=Paper=
{{Paper
|id=Vol-1425/paper2
|storemode=property
|title=Monitoring Short Term Changes of Malaria Incidence in Uganda with Gaussian Processes
|pdfUrl=https://ceur-ws.org/Vol-1425/paper2.pdf
|volume=Vol-1425
|dblpUrl=https://dblp.org/rec/conf/pkdd/PachecoMQL15
}}
==Monitoring Short Term Changes of Malaria Incidence in Uganda with Gaussian Processes==
https://ceur-ws.org/Vol-1425/paper2.pdf
Proceedings 1st International Workshop on Advanced Analytics and Learning on Temporal Data
AALTD 2015
Monitoring Short Term Changes of Malaria
Incidence in Uganda with Gaussian Processes
Ricardo Andrade-Pacheco1 , Martin Mubangizi2 , John Quinn2,3 , and Neil
Lawrence1
1
University of Sheffield, Department of Computer Science, UK
2
Makerere University, College of Computing and Information Science, Uganda
3
UN Global Pulse, Pulse Lab Kampala, Uganda
Abstract. A method to monitor communicable diseases based on health
records is proposed. The method is applied to health facility records of
malaria incidence in Uganda. This disease represents a threat for approx-
imately 3.3 billion people around the globe. We use Gaussian processes
with vector-valued kernels to analyze time series components individu-
ally. This method allows not only removing the effect of specific com-
ponents, but studying the components of interest with more detail. The
short term variations of an infection are divided into four cyclical phases.
Under this novel approach, the evolution of a disease incidence can be
easily analyzed and compared between different districts. The graphical
tool provided can help quick response planning and resources allocation.
Keywords: Gaussian processes, malaria, kernel functions, time series.
1 Introduction
More than a century after discovering its transmission mechanism, malaria has
been successfully eradicated from different regions of world [15]. However, it
is still endemic in 100 countries and represents a threat for 3.3 billion people
approximately [20]. In Uganda, malaria is among the leading causes of morbidity
and mortality [19]. Different types of interventions can be carried on to prevent
and treat malaria [20]. Their success depend on how well the disease can be
anticipated and how fast the population reacts to it. In this regard, mathematical
modelling can be a strong ally for decision-making and health services planning.
Spatiotemporal modelling for mapping and prediction of infection dynamics is a
challenging problem. First of all, because of the costs and difficulties of gathering
data. Second, because of the challenges of developing a sound theoretical model
that agrees with the data observed.
The Health Management Information System (HMIS) operated by the Uganda
Ministry of Health provides weekly records of the number of patients treated for
malaria in different hospitals across the country. Unfortunately, the number of
reporting hospitals is not consistent across time. This variation is prone to create
artificial trends in the observed data. Hence, the underreporting effect has to be
estimated to be removed.
A common approach for time series analysis is to decompose the observed
variation into specific patterns such as trends, cyclic effects or irregular fluctua-
tions [4, 3, 7]. Gaussian process (GP) models are a natural approach for analyzing
Copyright c 2015 for this paper by its authors. Copying permitted for private and academic
purposes.
2 R. Andrade-Pacheco, M. Mubangizi, J. Quinn, N. Lawrence
functions that represent time series. GPs provide a robust framework for non-
parametric probabilistic modelling [18]. The use of covariance kernels enable to
analyse non-linear patterns by embedding an inference problem into an abstract
space with a convenient structure[14]. By combining different covariance kernels
(via additions, multiplications or convolutions) into a single one, a GP is able to
describe more complex functions. Each of the individual kernels contributes by
encoding a specific set of properties or pattern of the resulting function [5].
We propose a monitoring system for communicable diseases based on Gaus-
sian processes. This methodology is able to isolate the relevant components of
the time series and study the short term variations of the disease. The output
of this system is a graphical tool that discretizes the disease progress into four
phases of simple interpretation.
2 Background
Say we are interested in learning the functional relation, between inputs and
output, based on a set of observations {(xi , yi )}ni=1 . GP models introduce an
additional latent variable fx , whose covariance kernel K is a function of the
input values. Usually, yi is considered a distorted version of the latent variable.
To deal with multiple outputs, GP models resort to generalizations of kernel
functions to the vector-valued case [1]. In time series literature, vector-valued
functions are commonly treated in the family of VAR models [12], while in geo-
statistics literature co-Kriging generalizations are used [8, 11]. These approaches
are equivalent. Let hx = (fx1 , . . . , fxd )> be a vector-valued GP, its corresponding
covariance matrix is given by
cov(hx , hz )ij = cov(fxi , fzj ) .
(1)
The diagonal elements of the correlation matrix cov(hx , hz )ii are just the
covariance functions of the real-valued GP elements. The non-diagonal elements
represent the cross-covariance functions between components [9, 10, 2].
3 Method Used
Suppose we have data generated from the combination of two independent sig-
nals (see Figure 1a). Usually, not only we are not able to observe the signals
separately, but the combined signal they yield is corrupted by noise in the data
collected (see Figure 1b). For the sake of this example, suppose that the two
signals of the example represent a long term trend (the smooth signal) and a
seasonal component (the sinusoidal signal). For an observer, the oscillations of
the seasonal component masks the behaviour of the long term trend. At some
point, however, the observer might want to know whether the trend is increasing
or decreasing. Similarly, there might be interest in studying only the seasonal
Alarm System for Malaria 3
component isolated from the trend. For example, in economics and finance, busi-
ness recession and expansion periods are determined by studying the cyclic com-
ponent of a set of indicators [16]. The cyclic component tells if an indicator is
above or below the trend, and its differences tell if it is increasing or decreasing.
We propose a similar approach for monitoring disease incidence time series,
but in our case, we will use a non-parametric approach. To extract the original
signals, the observed data can be modelled using a GP with a combination of
kernels, say exponentiated quadratics, one having a shorter lengthscale than the
other. Figures 1c and 1d shows a model of the combined and independent signals.
We also use a vector-valued GP to model directly the derivative of the time series,
rather than using simple differences of the observed trend. As a result, we are
able to provide uncertainty estimates about the speed of the changes around the
trend. Our approach is based on modelling linear functionals of an underlying
GP [13]. If hx = (fx , ∂fx /∂xi )> , its corresponding kernel is defined as
" ∂
#
K(xi , xj ) ∂xj K(xi , xj )
Γ (xi , xj ) = ∂ ∂2 . (2)
∂xi K(xi , xj ) ∂xi xj K(xi , xj )
In most multi-output problems, observations of the different outputs are
needed to learn their relation. Here, the relation between fx and its derivative is
known beforehand through the derivative of K. Thus ∂fx /∂xi can be learnt by
relying entirely on fx . For the signals described above, Figures 1e and 1f show
the corresponding derivatives computed using a kernel of the form of (2). The
derivatives of the long term trend are computed with high confidence, while the
derivatives of the seasonal component have more uncertainty. The last is due to
the magnitude of the seasonal component relative to the noise magnitude.
4 Uganda Case
In this exposition we focus on Kabarole district, but provide a snapshot of the
monitoring system for all the country. Our base assumption about the infection
process of malaria is that it evolves with some degree of smoothness across
time. Smooth functions can be represented by a kernel such that the closer the
observations in the input space, the more similar values of the output. The
Matérn kernel family satisfies this condition, as it defines dependence through
the distance between points with some exponential decay [18]. Different members
of this family encode different degrees of smoothness, being the limit case the
exponentiated quadratic kernel or RBF, which is infinitely differentiable. To
illustrate our method we will use an RBF kernel. Results with (rougher) Matérn
kernels do not differ much when used instead.
Despite malaria is a disease influenced by environmental factors like temper-
ature or water availability, we could not observe a seasonal effect in HMIS data
[6]. If that was the case, the model could be improved incorporating a periodic
kernel in the covariance structure. Yet, the model fit can be improved if a sec-
ond RBF kernel is added. In this case, one kernel has a short lengthscale and
4 R. Andrade-Pacheco, M. Mubangizi, J. Quinn, N. Lawrence
16 20
14 18
12 16
10 14
8 12
6 10
4 8
2 6
00 5 10 15 20 25 30 40 5 10 15 20 25 30
(a) Original signals (b) Combined and corrupted signals
20 16
18 14
16 12
14 10
12 8
10 6
8 4
6 2
40 5 10 15 20 25 30 00 5 10 15 20 25 30
(c) Latent variable samples (d) Components mean estimate
20 20
18 18
16 16
14 14
12 12
10 10
8 8
6 6
40 5 10 15 20 25 30 40 5 10 15 20 25 30
(e) Long term trend derivatives (f) Seasonal component derivatives
Fig. 1: Series decomposition. Panel (a) shows two independent signals. Panel (b) shows
the combination of both signals (dashed line) and a distorted signal after adding some
noise (solid line). Panel (c) shows latent variable samples representing the combined
signal (thin lines). Panel (d) compares the mean estimate of each component (solid
line) with the original signals (dashed line). Panels (e) and (f) show the components
derivatives. Tangent lines to the individual components are shown in red. The solid blue
lines represent the mean estimate of the composed signal. The gray lines are random
realizations of process derivative. For comparison, the estimates of the individual signals
(dashed lines) are shown below the composed signal.
therefore represents short term variations, while the other represents long term
changes.
An important factor to consider about HMIS data is that the number of
health facilities is highly variable. See Figure 2a. This variation is prone to
create artificial trends in the incidence of malaria reported. Such trends can be
removed by incorporating a linear kernel that describes the relation between
reporting facilities and malaria cases. Unlike the RBF kernels mentioned above,
Alarm System for Malaria 5
which take time as input, the linear kernel takes the number of health facilities as
input. In Table 1, we present a comparison of the model predictive performance,
when using different kernels, based on the leave-one-out predictive probabilities
[17]. The best predictive performance is achieved when considering short and
long term changes and a correction for misreporting facilities.
Table 1: Comparison of LOO-CV log predictive probabilities, when using differ-
ent kernels. The subindex ` refers to the lengthscale of the kernel (measured in
years).
Kernel LOO-CV (log)
RBF`=0.64 -40.54
RBF`=0.14 + RBF`=10 -16.26
RBF`=0.12 + RBF`=10 + Linear 41.21
Figure 2c shows the trend and short term component of the number of
malaria cases. Variations of a disease incidence around its trend represent short
term changes in the population health. Outbreak detection and control of non-
endemic diseases take place in this time frame. For some endemic diseases, this
variation can be associated to seasonal factors [6]. Quick response actions, such
as distribution of medicine and allocation of patients to health centres, have to
take place in this time regime to be effective. The short term variations can be
classified in four phases as shown in Figure 2d (values are standardized). The
upper left quadrant represents an incidence below the trend, but increasing; the
upper right quadrant represents an incidence above the trend and expanding;
the bottom right quadrant represents an incidence above the trend, but decreas-
ing; and the bottom left quadrant represents an incidence below the trend and
decreasing.
This tracking system of short term variations is independent of the order of
the disease counts, and can be used to monitor the infection progress in different
districts. It is easy to identify districts where the disease is being controlled
or where the infection is progressing at an unusual rate. Figure 2b shows the
monitoring system on the whole country. Those districts where the variation
coefficient of both the process and its derivative are less than 1 (meaning a weak
signal vs noise) were left in gray color.
5 Final Remarks
We have proposed a disease monitor based on vector-valued Gaussian processes.
Our approach is able to account for uncertainty in both the level of each com-
ponent and the direction of change. The simplicity for doing inference with this
6 R. Andrade-Pacheco, M. Mubangizi, J. Quinn, N. Lawrence
55
8 50
malaria cases (x 1000) 7 45
reporting facilities
6 40
5 35
4 30
3 25
2 20
1 15
0 10 Kabarole
5 7 9 1 3
200 200 200 201 201
(a) HMIS records from Kabarole (b) Phases on 2014/04/20
10.0
4
9.5
malaria cases (log scale)
9.0 2 B
rate of change
8.5 A
8.0 C 0
7.5 B
A D 2 D
7.0 C
6.5
4
6.0
5 7 9 1 3
200 200 200 201 201 4 2 0 2
variation around the trend
4
(c) Trend and short term variations (d) Cycle phases
Fig. 2: Malaria incidence tracker in Uganda. Panel (a) compares the number of malaria
cases (solid line) and the number of reporting health facilities (dashed line). Panel (b)
shows the disease phase in each district. The colors are assigned according to the
quadrants in panel (d). Panel (c) shows the long term trend (dashed line) and the
short term variations (solid line). Gray bullets represent the observed records. Panel
(d) shows a tracking system of the short term variations. The bullets A-D in panels (c)
and (d) correspond to the same time points.
model is not compromised by the use of a vector-valued approach. The model can
be benefited if spatial information is available and encoded in the kernel func-
tion. Further research is needed to explore the benefits of this model in practice.
We expect that an analysis from this perspective can add situational awareness
and contribute to interventions planning and resources allocation when facing
infectious diseases.
Acknowledgements
Ricardo Andrade-Pacheco is supported by CONACYT and SEP scholarships.
References
1. M. Álvarez, L. Rosasco, and N. D. Lawrence. Kernels for vector-valued functions:
A review. Foundations and Trends in Machine Learning, 4(3):195–266, 2012.
Alarm System for Malaria 7
2. L. Baldassarre, L. Rosasco, A. Barla, and A. Verri. Multi-output learning via
spectral filtering. Machine Learning, 87(3):259–301, 2012.
3. M. Baxter and R. G. King. Measuring business cycles: approximate band-pass
filters for economic time series. Review of economics and statistics, 81(4):575–593,
1999.
4. W. P. Cleveland and G. C. Tiao. Decomposition of seasonal time series: A model
for the census X-11 program. Journal of the American statistical Association,
71(355):581–587, 1976.
5. N. Durrande, J. Hensman, M. Rattray, and N. D. Lawrence. Gaussian process
models for periodicity detection. arXiv preprint arXiv:1303.7090, 2013.
6. S. I. Hay, R. W. Snow, and D. J. Rogers. From predicting mosquito habitat to
malaria seasons using remotely sensed data: practice, problems and perspectives.
Parasitology Today, 14(8):306–313, 1998.
7. A. Hyvärinen and E. Oja. Independent component analysis: algorithms and appli-
cations. Neural networks, 13(4):411–430, 2000.
8. G. Matheron. Pour une analyse krigeante de donnés régionalisées. Technical report,
École des Mines de Paris, Fontainebleau, France, 1982.
9. C. A. Micchelli and M. Pontil. Kernels for multi-task learning. In Advances in
Neural Information Processing Systems (NIPS). MIT Press, 2004.
10. C. A. Micchelli and M. Pontil. On learning vector–valued functions. Neural Com-
putation, 17:177–204, 2005.
11. D. E. Myers. Matrix formulation of co-Kriging. Journal of the International As-
sociation for Mathematical Geology, 14(3):249–257, 1982.
12. H. Quenouille. The analysis of multiple time-series. Griffin’s statistical monographs
& courses. Griffin, 1957.
13. S. Särkkä. Linear operators and stochastic partial differential equations in Gaussian
process regression. In Artificial Neural Networks and Machine Learning–ICANN
2011, pages 151–158. Springer, 2011.
14. J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cam-
bridge University Press, Cambridge, U.K., 2004.
15. P. I. Trigg and A. V. Kondrachine. Commentary: malaria control in the 1990s.
Bulletin of the World Health Organization, 76(1):11, 1998.
16. F. van Ruth, B. Schouten, and R. Wekker. The statistics Netherlands business
cycle tracer. Methodological aspects; concept, cycle computation and indicator
selection. Technical report, Statistics Netherlands, 2005.
17. A. Vehtari, V. Tolvanen, T. Mononen, and O. Winther. Bayesian leave-one-out
cross-validation approximations for Gaussian latent variable models. arXiv preprint
arXiv:1412.7461, 2014.
18. C. K. I. Williams and C. E. Rasmussen. Gaussian processes for Machine Learning.
MIT Press, 2006.
19. World Health Organization. World health statistics 2015. Technical report, WHO
Press, Geneva, 2015.
20. World Health Organization and others. World malaria report 2014. Technical
report, WHO Press, Geneva, 2014.