Biomedical Engineering Reference
In-Depth Information
the rst derivative of ().Iis an interval in the positive real axis containing
the closed interval [y
0
; y
s
] for y
0
= 0. The maximum penalized likelihood
estimate of (t) is a hyperbolic spline function (Schumaker, 1981) with
knots at y
i
, i = 0;:::;s, which is expressed as
^
(t) = a
s1
exp(!
s1
u
s1
) + b
s1
exp(!
s1
u
s1
); t2(y
0
; y
s
);
where (i) a
0
= b
0
= 0; (ii) a
i
= a
i1
i1
exp(!
i1
u
i1
) +
b
i1
i1
exp(!
i1
u
i1
) E
i
=(2!
i
), i = 1;:::;s1;
(iii) b
i
= a
i1
i1
exp(!
i1
u
i1
) + b
i1
i1
exp(!
i1
u
i1
) + E
i
=(2!
i
),
i = 1;:::;s1; (iv) a
s1
exp(2!
s1
u
s1
)b
s1
exp(2!
s1
u
s1
)
1=(!
s1
) = 0 for u
i
= y
i+1
y
i
,
i
= (!
i
+ !
i+1
)=(2!
i+1
),
i
=
q
P
s
j=i+1
(m
j
+ c
j
)=, i = 0; 1;:::;s1, and
E
i
= m
i
=f[a
i1
exp(!
i1
u
i1
) + b
i1
exp(!
i1
u
i1
)]g, i = 1;:::;s1.
Note that
^
() is continuous on [y
0
; y
s
], and the discontinuities of
^
0
() are
at the time points y
i
with m
i
6= 0. Therefore, for a given value of , the
estimate of h(t) is h(t) =
^
2
(t), t2I. The methods used to maximize
`
p
() (9) with respect to () are similar to those in [117]. The proposed
estimation method can be modied to estimate the intensity function of a
nonstationary Poisson process. This method is not applicable to estimation
from grouped data, for which the kernel estimates of Tanner and Wong
136
may be used.
Rosenberg
119
proposed a exible parametric procedure to model h(t) as
a linear combination of cubic B-splines as follows:
(!
i
!
i+1
)=(2!
i+1
) with !
i
=
X
K
h(t; a) =
exp(a
k
)B
k
(t); t2[y
min
;y
max
]
(10)
k=3
in which using exp(a
k
) as coecients insures that an estimate of h(t) is
nonnegative. Here a = (a
3
;:::;a
K
)
T
; K is the number of interior knots
1
<<
K
, to be used throughout this chapter unless stated otherwise;
and B
k
(t) are cubic B-spline functions
40
of t expressed as follows:
k+4
X
(
`
t)
3
+
B
k
(t) = (
k+4
k
)
Y
; k =3;:::;K;
(
m
`
)
`=k
m6=`
m=k;:::;k+4
for u
+
equal to u if u > 0 and 0 if u0. This can be constructed by fol-
lowing the parameterization in Atkinson
12
, letting
0
= y
min
and
K+1
=
y
max
, and dening six arbitrary \slack" knots such that
i
=
0
i, and
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