Forecasting of ATM cash demand Ayush Maheshwari Mathematics and scientific computing Department of Mathematics and Statistics Indian Institute of Technology (IIT), Kanpur ayushiitk@gmail.com Project guide: Dr. Jitendra Kumar Institute of Development and Research in Banking Technology (IDRBT) Road No. 1, Castle Hills, Masab Tank, Hyderabad 500 057 http://www.idrbt.ac.in/ July 6, 2012 1
CERTIFICATE This is to certify that project report titled Forecasting of ATM cash Demand submitted by Ayush Maheshwari of M.Sc(Integrated) 3rd year, Department of Mathematics and Scientific Computing, IIT Kanpur, is record of a bonafide work carried out by him under my guidance during the period 8 th may 2011 to 8 th july 2011 at Institute of Development and Research in Banking Technology, Hyderabad. The project work is a research study, which has been successfully completed as per the set objectives. Dr. Jitendra Kumar Assistant Professor Institute for Development and Research in Banking Technology Hyderabad-500057 2
DECLARATION I declare that the summer internship project report titled Forecasting of ATM cash Demand is my own work conducted under the supervision of Dr. Jitendra Kumar at the Institue of Development and Research in Banking Technology, Hyderabad. I have put in 60 days of my attendance with my supervisor at IDRBT and have been awarded project fellowship. I further declare that to the best of my knowledge, the report does not contain any part of any work which has been submitted for the award of any degree either in this institute or any other institute without proper citation. Ayush Maheshwari M.Sc(Integrated) 3 rd year Department of Mathematics and Scientific Computing IIT Kanpur 3
Introduction The time varying observation recorded in chronological order is called time series. The missing values are from the same times series model or appear because of some unobservable causes. Payment cards allowing for transactions in Automated teller Machines (ATM) became an indispensable tool in everyday life. Nevertheless, this phenomenon is far being understood, probably reflecting its minor importance in the past, before the strong increase in ATM usage in the early 80s. Research on payment systems has attracted increased academic attention over the last 15 years, merging monetary economics and banking theory with the study of mechanism of exchange. At the empirical front, information on ATM transaction is far from being explored. These databases have typically a large coverage and are timely, as they become typically available just a few days after the reference period. This report analyses ATM withdrawals for the period April 1 2007 to October 3 2009. A better understanding of consumer s habit is crucial for the logistics management of this type of services and may help other studies on the topic. The results presented may allow for a better understanding of ATM behaviour and consumer habits with respect to different days of week. This analysis is important to ensure the proper management of ATM. This report also deals with missing values of ATM s data corresponding to the days on which ATM failed or was in not working condition. These missing values give us the idea of ATM behaviour on particular days of week. We regressed various time series model on the given data, selected the best suitable model for that data and then used that model to find the missing values. So after finding missing values and making the data complete we forecast the future ATM withdrawals 4
Box Jenkins approach 5
Data Real-world data tends to be incomplete, noisy, and inconsistent and an important task when pre-processing the data is to fill in missing values, smooth out noise and correct inconsistencies. The data used in this report was provided by the random ATM of India and covers withdrawals made by local residents from April 1 2007 to October 3 2009(a sample of 917 observations). Figure 1 presents this time series both in levels and first differences. The series in levels is characterized by strong volatility while series in first differences has low volatility though both the series are having outliers. An outlier is an observation that is numerically distant from the rest of the data. Outlier can occur by chance in any distribution, but they are often indicative either of measurement error or that of a population has heavy tailed distribution. The breaks is both series and level plots correspond to missing values in data which is due to ATM failure on that day. The first figure below is time series in levels and second is time series in first difference. Daily ATM withdrawals (In levels) 6
(In difference) Figure 1 Methodology Handling missing values correctly is an important part of effective modelling. This section explains what missing values are, and describes the features provided in Analysis Services to work with missing values when building time series structures and models. The analysis of time series data constitutes an important area of statistics. Since, the data are records taken through time, missing observations in time series data are very common. This occurs because an observation may not be made at a particular time owing to faulty equipment, lost records, or a mistake, which cannot be rectified until later. When one or more observations are missing it may be necessary to estimate the model and also to obtain estimates of the missing values. By including estimates of missing values, a better understanding of the nature of the data is possible with more accurate forecasting. Different series may require different strategies to estimate these missing values. It is necessary to use these strategies effectively in order to obtain the best possible estimates. The objective of this report is to examine and compare the effectiveness of various techniques for the estimation of missing values in time series data models. The process of estimating missing values in time series data for univariate data involves analysis and modelling. Traditional time series analysis is commonly directed towards scalarvalued data, and can be represented by traditional Box-Jenkins autoregressive, moving 7
average, or autoregressive-moving average models. In some cases these models can be used to obtain missing values using an interpolation approach. In a recent development of time series analysis, the strategy has been to treat several variables simultaneously as vectorvalued variables, and to introduce an alternative way of representing the model, called state space modelling. Deterministic Modelling (also called Numerical Analysis Modelling) This method assumes the time series data corresponds to an unknown function and we try to fit the function in an appropriate way. The missing observation can be estimated by using the appropriate value of the function at the missing observation. Unlike traditional time series approaches, this method discards any relationship between the variables over time. The approach is based on obtaining the best fit for the time series data and is usually easy to follow computationally. In this approach there is a requirement for best fit process to be clearly defined. There are a variety of curves that can be used to fit the data. Stochastic Modelling (also called Time Series Modelling) Another common time series approach for modelling data is to use Box-Jenkins Autoregressive Integrated Moving Average (ARIMA) models. The ARIMA models are based on statistical concepts and principles and are able to model a wide range of time series patterns. These models use a systematic approach to identify the known data patterns and then select the appropriate formulas that can generate the kind of patterns identified. Once the appropriate model has been obtained, the known time series data can be used to determine appropriate values for the parameters in the model. The Box-Jenkins ARIMA models can provide many statistical tests for verifying the validity of the chosen model. In addition the statistical theory behind Box-Jenkins ARIMA models allows for statistical measurements of the uncertainty in a forecast to be made. However, a disadvantage of the Box-Jenkins ARIMA models is that it assumes that data is recorded for every time period. Often time series data with missing values require us to apply some intuitive method or appropriate interpolative technique to estimate those missing values prior to Box-Jenkins s ARIMA approach. Research Methodology To conduct our research, we will find index of missing values at various positions for data set. For estimation, the missing value positions for a 917 point data set are as follows 71, 225, 239, 250, 251, 253, 260, 265, 267, 269, 274, 400, 593, 701, 708, 734, 761, 762, 799, 809, 855, 860, and 876. In this report, our objective is to find the missing values at different positions within a time series data set. With minor adjustment, most of the approaches are able to be adapted to data sets with multiple missing values. However, if a data set consists of consecutive missing values, our estimates become less accurate due to lack of information. In this report, we are going to focus our analysis on a single missing value at various positions within the data set. 8
Analysis and Algorithms Analysis 1: (Analysis of Average Withdrawals) These analysis are being done days wise. All these calculation includes the days on which ATM was in not working condition. Starting from overall average withdraws of Weekdays. As we can see clearly that overall average withdraw is minimum on Sunday, increases up to maximum on Wednesday. Monday and Tuesday have almost same overall average withdraws. Now we analyze the ATM failure data which led to missing values in our data. This analysis gives us the possible reasons of many observations and will help us to draw various conclusions. Real world data tends to be incomplete and inconsistent. A missing value can signify a number of different things. Perhaps the field was not applicable, the event did not happen, or the data was not available. It could be that the person who entered the data did not know the right value, or did not care if a field was not filled in. 9
It appears that maximum number of ATM failures has been occurred on Sundays(13 Sundays) which can be due to the over usage of ATM s on weekdays and holidays on Sundays. Saturday and Monday are the days on which ATM never failed. Now we analyze the different range of withdraws with respect to days. Below is the table having days in rows and ranges in columns. The data in any cell is the total number of particular day on which average withdraw lied in particular range m = mean s = standard deviation Below is the Bar graph, in which every day has eight towers of different colours, where different colours represent different ranges and height of each tower represents total number of that particular day in that range. 10
In range(<m-3s), average withdraw never fell below m-3s In range(m-s to m), the height of Sunday tower is maximum which was expected from the lowest overall average withdraw of Sunday. In range(m to m+s), the height of Sunday s tower is minimum which was quite foreseeable from first bar graph as Sunday had the lowest overall average withdraw and Wednesday has the longest tower in range as it had the maximum overall average withdraw. In range(m+s to m+2s), the height of Monday s tower is maximum, the reason of which can be the maximum failures of ATM on Sundays caused customers to withdraw more money on Mondays. In range(m+2s to m+3s) withdraws happened only two times, one on Thursday(May 1,2008) and one on Friday(May 8,2009), both occurred in May. In range(>m+3s) withdraws happened only six times, two times each on Wednesday and Friday, one time each on Tuesday and Saturday. Below is the horizontal bar graph of Day v/s Range, which is just another bar graph representation of Data. 11
Below is the different graphs of Avg. withdraws v/s Different days As clear from above graph too that Sundays have maximum missing values. The above graph also shows outliers in time series. In data analysis, we deal with large number 12
of sampled or recorded variables. Some of observations are outlying from the rest, which are known as outliers. The extreme values from the same time series model or appear because of some unobservable causes having serious implications in the estimation and inference. This change deviate the error more. Methodical approach to find missing values Note: Data(i) refers to the data up to ith missing value. For example Data(1) means data up to 70 th position. Algorithm to Find missing value: 1) Find first position of first missing value. 2) Plot the data up to Data(1). 3) Obtain Sample Autocorrelation and Sample partial Autocorrelation of the Data(1). 4) Regress Data(1) on models up to ARMA(6,6). 5) Obtain SIC, AIC and R squared values for all the ARMA(p,q) combinations. Select the model with minimum SIC. Let model be ARMA(m,n). 6) Check if it is covariance stationary (If maximum value of absolute inverse AR roots is less than 1),if it is covariance stationary, forecast missing value using that model and for predicting further missing values we use the same ARMA(m,n) model by regressing the Data up to that missing value on the model. For example to predict the ith missing value we regress Data(i) on ARMA(m,n), then predict ith missing value using this model. 7) Else take first difference of Data(1). 8) Repeat process 4 th and 5 th on first difference of Data(1), let the model came be ARMA(j,k),do forecasting of missing value using that model. 9) For predicting further missing values we use the same ARMA(j,k) model by regressing the data up to that missing value on the model. For example to predict the ith missing value we regress first difference of Data(i) on ARMA(j,k), then predict ith missing value using this model. 10) Complete the data by finding all missing values. Discussion Now we apply above algorithm to our times series dataset. The first missing value occur at 71th position. So we first plot the data up to 70 th position i.e Data(1) and the first difference of Data(1). 13
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Now we fit forecasting models on Data(1). We begin by fitting deterministic-trend models to Data(1); we regress Data(1) on different ARMA models, allowing up to ARMA(6,6). Appendix 1.1 shows the AIC, SIC and R Squared values for all the ARMA(p,q) combinations. AIC selects an ARMA(5,1) model, whereas SIC selects an ARMA(1,1). We proceed with the more parsimonious model selected by SIC. The estimation results appear below and the residual plot in Figure 2;note, in particular, maximum value of absolute inverse AR roots is 1.0018 which is greater than 1. Discrete-time IDPOLY model: A(q)y(t) = C(q)e(t) A(q) = 1-1.002 q^-1 C(q) = 1-0.8504 q^-1 Estimated using ARMAX from data set z Loss function 3.04984e+006 and FPE 3.23192e+006 Sampling interval: 1 where y(t) = output at time t e(t) = white noise disturbance value at time t qis the delay operator Noise Variance: 3.1437e+006 AR root = -1.0018 Figure 3 show ARMA(1,1) model, residual sample autocorrelation with two standard error bands Figure 4 show ARMA(1,1) model, residual sample partial autocorrelation with two standard error bands Figure 2 15
Figure 3 Figure 4 The residual plot in Figure 4 clearly indicates that residuals are correlated. Moreover, the presence of unit root made Data(1) covariance non stationary. 16
In light of the suggestive nature of correlograms and unit root, we consider modelling Data(1). Now we fit forecasting models on Data(1). We regress Data(1) on different ARMA models, allowing up to ARMA(6,6). Appendix 1.1 shows the AIC, SIC and R Squared values for all the ARMA(p,q) combinations. SIC selects an ARMA(0,1). We proceed with the more parsimonious model selected by SIC. The estimation results appear below and the residual plot in Figure 5; Discrete-time IDPOLY model: y(t) = C(q)e(t) C(q) = 1-0.9605 q^-1 Estimated using ARMAX from data set z Loss function 2.64024e+006 and FPE 2.71906e+006 Sampling interval: 1 Figure 5 17
Figure 6 Figure 7 18
The residual plot in Figure 6 and Figure 7 clearly indicates that residuals are uncorrelated. So we can forecast now using ARIMA(0,1,1). The first forecasted value is 2620.374 at 71th position. So we have just filled first missing value. Similarly using our algorithm other missing values are found and once data is completed we can do forecasting of future. Conclusion Presence of important day s effect. A better understanding of consumer s habit is crucial for the logistic management of this type of services and may help other studies on the topic. This project would allow banks for a better understanding of ATM behavior and consumer habits with respect to different days of week. It would help banks to manage their ATM s properly by giving them a preliminary estimate of future transactions. References: en.wikipedia.org/wiki/automated_teller_machine Elements of Forecasting by Francis X.Diebold Jitendra kumar, Ashutosh shukla and Neeraj tiwari, Bayesian Analysis of a Stationary AR(1) model and outlier Appendix 1.1 19 AR MA AIC SIC R Squared 1 0 4628332.927 4779414.765-0.853461092 2 0 3842807.7 4097782.576-0.495543746 3 0 3643199.848 4011745.592-0.37792332 4 0 3410466.933 3878058.55-0.253566808 5 0 3393017.91 3984160.27-0.212024446 6 0 3433796.445 4163660.636-0.192041459 0 1 6680522.183 6898593.265-1.675280309 1 1 2875991.795 3066817.283-0.119278371 2 1 2940315.598 3237757.639-0.112079929 3 1 3009812.832 3422472.822-0.106300556 4 1 3058192.326 3591000.309-0.092420953 5 1 2843767.446 3448219.125 0.01278694 6 1 2917156.924 3652672.203 0.015834375 0 2 5606655.964 5978664.279-1.181998142 1 2 2940156.2 3237582.117-0.112019642 2 2 3024788.65 3439501.898-0.111805136 3 2 3081972.471 3618923.507-0.100915491 4 2 2848194.265 3453586.879 0.011250171 5 2 2850754.789 3569527.812 0.038236563
6 2 2923547.343 3780168.583 0.041460273 0 3 5335064.185 5874758.762-1.017816661 1 3 3009901.043 3422573.127-0.106332979 2 3 3090823.562 3629316.663-0.104077201 3 3 3087279.956 3743490.982-0.071748358 4 3 3212245.161 4022162.301-0.083719988 5 3 2873434.404 3715372.177 0.057890738 6 3 3265827.119 4360580.998-0.040602257 0 4 4914985.808 5588854.287-0.806574639 1 4 3059306.608 3592308.724-0.092818987 2 4 3140954.003 3808573.615-0.090381288 3 4 3108967.372 3892844.641-0.048877005 4 4 3299173.257 4265855.699-0.081695715 5 4 3266053.497 4360883.262-0.040674388 6 4 3337725.862 4602056.65-0.033555548 0 5 4675831.71 5490469.966-0.670260072 1 5 3112576.329 3774164.178-0.080529987 2 5 3553332.902 4449249.956-0.198793273 3 5 3179606.905 4111255.512-0.042493648 4 5 3286585.074 4388297.329-0.047216439 5 5 3414192.179 4707488.414-0.057234001 6 5 3729232.867 5309711.559-0.122261974 0 6 4673714.883 5667127.622-0.622478792 1 6 3175240.084 3975826.976-0.071235529 2 6 3725701.034 4817359.305-0.221540832 3 6 3248746.024 4337774.066-0.035159646 4 6 3429302.649 4728322.73-0.061913088 5 6 3571303.761 5084850.836-0.074735355 6 6 3194732.699 4697167.884 0.065668786 1.2 AR MA AIC SIC R-Squared 1 0 4004338 4136114 0.26881728 2 0 3702849 3950568 0.34318517 3 0 3424112 3773403 0.40998012 4 0 3391918 3860933 0.43222573 5 0 3425120 4027026 0.44304779 6 0 3419620 4152869 0.45982839 0 1 2613558 2699566 0.52277042 1 1 2677725 2856863 0.52502265 2 1 2751913 3032634 0.52580887 3 1 2814518 3203693 0.5288769 20
21 4 1 2888899 3396573 0.53024171 5 1 2967891 3604279 0.53118454 6 1 3033789 3805551 0.53446637 0 2 2677165 2856266 0.52512196 1 2 2758514 3039908 0.52467152 2 2 2775255 3159002 0.53544908 3 2 2894690 3403382 0.52929998 4 2 2976970 3615305 0.52975048 5 2 2898234 3635512 0.55526725 6 2 2974544 3854023 0.55659786 0 3 2746783 3026980 0.52669297 1 3 2768671 3151508 0.53655114 2 3 2780880 3269572 0.54780647 3 3 2966300 3602347 0.53143587 4 3 2856444 3583091 0.56167993 5 3 3030763 3926865 0.54821745 6 3 2989879 4001375 0.56704501 0 4 2803292 3190915 0.53075604 1 4 2886771 3394072 0.53058767 2 4 2958600 3592995 0.53265228 3 4 2912624 3653563 0.55305917 4 4 2997233 3883420 0.55321572 5 4 3173137 4246630 0.54050799 6 4 3182900 4399876 0.55226203 0 5 2883731 3390498 0.53108198 1 5 2968012 3604426 0.53116544 2 5 2933557 3679821 0.54984698 3 5 3115309 4036407 0.53561466 4 5 3193484 4273862 0.53756152 5 5 3245979 4487073 0.54338871 6 5 3191794 4557366 0.56383835 0 6 2952893 3586065 0.53355366 1 6 2895817 3632480 0.55563817 2 6 3036855 3934757 0.54730944 3 6 3062798 4098963 0.55648583 4 6 3123540 4317819 0.56061222 5 6 3151187 4499386 0.56938729 6 6 3399387 5013505 0.54874187