state space models
DESCRIPTION
State Space Models. Let { x t : t T } and { y t : t T } denote two vector valued time series that satisfy the system of equations:. y t = A t x t + v t (The observation equation) x t = B t x t- 1 + u t (The state equation). - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/1.jpg)
State Space Models
![Page 2: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/2.jpg)
Let { xt:t T} and { yt:t T} denote two vector valued time series that satisfy the system of equations:
yt = Atxt + vt (The observation equation)
xt = Btxt-1 + ut (The state equation)
The time series { yt:t T} is said to have state-space representation.
![Page 3: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/3.jpg)
Note: { ut:t T} and { vt:t T} denote two vector valued time series that satisfying:
1. E(ut) = E(vt) = 0.
2. E(utusˊ) = E(vtvsˊ) = 0 if t ≠ s.
3. E(ututˊ) = u and E(vtvtˊ) = v.
4. E(utvsˊ) = E(vtusˊ) = 0 for all t and s.
![Page 4: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/4.jpg)
Example: One might be tracking an object with several radar stations. The process {xt:t T} gives the position of the object at time t. The process { yt:t T} denotes the observations at time t made by the several radar stations.
As in the Hidden Markov Model we will be interested in determining position of the object, {xt:t T}, from the observations, {yt:t T} , made by the several radar stations
![Page 5: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/5.jpg)
Example: Many of the models we have considered to date can be thought of a State-Space models
Autoregressive model of order p:
tptpttt uyyyy 2211
![Page 6: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/6.jpg)
Define
Then tty x100
t
t
pt
t
y
y
y
1
1
x
and ttt uBxx 1 State equation
Observation equation
tt
p
u
1
0
0
100
010
1
21
x
![Page 7: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/7.jpg)
Hidden Markov Model: Assume that there are m states. Also that there the observations Yt are discreet and take on n possible values.
Suppose that the m states are denoted by the vectors:
1
0
0
,,
0
1
0
,
0
0
1
21
meee
![Page 8: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/8.jpg)
Suppose that the n possible observations taken at each state are
1
0
0
,,
0
1
0
,
0
0
1
21
nfff
![Page 9: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/9.jpg)
Let
ijmm
itjtij XXP , 1 ee
and
ijnm
itjtij XYP Βef ,
Note
i
im
i
i
itt XXE eΠe
2
1
1
![Page 10: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/10.jpg)
Let
So that
itttt XXEX eu 1
itX eΠ
1 tt XX Π
ttt XX uΠ 1 The State Equation
with
0, 121 ttttt XEXXE uu
![Page 11: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/11.jpg)
Also
Hence
tttttt XXXX uΠuΠ 11
and
tttttttt XXXX uuΠuuΠΠΠ 1111
1111 tttttttttt XXXXXX ΠuuΠΠΠuu
1111 ttttttttt XXXXXEXE ΠΠuuΣu
ΠΠ 11 diagdiag ttt XXXE
where diag(v) = the diagonal matrix with the components of the vector v along the diagonal
![Page 12: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/12.jpg)
then
Since
ttt XX uΠ 1
and
ttt XX uΠ diagdiagdiag 1
11 diagdiag ttt XXXE Π
Thus
ΠΠΠΣu 11 diagdiag tt XX
![Page 13: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/13.jpg)
We have defined
ijnm
itjtij XYP Βef ,
Hence
i
in
i
i
itt XYE eΒe
2
1
Let
tttt XYEY v
tt XY Β
![Page 14: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/14.jpg)
Then
with
The Observation Equation
0v tt XE
ttt XY vΒ
and
ΒΒΒvvΣv ttttt XXXE diagdiag
![Page 15: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/15.jpg)
Hence with these definitions the state sequence of a Hidden Markov Model satisfies:
with
The Observation Equation
0v tt XE
ttt XY vΒ
and ΒΒΒvvΣv ttttt XXXE diagdiag
ttt XX uΠ 1 The State Equation
with 0u tt XE
and ΠΠΠuuΣu 111 diagdiag ttttt XXXE
The observation sequence satisfies:
![Page 16: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/16.jpg)
Kalman Filtering
![Page 17: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/17.jpg)
We are now interested in determining the state vector xt in terms of some or all of the observation vectors y1, y2, y3, … , yT.We will consider finding the “best” linear predictor. We can include a constant term if in addition one of the observations (y0 say) is the vector of 1’s.
We will consider estimation of xt in terms of 1. y1, y2, y3, … , yt-1 (the prediction problem)
2. y1, y2, y3, … , yt (the filtering problem)
3. y1, y2, y3, … , yT (t < T, the smoothing problem)
![Page 18: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/18.jpg)
For any vector x define:
where
sxsxsxs pˆˆˆˆ 21 x
is the best linear predictor of x(i), the ith component of x, based on y0, y1, y2, … , ys.
sx iˆ
The best linear predictor of x(i) is the linear function that of x, based on y0, y1, y2, … , ys that minimizes
2ˆ sxxE ii
![Page 19: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/19.jpg)
Remark: The best predictor is the unique vector of the form:
Where C0, C1, C2, … ,Cs, are selected so that:
sss yCyCyCx 1100ˆ
sis i ,,2,1,0 ˆ yxx
sisE i ,,2,1,0 ˆ i.e. 0yxx
![Page 20: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/20.jpg)
Remark: If x, y1, y2, … ,ys are normally distributed then:
sEs yyyxx ,,,ˆ 21
![Page 21: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/21.jpg)
Cvuv ˆ
Let u and v, be two random vectors than
is the optimal linear predictor of u based on v if
1 vvvuC EE
Remark
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State Space Models
![Page 23: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/23.jpg)
Let { xt:t T} and { yt:t T} denote two vector valued time series that satisfy the system of equations:
yt = Atxt + vt (The observation equation)
xt = Btxt-1 + ut (The state equation)
The time series { yt:t T} is said to have state-space representation.
![Page 24: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/24.jpg)
Note: { ut:t T} and { vt:t T} denote two vector valued time series that satisfying:
1. E(ut) = E(vt) = 0.
2. E(utusˊ) = E(vtvsˊ) = 0 if t ≠ s.
3. E(ututˊ) = u and E(vtvtˊ) = v.
4. E(utvsˊ) = E(vtusˊ) = 0 for all t and s.
![Page 25: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/25.jpg)
Let { xt:t T} and { yt:t T} denote two vector valued time series that satisfy the system of equations:
yt = Atxt + vt
xt = Bxt-1 + ut
Let
Kalman Filtering:
stt Es yyyxx ,,,ˆ 21
and
suutt
stu ssE yyyxxxxΣ ,,,ˆˆ 21
![Page 26: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/26.jpg)
Then
1ˆ1ˆ 1 tt tt xΒx
1ˆ1ˆˆ ttt tttttt xAyKxx
111 vΣAΣAAΣK tt
ttttt
ttt
where
One also assumes that the initial vector x0 has mean and covariance matrix an that
μx 0ˆ 0
![Page 27: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/27.jpg)
The covariance matrices are updated
uΣBΣBΣ
11,1
1 ttt
ttt
with
11 tttt
ttt
ttt AΣKΣΣ
ΣΣ 000
![Page 28: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/28.jpg)
Summary: The Kalman equations
uΣBΣBΣ
11,1
1 ttt
ttt1.
11 tttt
ttt
ttt AΣKΣΣ
1ˆ1ˆˆ ttt tttttt xAyKxx
111 vΣAΣAAΣK tt
ttttt
ttt
1ˆ1ˆ 1 tt tt xΒx
2.
3.
4.
5.
μx 0ˆ 0with ΣΣ 0
00and
![Page 29: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/29.jpg)
Now stt Es yyyxx ,,,ˆ 21
hence
Proof:
121 ,,,1ˆ ttt Et yyyxx
1ˆ 1 ttxΒ
1211 ,,, tttE yyyuΒx
1211 ,,, ttE yyyxΒ
Note 121 ,,,1ˆ ttt Et yyyyy 121 ,,, ttttE yyyvxA
1ˆ,,, 121 tE ttttt xAyyyxA
proving (4)
![Page 30: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/30.jpg)
Let 1ˆ tttt yye
1ˆ tttttt xAvxA tttt t vxxA 1ˆ
1ˆ tttt xAy
Let 1ˆ tttt xxd
Given y0, y1, y2, … , yt-1 the best linear predictor of dt using et is:
ttttt EE eeeed 1 tttE eyyyd ,,,, 110 tttE yyyyd ,,,, 110
![Page 31: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/31.jpg)
tttt tt eKxx 1ˆˆHence
tttttttt ttEE vxxAxxed 1ˆ1ˆ
1ˆ ttttt xAyewhere
1ˆˆ tt tt xx
1 ttttt EE eeedKand
Now
ttttt ttE Axxxx 1ˆ1ˆ
t
ttt AΣ 1
(5)
![Page 32: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/32.jpg)
tttttt tEE vxxAee 1ˆ
Also
tttt t vxxA 1ˆ
tttttt ttE AxxxxA 1ˆ1ˆ
tttt tE vxxA 1ˆ
tttttt EtE vvAxxv 1ˆ
111 vΣAΣAAΣK tt
ttttt
tut
hence
vΣAΣA
tt
ttt1
(2)
![Page 33: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/33.jpg)
Thus
1ˆ1ˆ 1 tt tt xΒx
1ˆ1ˆˆ ttt tttttt xAyKxx
111 vΣAΣAAΣK tt
ttttt
ttt
where
101 ,,1ˆ1ˆ
tttttt
tt ttE yyxxxxΣ Also
1ˆ 11 tE ttt xΒuΒx
10 ,,1ˆ tttt t yyxΒuΒx
(4)
(5)
(2)
![Page 34: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/34.jpg)
uΣBΣBΣ
1
1,11 t
ttt
tt
11 tttt
ttt
ttt AΣKΣΣ
The proof that
will be left as an exercise.
Hence(3)
(1)
![Page 35: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/35.jpg)
Example:
What is observe is the time series
tttt uxxx 2211 Suppose we have an AR(2) time series
ttt vxy
{ut|t T} and {vt|t T} are white noise time series with standard deviations u and v.
![Page 36: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/36.jpg)
then
0,
01, 21
1
tt
t
tt
u
x
xuΒx
This model can be expressed as a state-space model by defining:
001 2
121
1
t
t
t
t
t u
x
x
x
x
ttt uΒxx 1or
![Page 37: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/37.jpg)
can be written
The equation:
ttt vxy
tttt
tt vv
x
xy
Ax1
0,1
2vvΣ
Note:
00
02u
uΣ
![Page 38: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/38.jpg)
The Kalman equations
uΣBΣBΣ
11,1
1 ttt
ttt1.
11 tttt
ttt
ttt AΣKΣΣ
tt
ttt
ttss
ss
2212
12111Σ
tt
ttt
ttrr
rr
2212
1211Σ
1ˆ1ˆˆ ttt tttttt xAyKxx
111 vΣAΣAAΣK tt
ttttt
ttt
1ˆ1ˆ 1 tt tt xΒx
2.
3.
4.
5.
Let
![Page 39: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/39.jpg)
The Kalman equations
uΣBΣBΣ
11,1
1 ttt
ttt1.
00
0
0
1
01
2
2
1
122
112
112
11121
2212
1211 utt
tt
tt
tt
rr
rr
ss
ss
1 2 1 1 2 211 11 1 12 1 2 22 2
1 112 11 1 12 2
122 11
2t t t tu
t t t
t t
s r r r
s r r
s r
![Page 40: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/40.jpg)
1
2212
1211
2212
1211
0
101
0
1
vtt
tt
tt
tt
tss
ss
ss
ssK
111 vΣAΣAAΣK tt
ttttt
ttt2.
vt
tv
t
t
vt
t
t
s
ss
s
ss
s
11
12
11
11
1
11
12
11
![Page 41: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/41.jpg)
11 tttt
ttt
ttt AΣKΣΣ
tt
tt
ttt
tt
tt
tt
ss
ss
ss
ss
rr
rr
2212
1211
2212
1211
2212
1211 01K
3.
tt
vt
tv
t
t
tt
tt
ss
s
ss
s
ss
ss1211
211
12
211
11
2212
1211
2
11
2
111111
vt
ttt
s
ssr
211
12111212
vt
tttt
s
sssr
2
11
2
122222
vt
ttt
s
ssr
![Page 42: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/42.jpg)
1ˆ
1ˆ
011ˆ
1ˆ
2
121
1 tx
tx
tx
tx
t
t
t
t
1ˆ1ˆ 1 tt tt xΒx4.
1ˆ1ˆ1ˆ 2211 txtxtx ttt
![Page 43: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/43.jpg)
1ˆ1ˆˆ ttt tttttt xAyKxx5.
1ˆ
1ˆ01
1ˆ
1ˆ
ˆ
ˆ
111 tx
txy
tx
tx
tx
tx
t
ttt
t
t
t
t K
1ˆ
1ˆ
1ˆ
ˆ
ˆ
211
12
211
11
11
txy
s
ss
s
tx
tx
tx
txtt
vt
tv
t
t
t
t
t
t
![Page 44: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/44.jpg)
1ˆ1ˆˆ2
11
11
txys
stxtx tt
vt
t
tt
1ˆ1ˆˆ2
11
1211
txy
s
stxtx tt
vt
t
tt
![Page 45: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/45.jpg)
Now consider finding
These can be found by successive backward recursions for t = T, T – 1, … , 2, 1
Kalman Filtering (smoothing):
Ttt ET yyyxx ,,,ˆ 21
where
suutt
stu ssE yyyxxxxΣ ,,,ˆˆ 21
1ˆˆ1ˆˆ 111 tTtT ttttt xxJxx
1111,11
t
ttt
ttt ΣΒΣJ
![Page 46: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/46.jpg)
The covariance matrices satisfy the recursions
11
11
1,11,1
tt
ttT
tttt
ttT
tt JΣΣJΣΣ
![Page 47: State Space Models](https://reader035.vdocuments.site/reader035/viewer/2022062301/5681306c550346895d964c13/html5/thumbnails/47.jpg)
1.
The backward recursions
1ˆˆ1ˆˆ 111 tTtT ttttt xxJxx
1111,11
t
ttt
ttt ΣΒΣJ
11
11
1,11,1
tt
ttT
tttt
ttT
tt JΣΣJΣΣ
2.
3.
tt
ttt
ttss
ss
2212
12111Σ
tt
ttt
ttrr
rr
2212
1211Σ
In the example:
0121
Β
txtx ttt
ttt
tt ˆ and 1ˆ,, 11 ΣΣ
- calculated in forward recursion