Advanced time-series analysis

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1 UCL DEPARTMENT OF SECURITY AND CRIME SCIENCE Advanced time-series analysis Lisa Tompson Research Associate UCL Jill Dando Institute of Crime Science

2 Overview Fundamental principles of time-ordered data Introduction to time-series analysis Revealing trends Forecasting Evaluating interventions Where to find out more

3 Time-ordered 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 [proh-ping-kwi-tee]

4 Time-series: an introduction Time-series 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 Time-series analysis for revealing trends

6 Time-series analysis for revealing trends All time-series 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 Time-series analysis for revealing trends Two types of patterns are important: Linear or non-linear 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 Time-series 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 Time-series 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 time-series: 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 Time-series 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 Time-series analysis for forecasting

14 Time-series 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) Short-term predictions will be more reliable that long-term predictions

15 Time-series analysis for forecasting You could use a moving-average 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 Time-series analysis for evaluating interventions

17 Time-series analysis for evaluating interventions Pre-test Treatment applied Post-test 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 Jan-03 Apr-03 Jul-03 Oct-03 Jan-04 Apr-04 Jul-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-06 Apr-06 Jul-06 Oct-06 Jan-07 Apr-07 Jul-07 Oct-07 Number of Referrals Jan-08 Apr-08

19 Before-after design? 30 Number of Referrals Average crime rate 12 months pre-test (Jan06-Dec06): Jan-03 Apr-03 Jul-03 Oct-03 Jan-04 Apr-04 Jul-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-06 Apr-06 Jul-06 Oct-06 Jan-07 Apr-07 Jul-07 Oct-07 Jan-08 Apr-08 Average crime rate 12 months post-test (Apr07-Mar-08): 20.2

20 Why can t we just use a before-after design? It might be tempting to use a t-test to assess whether there was any statistical evidence of a change in group means HOWEVER one of the assumptions of a t-test is that the groups consist of independent observations with equal variances This is violated by time-series 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 Auto-Regressive, Integrated, Moving-Average (p, d, q) p d q Auto-regressive element represents the lingering effects of previous scores Integrated element represents the trends present in the data Moving-average 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 time-series 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 time-series - 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 time-series - 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 time-series autocorrelation The values of P and Q (p, d, q) are investigated second p relates to autoregressive components, q relates to moving-average components Usually a model has either an autoregressive component OR a moving-average 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 time-series 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 time-series model Models are estimated through patterns in their ACFs and PACFs Auto-regressive Moving-average

30 Estimating a time-series 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 moving-average 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 = pre-intervention, 1 = post-intervention Or can use other IVs E.g. number of police officers, number of ex-prisoners released Need to identify, estimate and diagnose your model based on pre-intervention 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 time-series analysis Determining whether crime patterns vary in specific ways over time has implications for our understanding of the problem Time-series 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 Quasi-Experimental 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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