Download Discussion of Should we sample a time series more frequently

Discussion of Should we sample a time series more
frequently?: decision support via multirate spectrum
estimation by Nason, Powell, Elliot and Smith
Yi Yu and Ivor Cribben
September 8, 2016
We congratulate the authors on their paper. The paper proposes a new Bayesian spectral
estimation method that allows for the coherent fusion of data sampled at different rates. The
method can also be used to ascertain the appropriate sampling rate for a data set. Finally, the
authors develop an R package, regspec, which provides a fast implementation of the method and
good visualisation tools.
The paper concentrates on the setting of a data set that is collected at a slow-then-fast rate.
However, we are interested in more general settings where there are possibly more than one region
of fast- and/or slow-sampled data, and in the flexibility of the regspec package. In the following,
we use the Dfexample data example provided in regspec to investigate the setting where the
data is sampled at a fast rate, then a slow rate, and finally at a fast rate. Specifically, we sample
the first 51 and the last 51 time points at every integer with the data in between sampled at every
3 time points. To obtain an estimator of the spectrum, we compare four different settings:
1. Slow then fast. We only use the slow-sampled period and the last fast-sampled period, as
proposed in the paper.
2. Average the two fast periods. We obtain estimators from two fast-sampled periods separately
and use their average to update the slow-sampled period.
3. Fast then slow. We only use the slow-sampled period and the first fast-sampled period.
4. Updated fast. We use the 1st fast-sampled period to update the 2nd, and use the updated
estimators to estimate the slow-sampled period.
5. Updated fast. We use the 2nd fast-sampled period to update the 1st, and use the updated
estimators to estimate the slow-sampled period.
6. Split the 2nd fast. We split the 2nd fast-sampled period into two, and use one to update the
other, then use the updated estimators to estimate the slow-sampled period.
PNω
Figures 1-6 and Table 1 show the plots and the discrepancy criterion, d = Nω−1 i=1
[log{f (ωi )}−
2
ˆ
log{f (ωi )}] , used in the paper. Interestingly, in terms of the discrepancy values, the best results
are obtained by splitting one fast-sampled region, using one to update the other, and then updating
1
Settings
d
1
0.433
2
0.268
3
0.268
4
0.221
5
0.221
6
0.165
0.4
0.5
0.0
0.1
Frequency
Inf
10
5 wavelength,3.3
0.4
0.0
0.5
2.5
2
Inf
10
5 wavelength,3.3
2.5
Inf
2
Spectrum
80
0.4
0.5
0.0
0.1
Frequency
Inf
10
5 wavelength,3.3
10
0.2 frequency 0.3
0.4
0.5
0.0
0.1
Figure 4: Setting 4
2
5 wavelength,3.3
2.5
2
Inf
10
5 wavelength,3.3
0.2 frequency 0.3
0.4
0.5
2.5
2
Frequency
Frequency
2.5
0.5
0
40
Spectrum
0
0.2 frequency 0.3
0.4
Figure 3: Setting 3
120
120
80
40
0.1
0.2 frequency 0.3
Frequency
Figure 2: Setting 2
0
0.0
0.1
Frequency
Figure 1: Setting 1
Spectrum
0.2 frequency 0.3
80 120
0.2 frequency 0.3
40
0.1
80
0
0
0.0
40
Spectrum
80
40
Spectrum
80
40
0
Spectrum
Table 1: Discrepancy criterion values for different settings.
2.5
Figure 5: Setting 5
2
Inf
10
5 wavelength,3.3
Figure 6: Setting 6
the slow-sample period. We also attained similar results when we increased the fast sample size
and when the slow rate data was sampled at every 4, 5 and 6 time points.
In the paper, the authors focus on the slow-then-fast phenomenon in survey data. We are also
aware of other research areas that face the challenge of multirate sampled data, and we believe
the method can make a significant contribution to them. For example, in human neuroscience, a
great challenge is determining causal pathways in which different brain areas interact to support
cognition and behavior. While Granger causality has been applied to functional magnetic resonance
imaging (fMRI) data to reveal causal influences among cortical areas, the main challenge in this
estimation is the problematic low sampling rate (1-2 sec). However, this slow sampling rate is
necessary to achieve the high spatial resolution of fMRI. The sensitivity and stability of Granger
causality could be critically improved if the temporal sampling rate is high enough. We hope that
the newly proposed method could be used on this data type and on data collected using the recently
developed dynamic functional magnetic resonance inverse imaging (InI), which achieves an order
of magnitude faster sampling rate. In addition, the proposed method could also play a role in the
simultaneous recording and analysis of electroencephalography (EEG) and fMRI data. The EEGfMRI combination allows researchers to achieve both high temporal and spatial resolution in the
recording of human brain function. While this data combination is currently limited by a number
of issues, both modalities overlap and exhibit a linear association.
Finally, econometric models that incorporate variables sampled at different frequencies have
recently attracted substantial interest. For example, the Gross Domestic Product (GDP), a very
important indicator of macroeconomic activity, is released quarterly and is subject to subsequent
revisions, while a range of leading and coincident indicators are available more frequently or are
more timely (monthly or an even higher frequency). Policy-makers, need to assess the current
2
state of the economy and its expected developments in real-time with incomplete data. The most
common techniques for mixed frequency data or Nowcasting include bridge equations, MIxed DAta
Sampling (MIDAS) models, mixed frequency VARs, and mixed frequency factor models. Bridge
equations (Baffigi et al., 2004) focus on forecasting and link the low-frequency variables and timeaggregated indicators through equations. Forecasts of the high frequency variables are provided
by specific high-frequency time series models, then the forecast values are aggregated and plugged
into the bridge equations to obtain the forecast of the low-frequency variable. Mixed-data sampling
(MIDAS: Ghysels et al., 2004) models handle time series sampled at different frequencies, where
distributed lag polynomials are used to ensure parsimonious specifications. This method is employed
to forecast macroeconomic time series, where typically quarterly GDP growth is forecasted by
monthly macroeconomic and financial indicators (Clements and Galvão, 2008). We hope to see the
proposed method making a new contribution to this area.
References
Baffigi, A., Golinelli, R., and Parigi, G. (2004). Bridge models to forecast the euro area GDP.
International Journal of Forecasting, 20(3):447–460.
Clements, M. P. and Galvão, A. B. (2008). Macroeconomic forecasting with mixed-frequency data:
Forecasting output growth in the united states. Journal of Business & Economic Statistics,
26(4):546–554.
Ghysels, E., Santa-Clara, P., and Valkanov, R. (2004). The MIDAS touch: Mixed data sampling
regression models. Finance.
3