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Forecasting Intraday Call
Arrivals Modelling Special Days
Devon K. Barrow
Nikolaos Kourentzes
27th European Conference on Operational Research
12-15 July 2015
University of Strathclyde
1. Research Questions
2. Call Arrival Data and Challenges
3. Experimental Design
4. Results
5. Conclusions and future work
Outline
Forecasting Intraday Call Arrivals Outline 2
• How do different forecasting methods
perform when forecasting (high frequency)
call centre arrivals?
• What is the impact of coding or not coding
of (functional) outlying periods on
forecasting performance?
Research Questions
Research Questions 3 Forecasting Intraday Call Arrivals
1. Research Questions
2. Call Arrival Data and Challenges
3. Experimental Design
4. Results
5. Conclusions and future work
Outline
Forecasting Intraday Call Arrivals Call Centre Data 4
• High dimensional and sampled at a high
frequency (Kourentzes and Crone, 2010).
• Complex seasonal patterns
– Intraday
– Intraweek
– interyear dependencies (De Livera et al., 2011)
Call Arrival Data and Challenges Challenges
5 Call Centre Data Forecasting Intraday Call Arrivals
Call Arrival Data and Challenges Some call centre data
6
• Two series from a large UK service provider call centre
• A leading entertainment company in Europe
• Data is sampled at half-hourly intervals
• Consists of 103 weeks and 3 days from 29 June 2012 to
23 June 2014 inclusive including bank holidays and
weekends.
Call Centre Data Forecasting Intraday Call Arrivals
Call Arrival Data and Challenges Complex seasonal patterns
7
Intraday Intraweek
Mean
Median
Monday
Intrayear
Call Centre Data Forecasting Intraday Call Arrivals
• Development of new methods
– Double seasonal exponential smoothing (Taylor, 2008)
– Multiplicative double seasonal ARMA model (Taylor, 2003)
– Exponential weighting (Taylor, 2008, 2010)
– Regression (Tych et al., 2002; Taylor, 2010)
– Singular vector decomposition (Shen and Huang, 2005, 2008;
Shen, 2009)
– Gaussian linear mixed-effects models (Aldor-Noiman et al.,
2009; Ibrahim and L’Ecuyer, 2013)
• Intraweek seasonal moving average performs well at medium to long horizons (Tandberg et al. 1995; Taylor, 2008; Taylor,
2010; Ibrahim and L’Ecuyer, 2013)
Call Arrival Data and Challenges Handling high frequency complex seasonality
8 Call Centre Data Forecasting Intraday Call Arrivals
!
• Data is context sensitive
– Effects of holidays, special events and
promotional activities
• Prone to quite sizeable unexplained
variations (outliers)
– E.g. due to system failures and data
processing
Call Arrival Data and Challenges Challenges
9 Call Centre Data Forecasting Intraday Call Arrivals
Call Arrival Data and Challenges Anomalies (outliers)
10
Series 1
Series 1I
Call Centre Data Forecasting Intraday Call Arrivals
Call centre data Seasonal profile
12
4 8 12 16 200
100
200
300
400
Calls
Monday
4 8 12 16 20
Tuesday
4 8 12 16 20
Wednesday
4 8 12 16 20
Hour
Thursday
4 8 12 16 20
Friday
4 8 12 16 20
Saturday
4 8 12 16 20
Sunday
Median
25-75%
10-90%
05-95%
4 8 12 16 200
10
20
30
40
Calls
Monday
4 8 12 16 20
Tuesday
4 8 12 16 20
Wednesday
4 8 12 16 20
Hour
Thursday
4 8 12 16 20
Friday
4 8 12 16 20
Saturday
4 8 12 16 20
Sunday
Median
25-75%
10-90%
05-95%
Series 1
Series 1I Large variation from the middle Call Centre Data Forecasting Intraday Call Arrivals
Call centre data Outliers vs. median pattern
Call Centre Data 13
4 8 12 16 200
100
200
300
400
Monday
Calls
4 8 12 16 20
Tuesday
4 8 12 16 20
Wednesday
4 8 12 16 20
Thursday
Hour
4 8 12 16 20
Friday
4 8 12 16 20
Saturday
4 8 12 16 20
Sunday
Profile
Outliers
4 8 12 16 200
10
20
30
40
Monday
Calls
4 8 12 16 20
Tuesday
4 8 12 16 20
Wednesday
4 8 12 16 20
Thursday
Hour
4 8 12 16 20
Friday
4 8 12 16 20
Saturday
4 8 12 16 20
Sunday
Profile
Outliers
Series 1
Series 1I Regular profile very different from outliers
Forecasting Intraday Call Arrivals
• Approaches to handling ‘special days’: – Information is either available and/or data is pre-
cleansed
– The forecaster has an external methodology for tackling ‘special days’ (Jongbloed and Koole, 2001; Avramidis et al., 2004; Taylor, 2008a; Pacheco et al., 2009; Taylor, 2010b)
– Removing such days altogether (Taylor et al. 2006)
– Singular vector decomposition for automatic outlier detection (Shen and Huang, 2005)
• Kourentzes (2011) demonstrates that there are substantial accuracy benefits to be had from modelling irregular load patterns.
Call Arrival Data and Challenges Handling ‘special days’
14 Call Centre Data Forecasting Intraday Call Arrivals
!
1. Research Questions
2. Call Arrival Data and Challenges
3. Experimental Design
4. Results
5. Conclusions and future work
Outline
Forecasting Intraday Call Arrivals Experimental Design 15
Experimental Design Functional outlier modelling
16
• We evaluate seven alternative methodologies for
modelling functional outliers
• These are based on extensions of conventional outlier
modelling and novel approaches
• We assume that the data generating process of normal
observations is captured adequately and the outliers are
already labelled
Control
Single Binary
Dummy
Multiple Binary
Dummy
Single Integer
Profile Dummy
Trigonometric
Dummy
Model
Separately
Experimental Design Forecasting Intraday Call Arrivals
Experimental Design Functional outlier modelling
17
• Control/benchmark – A set of autoregressive lagged inputs (past values) are identified
using stepwise regression
– Outliers are not modelled
• Single Binary Dummy Variable – Indicator variable s.t. one = outlier; zero otherwise
• Multiple Binary Dummy Variable – S is the seasonal length (S=48)
– 48 (S) dummy variables to code each observation
– 47 (S – 1) dummy variables to code each observation
– Stepwise selection, s = {1,…,S=48} s.t. s is significantly different from normal observations
Experimental Design Forecasting Intraday Call Arrivals
Experimental Design Functional outlier modelling
18
• Single Integer Dummy Variable – Monotonically increasing variable from 1 to 48 if outlier; zero
otherwise
• Profile Dummy Variable – This variable is equal to the profile if there is an outlier; zero
otherwise
• Trigonometric Dummy Variables – Sine and cosine if there is an outlier; zero otherwise
• Model Separately – Create a new series containing only outliers
– Replace outliers in original series with normal observations
Experimental Design Forecasting Intraday Call Arrivals
Experimental Design Experimental setup
Experimental design 19
• Neural network setup – Inputs based on backwards regression + dummies + seasonal
dummies.
– Mode ensemble of 50 networks trained by scaled conjugate gradient descent
– Hidden nodes identified experimentally for each set of inputs
• Forecast creation – Forecast horizon is set to 1 day ahead (1-48 half hourly steps
ahead)
– Test set of 100 days (4800 data points x 48 forecasted horizons)
• Forecast evaluation – Mean Absolute Error (MAE)
– Relative Mean Absolute Error (RMAE) i.e. MAE_{Method} / MAE_{Control} AE = |Yt - Ft| Forecasting Intraday Call Arrivals
1. Research Questions
2. Call Arrival Data and Challenges
3. Experimental Design
4. Results
5. Conclusions and future work
Outline
Forecasting Intraday Call Arrivals Results 20
Results Neural networks versus benchmarks
21
Time Series 1 Overall Outlier Normal
Naïve 1 3.930 2.711 4.269
Naïve Day 1.417 1.211 1.475
Naïve Week 1.333 1.057 1.410
MA Day 1.238 1.278 1.226
MA Week 1.233 0.845 1.341
ETS Day 1.887 1.438 2.013
ETS Week 1.529 1.165 1.630
ETS Double 1.086 0.857 1.149
MAPA Day 1.522 1.362 1.566
MAPA Week 1.484 1.274 1.542
NN Control 1.000 1.000 1.000 Results Forecasting Intraday Call Arrivals
Results Neural networks versus benchmarks
Time Series II Overall Outlier Normal
Naïve 1 2.236 1.770 2.500
Naïve Day 1.415 1.119 1.582
Naïve Week 1.433 1.157 1.590
MA Day 1.415 1.119 1.582
MA Week 1.129 0.973 1.217
ETS Day 1.600 1.368 1.731
ETS Week 1.479 1.288 1.587
ETS Double 1.151 0.990 1.242
MAPA Day 1.347 1.242 1.406
MAPA Week 1.280 1.201 1.325
NN Control 1.000 1.000 1.000 22 Results Forecasting Intraday Call Arrivals
Results Outlier modelling versus control
Time Series 1 Overall Outlier Normal
NN Control 1.000 1.000 1.000
NN Bin1 0.996 0.933 1.013
NN BinS 0.884 0.822 0.901
NN BinS-1 0.873 0.831 0.885
NN Bin Step 0.897 0.858 0.908
NN Bin Back 0.895 0.850 0.907
NN Int 0.935 0.913 0.941
NN SinCos 0.980 0.865 1.012
NN Profile 0.986 0.953 0.995
NN Replace 1.002 1.019 0.997 23 Results Forecasting Intraday Call Arrivals
Results Outlier modelling versus control
Time Series II Overall Outlier Normal
NN Control 1.000 1.000 1.000
NN Bin1 0.956 0.930 0.970
NN BinS 0.922 0.866 0.954
NN BinS-1 0.923 0.870 0.954
NN Bin Step 0.929 0.876 0.958
NN Bin Back 0.931 0.875 0.963
NN Int 0.957 0.940 0.966
NN SinCos 0.951 0.912 0.972
NN Profile 0.963 0.936 0.979
NN Replace 1.030 1.090 0.996 Results 24 Forecasting Intraday Call Arrivals
1. Research Questions
2. Call Arrival Data and Challenges
3. Experimental Design
4. Results
5. Conclusions and future work
Outline
Forecasting Intraday Call Arrivals Conclusion and future work 25
Conclusion and future work
Conclusion and future work 26
• Conclusion
– Neural networks are good for call centre data for two reasons:
• They can do complex structures
• They can do complex outliers with relatively simple modelling
• Future work
– Automatic functional outlier detection and modelling for call centre arrival data
Forecasting Intraday Call Arrivals
Devon K. Barrow Coventry Business School
Coventry University, Priory Street, Coventry, CV1
5FB
Direct line: + 44 024 7765 7413
Skype: devon.k.barrow
Email: [email protected]