Advanced timeseries analysis


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1 UCL DEPARTMENT OF SECURITY AND CRIME SCIENCE Advanced timeseries analysis Lisa Tompson Research Associate UCL Jill Dando Institute of Crime Science
2 Overview Fundamental principles of timeordered data Introduction to timeseries analysis Revealing trends Forecasting Evaluating interventions Where to find out more
3 Timeordered data Observations are never pure measures of a phenomena Crime rates can fluctuate over time for a variety of reasons Such variation is known as random noise Observations close together in time will be more closely related than observations further apart This is known as propinquity [prohpingkwitee]
4 Timeseries: an introduction Timeseries analysis is used when observations (e.g. counts of crime) are made repeatedly should be over 50 if you want to account for seasonal trends Three main aims of this type of analysis To help reveal or clarify trends To forecast future patterns of events To test the impact of interventions (IVs)
5 Timeseries analysis for revealing trends
6 Timeseries analysis for revealing trends All timeseries data have three basic parts: A trend component; A seasonal component; and A random component. Trends are often masked because of the combination of these three components Especially the random noise!
7 Timeseries analysis for revealing trends Two types of patterns are important: Linear or nonlinear trends  i.e., upwards or downwards or both (quadratic); and Seasonal effects which follow the same overall trend but repeat themselves in systematic intervals over time.
8 Timeseries analysis for revealing trends Smoothing techniques can be used to filter out the random noise Moving average is the simplest form of smoothing technique
9 Timeseries analysis for revealing trends More advanced smoothing techniques are available in statistical packages (such as SPSS, Stata, R) These can estimate three parameters which are used to generate an equation that best describes the timeseries: The parameter gamma (γ) determines how the trend is modelled The parameter alpha (α) determines how much weighting should be assigned to recent and distant observations The parameter delta (δ) models the seasonal component Need to vary the parameters to see which provides the best fit against your data (e.g. most accurate model) For this you can use the Sum of Squared Errors (SSE)
10 Timeseries analysis for revealing trends A sophisticated method for revealing trends is known as seasonal decomposition
11 R screenshot
12 The power of visualisation remainder trend seasonal data time
13 Timeseries analysis for forecasting
14 Timeseries analysis for forecasting Where crime rates follow regular patterns, this can be useful for predicting future levels of crime Forecasts are commonly produced so that the workload of criminal justice agencies can be managed The accuracy of forecasts is likely to be limited, given the complexity of influencing factors Predictions rely on the assumption that evident trends and seasonal effects will continue (i.e. it is a stable pattern) Shortterm predictions will be more reliable that longterm predictions
15 Timeseries analysis for forecasting You could use a movingaverage model to forecast future patterns Commonly you would ascertain the value of doing this with data you already have (i.e. can use as a test against the forecast scores) The value would be derived from comparing the error scores between the predicted data points and the real data points If the model is sufficiently accurate you could then use it to make a forecast. However, if it is not very accurate, it would be unwise to use the model. As you know, more sophisticated models exist (e.g. using gamma (γ) alpha (α) delta (δ))
16 Timeseries analysis for evaluating interventions
17 Timeseries analysis for evaluating interventions Pretest Treatment applied Posttest Experimental O 1 O 2 O 3 O 4... X O 6 O 7 O 8 O 9... Control O 1 O 2 O 3 O 4... O 6 O 7 O 8 O 9...
18 Evaluation in the real world Jan03 Apr03 Jul03 Oct03 Jan04 Apr04 Jul04 Oct04 Jan05 Apr05 Jul05 Oct05 Jan06 Apr06 Jul06 Oct06 Jan07 Apr07 Jul07 Oct07 Number of Referrals Jan08 Apr08
19 Beforeafter design? 30 Number of Referrals Average crime rate 12 months pretest (Jan06Dec06): Jan03 Apr03 Jul03 Oct03 Jan04 Apr04 Jul04 Oct04 Jan05 Apr05 Jul05 Oct05 Jan06 Apr06 Jul06 Oct06 Jan07 Apr07 Jul07 Oct07 Jan08 Apr08 Average crime rate 12 months posttest (Apr07Mar08): 20.2
20 Why can t we just use a beforeafter design? It might be tempting to use a ttest to assess whether there was any statistical evidence of a change in group means HOWEVER one of the assumptions of a ttest is that the groups consist of independent observations with equal variances This is violated by timeseries data due to the temporal dependency (known as autocorrelation) For this reason we need to use a method known as autoregression (AR) which accounts for the autocorrelation
21 ARIMA modelling AutoRegressive, Integrated, MovingAverage (p, d, q) p d q Autoregressive element represents the lingering effects of previous scores Integrated element represents the trends present in the data Movingaverage element represents lingering effects of the preceding random shocks A simpler version of this is ARMA (sometimes called a Box Jenkins model)
22 ARIMA modelling First, you have to perform some preliminary analyses before you fit an ARIMA model to your data These steps can be summarised as: 1. Identification 2. Estimation 3. Diagnosis Identification and estimation of a timeseries is the process of finding the integer values of p, d, and q. These are typically small (0, 1, 2) When these values are 0, the element is not needed in the model
23 Identification of a timeseries  transformation The middle element of (p, d, q) is investigated first The goal is to determine if the data are stationary Stationary refers to a constant mean and variance If the mean is changing (i.e. the trend is going up or down), the trend can be removed by differencing
24 Identification of a timeseries  transformation Stationary refers to a constant mean and variance If the variability is changing, the data may be made stationary by logarithmic transformation (the LOG formula in Excel) You need to plot it again to ascertain if the variance has stabilised after transformation Once you are happy that the mean and variance are as constant as possible you can assign d a value If d = 0 the data are stationary and there is no trend; If d = 1 the data need to be differenced once so the linear trend is removed; If d = 2 the data need to be difference twice so that the linear and quadratic trends are removed.
25 Identification of a timeseries autocorrelation The values of P and Q (p, d, q) are investigated second p relates to autoregressive components, q relates to movingaverage components Usually a model has either an autoregressive component OR a movingaverage component, but very occasionally, a series has both How do we tell? We look at autocorrelation patterns
26 Assessing autocorrelation patterns You can use a correlogram to assess whether autocorrelation are present in the timeseries data This is an image of correlation statistics It is commonly used for checking randomness in a data set
27 Measuring autocorrelation If adjacent observations are highly correlated, then so too will those that are  say  two or three observations apart We therefore need to consider the lag of the autocorrelation we are measuring To control for this we use an autocorrelation function (ACF) in conjunction with a partial autocorrelation function (PACF)
28 ACFs and PACFs An autocorrelation function (ACF) shows the serial correlation coefficients for consecutive lags A partial autocorrelation function (PACF) partials out the immediate autocorrelations and estimates the autocorrelation at a specific lag (e.g. 2 lags)
29 Estimating a timeseries model Models are estimated through patterns in their ACFs and PACFs Autoregressive Movingaverage
30 Estimating a timeseries model AR: first order regressive process (p) Large value for lag 1, with the ACF decaying over greater lags and the PACF only being significant for lag 1 When p = 0 there is no relationship between adjacent observations When p = 1 there is a relationship between observations at lag 1 When p = 2 there is a relationship between observations at lag 2 MA: a movingaverage process (q) Large value for lag 1, and the ACF will only be significant for this lag whilst the PACF decays over greater lags When q = 0 there are no moving average components When q = 1 there is a relationship between the current score and the random shock at lag 1 When q = 2 there is a relationship between the current score and the random shock at lag 2
31 Time to run a model or ten arima count_4_10, arima (1,0,0) nolog
32 Diagnosing a model This involves assessing to see how well your model fits your data E.g. Are the values of the observations predicted from the model close to actual ones? We assess this through examining the residuals (differences between actual & predicted values) If the model is good, the residuals will be a series of random errors We assess this through ACFs and PACFs
33 ARMAX? Simply an ARIMA model with independent variables (IVs) Can dummy code an intervention variable So that 0 = preintervention, 1 = postintervention Or can use other IVs E.g. number of police officers, number of exprisoners released Need to identify, estimate and diagnose your model based on preintervention data before adding the IVs
34 Add in your IV to the model The model fit The beta coefficient The p value
35 Then diagnose once more
36 Types of questions you may want to answer What kind of trend do I have going on? Do these trends differ by area? Are there distinct seasonal patterns in my data? Are they stable enough to be able to predict the next month, quarter, year? What has been the impact of an intervention? Are the before and after trends different? What relationship does an IV (e.g. no of Police Officers on the beat) have with my DV (e.g. count of crime)?
37 The value of using timeseries analysis Determining whether crime patterns vary in specific ways over time has implications for our understanding of the problem Timeseries analysis helps to reveal the patterns and to establish the statistical significance of them Predictable patterns create opportunities for crime prevention and detection Deployment of policing resources Designing crime prevention initiatives
38 Where to go for more information
39 Some further reading Campbell, D.T. & Stanley, J.C (1963). Experimental and Quasi Experimental Designs for Research. Boston: Houghton Mifflin Company. Shadish, W.R.; Cook, T.D. & Campbell, D.T. (2002). Experimental and QuasiExperimental Designs for Generalized Causal Inference. Boston: Houghton Mifflin Company. Tabachnick, B. & Fidel, L (2007). Using multivariate statistics. 5th ed. Boston: Allyn and Bacon. Chapter 17
40 Thank you Lisa Tompson Research Associate UCL Jill Dando Institute of Crime Science
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