Upload
vuongmien
View
214
Download
0
Embed Size (px)
Citation preview
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 2 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 3 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 4 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 5 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 6 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 6 / 26
Backshift
Backshift Operator
B̂kYt = Yt−k (2)
B̂kεt = εt−k (3)
Yt =
p∑i=1
φi B̂iYt + εt −
q∑j=1
θj B̂jεt
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 7 / 26
Backshift
Backshift Operator
B̂kYt = Yt−k (2)
B̂kεt = εt−k (3)
Yt =
p∑i=1
φi B̂iYt + εt −
q∑j=1
θj B̂jεt
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 7 / 26
ARMA Characteristic Equations
Yt =
p∑i=1
φi B̂iYt + εt −
q∑j=1
θj B̂jεt
φ(B̂)Yt = θ
(B̂)εt (4)
ARMA Characteristic Equations
φ(x) = 1−p∑
i=1
φixi (5)
θ(x) = 1−q∑
j=1
θjxj (6)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 8 / 26
ARMA Characteristic Equations
Yt =
p∑i=1
φi B̂iYt + εt −
q∑j=1
θj B̂jεt
φ(B̂)Yt = θ
(B̂)εt (4)
ARMA Characteristic Equations
φ(x) = 1−p∑
i=1
φixi (5)
θ(x) = 1−q∑
j=1
θjxj (6)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 8 / 26
ARMA Characteristic Equations
Yt =
p∑i=1
φi B̂iYt + εt −
q∑j=1
θj B̂jεt
φ(B̂)Yt = θ
(B̂)εt (4)
ARMA Characteristic Equations
φ(x) = 1−p∑
i=1
φixi (5)
θ(x) = 1−q∑
j=1
θjxj (6)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 8 / 26
SARMA Characteristic Equations
Introduce seasonality of lag s.
SARMA Characteristic Equations
Φ(x) = 1−P∑i=1
Φixis (7)
Θ(x) = 1−Q∑j=1
Θjxjs (8)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 9 / 26
SARMA Characteristic Equations
Introduce seasonality of lag s.
SARMA Characteristic Equations
Φ(x) = 1−P∑i=1
Φixis (7)
Θ(x) = 1−Q∑j=1
Θjxjs (8)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 9 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 10 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 10 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 11 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 11 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 12 / 26
Results
Figure: Fit SARMA on activity time series
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 13 / 26
Results
Figure: Fit SARMA on temperature time series
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 14 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 15 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 16 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 16 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 16 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 17 / 26
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)
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 17 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 18 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 19 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 20 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 21 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 22 / 26
Forecasting: Temperature
Figure: 24 hour forecast for temperature of patient 8, given by averaging thefitted model of 4 days.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 23 / 26
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.
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 24 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 25 / 26
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
B.Hadj-Amar, N.Cunningham, S.Ip Time Series Analysis March 11, 2016 26 / 26