EE363
Prof. S. Boyd
EE363 homework 4
1.
Estimating an unknown constant from repeated measurements.
We wish to estimate
x
∼ N
(0
,
1) from measurements
y
i
=
x
+
v
i
,
i
= 1
, . . ., N
, where
v
i
are IID
N
(0
, σ
2
),
uncorrelated with
x
. Find an explicit expression for the MMSE estimator ˆ
x
, and the
MMSE error.
2.
Estimator error variance and correlation coeFcient.
Suppose (
x, y
)
∈
R
2
is Gaussian,
and let ˆ
x
denote the MMSE estimate of
x
given
y
, and ¯
x
denote the expected value of
x
. We de±ne the relative mean square estimator error as
η
=
E
(ˆ
x
−
x
)
2
/
E
(¯
x
−
x
)
2
.
Show that
η
can be expressed as a function of
ρ
, the correlation coe²cient of
x
and
y
.
Does your answer make sense?
3.
MMSE predictor and interpolator.
A scalar time series
y
(0)
, y
(1)
, . . .
is modeled as
y
(
t
) =
a
0
w
(
t
) +
a
1
w
(
t
−
1) +
···
+
a
N
w
(
t
−
N
)
,
where
w
(
−
N
)
, w
(
−
N
+ 1)
, . . .
are IID
N
(0
,
1). The coe²cients
a
0
, . . ., a
N
are known.
(a)
Predicting next value from current value.
Find the MMSE predictor of
y
(
t
+ 1)
based on
y
(
t
). (Note: we really mean based on just
y
(
t
), and not based on
y
(
t
)
, y
(
t
−
1)
, . . .
) Your answer should be as explicit as possible.
(b)
MMSE interpolator.
Find the MMSE predictor of
y
(
t
) (for
t >
1) based (only)
on
y
(
t
−
1) and
y
(
t
+1) (for
t
≥
1). Your answer should be as explicit as possible.
4.
Estimating initial subpopulations from total growth observations.
A sample that con-
tains three types of bacteria (called A, B, and C) is cultured, and the total bacteria
population is measured every hour. The bacteria populations grow, independently of
each other, exponentially with di³erent growth rates: A grows 2% per hour, B grows
5% per hour, and C grows 10% per hour. The goal is to estimate the initial bacteria
populations based on the measurements of total population.
Let
x
A
(
t
) denote the population of bacteria A after
t
hours (say, measured in grams),
for
t
= 0
,
1
, . . .
, and similarly for
x
B
(
t
) and
x
C
(
t
), so that
x
A
(
t
+ 1) = 1
.
02
x
A
(
t
)
,
x
B
(
t
+ 1) = 1
.
05
x
B
(
t
)
,
x
C
(
t
+ 1) = 1
.
10
x
C
(
t
)
.
The total population measurements are