Trends, seasonality and anomalies: making your time-series talk

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Trends, seasonality and anomalies making your
time-series talk
Wladimir J. Alonso Fogarty International Center /
NIH
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Goals for of this talk

1. Learn how to extract the basic components of
   epidemiological relevance from a time-series
2. Learn how to explore the spatial patterns of
   those components
3. Introduce the modeling tool Epipoi
   (www.epipoi.info)

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But before this
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A parenthesis for Graphical Excellence

• well-designed presentation of interesting data
  a matter of substance, statistics and design
• consists of complex ideas communicated with
  clarity, precision and efficiency
• is nearly always multivariate
• requires telling the truth about the data
• Provides the viewer with the greatest number of
  ideas in the shortest time with the least ink in
  the smallest space

Edward Tufte (1983)
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Napoleon's Retreat from Moscow, 1812by Illarion
Pryanishnikov
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Charles Joseph Minard (1861) Losses suffered by
Napoleon's army in the Russian campaign of 1812
"It may well be the best statistical graphic ever
drawn (Edward R. Turfte, 1983)
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First Organize your dataset in a meaningful way
A typical mortality dataset
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Structured spreadsheet as a source of
instantaneous analysis

• - Age groups
• - Causes of deaths
• Longitude
• Latitude

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So you can plot in this way

• Trends, anomalies, seasonality and even spatial
  can be seen

Alonso et al 2011 Spatio-temporal patterns of
diarrhoeal mortality in Mexico. Epidemiol. Infect
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We can use this display to see the shift in the
timing of RSV circulation in São Paulo city and
its implications for immunoprophylaxis
period of palivizumabe prophylaxis
Paiva et al 2012 JMV
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And then we can use a different plot for
displaying the epidemiologic and putative
explanatory series
Paiva et al 2012 JMV
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In fact, sometimes a simple organization of data
in space can generate all the information we
need!
This is a quick example on how we found that
(surprisingly!) the Northern hemisphere timing of
the vaccine would be more efficient than the
current Southern timing for Brazil Mello et al
2010 PLoS One
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Influenza virus isolated plotted exactly in their
time of collection
Mello et al (2010)
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Now we overlap the Southern and Northern
Hemisphere recommendations
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And count first the matches obtained with the
Southern Hemisphere recommendation
11 matches
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And compare with the matches if the Northern
Hemisphere timing of the vaccine and composition
were applied
24 matches!
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Part 1 How to extract the basic components of
epidemiological relevance from a time-series?
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Brazilian dataset of deaths coded as pneumonia
and influenza

• We are going to extract as much information as
  possible from this series

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Brazilian dataset of deaths coded as pneumonia
and influenza

• Example of analyses performed in Schuck-Paim et
  al 2012 Were equatorial regions less affected by
  the 2009 influenza pandemic? The Brazilian
  experience. PLoS One.
• Data source Department of Vital Statistics from
  the Brazilian Ministry of Health

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Series to be analyzed

• Typical epidemiological time series from where to
  obtain as many meaningful and useful parameters
  as possible

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Average
mortality at time t

• Many times this information is
• all we need!

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Average

• But, it still leaves much of the variation
  (residuals) of the series unexplained
• the first of which seems to be an unbalanced
  between the extremities

mortality at time t
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Linear trend

• Better now!

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Trend (linear)
Mortality at time t

• We can use this information
• (e.g. is the disease increasing/decreasing? -
  but then the data needs to be incidence)

Mean Mortality
Linear trends
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Trend (with quadratic term too)

• Better definition
• It gets more complicated as a parameter to be
  compared across time-series
• But better if our purpose is eliminate the
  temporal trend

Mortality at time t
Quadratic trends
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Getting rid of the trend

• Blue line detrended series

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But lets keep the graphic of the original series
for illustrative purposes

• Clearly, there are still other interesting
  epidemiological patterns to describe

Mortality at time t
Mean Mortality
Linear and quadratic trends
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We can see some rhythm

• The block of residuals alternates cyclically
• Therefore this is something that can be
  quantified using few parameters

Mortality at time t
Mean Mortality
Linear and quadratic trends
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Jean Baptiste Joseph Fourier(1768 1830)
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The Fourier theorem states that any waveform can
be duplicated by the superposition of a series of
sine and cosine waves

• As an example, the following Fourier expansion
  of sine waves provides an approximation of a
  square wave
• Source http//www.files.chem.vt.edu/chem-ed/data/
  fourier.html

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Fourier decomposition

• the periodic variability of the monthly mortality
  time-series is partitioned into harmonic
  functions.
• By summing the harmonics we obtain what can be
  considered as an average seasonal signature of
  the original series, where year-to-year
  variations are removed but seasonal variations
  within the year are preserved
• This method is not always appropriate when
  dealing with complex population time series,
  since it cannot take into account the
  often-observed changes in the periodic behavior
  of such series (i.e., they are not stationary).

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Before modeling cycles
Mortality at time t

• so, remembering, these are the residuals before
  Fourier

Mean Mortality
Linear and quadratic trends
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and now with the incorporation of the annual
harmonic
Mortality at time t
Annual harmonic
Mean Mortality
trends
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or with the semi-annual harmonic only?
Mortality at time t
semiannual harmonic
Mean Mortality
trends
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Much better when the annual semi-annual
harmonics are considered together!
Mortality at time t
Annual and semi-annual harmonics
Mean Mortality
trends
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Although not much difference when the quarterly
harmonic is added
Mortality at time t
Periodic (seasonal) components
Mean Mortality
trends
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average seasonal signature of the original series

• We obtained therefore the average seasonal
  signature of the original series (where
  year-to-year variations are removed but seasonal
  variations within the year are preserved)
• Now, lets extract some interest parameters
  (remember, we always need a number to compare,
  for instance, across different sites)

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Timing and Amplitude
average seasonal signature of the original series
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Variations in relative peak amplitude of
pneumonia and influenza coded deaths with
latitude Alonso et al 2007 Seasonality of
influenza in Brazil a traveling wave from the
Amazon to the subtropics. Am J Epidemiol
Latitude (degrees)
(p lt 0.001)
0 10 20 30 40 50 60
70 80 90
Amplitude of the major peak ()
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The seasonal component was found to be most
intense in southern states, gradually attenuating
towards central states (15oS) and remained low
near the Equator
Latitude (degrees)
(p lt 0.001)
0 10 20 30 40 50 60
70 80 90
Amplitude of the major peak ()
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Variations in peak timing of influenza with
latitude
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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Peak timing was found to be structured
spatio-temporally annual peaks were earlier in
the north, and gradually later towards the south
of Brazil
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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Such results suggest southward waves of influenza
across Brazil, originating from equatorial and
low population regions and moving towards
temperate and highly populous regions in 3
months.
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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But can we still improve the model?
Yes, and in some cases we should, Mostly to model
excess estimates
e.g. pandemic year
Mortality at time t
Periodic (seasonal) components
Mean Mortality
trends
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Residuals after excluding atypical (i.e.
pandemic) years from the model

• To define what is normal it is necessary to
  exclude the year that we suspect might be
  abnormal from the model

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Ok, so now we can count what was the impact of
the pandemic here right?
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No! (unless you consider all the other anomalies
pandemics (and anti-pandemics)

• That is why we need to include usual residual
  variance in the model, and calculate excess
  BEYOND usual variation

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Residuals after modeling year to year variance
(1.96 SD above model)
Mortality at time t
Periodic (seasonal) components
error term
Mean Mortality
trends
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This is a measure of excess that is much closer
to the real impact of the pandemic
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Geographical patterns in the severity of pandemic
mortality in a large latitudinal range
Schuck-Paim et al 2012 PLoS One
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Program available at www.epipoi.info
Paper explaing the program available at
http//www.biomedcentral.com/content/pdf/1471-2458
-12-982.pdf
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Example from diarrhea mortality in Mexico
(1979-1988)
Alonso WJ et al Spatio-temporal patterns of
diarrhoeal mortality in Mexico. Epidemiol Infect
2011 Apr1-9.
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quantitative and qualitative change of diarrhea
in Mexico 1917-2001
Summer peaks
Winter peaks
Gutierrez et al. Impact of oral rehydration and
selected public health interventions on reduction
of mortality from childhood diarrhoeal diseases
in Mexico. Bulletin of the WHO 1996 Velazquez et
al. Diarrhea morbidity and mortality in Mexican
children impact of rotavirus disease. Pediatric
Infectious Disease Journal 2004 Villa et al.
Seasonal diarrhoeal mortality among Mexican
children. Bulletin of the WHO 1999
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State-specific rates, sorted by the latitude of
their capitals, from north to south (y axis)
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Timing of annual peaks (1979-1988)
First peak in the Mexican capital !
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Major Annual Peaks of diarrhea of the period
1979-1988 in Mexican states, sorted by their
latitude
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Climatologic factors
Monthly climatic data were obtained from
worldwide climate maps generated by the
interpolation of climatic information from
ground-based meteorological stations
Mitchell TD, Jones PD. An improved method of
constructing a database of monthly climate
observations and associated high-resolution
grids. International Journal of Climatology
200525693-712. (data at http//www.cru.uea.ac.
uk/cru/data/hrg/)
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Early peaks in spring in the central region of
Mexico (where most of the people lives) followed
by a decrease in summer
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Early peaks in the monthly average maximum
temperature in the central region of Mexico
followed by a decrease in summer too !
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The same climatic factors that enabled a dense
and ancient human occupation in the central part
of Mexico prevent a strong presence of bacterial
diarrhea and the observed early peaks
Mild summers - with average maximum temperatures
below 24 oC
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Thanks! alonsow_at_mail.nih.gov