1. 'Trends, time series and forecasting
Paul Fryers
East Midlands KIT

2. 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

3. 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

4. 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

5. Continuous time : electrocardiogram trace

6. Monthly emphysema deaths

7. 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!)

8. Is the time series consistent? – change in trend

9. Is the time series consistent? – step changes

10. 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...

11. Is the time series consistent? – step changes?

12. Is the time series consistent? – outlier

13. 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)

14. Is there any visual evidence of any pattern or trend?

15. Is there any visual evidence of any pattern or trend?

16. Is there any visual evidence of any pattern or trend?

17. Is there any visual evidence of any pattern or trend?

18. Is there any visual evidence of any pattern or trend?

19. Graph of an indicator showing a seasonal pattern plus rising trend
Is there any visual evidence of any pattern or trend?
Graph of an indicator showing a seasonal pattern plus rising trend

20. 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

21. Is the trend linear?

22. Example of a falling rate – straight line

23. Example of a falling rate – exponential curve

24. 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%)

25. 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

26. 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...

27. 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 studied
a = intercept on y-axis (nominal value of indicator at time 0)
t = time value
b = ‘gradient’ (amount y is multiplied by for each increase of 1 in time)
ln(0) = –∞
ln(∞) = ∞

28. 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 could subtract from 1 and use log

29. Transformations – logit-transform proportions
The logit function: logit(y) = ln(y/(1–y)) = ln(y) – ln(1–y)
logit(0) = –∞
logit(½) = 0
logit(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

30. 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

31. Linear regression

32. Confidence intervals for the gradient

33. Moving average Familiar as a method of presenting data
For annual data, rather than presenting data for 2004, 2005, 2006, and 2008, we may present three-year figures: , and
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]

34. Monthly emphysema deaths

35. 3-point moving average, highlighting seasonality

36. 13-point moving average, highlighting trend

37. 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

38. 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

39. 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

40. 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

41. 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’)

42. 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

43. Teenage conceptions – England

44. Teenage conceptions – London GOR

45. Teenage conceptions – Newham

46. Teenage conceptions – Harrow

47. Alcohol-related admission rates – Bassetlaw PCT
Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories

48. Stroke mortality rates – Nottinghamshire PCT
Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories

49. Fractured neck of femur admission rates – Nottinghamshire PCT
Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories

50. Deaths occurring at home – Bassetlaw PCT
Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories

51. Emergency admission rates for stroke/TIA – East Midlands – males
Report to East Midlands Cardiac & Stroke Network

52. Emergency admission rates for acute coronary syndrome – East Midlands – males
Report to East Midlands Cardiac & Stroke Network

53. Emergency admission rates for acute coronary syndrome – East Midlands – females
Report to East Midlands Cardiac & Stroke Network

54. 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 = axt + (1–a)(Lt–1 + Tt–1)
Tt = g(Lt–Lt–1) + (1–g)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)
a is the first smoothing constant, used to smooth the level
g 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 a and g such that the squared differences between the observed values and the forecasts are minimised

55. 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

56. 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

57. 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

58. 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

59. Using forecasts to set and monitor targets – 1
Our Healthier Nation set targets to reduce circulatory disease death rates by 40% between and 2010
This seemed reasonable at the time
However, if everywhere reduces circulatory disease by 40%, the gap between affluent and deprived parts of the country remains the same (e.g. Doncaster’s SMR will be the same in 2010 as it was in 1997)

60. Using forecasts to set and monitor targets – 2
In 2004, there was a view that the OHN targets were being achieved more easily in more affluent areas
New Spearhead targets were set to reduce the gap between the Spearhead group of local authorities and the national average by 40% between and
In fact rates were dropping much faster than the OHN targets required

61. Using forecasts to set and monitor targets – 3
Spearhead targets are set relative to national rates so they have to be updated annually, taking into account national forecasts
We have to forecast the England rate in 2010 (current forecast: 61 deaths per 100,000 population)

62. Using forecasts to set and monitor targets – 4
Spearhead targets are set relative to national rates so they have to be updated annually, taking into account national forecasts
We have to forecast the England rate in 2010 (current forecast: 61 deaths per 100,000 population)
Then set the local Spearhead target to give the required % narrowing of the gap (target is 65 deaths per ,000)

63. Using forecasts to set and monitor targets – 5
Spearhead targets are set relative to national rates so they have to be updated annually, taking into account national forecasts
We have to forecast the England rate in 2010 (current forecast: 61 deaths per 100,000 population)
Then set the local Spearhead target to give the required % narrowing of the gap (target is 65 deaths per ,000)
We can forecast the local rate to assess whether we’re on target

64. Within district target
In Doncaster’s deprived communities had an all age all cause mortality rate 16.7% greater than Doncaster as a whole
To reduce the gap by 10% it must only be 15.0% above the Doncaster average by 2010
Doncaster forecast to be 713 deaths per ,000 person years in 2010
Target for the deprived quintile is 821
This represents a reduction in death rates of 18%

65. Within district target
Using forecasting with realistic forecast intervals often serves to demonstrate the impossibility of measuring outcomes at PCT/local authority level or below
It is essential always to plot as long a time series as possible to illustrate the wide year-on-year fluctuations
Forecast intervals, even for the next year, are very broad

66. Within district target
Clearer to graph the annual excess mortality as a percentage
Is this measurable?
One weakness of this method is that it only uses the deprived quintile – it ignores most of the distribution

67. 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 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

68. Example – rapidly changing rates
Circulatory disease death rates are falling dramatically
average rate was 91 deaths per 100,000 population-years
2008 forecast was 74
In 2008, by taking the average of as our ‘current’ rate we were potentially overestimating the impact of interventions by 23%

69. 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

70. Contact
Paul Fryers