative. For any 5. For any
x,y∈ S, x+y∈ S
0 x∈ S, x+0 = 0+x = x
x∈ S y∈ S x+y = 0
x,yandz∈ S, (x+y)+z = x+ (y+z)
a,b∈ R and x, y and z∈ S,Closeness: ax ∈ SAssociativity : a(bx) = (ab)xDistributivity : (a+ b)x = ax+ bx
a(x+ y) = ax+ ayαu
βv
αu+βv
x = x(0), x(1), ! x(n), !{ }T
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Vectors and Matrices
• Norm function
• Inner product
x ≥ 0""for"any""x∈ S
x = 0"if and only ifx = 0
ax = a x ,where aisa scalar
x+y ≤ x + y , (triangle inequality)
1) x,y = y,x
2) ax,y =a x,y
3) x+y,z = x,z + y,z
4) x,x > 0ifx ≠ 0,and x,x = 0ifandonlyifx = 0
• The inner product operation can be used to defined a norm function, called in this case, an induced norm.
• Theorem (Cauchy-Schwartz inequality) In an inner product space S with induced norm for any with equality if and only if for some scalar a.
x = x,x1/2
x,y ≤ x . y
x,y∈ S y =ax
• The vectors from a vector space with inner product are said orthogonal if
• The null vector is orthogonal to all other vectors
yxand
x,y = 0x,y = xkykk=1
n
∑ = yHx
The inner product in Cn is defined as
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Lp norm
1. l1, called Manhattan distance, …
2. l2, called Euclidean distance, ……
3. lp, ……………………………………
4. l∞, ……………………………………
5. l0,……………………
d1(x,y)= xi − yii=1
n
∑
d2(x,y)= xi − yi2
i=1
n
∑
dp(x,y)= xi − yip
i=1
n
∑#
$%%
&
'((
1/p
d∞(x,y)=max xi − yi
i=1,2,..,n
d0(x,y)= limp→0dp(x,y)"#
$%p= # xi − yi > 0
"#
$%
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Metrics of continuous signals
1. l1, called Manhattan distance,
2. l2, called Euclidean distance,
3. lp, ……………………………
4. l∞, ……………………………
d1(x,y)= x(t)− y(t) dta
b
∫
d2(x,y)= x(t)− y(t)2dt
a
b
∫
dp(x,y)= x(t)− y(t)pdt
a
b
∫#
$%%
&
'((
1/p
d∞(x,y)= sup x(t)− y(t) :a ≤ t ≤ b{ }
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Representation and approximation in spaces
• The goal is the computation of the coefficients such that, given , the vector is the closer vector of x, that is,
• Orthogonallity pinciple
Let be a set of vectors in S and let In the representation The induced norm of the error vector is minimized when the error is orthogonal to each is
c1 c2 cmx∈ S
x̂ = cipik=1
m
∑
c* = argmincx̂(c)− x
2
2
x̂
x
x = cipii=1
m
∑ +e= x̂+e
p1 ,p2 ,...pmx∈ S
ee= x̂−x
pi
e,pi = 0, i =1,2,...,m
x̂
x
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Coefficients estimation
J(c) = e 2= e, e
= x− cipii=1
m
∑ , x− cjp jj=1
m
∑
= x, x − x, cjp jj=1
m
∑ − cipii=1
m
∑ , x + cipii=1
m
∑ , cjp jj=1
m
∑
= x, x − cj* x,p j
j=1
m
∑ − ci pi,xi=1
m
∑ + cicj pi,p ji, j=1
m
∑
= x, x − ci* x,pi + ci x,p j
*
i=1
m
∑ + cicj* pi,p j
i, j=1
m
∑
= x, x − 2Re ci* x,pi
i=1
m
∑ + cicj* pi,p j
i, j=1
m
∑
= x 2− 2Re(cHp)+ cHRc
x− cipii=1
m
∑
e! "# $#
, p j = 0, j =1,2,...,m
p1,p1 p2,p1 % pm,p1p1,p2 p2,p2 % pm,p2% % % %p1,pm p2,pm % pm,pm
#
$
%%%%%
&
'
(((((
Graminian! "####### $#######
R& '####### (#######
c1c2%cm
#
$
%%%%%
&
'
(((((
c&'(
=
x,p1x,p2%x,pm
#
$
%%%%%
&
'
(((((
p& '# (#
Rc = p
c =R−1p
x̂
x
e∇J(c) = −2p+ 2Rc = 0
Rc = pc =R−1p
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Example 2: Linear Regression
xi ,yi( ) : yi =axi +b
y1y2yn
!
"
#####
$
%
&&&&&
y
=
ax1 +b
ax2 +b
axn +b
!
"
#####
$
%
&&&&&
+
e1e2en
!
"
#####
$
%
&&&&&
e
y = Ac+e
c = a b!"#
$%&T
, A=
x1 1
x2 1
1xn 1
!
"
#####
$
%
&&&&&
c = AHA( )−1AHy
Signals and Systems in Bioengineering, SSB, João Miguel Sanches, DBE/IST, 1ºSem, 2014/2015
Introdução à Engenharia Biomédica, MTBiom, 2014/2015
Representation of continuous signals
• Let {pi(t)} be a set of continuous basis functions. Find the optimal set of coefficients ci that minimize the residue e(t)
x(t) = ciψi (t)i=1
m
∑f (t )
!"# $#+ e(t)
J(c) = x(t)− ψi pi (t)i=1
m
∑2
2