Delivered by Ivor Cribben (Assistant Professor, University of Alberta) at the 2016 New York R Conference on April 8th and 9th at Work-Bench.

1. R packages for time-varying
networks and extremal dependence
Ivor Cribben
Department of Finance and Statistical Analysis
Alberta School of Business
NY R Conference
April 9th, 2016
Joint work with: Yi Yu, University of Cambridge
Nadia Frolova, University of Alberta

2. Correlation matrix
Time series
Precision matrix
Undirected graph

3. Time series
• Recently, there has been an increased interest in estimating time varying
dependence/undirected graphs.
• The static methods (covariance, correlation and precision matrices) all
have a natural time-varying analogue in conjunction with a sliding window.

4. Allen et al. (2012)
Time-varying dependence

5. • Hutchinson et al. (2013) and Lindquist et al. (2014) recently discussed the
limitations of the sliding window technique.
• We introduce a new method for estimating time varying dependence
without the use of a sliding window.
Time-varying dependence

6. Community detection

7. Time series

8. • We consider a data-driven technique, called Network Change Point
Detection (NCPD), for detecting change points in the community network
structure.
• NCPD
 detects temporal change points in the community network structure
 estimates the community network structure for data in the temporal
partition that falls between pairs of change points.
• It is assumed that the network remains constant within each partition.
Network Change Point Detection

9. NCPD

10. • Functional magnetic resonance imaging (fMRI) is a type of specialized MRI scan. It
is a non-invasive technique for studying brain activity.
• During the course of an fMRI experiment, a series of brain images are acquired
(one every 2s or so) while the subject performs a set of tasks.
fMRI data

11. Functional connectivity

12. Functional connectivity

13. Resting state fMRI data

14. • We evaluate the amount of change for each candidate change point by the
rotation from one subspace to another or the cosine of the principal
angles between these two subspaces.
• In the same spirit, when estimating the similarity of the networks across
different subjects, we can use the cosine of the principal angles between
their two partitions.
• The pair with the largest criterion value are the most similar networks.
Resting state fMRI data

15. Resting state fMRI data
• Future work - controls and subjects with brain disorders such as depression,
Alzheimer’s disease and schizophrenia.
• It is hoped that the large-scale temporal features resulting from the
description of connectivity from our novel method will lead to better
o diagnostic and
o prognostic indicators of the brain disorders.

16. • Parallel computing. Code available to download locally from
http://www.statslab.cam.ac.uk/yy366/
• The average time to run the algorithm on time series (T = 285, P = 116) was
132s on a dual core processor.
Computation

18. • There has been renewed interest in
understanding and modeling extreme events
– Climate
– Finance
• Extreme events are observations that occur in
the (left and right) tails of a distribution .
• By their nature they occur infrequently.
Extreme events

19. • The study of extreme events is very interesting.
• Here, we consider the temporal dependence between extreme events.
• This temporal dependence can either occur within a single time series or
between two or more time series.
Extreme events

20. • The extremogram is a flexible quantitative tool for measuring various
forms of extremal dependence in a stationary time series.
• In simple terms, the extremogram computes the extent to which a large
value of the time series has on a future value of the same time series or
another time series, h time-lags ahead.
• The application of the extremogram to financial time series data is apt
– heavy tails (extreme values) and
– clustering of extreme events over time.
The extremogram

21. • The data the extremogram considers are naturally rare, hence accurate
estimation of the extremogram requires large sample sizes.
• Fortunately, long time series are available in the field of finance.
• The extremogram and its derivatives can easily describe graphically and
quantitatively the size and persistence of the extreme value clusters.
The extremogram

22. • Assuming 𝑎 𝑚 is the (1 − 1/m)-quantile of the stationary distribution of a
time series 𝑋𝑡 , the sample extremogram based on the observations
𝑋1,…, 𝑋 𝑛 is given by
𝜌 𝐴,𝐵(ℎ) =
𝐼{𝑎 𝑚
−1 𝑋 𝑡+ℎ ∈ 𝐵, 𝑎 𝑚
−1 𝑋𝑡 ∈ 𝐵}𝑛−ℎ
𝑡=1
𝐼{𝑎 𝑚
−1 𝑋𝑡 ∈ 𝐴}𝑛
𝑡=1
• In practice, 𝑎 𝑚 is replaced by an empirical quantile of the time series 𝑋𝑡 .
For example, it could be replaced by the 0.9th, the 0.95th or the 0.99th
empirical quantile.
The sample extremogram

23. • We study the
1) persistence of extreme price outcomes in electricity markets and
2) the relationship between extreme events
across 5 regional electricity spot markets in Australia.
• Our study yields important insights into
1) extreme spot electricity prices
2) the persistence of such events
3) spillover effects and
4) the impact of interconnection within Australian electricity markets.
Spot electricity prices

24. Spot electricity prices
SA NSW
QLD TAS

25. The cross-extremogram
• We define the sample cross-extremogram for the bivariate time series
(𝑋𝑡, 𝑌𝑡) 𝑡∈ℤ
by
𝜌 𝐴,𝐵(ℎ) =
𝐼{𝑎 𝑚,𝑌
−1
𝑌 𝑡+ℎ ∈ 𝐵, 𝑎 𝑚,𝑋
−1
𝑋𝑡 ∈𝐴}𝑛−ℎ
𝑡=1
𝐼{𝑎 𝑚,𝑋
−1 𝑋𝑡 ∈ 𝐴}𝑛
𝑡=1
where 𝑎 𝑚,𝑋 and 𝑎 𝑚,𝑌 are replaced by the respective empirical
quantiles computed from (𝑋𝑡) 𝑡=1,…,𝑛 and (𝑌𝑡) 𝑡=1,…,𝑛, respectively.

26. Spot electricity prices
Sample cross-extremograms of half-hourly spot
electricity prices conditioning on price spikes in NSW.
(A) NSW → QLD, (B) NSW → SA, (C) NSW → TAS and
(D) NSW → VIC based on a sample period from
January 1, 2009 to March 31, 2015.

27. Spot electricity prices
• Our results are important for risk management and hedging decisions.
• In particular, market participants operating in several regional markets
simultaneously, such as large producers, retailers or merchant
interconnectors, clearly should apply extremograms for their planning, trading
and risk management decisions.

28. Trivariate extremogram
• For a stationary trivariate time series (𝑋𝑡, 𝑌𝑡, 𝑍𝑡) 𝑡∈ℤ
, many different variations of
the cross-extremogram can be defined depending on the context.
• For example, we may consider,
𝜌1(ℎ) =
𝐼{𝑋𝑡 > 𝑎 𝑚,1 and (𝑌 𝑡+ℎ > 𝑎 𝑚,2 or 𝑍 𝑡+ℎ > 𝑎 𝑚,3)}𝑛−ℎ
𝑡=1
𝐼{𝑋𝑡 >𝑎 𝑚,1}𝑛
𝑡=1
𝜌2(ℎ) =
𝐼{(𝑋𝑡 > 𝑎 𝑚,1 or 𝑌𝑡 > 𝑎 𝑚,2) and 𝑍 𝑡+ℎ > 𝑎 𝑚,3}𝑛−ℎ
𝑡=1
𝐼{𝑋𝑡 > 𝑎 𝑚,1 or 𝑌𝑡 > 𝑎 𝑚,2}𝑛
𝑡=1
where 𝑎 𝑚,1, 𝑎 𝑚,2 and 𝑎 𝑚,3 are chosen as the corresponding
empirical quantiles of 𝑋𝑡, 𝑌𝑡, and 𝑍𝑡, respectively.

29. Equity indices
𝜌1(ℎ) =
𝐼{𝐵𝐴𝐶𝑡 and (𝐶 𝑡+ℎ > 𝑎 𝑚,2 or 𝑀𝑆𝐹𝑇 𝑡+ℎ > 𝑎 𝑚,3)}𝑛−ℎ
𝑡=1
𝐼{𝐵𝐴𝐶𝑡 >𝑎 𝑚,1}𝑛
𝑡=1

30. Extremogram R package

31. Cribben, I., & Yu, Y. (2015) Estimating whole brain dynamics using spectral clustering. arXiv
preprint arXiv:1509.03730.
Davis, R.A., Mikosch, T., Cribben, I. (2012) Towards Estimating Extremal Serial Dependence via the
Bootstrapped Extremogram. Journal of Econometrics 170, 142–152.
Davis, R.A., Mikosch, T., Cribben, I. (2012) Estimating Extremal Dependence in Univariate and
Multivariate Time Series via the Extremogram. arXiv:1107.5592 [stat.ME]
Frolova, N., Cribben, I. (2015) extremogram: estimating extreme value dependence. R package
version 3.1.0.
References