Presentation on theme: "'Trends, time series and forecasting"— Presentation transcript:
1'Trends, time series and forecasting Paul FryersEast Midlands KIT
2Overview Introduction Understanding trends and time series – Seasonality– TransformationsMethods for analysing time series – Regression– Moving averages– AutocorrelationOverview of forecastingForecasting methods – Extrapolation of regression– Holt’s methodUses for forecasting – Setting and monitoring targets– Estimating current valuesGeneral methodological points
3What is a time series?A set of well defined measures collected through time:MortalityDiagnosesTemperatureRainfallShare priceSunspotsIce cream salesAir passengersRoad accidents
4What is special about time series data? There is an implicit order to the data with a first, second, third,..., nth valuePrevious observations may be important determinants of later observationsthis has implications for analysisTrend and/or seasonal effects may be presenta trend is a tendency for observations to fall or rise over timeseasonal effects are regular repeating patterns of rises or fallsDifferent techniques are needed for analysis of historical data and for producing forecasts
7Understanding trends and time series First plot the dataIs the time series consistent?Look for step changes in level or trendIs 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!)
10Handling inconsistency Usually, we will simply break the time series at the point where the trend changes, or the step change occursAnalyse only the data since that point, or analyse the different parts of the time series separatelyOr use a method/software that will do that automatically, eg by weighting more recent points more heavilyWe 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 consistenteg adjusting for a coding change (ICD coding, definition of unemployment, etc.)But it’s not always clear cut...
13Handling outliersNormally, we ignore outliers, ie exclude them from the analysisthis can be a nuisance for some analysesBut 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)
14Is there any visual evidence of any pattern or trend?
15Is there any visual evidence of any pattern or trend?
16Is there any visual evidence of any pattern or trend?
17Is there any visual evidence of any pattern or trend?
18Is there any visual evidence of any pattern or trend?
19Graph 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
20Handling 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 levelWe can adjust the data by applying the inverse factor to each periodEasier to use an integrated method that adjusts for the seasonality within the analysis
24Transformations – non-linear trends In many cases, it is meaningless for the forecasts to fall below zeroIn public health we are most commonly dealing with counts, rates or proportionsWe routinely transform the data in order to ‘make the data linear’ and constrain them to be no less than zeroBy 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 decreaseWe should use a logit-transformation for proportions (or percentages), which constrains the variable to be between 0 and 1 (or 0% and 100%)
25Transformations – falling exponential curve A rapidly falling trendThe 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 thereIt 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
26Transformations – rising exponential curve A rapidly increasing trendFor 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 rateIf the continued rise in the count or rate is implausible then it is better to use a linear model or logit...
27Transformations – log-transform counts and rates Fitting an exponential curve:Equation of curve: ln(y) = ln(a) + ln(b)t or y = a × btwhere 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(∞) = ∞
28Transformations – logistic curve Proportions can not go below zero or above 1The tails are equivalent: e.g. proportion surviving = 1 – proportion dyingParticularly important for proportions that span a large range, from under 0.5 to nearly 1, e.g. percentage achievement on QOF scoresFor proportions or percentages close to zero, the logit is equivalent to the logFor proportions always close to could subtract from 1 and use log
29Transformations – 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 dataFor 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
30Methods for analysing time series RegressionMost common method: simply fit a line or curve to the data, treating ‘time’ as any other explanatory variableGives equal weight to all points in the time seriesAssumes points are independent, identically distributed, observationsGradient has confidence intervals: if CIs don’t include zero, the gradient is signicantTwo other concepts that are used as the basis for analysing time series:Moving averageAutocorrelation
33Moving 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: , andSmoothes out fluctuations in the data, making trends easier to seeAlso 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]
37AutocorrelationIn time series, observations can often be predicted by combinations of previous observationsIf the observations are correlated with their immediate predecessors, we can calculate the Pearson correlation coefficient between themThis is called autocorrelation of lag 1Observations 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 observationsSmoothes out fluctuations in the data, making trends easier to see
38Forecasting Why do we need to forecast? Extrapolating Forecasting methodsExamplesHolt’s MethodInterval forecastsHow far back and how far forward?Using forecaststo set and monitor progress against targetsto estimate current health outcomes/indicators
39Why do we need to forecast? To inform planning by estimating future needsthe health of the population tends to change slowly and react slowly to public health interventions so we need to look aheadTo anticipate future major eventse.g. outbreaksTo set and monitor progress against targetswhere are we likely to be on current progress?are we on track to meet targets?To estimate current health outcomesour 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
40Forecasting 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 valueseg:mortality ratesteenage pregnancy rateshospital activity ratesprevalence estimatesAssumes:consistent definitions and measurement, past and futureeither that nothing significant changes, or that changes/ improvements continue at the same rate
41Extrapolating from regression lines A common method is to fit a regression line (or curve) to the historic data and extrapolate it to the futureThis is OK for a short time into the future as long as the historic data are stable, ie changing at a steady rateBut: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’)
42Forecasting methodsThere is a range of methods which are intended for forecasting, eg moving average methods, autocorrelation methods, Box-Jenkins methodsThese methods take into account fluctuations from year to year, trends (ie gradual changes over time) and seasonal variationsThey 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 futureThe 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
47Alcohol-related admission rates – Bassetlaw PCT Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
48Stroke mortality rates – Nottinghamshire PCT Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
49Fractured neck of femur admission rates – Nottinghamshire PCT Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
50Deaths occurring at home – Bassetlaw PCT Data provided to Nottinghamshire and Bassetlaw PCTs for WCC trajectories
51Emergency admission rates for stroke/TIA – East Midlands – males Report to East Midlands Cardiac & Stroke Network
52Emergency admission rates for acute coronary syndrome – East Midlands – males Report to East Midlands Cardiac & Stroke Network
53Emergency admission rates for acute coronary syndrome – East Midlands – females Report to East Midlands Cardiac & Stroke Network
54Holt’s MethodHolt’s exponential smoothing (aka double exponential smoothing) is a moving average methodThere are two equations involved in fitting the model:Lt = axt + (1–a)(Lt–1 + Tt–1)Tt = g(Lt–Lt–1) + (1–g)Tt–1where xt is the observed value at time tLt 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 levelg is the second smoothing constant, used to smooth the trendThe model is fitted iteratively from the start of the time series, usually setting L1 initially to x1 and T1 to x2 – x1A software package optimises the constants a and g such that the squared differences between the observed values and the forecasts are minimised
55Holt’s Method in practice Several statistical packages will do this:ForecastPro – not free but very easy to useStata – not free and needs code, but PHE has a corporate licenceR – open source software which requires codeExcel – you can put the equations into Excel but have to optimise the parameters manuallyIf you use Stata, R or Excel, you need to put some effort into optimising the parameters, which requires some expertise and timeForecastPro has very clever optimisation routines, which always seem to result in sensible forecasts and forecast intervalsBUT!!!Every forecast should be graphed and checked – even the most expert of automated ‘expert systems’ cannot and should not be totally relied on
56Interval 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
57Interval 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 valueHowever, of course there is uncertainty about that predictionForecast intervals give an indication of the degree of uncertainty, and are far more valuable than the actual point forecastsThese forecast intervals are calculated by the forecasting software
58How 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 trendIf 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 trendIf 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 appliedThe 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
59Using forecasts to set and monitor targets – 1 Our Healthier Nation set targets to reduce circulatory disease death rates by 40% between and 2010This seemed reasonable at the timeHowever, 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)
60Using 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 areasNew Spearhead targets were set to reduce the gap between the Spearhead group of local authorities and the national average by 40% between andIn fact rates were dropping much faster than the OHN targets required
61Using 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 forecastsWe have to forecast the England rate in 2010 (current forecast: 61 deaths per 100,000 population)
62Using 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 forecastsWe 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)
63Using 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 forecastsWe 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
64Within district target In Doncaster’s deprived communities had an all age all cause mortality rate 16.7% greater than Doncaster as a wholeTo reduce the gap by 10% it must only be 15.0% above the Doncaster average by 2010Doncaster forecast to be 713 deaths per ,000 person years in 2010Target for the deprived quintile is 821This represents a reduction in death rates of 18%
65Within district target Using forecasting with realistic forecast intervals often serves to demonstrate the impossibility of measuring outcomes at PCT/local authority level or belowIt is essential always to plot as long a time series as possible to illustrate the wide year-on-year fluctuationsForecast intervals, even for the next year, are very broad
66Within district target Clearer to graph the annual excess mortality as a percentageIs this measurable?One weakness of this method is that it only uses the deprived quintile – it ignores most of the distribution
67Using 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 dateFor 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 arbitraryWe 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 agoForecasts of ‘current’ periods can give us robust ‘underlying’ values or rates
68Example – rapidly changing rates Circulatory disease death rates are falling dramaticallyaverage rate was 91 deaths per 100,000 population-years2008 forecast was 74In 2008, by taking the average of as our ‘current’ rate we were potentially overestimating the impact of interventions by 23%
69Summary – key pointsLook 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 changesDecide how far back you should startTransform data to ensure that the shape of the graph and any logical limits on variability (e.g. >0, <100%) are reflected in the mathematical assumptionsUse regression to analyse past changesUse forecasting methods such as Holt’s Method (or Holt-Winters for seasonal data) to make predictions of future rates with realistic forecast intervalsEnsure that data are independent of one another: no rolling averagesAlways graph the results, to ensure that the maths hasn’t had an off day