Time series analysis of activity and temperature data of ...sip/work/presentation_timeSeries.pdf · Time series analysis of activity and temperature data of four healthy individuals

Time series analysis of activity and temperature data offour healthy individuals

B.Hadj-Amar N.Cunningham S.Ip

March 11, 2016

B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 1 / 26

Aims

Fit time series models to activity data

Identify periodicity in the data

Produce a 24-hr ahead forecast for activity and temperature

Medical applications


Data

Healthy individuals wore devices recording physical activity and skintemperature

Four individuals recorded for approximately four days

Physical activity recorded minutely, skin temperature every tenminutes — hourly median taken

Missing data


Data

0

20

40

60

2013−10−04 2013−10−05 2013−10−06 2013−10−07

Res

t Act

ivity

33

34

35

36

37

2013−10−04 2013−10−05 2013−10−06 2013−10−07

Tem

pera

ture


Autocorrelation function

0 10 20 30 40 50 60 70

−0.

50.

00.

51.

0

Lag

AC

F

Activity

0 10 20 30 40 50 60 70

−0.

40.

00.

20.

40.

60.

81.

0Lag

Temperature


ARMA

{Y1,Y2, ···,YT} is a time series with zero mean.εt is some noise with zero mean and variance σ2.

ARMA

ARMA(p, q) model has the form

Yt =

p∑i=1

φiYt−i + εt −q∑

j=1

θjεt−j (1)


ARMA

{Y1,Y2, ···,YT} is a time series with zero mean.εt is some noise with zero mean and variance σ2.

ARMA

ARMA(p, q) model has the form

Yt =

p∑i=1

φiYt−i + εt −q∑

j=1

θjεt−j (1)


Backshift

Backshift Operator

B̂kYt = Yt−k (2)

B̂kεt = εt−k (3)

Yt =

p∑i=1

φi B̂iYt + εt −

q∑j=1

θj B̂jεt


Backshift

Backshift Operator

B̂kYt = Yt−k (2)

B̂kεt = εt−k (3)

Yt =

p∑i=1


q∑j=1

θj B̂jεt


ARMA Characteristic Equations

Yt =

p∑i=1


q∑j=1

θj B̂jεt

φ(B̂)Yt = θ

(B̂)εt (4)


φ(x) = 1−p∑

i=1

φixi (5)

θ(x) = 1−q∑

j=1

θjxj (6)



Yt =

p∑i=1


q∑j=1

θj B̂jεt

φ(B̂)Yt = θ

(B̂)εt (4)


φ(x) = 1−p∑

i=1

φixi (5)

θ(x) = 1−q∑

j=1

θjxj (6)



Yt =

p∑i=1


q∑j=1

θj B̂jεt

φ(B̂)Yt = θ

(B̂)εt (4)


φ(x) = 1−p∑

i=1

φixi (5)

θ(x) = 1−q∑

j=1

θjxj (6)


SARMA Characteristic Equations

Introduce seasonality of lag s.


Φ(x) = 1−P∑i=1

Φixis (7)

Θ(x) = 1−Q∑j=1

Θjxjs (8)



Introduce seasonality of lag s.


Φ(x) = 1−P∑i=1

Φixis (7)

Θ(x) = 1−Q∑j=1

Θjxjs (8)


SARMA

SARMA

SARMA(p, q)× (P,Q)s model has the form

φ(B̂)

Φ(B̂)Yt = θ

(B̂)

Θ(B̂)εt (9)

Assume Normal noise, parameters can be estimated using maximum loglikelihood.


SARMA

SARMA

SARMA(p, q)× (P,Q)s model has the form

φ(B̂)

Φ(B̂)Yt = θ

(B̂)

Θ(B̂)εt (9)

Assume Normal noise, parameters can be estimated using maximum loglikelihood.


Information Criterions

AIC and BIC can be used to select what p, q,P,Q, s to use.

AIC and BIC

Select p, q,P,Q, s which minimizes one of these

AIC = T ln σ̂2 + (p + q + P + Q)2 (10)

BIC = T ln σ̂2 + (p + q + P + Q) lnT (11)


Information Criterions

AIC and BIC can be used to select what p, q,P,Q, s to use.

AIC and BIC

Select p, q,P,Q, s which minimizes one of these

AIC = T ln σ̂2 + (p + q + P + Q)2 (10)

BIC = T ln σ̂2 + (p + q + P + Q) lnT (11)


Experiment

3 day fit, 1 day forecast

Select s = 24 hours.

Fit ARMA(p, q) and select the best p, q pair.

Fix p, q, fit SARMA(p, q)× (P,Q)24 and select the best P,Q pair.

Fit and then assess 24 hour forecast


Results

Figure: Fit SARMA on activity time series


Results

Figure: Fit SARMA on temperature time series


Results

AICData p q P Q MSE (◦C2)

Temp2 2 4 2 0 1.71± 0.09Temp8 2 1 2 0 1.4± 0.1

Temp24 4 4 1 1 0.43± 0.04Temp26 5 2 2 0 0.7± 0.1

BICData p q P Q MSE (◦C2)

Temp2 0 2 2 0 1.72± 0.08Temp8 2 1 2 0 1.4± 0.1

Temp24 2 4 1 1 0.36± 0.03Temp26 2 0 2 0 0.69± 0.06

Table: Selected SARMA models for temperature with mean squared error


Harmonic Regression

Let us consider the periodic model:

Xt = Acos(2πωt + φ) + Zt ,

where A amplitude, φ phase shift, ω fixed frequency, Zt ∼WN(0, σ2)

Using, trigonometric identities we re-write:

Xt = β1cos(2πωt) + β2sin(2πωt) + Zt

where β1 = Acos(φ) and β2 = −Asin(φ).

Linear in β1 and β2 ⇒ Linear regression


Harmonic Regression









Harmonic Regression









Harmonic Regression

Spectral Representation Theorem states that any (weakly) stationary timeseries can be approximated:

Xt = µ+K∑

k=1

{βk1cos(2πωkt) + βk2sin(2πωkt)

}

Issue: Find this collection of frequencies {ωk}Kk=1 that drive the data

Therefore, we use the periodogram I (ωj), estimator of thefrequency spectrum f (ωj)

E[I (ωj)] = f (ωj)


Harmonic Regression

Spectral Representation Theorem states that any (weakly) stationary timeseries can be approximated:

Xt = µ+K∑

k=1

{βk1cos(2πωkt) + βk2sin(2πωkt)

}

Issue: Find this collection of frequencies {ωk}Kk=1 that drive the data

Therefore, we use the periodogram I (ωj), estimator of thefrequency spectrum f (ωj)

E[I (ωj)] = f (ωj)


Periodogram

However, the periodogram is not a consistent estimator.

Generating periodogram of AR(1), for different values of T :

Figure: Showing not consistency of the periodogram: black line is theperiodogram, red line is the true spectrum.


Smoothed Periodogram

The smoothed periodogram is instead a consistent estimator of thefrequency spectrum:

f̂ (ωj) =1

2M + 1

M∑k=−M

hk I (ωj +k

T)

Figure: Showing not consistency of the periodogram: black line is theperiodogram, red line is the true spectrum.


Temperature: Finding the frequencies that drive the data

Figure: Periodogram and smoothed periodograms (uniform and Daniell weights)for temperature of patient 8. This figure is best viewed in colours.


Periodogram and Spectrum: AR(p) approach

The spectrum of any (weakly) stationary time series can be approximatedby the spectrum of an AR(p) process.

(a) AIC and BIC (b) Spectrum AR(2), AR(9)

Figure: AR(p) approach to obtain the correct frequency representation. Mainfrequencies that drive the data are around 1/24, and 1/8


Harmonic Regression: Temperature

Figure: Harmonic regression for temperature of patient 8, using 5 differentharmonics. Dotted lines represent the fitting for single harmonics; thick red line isthe superposition of the dotted harmonics, which is the final fit


Forecasting: Temperature

Figure: 24 hour forecast for temperature of patient 8, given by averaging thefitted model of 4 days.


Harmonic Regression: Rest Activity

Figure: Harmonic regression for rest activity of patient 8, using 5 harmonics. Ontop it is shown the classic harmonic fitting, whereas on the bottom the alternativemodel, where negative values are taken to be zero.


Conclusions and future work

Harmonic regression provided a better fit for the data

In both cases skin temperature was more accurately modeled

Further work

Examine larger datasetsConsider weekday vs weekend effectsModel dependence structure between skin temperature and activitylevels


References

Mark Fiecas - Spectral Analysis of Time Series Data

University of Warwick - ST414

2015

Jenkins, GM and Reinsel, GC - Time series analysis: forecasting and control

Holden-Day, San Francisco

1976

Shumway, Robert H and Stoffer, David S - Time series analysis and its applications

Springer Science & Business Media

2013


Documents

Time series analysis of activity and temperature data of ...sip/work/presentation_timeSeries.pdf · Time series analysis of activity and temperature data of four healthy individuals