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

Wladimir J. AlonsoFogarty International Center / NIH

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)

But before this…

• 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

A parenthesis for “Graphical Excellence”

Edward Tufte (1983)

Napoleon's Retreat from Moscow, 1812by Illarion Pryanishnikov

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)

First: Organize your dataset in a

meaningful way

A typical mortality dataset

Time in a meaningful sequence

Variables in meaningful sequence

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

Structured spreadsheet as a source of instantaneous analysis

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

We can use this display to see the shift in the timing of RSV circulation in São Paulo city and its implications for

immunoprophylaxis

Paiva et al 2012 JMVperiod of palivizumabe prophylaxis

And then we can use a different plot for displaying the epidemiologic and putative explanatory series

Paiva et al 2012 JMV

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

Influenza virus isolated plotted exactly in their time of collection

Mello et al (2010)

Belém

São Paulo

influenza viruses isolated monthly from 1999 to 2007 in Belém and São Paulo

Now we overlap the Southern and Northern Hemisphere recommendations

And count first the matches obtained with the Southern Hemisphere recommendation…

11 matches

And compare with the matches if the Northern Hemisphere timing of the vaccine and composition were applied

24 matches!

Part 1: How to extract the basic components

of epidemiological relevance from a time-series?

Brazilian dataset of deaths coded as pneumonia and influenza

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

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

Series to be analyzed

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

Average

Many times this information is

all we need!

mortality at time t

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

Linear trend

• Better now!

Trend (linear)

We can use this information (e.g. is the disease increasing/decreasing? -

but then the data needs to be incidence)

Mortality at time t

Linear trendsMean Mortality

Trend (with quadratic term too)

Mortality at time t

Quadratic trends

2

210ttY

t

• 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

Getting rid of the trend

Blue line: “detrended series”

But let’s keep the graphic of the original series for illustrative purposes

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

Mortality at time t

Linear and quadratic trends

2

210ttY

t

Mean Mortality

We can see some rhythm…

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

Linear and quadratic trends

2

210ttY

t

Mean Mortality

Mortality at time t

Jean Baptiste Joseph Fourier(1768 –1830)

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

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”).

Before modeling cycles:

…so, remembering, these are the residuals before Fourier

Linear and quadratic trends

2

210ttY

t

Mean Mortality

Mortality at time t

… and now with the incorporation of the annual harmonic

Mortality at time t

trends

Annual harmonicMean

Mortality

or with the semi-annual harmonic only?

Mortality at time t

trends

semiannual harmonicMean

Mortality

Much better when the annual + semi-annual harmonics are considered together!

Mortality at time t

trends

Annual and semi-annual harmonicsMean

Mortality

Although not much difference when the quarterly harmonic is added…

Mortality at time t

trends

Periodic (seasonal) componentsMean

Mortality

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, let’s extract some interest parameters (remember, we always need a “number” to compare, for instance, across different sites)

Timing and Amplitude

average seasonal signature of the original

series

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

Latit

ude

(deg

rees

)

5

0

-5

-10

-15

-20

-25

-30

-35

Amplitude of the major peak (%)0 10 20 30 40 50 60 70 80 90

(p < 0.001)

The seasonal component was found to be most intense in southern states, gradually attenuating towards central states (15oS) and remained low near the Equator

Latit

ude

(deg

rees

)

5

0

-5

-10

-15

-20

-25

-30

-35

Amplitude of the major peak (%)0 10 20 30 40 50 60 70 80 90

(p < 0.001)

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

Variations in peak timing of influenza with latitude

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

Peak timing was found to be structured spatio-temporally: annual peaks were earlier in the north, and gradually later

towards the south of Brazil

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

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.

But can we still improve the model?

Mortality at time t

trends

Periodic (seasonal) componentsMean

Mortality

Yes, and in some cases we should,Mostly to model excess estimates

e.g. pandemic year

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

Ok, so now we can count what was the impact of the pandemic here right?

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

Residuals after modeling year to year variance

(1.96 SD above model)

Mortality at time t

trends

Periodic (seasonal) components error term

Mean Mortality

)()3

2sin()

3

2cos()

6

2sin()

6

2cos()

12

2sin()

12

2cos( 332211

2210 t

ttttttttYt

This is a measure of excess that is much closer to the real impact of the pandemic

Geographical patterns in the severity of pandemic mortality in a large latitudinal range

Schuck-Paim et al 2012 PLoS One

Program available at www.epipoi.info

Paper explaing the program available at http://www.biomedcentral.com/content/pdf/1471-2458-

12-982.pdf

Example from diarrhea mortality in Mexico (1979-1988)

Alonso WJ et al Spatio-temporal patterns of diarrhoeal mortality in Mexico. Epidemiol Infect 2011 Apr;1-9.

quantitative and qualitative change of diarrhea in Mexico 1917-2001

Winter peaks

Summer 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

State-specific rates, sorted by the latitude of their capitals, from north to south (y axis)

Timing of annual peaks (1979-1988)

First peak in the Mexican capital !

Major Annual Peaks of diarrhea of the period 1979-1988 in Mexican states, sorted by their latitude

Monthly climatic data were obtained from worldwide climate maps generated by the interpolation of climatic information from ground-based meteorological stations

Climatologic factors

Mitchell TD, Jones PD. An improved method of constructing a database of monthly climate observations and associated high-resolution grids. International Journal of Climatology 2005;25:693-712. (data at: http://www.cru.uea.ac.uk/cru/data/hrg/)

Early peaks in spring in the central region of Mexico (where most of the people lives) followed by a decrease in summer

Early peaks in the monthly average maximum temperature in the central region of Mexico followed by a decrease in summer too !

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

Thanks! alonsow@mail.nih.gov