state-space models - analysis of time series (theory ... · intro revision n(m;) theory t + njt,tjt...

Intro Revision N (M,Σ) Theory t + n|t, t|T θ estimation SKF Limitations Summary

State-Space Models - Analysis of time series(Theory)

(CIV6540 - Machine Learning for Civil Engineers)

Professor: James-A. Goulet

Departement des genies civil, geologique et des mines

Polytechnique Montreal

Chapter 11 – Goulet (2018).Probabilistic Machine Learning for Civil Engineers.

Chapter 18 – Murphy, K. (2012).Machine learning: A probabilistic perspective.The MIT Press.

Professor: J-A. Goulet Polytechnique Montreal

State-Space Models - Analysis of time series (Theory) | V1.2 | Machine Learning for Civil Engineers 1 / 61


Time-seriesHow do we interpret data when it is time-dependent?

17-01 17-04 17-07 17-10 17-12

9

10

11

12

Time [YY-MM]

Temperature[C]-y t

E [Yt|y1:t] ± σE[Yt|y1:t]E[Yt|y1:t]t|t

Possible tasks

1. Filter the observation noise

2. Forecast future states or observations

3. Estimate hidden state variables

4. Detect anomaliesProfessor: J-A. Goulet Polytechnique Montreal




17-01 17-04 17-07 17-10 17-12·105

9

10

11

12

Time [YY-MM]

Tem

pera

ture

[C]-

y tE[Yt|y1:t]± σE[Yt|y1:t] E[Yt|y1:t]t|t yt ± σV yt

Possible tasks








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11

12

Time [YY-MM]

xLL tµt|t ± σt|t µt|t xt

Possible tasks








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0

0.2

0.4

0.6

0.8

1

Time [YY-MM]

Pr(S)

StationaryNon-stationary

Possible tasks




4. Detect anomalies




Methodology overview

I Machine Learning: State-Space Models

I Control theory: Kalman Filter

I Statistics: Dynamic Linear Models

These methods are all based on Gaussian conditionals

1000

x1500

00

700x2

1000

f X1,

X2(

x 1,x

2)

1000

x1

µ1|2 =660 σ1|2 =90

50000

700x2

1000

f X1|

x 2(x

1|x2)




Kalman filter – Origin

Developped for NASA as the guiding system forthe Appolo mission (1960)

Still employed today in Space X Falcon 9(2018+)...

“Rocket science” applied to civil engineering



[Wikipedia][Wikipedia, SapceX]


Module #7 Outline

Revision N (M,Σ)

Theory

t + n|t, t|Tθ estimation

SKF

Limitations

Topics organization

Probabilitytheory

{1 Revision probability & linear algebra

2 Probability distributions −4 −2 0 2 40

0.2

0.4 µ µ+σµ−σ

x−4 −2 0 2 40

0.5

1µ µ+σµ−σ

x

F X(x)

f X(x)

MachineLearning

Basics

0 Introduction

3 Bayesian Estimation p(A|B) =p(B|A)p(A)

p(B)

4 MCMC sampling & Newton −5 0 5−5

0

5

θ1

θ 2

−50

5

−50

50.00

0.05

0.10

θ1θ2

s = 1000

Data-drivenmethods

5 Regression [ ] cl+

Pato

gens

[ppm

]

6 Classification0 0.2 0.4 0.6 0.8 10

10

x

f X|D

(x|d

)

d=0 d=1 d=2 d=3 d=4 d=5

0 0.2 0.4 0.6 0.8 10

0.5

1

P(D

S)

S={4,5} S={2,3} S={0,1}

7 State-space model for time-series

Model drivenmethods

{8 Model Calibration −2

02

−20

2

y1y2

−2 0 2

−2

0

2

y1

y 2

DecisionMaking

{9 Decision Theory

E[U] = 7.0

E[U] = 3.7

10 AI & Sequential decision problems




Section Outline

Revision N (M,Σ)1.1 Fonctions of Gaussian Random Variables1.2 Transformation Rule1.3 Conditional Gaussian PDF




Fonctions of Gaussian Random Variables

Fonctions of Gaussian Random Variables – g(X) (Revision)Given X = [X1,X2, · · · ,Xn]ᵀ defined by the PDF fX(x), and givenY = [Y1,Y2, · · · ,Ym]ᵀ obtained from a linear function

Y = g(X) =

g1(X)g2(X)

...gn(X)

Question: what is the PDF fY(y),

2 cases:1. m = n = 1

2. m, n ≥ 1

We consider onlylinear fonctions

i.e. Y = g(X) = AX + b

! Applicable seulement pour le cas ou il y ameme nombre d’elements dans X et Y




Transformation Rule

Y = g(X ) : Case 1, m = n = 1

fY (y)dy = fX (x)dx

fY (y) = fX (x)

∣∣∣∣dx

dy

∣∣∣∣︸︷︷︸

fX (x)≥0

For Y = aX + b∣∣∣∣dy

dx

∣∣∣∣ = |a|

X ∼ N (x ;µX , σ2X )→ Y ∼ N

(y ; aµX + b, (aσX )2

)




Transformation Rule

Example: Case 1, Y = 2X [ ]

X ∼ N ( 10︸︷︷︸µX

, 52︸︷︷︸σ2X

)

Y ∼ N (2µX︸︷︷︸µY

, (2σX︸︷︷︸σY

)2)

x-10 0 10 20 30 40 50

y-10

0

10

20

30

40

50

x-10 0 10 20 30 40 50

f X(x)

00.020.040.060.08

y

0 0.02 0.04fY(y)

-10

0

10

20

30

40

50



[CIV8530/M0 linearisation.m]


Transformation Rule

Y = g(X) : Case 2, m = n > 1

fX(x)dx = fY(y)dy

fX(x)dxdy = fY(y)

Jk,i = Jyk ,xi =∂yk∂xi

Jacobian (Jy,x): size of the neighbourhoodfrom dy to dx

Jy,x =

Jy1,x1 · · · Jy1,xn. . .

...sym. Jym,xn

=

∇g1(x)

...∇gn(x)

dx

dy= |det Jy,x|−1

dy

dx= |det Jy,x|

fY(y) = fX(x) |det Jy,x|−1




Transformation Rule

Y = g(X) : Cas 2, m = n > 1 (cont.)Given a linear function Y = g(X) = AX + b where

X =

X1

...Xn

, MX =

µX1

...µXn

, ΣX =

σ2X1

· · · ρ1nσX1σXn

.........

...sym. σ2

Xn

n×1︸︷︷︸

Y

=

n×n︸︷︷︸

A=Jy,x

×

n×1︸︷︷︸

X

+

n×1︸︷︷︸

b

MY = AMX + b

ΣY = AΣXAᵀ

ΣXY = AΣX

[XY

],

[MX

MY

],

[ΣX ΣXY

ΣYX ΣY

]




Conditional Gaussian PDF

Gaussian Conditional PDF (Revision)

Conditional distributions of a multivariate Gaussian are Gaussian

X =

{X1

X2

}, M =

{M1

M2

}, Σ =

[Σ1 Σ12

Σ21 Σ2

]

fX1|X2(x1|X2 = x2︸︷︷︸

observations

) =fX1X2(x1, x2)

fX2(x2)= N (x1; M1|2,Σ1|2)

M1|2 = M1 + Σ12Σ−12 (x2 −M2), Σ1|2 = Σ1 −Σ12Σ−1

22 Σ21

2D: µ1|2 = µ1 + ρσ1x2 − µ2

σ2, σ1|2 = σ1

√1− ρ2





Example – Multivariate Gaussian Conditional

The prior knowledge of the true resistance (X1,X2) of two beamsis

X1 ∼ N (x1; 500, 1502) [kN ·m]X2 ∼ N (x2; 500, 1502) [kN ·m]

Knowing that the resistance of beams is correlated: ρ12 = 0.8,what is the resistance of X1,if we observe that x2 = 700 kN ·m?

fX1|x2(x1|x2) = N (x1;µ1|2, σ

21|2) = N (x1; 660, 902)



[Radio-Canada]



Example – Multivariate Gaussian Conditional (cont.)

fX1X2(x1, x2) = N (MX,ΣX)

MX =

[500500

]

ΣX =

[1502 0.8 · 1502

0.8 · 1502 1502

]

fX1|x2(x1|x2) =

fX1X2(x1, x2)

fX2(x2)= N (µ1|2, σ

21|2)

{µ1|2 = µ1 + ρσ1

x2−µ2σ2

σ1|2 = σ1

√1− ρ2

µ1|2 = 500 + 0.8× 150

observation︷︸︸︷700 −500

150= 660 kN ·m

σ1|2 = 150√

1− 0.82 = 90 kN ·m





Example – Multivariate Gaussian Conditional (cont.) [ ]

fX1|x2(x1|x2) =

fX1X2(x1, x2)

fX2(x2)= N (µ1|2, σ

21|2)

{µ1|2 = µ1 + ρσ1

x2−µ2σ2

σ1|2 = σ1

√1− ρ2

1000

x1500

00

700x2

1000

f X1,

X2(

x 1,x

2)

1000

x1

µ1|2 =660 σ1|2 =90

50000

700x2

1000f X

1|x 2

(x1|x

2)



[CIV8530/M1 poutres conditionnelles.m]


Summary – Multivariate Gaussian

For Y = AX + b, where X ∼ N (x; MX,ΣX), and a vector ofobservations y

MY = AMX + b

ΣY = AΣXAᵀ

ΣXY = ΣXAᵀ

[XY

],

[MX

MY

],

[ΣX ΣXY

ΣYX ΣY

]

posterior knowledge︷︸︸︷fX|Y(x|Y = y︸︷︷︸

observations

) =

prior knowledge︷︸︸︷fXY(x, y)

fY(y)= N (x; MX|Y,ΣX|Y)

MX|Y = MX + ΣXYΣ−1Y (y −MY), ΣX|Y = ΣX −ΣXYΣ−1

Y ΣYX




Section Outline

Theory2.1 Basic problem setup2.2 Notation2.3 Prediction and update steps2.4 Example #12.5 General formulation




Basic problem setup

Example #1 – Temperature time-series

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Time - t

Stat

e-x

t

E[Xt|y1:t]E[Xt|y1:t]± σ[Xt|y1:t]yttrue values




Basic problem setup

Basic problem setupt = 0, 1, · · · ,T : Discrete time stampsxt : State of a system at a time step t (deterministic, yet unknown)yt : Imperfect observation of xt at a time step t

yt = xt + vt︸︷︷︸observation model

,

observation error︷︸︸︷vt : V ∼ N (v ; 0, σ2

V )

xt = xt−1 + wt︸︷︷︸transition model

,

model error︷︸︸︷wt : W ∼ N (w ; 0, σ2

W )

Our knowledge of xt conditional on the all available data y1:t is

fXt |y1:t(xt |y1:t) = N (xt ;E[Xt |y1:t ], Var[Xt |y1:t ])




Notation

Compact notation

Long form ≡ Short form

{y1, y2, · · · , yt} ≡ y1:t

E[Xt |y1:t ] ≡ µt|t

E[Xt |y1:t−1] ≡ µt|t−1

Var[Xt |y1:t ] ≡ σ2t|t

fX (x) ≡ f (x)




Prediction and update steps

Prediction step: t = 0→ t = 1At time t = 0 only our prior knowledge E[X0] = µ0 andVar[X0] = σ2

0 is available. Note: no knowledge, Var[X0]→∞

We compute our prior knowledge at t = 1 from our knowledgeat t = 0.

Transition model: x1 = x0 + w1, w1 : W ∼ N (0, σ2W )

Observation model: y1 = x1 + v1, v1 : V ∼ N (0, σ2V )

E[X1|X0] ≡ µ1|0 = µ0

Var[X1|X0] = σ21|0 = σ2

0 + σ2W

E[Y1|X0] = µ1|0 = µ0

Var[Y1|X0] = σ21|0 + σ2

V = σ20 + σ2

W + σ2V

Cov[X1,Y1|X0] = σ21|0 = σ2

0 + σ2W




Example #1


0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Time - t

Stat

e-x

t





Example #1


We want to estimate the temperature xt over a period of 5 hoursusing imprecise measurements made every hour, t = 0 : 5.

Our prior knowledge is that E[X0] = 10◦C and σ[X0] = 2◦C .


,

observation error︷︸︸︷vt : V ∼ N (0, 22)


,

model error︷︸︸︷wt : W ∼ N (0, 0.52)




Example #1


0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20t = 2 | yt = 12.3

E[Xt−1|t−1] = E[Xt−1|y1:t−1] = 8.5

Var [Xt−1|t−1] = 2.1

Time - t

Stat

e-x

t






Example #1


0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20t = 3 | yt = 9

E[Xt−1|t−1] = E[Xt−1|y1:t−1] = 9.9

Var [Xt−1|t−1] = 1.5

Time - t

Stat

e-x

t






Example #1


0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20t = 4 | yt = 7.6

E[Xt−1|t−1] = E[Xt−1|y1:t−1] = 9.6

Var [Xt−1|t−1] = 1.2

Time - t

Stat

e-x

t






Example #1


0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20t = 5 | yt = 10.2

E[Xt−1|t−1] = E[Xt−1|y1:t−1] = 9.1

Var [Xt−1|t−1] = 1.1

Time - t

Stat

e-x

t






General formulation

Example #2 – Temperature time-series (constant speed)

0 1 2 3 4 5 6 7 8 9 10

−10

0

10

Time - t

Stat

e-

x tE[Xt|y1:t]E[Xt|y1:t]± σ[Xt|y1:t]yttrue values

0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

Time - t

Rat

eof

chan

ge-

x’t E[X ′t|y1:t]

E[X ′t|y1:t]± σ[Xt|y1:t]true values




General formulation

Example #2 – Temperature time-series (constant speed)We want to estimate xt = [xt , xt ]

ᵀ, the temperature and its rateof change over a period of 5 hours using imprecise measurementsmade every hour, t = 0 : 5.

yt = xt + vt ≡F=[1, 0]︷︸︸︷

F xt + vt︸︷︷︸observation model

xt = xt−1 + ∆txt−1 + wt

xt = xt−1 + wt

}≡ xt =

A=

[1 ∆t0 1

]︷︸︸︷

A xt−1 +

wt=

[wt

wt

]︷︸︸︷wt

︸︷︷︸transition model

We need a general formulationapplicable for any linearobservation/transition model

yt = Fxt + vtxt = Axt−1 + wt




General formulation


0 1 2 3 4 5 6 7 8 9 10

−10

0

10

Time - t

Stat

e-


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

Time - t

Rat

eof

chan

ge-






General formulation


We want to estimate xt = [xt , xt ]ᵀ, the temperature and its rate

of change over a period of 5 hours using imprecise measurementsmade every hour, t = 0 : 5.

yt = xt + vt ≡[1, 0]︷︸︸︷F xt +

V∼N (0,R)︷︸︸︷vt︸︷︷︸

observation model

xt = xt−1 + ∆txt−1 + wt

xt = xt−1 + wt

}≡

[xtxt

]︷︸︸︷xt =

[1 ∆t0 1

]︷︸︸︷

A xt−1 +

W∼N (0,Q)︷︸︸︷[wt

wt

]︷︸︸︷wt

︸︷︷︸transition model




General formulation


yt =

[1, 0]︷︸︸︷F xt +

V∼N (0,R)︷︸︸︷vt︸︷︷︸

observation model

[xtxt

]︷︸︸︷xt =

[1 ∆t0 1

]︷︸︸︷

A xt−1 +

W∼N (0,Q)︷︸︸︷[wt

wt

]︷︸︸︷wt︸︷︷︸

transition model

Prior knowledgeE[X0] = 10◦C , σ[X0] = 2◦C

E[X0] = 0◦C , σ[X0] = 0.5◦C

X0 ⊥⊥ X0 → ρX0X0= 0

µ0 =

[100

], Σ0 =

[22 00 0.52

]

Observation errors

R = σ2V = 22

Model errors

Q = σ2w

[∆t3

3∆t2

2∆t2

2 ∆t

], σ2

w = 0.1




General formulation

Example #2 – Temperature time-series (const. speed) [ ]



[CIV ML/KF example LT.m]


General formulation

Example #2 – Temperature time-series (const. speed) [ ]

0 1 2 3 4 5 6 7 8 9 10

−10

0

10

Time - t

Stat

e-


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

Time - t

Rat

eof

chan

ge-







Section Outline

t + n|t, t|T3.1 Introduction to forecasting and smoothing3.2 Forecasting: f (xt+n|y1:t)3.3 Smoothing: f (xt |y1:T )




Introduction to forecasting and smoothing

Filtering/Smoothing/Forecasting

TFiltering

0 t

TSmoothing

0 t

TForecasting

0 t t+n





Example #2 (cont.) – Forecasting temperature

yt =

[1, 0]︷︸︸︷F xt +

V∼N (0,R)︷︸︸︷vt︸︷︷︸

observation model

[xtxt

]︷︸︸︷xt =

[1 ∆t0 1

]︷︸︸︷

A xt−1 +

W∼N (0,Q)︷︸︸︷[wt

wt

]︷︸︸︷wt︸︷︷︸

transition model

Prior knowledgeE[X0] = 10◦C , σ[X0] = 2◦C

E[X0] = 0◦C , σ[X0] = 0.5◦C

X0 ⊥⊥ X0 → ρX0X0= 0

µ0 =

[100

], Σ0 =

[22 00 0.52

]

Observation errors

R = σ2V = 22

Model errors

Q = σ2w

[∆t3

3∆t2

2∆t2

2 ∆t

], σ2

w = 0.1





Example #2 (cont.) – Forecasting temperature [ ]






Example #2 (cont.) – Forecasting temperature [ ]

0 1 2 3 4 5 6 7 8 9 10

−10

0

10

20

Time - t

Stat

e-


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

Time - t

Rat

eof

chan

ge-







Smoothing: f (xt |y1:T )


Filter:

Smooth:

t

µ0

Σ0

0

µ1|1Σ1|1

...

1

µt|tΣt|t

t

... µT−1|T−1

ΣT−1|T−1

T − 1

µT |TΣT |T

T

µT−1|TΣT−1|T

...µt|TΣt|T

...µ1|TΣ1|T

µ0|TΣ0|T

µT |T and ΣT |T is treated as an observation

to update µT−1|T−1 and ΣT−1|T−1

leading to the smoothed estimate µT−1|T and ΣT−1|T





General formulation – RTS Kalman Smoother equations

Filter:

Smooth:

t

µ0

Σ0

0

µ1|1Σ1|1

...

1

µt|tΣt|t

t

... µT−1|T−1

ΣT−1|T−1

T − 1

µT |TΣT |T

T

µT−1|TΣT−1|T

...µt|TΣt|T

...µ1|TΣ1|T

µ0|TΣ0|T

RTS Kalman Smoother

f (xt |y1:T ) = N (µt|T ,Σt|T )

µt|T = µt|t + Jt(µt+1|T − µt+1|t

)

Σt|T = Σt|t + Jt(Σt+1|T −Σt+1|t

)Jᵀt

Jt , Σt|tAᵀΣ−1

t+1|t





Example #1 (cont.) – Smoothing temperature




,



,






Example #1 (cont.) – Smoothing temperature [ ]

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Time - t

Stat

e-x

t






Section Outline

θ estimation4.1 Measuring model’s performance: Log-likelihood4.2 Newton-Raphson algorithm




Measuring model’s performance: Log-likelihood

Measuring a model performanceHow do we measure the performance of a model?

With the marginal prior probability of each observation.

f (yt |y1:t−1) = N (yt ; Fµt|t−1,FΣt|t−1Fᵀ + R)

Likelihood

f (y1:T ) =T∏

t=1

f (yt |y1:t−1) [note: yi ⊥⊥ yj ,∀i 6= j ]

Since f (yt |y1:t−1)� 1,∏T

t=1 f (yt |y1:t−1) ≈ 0

Log-likelihood

log f (y1:T ) =T∑

t=1

log f (yt |y1:t−1)





Example #1 (cont.) – Log-likelihood




,



,





Newton-Raphson algorithm

Learning model parameters θ

How do we learn the parameters θ = [σW , σV , . . .]ᵀ of a model?

When a large amount of data is available→ Maximum Likelihood Estimate (MLE)

θ∗ = arg maxθ

log-likelihood︷︸︸︷log f (y1:T )

≡ arg maxθ

log f (y1:T |θ)

d

dθjlog p(y1:T |θ) = 0→ Newton-Raphson




Section Outline

SKF5.1 Introduction5.2 Switching Kalman Filter5.3 SKF example




Introduction

Non-stationary time series

The Kalman filter and smoother are applicable when thetime-serie analyzed is stationary or trend-stationary

Switching Kalman Filter: Estimate at each time step t, what isthe probability of each model structure.




Introduction

Switching Kalman Filter

17-01 17-04 17-07 17-10 17-12

9

10

11

12

Time [YY-MM]

Temperature[C]-y t





Introduction


17-01 17-04 17-07 17-10 17-12·105

9

10

11

12

Time [YY-MM]

Tem

pera

ture

[C]-

y t

E[Yt|y1:t]± σE[Yt|y1:t] E[Yt|y1:t]t|t yt ± σV yt




Introduction


17-01 17-04 17-07 17-10 17-12·105

9

10

11

12

Time [YY-MM]

xLL t

µt|t ± σt|t µt|t xt




Introduction


17-01 17-04 17-07 17-10 17-12·105

0

0.2

0.4

0.6

0.8

1

Time [YY-MM]

Pr(S)






Switching Kalman Filter – Algorithm

t-1model 1






t-1model 1

model 2






Filter withmodel 1

Filter withmodel 2

t-1model 1

model 2






Filter withmodel 1

Filter withmodel 2

t-1 t

Merge

Filter withmodel 1

Filter withmodel 2

Merge

model 1

model 2

model 1

model 2





Switching Kalman Filter – Merge step [ ]t

Merge

model 1

The merge step consistsin mixing several Gaussian

random variables

−2 −1 0 1 2 3 4 5 6 7 8 90

5 · 10−2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 µ1 =2 µ2 =5µ12 =4.1

π1 =0.3 π2 =0.7

σ1 =1 σ2 =1

σ12 =2.494

x

PDF

:f()

fX1(x1) fX2(x2) fX12(x12)

µ12 = π1µ1 + π2µ2

m = [µ1, µ2]− µ12

σ212 = π1σ

21 + π2σ

22 + mmᵀ



[Murphy (2012), §11.2.1 | CIV ML/Gaussian mixture demo.m]




Merge

model 1


random variables

−2 −1 0 1 2 3 4 5 6 7 8 9 10 11 120

5 · 10−2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 µ1 =2 µ2 =8µ12 =2.6

π1 =0.9 π2 =0.1

σ1 =1 σ2 =1

σ12 =5.5245

x

PDF

:f()

fX1(x1) fX2(x2) fX12(x12)

µ12 = π1µ1 + π2µ2

m = [µ1, µ2]− µ12

σ212 = π1σ

21 + π2σ

22 + mmᵀ







Merge

model 1


random variables

−2 −1 0 1 2 3 4 5 6 70

5 · 10−2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 µ1 =2µ2 =3µ12 =2.1

π1 =0.9 π2 =0.1

σ1 =1 σ2 =1σ12 =1.3491

x

PDF

:f()

fX1(x1) fX2(x2) fX12(x12)

µ12 = π1µ1 + π2µ2

m = [µ1, µ2]− µ12

σ212 = π1σ

21 + π2σ

22 + mmᵀ







Merge

model 1


random variables

−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

5 · 10−2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 µ1 =2µ2 =2µ12 =2

π1 =0.3 π2 =0.7

σ1 =1 σ2 =1σ12 =1

x

PDF

:f()

fX1(x1) fX2(x2) fX12(x12)

µ12 = π1µ1 + π2µ2

m = [µ1, µ2]− µ12

σ212 = π1σ

21 + π2σ

22 + mmᵀ





SKF example

Example #4 – non-stationary temperature data

17-01 17-04 17-07 17-10 17-12

9

10

11

12

Time [YY-MM]

Temperature[C]-y t





SKF example

Example #4 – non-stationary data

yt =

[1, 0]︷︸︸︷F xt +

V∼N (0,0.52)︷︸︸︷vt︸︷︷︸

observation model

µ0 =

[100

], Σ0 =

[12 00 0.12

]

Z =

[0.99999 0.000010.00001 0.99999

]

Model 1: Constant

[xtxt

]︷︸︸︷xt =

[1 00 0

]︷︸︸︷

A xt−1 +

W∼N (0,Q)︷︸︸︷[wt

wt

]︷︸︸︷wt︸︷︷︸

transition model

Model 2: Non-constant

[xtxt

]︷︸︸︷xt =

[1 ∆t0 1

]︷︸︸︷

A xt−1 +

W∼N (0,Q)︷︸︸︷[wt

wt

]︷︸︸︷wt︸︷︷︸

transition model

Q11 = Q22 = Q21 = 0∗, Q12 =

[0 00 σ2

W

]




SKF example

Example #4 – non-stationary data

17-01 17-04 17-07 17-10 17-12·105

9

10

11

12

Time [YY-MM]

Tem

pera

ture

[C]-

y t

E[Yt|y1:t]± σE[Yt|y1:t] E[Yt|y1:t]t|t yt ± σV yt

17-01 17-04 17-07 17-10 17-12·105

9

10

11

12

Time [YY-MM]

xLL t

µt|t ± σt|t µt|t xt

17-01 17-04 17-07 17-10 17-12·105

−1

−0.5

0

0.5

1

1.5

2

·10−2

Time [YY-MM]

xLT t

17-01 17-04 17-07 17-10 17-12·105

0

0.2

0.4

0.6

0.8

1

Time [YY-MM]

Pr(S)





LimitationsMLE Parameter calibration:

I Sensitive to θ0 (Several local maxima)

I Does not account for uncertainty in parameter estimates→ you need a lot of data

Numerical errors issues:

I The covariance is rapidly reduced in the measurement step(e.g. accurate measurements are used / after a period withmissing data

I There is a large difference between the variance of severalstate variables contributing to the same observation.

I Alternative approaches:I UD FilterI Square-Root Filter




Limitations

Linear models:

I The Kalman Filter and Smoother is only applicable for linearmodels

I Alternative approaches:I Particle FilterI Unscented Kalman FilterI Extended Kalman Filter

Non-constant time-steps:

I If times steps are not constant across the dataset, parametervalues must be time-step-size dependent.

{At ,Ft ,Qt ,Rt}




SummaryTransition model

xt = Axt−1 + wt︸︷︷︸W∼N (0,Q)

Observation model

yt = Fxt + vt︸︷︷︸V∼N (0,R)

I A: Model transition matrix

I F: Model observation matrix

I Q: Model errors covariance matrix

I R: Measurement errors covariancematrix

Filtering/Smoothing/Forecasting

TFiltering

0 t

TSmoothing

0 t

TForecasting

0 t t+n

(µt|t ,Σt|t ,Lt ) = Filter(µt−1|t−1,Σt−1|t−1, yt ; A, F,Q, R)

Prediction stepµt|t−1 = Aµt−1|t−1


Measurement step

f (xt |y1:t ) = N (xt ; µt|t ,Σt|t )

µt|t = µt|t−1 + Kt rt Posterior expected value

Σt|t = (I− KtF)Σt|t−1 Posterior covariance

Optimal parameter θ∗ must be identified using a maximizationalgorithm (e.g. Newton-Raphson)

θ∗ = arg max

θ

T∑t=1

lnLt

Marginal prior probability of each observation

Lt = f (yt |y1:t−1) = N (yt ; Fµt|t−1, FΣt|t−1Fᵀ + R)



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