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'Trends, time series and forecasting
Paul FryersEast Midlands KIT
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Overview
Introduction
Understanding trends and time series – Seasonality– Transformations
Methods for analysing time series – Regression– Moving averages– Autocorrelation
Overview of forecasting
Forecasting methods – Extrapolation of regression – Holt’s method
Uses for forecasting – Setting and monitoring targets – Estimating current values
General methodological points
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What is a time series?
A set of well defined measures collected through time:– Mortality– Diagnoses– Temperature– Rainfall– Share price– Sunspots– Ice cream sales– Air passengers– Road accidents
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What is special about time series data?
There is an implicit order to the data with a first, second, third,..., nth value
Previous observations may be important determinants of later observations– this has implications for analysis
Trend and/or seasonal effects may be present– a trend is a tendency for observations to fall or rise over time– seasonal effects are regular repeating patterns of rises or falls
Different techniques are needed for analysis of historical data and for producing forecasts
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Continuous time : electrocardiogram trace
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Monthly emphysema deaths
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Understanding trends and time series
First plot the data Is the time series consistent?
– Look for step changes in level or trend Is there any visual evidence of any pattern or trend? Is there evidence of a regular ‘seasonal’ pattern? If there is a trend, is it linear? (probably not!)
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Is the time series consistent? – change in trendIn
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Time
Graph of an indicator over time demonstrating aclear change in trend
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Is the time series consistent? – step changes
Number of recorded 'violence against the person' crimes in England & Wales1980- 2006/07
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Introduction of the National Crime Recording Standard (NCRS)
Change to the Home Office counting rules for recorded crime
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Handling inconsistency
Usually, we will simply break the time series at the point where the trend changes, or the step change occurs
Analyse only the data since that point, or analyse the different parts of the time series separately
Or use a method/software that will do that automatically, eg by weighting more recent points more heavily
We may be able to adjust or transform the data prior to a step change but only if we understand the reason for the change and are confident that the adjustment makes the data consistent
– eg adjusting for a coding change (ICD coding, definition of unemployment, etc.)
But it’s not always clear cut...
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Is the time series consistent? – step changes?
Mortality rate per 100,000 population-years at risk
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Is the time series consistent? – outlier
Mortality rate per 100,000 population-years at risk
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Handling outliers
Normally, we ignore outliers, ie exclude them from the analysis– this can be a nuisance for some analyses
But again, it’s not always clear cut:– we need to identify plausible reasons for the outlier/s (eg known issues
with data collection, or a specific factor that has influenced the outcome)
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Is there any visual evidence of any pattern or trend?In
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Time
Graph of an indicator over time demonstrating asteady increasing trend
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Is there any visual evidence of any pattern or trend?In
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Time
Graph of an indicator over time demonstrating anincreasing trend with greater variability
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Is there any visual evidence of any pattern or trend?In
dica
tor V
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Time
Graph of an indicator over time demonstrating ansteady decreasing trend
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Is there any visual evidence of any pattern or trend?In
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alue
Time
Graph of an indicator over time demonstratingno change over time
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Is there any visual evidence of any pattern or trend?In
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tor V
alue
Time
Graph of an indicator over time demonstrating aseasonal pattern
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Is there any visual evidence of any pattern or trend?
Mortality rate per 100,000 population-years at risk
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Graph of an indicator showing a seasonal pattern plus rising trend
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Handling seasonality
Seasonality can be additive or multiplicative– ie each different period in the cycle has an extra factor added to (or
subtracted from) or multiplied by the overall average level We can adjust the data by applying the inverse factor to each period Easier to use an integrated method that adjusts for the seasonality within
the analysis
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Is the trend linear?In
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Time
Graph of an indicator over time demonstrating ansteady decreasing trend
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Indi
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Time
Graph of an indicator over time demonstrating ansteady decreasing trend
Example of a falling rate – straight line
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Indi
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Graph of an indicator over time demonstrating ansteady decreasing trend
Example of a falling rate – exponential curve
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Transformations – non-linear trends
In many cases, it is meaningless for the forecasts to fall below zero In public health we are most commonly dealing with counts, rates or
proportions We routinely transform the data in order to ‘make the data linear’ and
constrain them to be no less than zero By default, we should use a log-transformation for counts or rates, fitting
an exponential curve which assumes a constant rate of change, rather than a constant numerical increase or decrease
We should use a logit-transformation for proportions (or percentages), which constrains the variable to be between 0 and 1 (or 0% and 100%)
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Transformations – falling exponential curve
A rapidly falling trend The indicator looks to be heading rapidly towards zero, but the log
transformation ensures that it stays positive: the rate or count is ‘tending towards’ zero but can never quite get there
It represents a constant rate of change (i.e. reducing by x% each year rather than reducing by a set amount each
year) This should be the
default option for analysis of counts or rates
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Graph of an indicator over time demonstrating aconstant rate of decrease (exponential curve)
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Transformations – rising exponential curve
A rapidly increasing trend For a count or rate, mathematically it is preferable to use an exponential
curve, but need to beware of other practical constraints: there will usually be some practical limit to a count or rate
If the continued rise in the count or rate is implausible then it is better to use a linear model or logit...
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Transformations – log-transform counts and rates
Fitting an exponential curve:
Equation of curve: ln(y) = ln(a) + ln(b)t or y = a × bt
where y = value of variable being studieda = intercept on y-axis (nominal value of
indicator at time 0)t = time valueb = ‘gradient’ (amount y is multiplied by
for each increase of 1 in time)
ln(0) = –∞ln(∞) = ∞
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Transformations – logistic curve
Proportions can not go below zero or above 1 The tails are equivalent: e.g. proportion surviving = 1 – proportion dying Particularly important for proportions that span a large range, from
under 0.5 to nearly 1, e.g. percentage achievement on QOF scores For proportions or
percentages close to zero, the logit is equivalent to the log
For proportions always close to 1 could subtract from 1 and use log
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Transformations – logit-transform proportions
The logit function: logit(y) = ln(y/(1–y)) = ln(y) – ln(1–y)
logit(0) = –∞logit(½) = 0logit(1) = ∞
We transform proportions by applying the logit function, then fit a regression line to the transformed data
For rates or counts which have a practical limit, if we have a sound basis for estimating that realistic maximum then we could do so and treat the rate or count as a proportion of that upper limit
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Methods for analysing time series
Regression– Most common method: simply fit a line or curve to the data, treating
‘time’ as any other explanatory variable– Gives equal weight to all points in the time series– Assumes points are independent, identically distributed, observations– Gradient has confidence intervals: if CIs don’t include zero, the gradient is
signicant Two other concepts that are used as the basis for analysing time series:– Moving average– Autocorrelation
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Linear regression
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Confidence intervals for the gradient
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Moving average
Familiar as a method of presenting data– For annual data, rather than presenting data for 2004, 2005, 2006, 2007
and 2008, we may present three-year figures: 2004-06, 2005-07 and 2006-08
– Smoothes out fluctuations in the data, making trends easier to see Also called ‘rolling averages’ Moving averages of different periods can be used to highlight different
features of a time series (example follows) BUT!!!
Moving averages must not be used as the basis for regression, time series analysis or forecasting as they are not independent observations (they share their data with their neighbours)
[Note: time series methods such as Holt’s Method and Box-Jenkins (ARIMA) models use moving averages within the analysis, but the data from which the model is derived should not be moving averages]
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Monthly emphysema deaths
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3-point moving average, highlighting seasonality
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13-point moving average, highlighting trend
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Autocorrelation
In time series, observations can often be predicted by combinations of previous observations
If the observations are correlated with their immediate predecessors, we can calculate the Pearson correlation coefficient between them
This is called autocorrelation of lag 1 Observations can also be correlated with predecessors from further back
in the time series – autocorrelation of lag k (where k is number of observations back in the series)
– In time series, observations can be predicted by combinations of previous observations
– Smoothes out fluctuations in the data, making trends easier to see
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Forecasting
Why do we need to forecast? Extrapolating Forecasting methods
– Examples– Holt’s Method– Interval forecasts– How far back and how far forward?
Using forecasts– to set and monitor progress against targets– to estimate current health outcomes/indicators
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Why do we need to forecast?
To inform planning by estimating future needs– the health of the population tends to change slowly and react slowly
to public health interventions so we need to look ahead To anticipate future major events
– e.g. outbreaks To set and monitor progress against targets
– where are we likely to be on current progress?– are we on track to meet targets?
To estimate current health outcomes– our most recent data tend to be a year or more out of date so if we
want to know where we are now or even where we were last year we have to forecast
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Forecasting from past trends
If we have time series for a health outcome, health service output indicator or risk factor, we can use this to forecast future values
eg:– mortality rates– teenage pregnancy rates– hospital activity rates– prevalence estimates
Assumes:– consistent definitions and measurement, past and future– either that nothing significant changes, or that changes/
improvements continue at the same rate
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Extrapolating from regression lines
A common method is to fit a regression line (or curve) to the historic data and extrapolate it to the future
This is OK for a short time into the future as long as the historic data are stable, ie changing at a steady rate
But:The regression line is fitted across the whole of the historic data, and gives equal weight to all points: e.g. the value for last year is given the same weight as one from 20 years ago – it doesn’t give the best estimate of ‘current trends’
We cannot give realistic confidence intervals for future values (‘prediction intervals’ or ‘forecast intervals’)
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Forecasting methods
There is a range of methods which are intended for forecasting, eg moving average methods, autocorrelation methods, Box-Jenkins methods
These methods take into account fluctuations from year to year, trends (ie gradual changes over time) and seasonal variations
They tend to give greater weight to more recent values, hence ‘start from where we are’
They give confidence intervals for forecasts, which tend to get wider as we move further into the future
The most useful methods for public health applications tend to be Holt’s Method (which includes a trend component) and Holt-Winters (which adds a seasonal component)
Note, as with regression analysis, the points in the time series must be independent of each other: rolling averages must never be used for forecasting
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Teenage conceptions – England
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Observed Rate Target Forecast Rate Forecast Confidence Interval
http://www.empho.org.uk/pages/viewResource.aspx?id=11285
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Teenage conceptions – London GOR
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Observed Rate Target Forecast Rate Forecast Confidence Interval
http://www.empho.org.uk/pages/viewResource.aspx?id=11285
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Teenage conceptions – Newham
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Teenage conceptions – Harrow
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http://www.empho.org.uk/pages/viewResource.aspx?id=11285
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Alcohol-related admission rates – Bassetlaw PCT
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Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
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Fractured neck of femur admission rates – Nottinghamshire PCT
Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
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Emergency admission rates for acute coronary syndrome – East Midlands – males
Report to East Midlands Cardiac & Stroke Network
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Emergency admission rates for acute coronary syndrome – East Midlands – females
Report to East Midlands Cardiac & Stroke Network
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Holt’s Method
Holt’s exponential smoothing (aka double exponential smoothing) is a moving average method
There are two equations involved in fitting the model:Lt = xt + (1–)(Lt–1 + Tt–1)
Tt = (Lt–Lt–1) + (1–)Tt–1
where xt is the observed value at time t
Lt is the forecast at time t (the ‘level’ parameter)
Tt is the estimated slope at time t (the ‘trend’ parameter)
is the first smoothing constant, used to smooth the level is the second smoothing constant, used to smooth the trend
The model is fitted iteratively from the start of the time series, usually setting L1 initially to x1 and T1 to x2 – x1
A software package optimises the constants and such that the squared differences between the observed values and the forecasts are minimised
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Holt’s Method in practice
Several statistical packages will do this:– ForecastPro – not free but very easy to use– Stata – not free and needs code, but PHE has a corporate licence– R – open source software which requires code– Excel – you can put the equations into Excel but have to optimise the
parameters manually If you use Stata, R or Excel, you need to put some effort into optimising
the parameters, which requires some expertise and time ForecastPro has very clever optimisation routines, which always seem to
result in sensible forecasts and forecast intervals BUT!!!
Every forecast should be graphed and checked – even the most expert of automated ‘expert systems’ cannot and should not be totally relied on
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Interval forecasts not point forecasts
When we forecast the future we give a single figure for each forecast that is our best estimate of the future value
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Interval forecasts not point forecasts
When we forecast the future we give a single figure for each forecast that is our best estimate of the future value
However, of course there is uncertainty about that prediction Forecast intervals give an indication of the degree of uncertainty, and are
far more valuable than the actual point forecasts
These forecast intervals are calculated by the forecasting software
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80
1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Rate
per
100
,000
resi
dent
pop
ulati
on
Year
Teenage Conception Rates, 1992-2012East Midlands
Observed values Forecasts 95% Forecast Interval Target
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How far back and how far forward?
As discussed earlier, if the graph shows a distinct change in trend or step change then we should ignore the data before the current trend
– If we use Holt’s Method or similar, it is less critical because the method tends to give more weight to recent data and largely ignores earlier points, but if it is clear from the graph, it is still wise to use only the data which exhibit the current trend
– If the change is very recent, then we probably don’t have a sound basis for forecasting – this would be reflected in the forecast intervals (covered later)
How far ahead can we forecast – the ‘forecast horizon’?– A rule of thumb is quoted, that you can forecast around half as far forwards
as you have data going back, however it depends on the stability of the series, and common sense should be applied
– The question is less critical if you present forecast intervals: these will become extremely wide as you get further into the future, demonstrating that the forecasts are meaningless
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Using forecasting to estimate ‘current’ rates
By ‘current’ we normally mean ‘the average of the last three years for which data are available’
For deaths, the ‘current’ values, used for analysing our current mortality rates, for example, are based on 2010-2012 data, i.e. data from between 4½ and 1½ years ago: on average 3 years out of date
For small areas, even with 3 years’ data, we still have very few deaths or cases to work with and hence our baseline can be pretty arbitrary
We may be able to use forecasting methodology to help with both of these problems:If we forecast 2014 values based on a time series from 2000 to 2012 then we have a) a more robust baseline, based on 13 years’ data not 3
b) a baseline which reflects ‘now’ rather than 3 years ago Forecasts of ‘current’ periods can give us robust ‘underlying’ values or
rates
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0
20
40
60
80
100
120
140
160
180
200
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Rate
per
100
,000
resi
dent
pop
ulati
on
Year
Circulatory Disease Mortality Rates, 1993-2010Doncaster and England
England Rate Doncaster Rate Forecast95% Forecast Interval Upper Limit England Forecast'Current' Values
Example – rapidly changing rates
Circulatory disease death rates are falling dramatically 2004-06 average rate was 91 deaths per 100,000 population-years 2008 forecast was 74 In 2008, by taking the average of 2004-06 as our ‘current’ rate we were
potentially overestimating the impact of interventions by 23%
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Summary – key points
Look at a graph of the data, and think about the data you are working with, considering whether there are reasons why past trends may not be a sound basis for future changes
Decide how far back you should start Transform data to ensure that the shape of the graph and any logical
limits on variability (e.g. >0, <100%) are reflected in the mathematical assumptions
Use regression to analyse past changes Use forecasting methods such as Holt’s Method (or Holt-Winters for
seasonal data) to make predictions of future rates with realistic forecast intervals
Ensure that data are independent of one another: no rolling averages Always graph the results, to ensure that the maths hasn’t had an off day