State Space Models

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State Space Models

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.

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.

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

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

Define

Then tty x100

and ttt uBxx 1 State equation

Observation equation

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:

Suppose that the n possible observations taken at each state are

itjtij XXP , 1 ee

itjtij XYP Βef ,

itt XXE eΠe

So that

itttt XXEX eu 1

itX eΠ

1 tt XX Π

ttt XX uΠ 1 The State Equation

0, 121 ttttt XEXXE uu

tttttt XXXX uΠuΠ 11

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

ttt XX uΠ 1

ttt XX uΠ diagdiagdiag 1

11 diagdiag ttt XXXE Π

ΠΠΠΣu 11 diagdiag tt XX

We have defined

itjtij XYP Βef ,

itt XYE eΒe

tttt XYEY v

tt XY Β

The Observation Equation

0v tt XE

ttt XY vΒ

ΒΒΒvvΣv ttttt XXXE diagdiag

Hence with these definitions the state sequence of a Hidden Markov Model satisfies:

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:

Kalman Filtering

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)

For any vector x define:

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

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

Remark: If x, y1, y2, … ,ys are normally distributed then:

sEs yyyxx ,,,ˆ 21

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

State Space Models

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.

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.

yt = Atxt + vt

xt = Bxt-1 + ut

Kalman Filtering:

stt Es yyyxx ,,,ˆ 21

stu ssE yyyxxxxΣ ,,,ˆˆ 21

1ˆ1ˆ 1 tt tt xΒx

1ˆ1ˆˆ ttt tttttt xAyKxx

111 vΣAΣAAΣK tt

One also assumes that the initial vector x0 has mean and covariance matrix an that

μx 0ˆ 0

The covariance matrices are updated

uΣBΣBΣ

11 tttt

ttt AΣKΣΣ

ΣΣ 000

Summary: The Kalman equations

uΣBΣBΣ

11 tttt

ttt AΣKΣΣ

111 vΣAΣAAΣK tt

1ˆ1ˆ 1 tt tt xΒx

μx 0ˆ 0with ΣΣ 0

Now stt Es yyyxx ,,,ˆ 21

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)

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

tttt tt eKxx 1ˆˆHence

tttttttt ttEE vxxAxxed 1ˆ1ˆ

1ˆ ttttt xAyewhere

1ˆˆ tt tt xx

1 ttttt EE eeedKand

ttttt ttE Axxxx 1ˆ1ˆ

ttt AΣ 1

tttttt tEE vxxAee 1ˆ

tttt t vxxA 1ˆ

tttttt ttE AxxxxA 1ˆ1ˆ

tttt tE vxxA 1ˆ

tttttt EtE vvAxxv 1ˆ

111 vΣAΣAAΣK tt

vΣAΣA

1ˆ1ˆ 1 tt tt xΒx

111 vΣAΣAAΣK tt

101 ,,1ˆ1ˆ

tttttt

tt ttE yyxxxxΣ Also

1ˆ 11 tE ttt xΒuΒx

10 ,,1ˆ tttt t yyxΒuΒx

uΣBΣBΣ

1,11 t

11 tttt

ttt AΣKΣΣ

The proof that

will be left as an exercise.

Hence(3)

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.

01, 21

This model can be expressed as a state-space model by defining:

ttt uΒxx 1or

can be written

The equation:

ttt vxy

The Kalman equations

uΣBΣBΣ

11 tttt

ttt AΣKΣΣ

12111Σ

1211Σ

111 vΣAΣAAΣK tt

1ˆ1ˆ 1 tt tt xΒx

The Kalman equations

uΣBΣBΣ

1211 utt

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

s r r r

111 vΣAΣAAΣK tt

11 tttt

ttt AΣKΣΣ

1211 01K

ss1211

111111

12111212

122222

1ˆ1ˆ 1 tt tt xΒx4.

1ˆ1ˆ1ˆ 2211 txtxtx ttt

1ˆ1ˆˆ ttt tttttt xAyKxx5.

111 tx

1ˆ1ˆˆ2

stxtx tt

1ˆ1ˆˆ2

stxtx tt

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

stu ssE yyyxxxxΣ ,,,ˆˆ 21

1ˆˆ1ˆˆ 111 tTtT ttttt xxJxx

1111,11

ttt ΣΒΣJ

The covariance matrices satisfy the recursions

1,11,1

tt JΣΣJΣΣ

The backward recursions

1ˆˆ1ˆˆ 111 tTtT ttttt xxJxx

1111,11

ttt ΣΒΣJ

1,11,1

tt JΣΣJΣΣ

12111Σ

1211Σ

In the example:

txtx ttt

tt ˆ and 1ˆ,, 11 ΣΣ

- calculated in forward recursion

State Space Models

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