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KITUniversity of the State of Baden-Wuerttemberg and
National Research Center of the Helmholtz Association
KNOWLEDGE MANAGEMENT GROUP
INSTITUTE OF APPLIED INFORMATICS AND FORMAL DESCRIPTION METHODS , FACULTY OF ECONOMICS AND BUSINESS ENGINEERING
www.kit.edu
Complex Time Series AnalysisBinh Luong
Presentation of Diploma Thesis
Supervisors: Prof. Dr. Rudi Studer (KIT)
Dr. Christoph Lingenfelder and Dr. Boris Charpiot (IBM Deutschland)
Dr. Achim Rettinger and Dipl.
Inform. Benedikt Kmpgen (KIT)
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Overview
IntroductionTime Series
Seasonality
Time Series Modeling TechniquesARIMA
Exponential SmoothingProblems Analysis and Approaches
Unstable Seasonal Pattern
Multiple Seasonal Patterns
Non Integer Periodicity
Evaluation ResultsConclusions and Future Works
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INTRODUCTION
Complex Time Series Analysis
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Evaluation Results
Time Series - Definition
Motivation: How to plan for the future?
Need of tool to analyze past data and predict future data
A time series (TS) is an ordered sequence ofnumeric values, observed at successive points of
time.Time series are overall:Stock price
Exchange rate, interest rate, inflation rate, national GDP
Retail sales
Electric power consumption
Temperatures at a weather station
Number of unemployment figures for a region
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Conclusions and Future Works
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Time Series - Components
A TS is a combination of 4 components: trend, seasonal, cycle, error
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Conclusions and Future Works
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Seasonality in Time Series
IBM Netezza Analytics (INZA) determines the seasonality
period as follows:
Run Fast Fourier Transformation
Find peaks in the frequency diagram
Calculate weight of each peak
The nearest integer of the peak with the highest weight is set to be
the periodicity of the time series
Binh LuongComplex Time Series Analysis
Problems Analysis and ApproachesIntroduction
Conclusions and Future Works
Kernel-Run Analysis: Detected seasonsSeason: 10.1 Weight: 0.67Season: 4.9 Weight: 0.47Detected periodicity = 10
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Time Series Modeling Techniques
The most two common TS modeling techniques are:
ARIMA
Exponential Smoothing
ARIMA [1]:
The forecast for a period is calculated as a weighted linearcombination of its own past values and past errors
= + = Exponential Smoothing [2,3]:
Each component of a time series (trend, seasonal, error) isrepresented as a weighted moving average of all past valueswith the weights decreasing exponentially
+ 1
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PROBLEMS ANALYSIS AND
APPROACHES
Complex Time Series Analysis
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Overview of the problems
In this work approaches are designed for 3 separate problems:
Time series with unstable seasonal pattern
Time series with non-integer periodicity
Time series with multiple seasonal patterns
Our approaches work as a pre-processing and post-processing
steps to solve those issues. ARIMA and Exponential Smoothingare still applied to model and forecast time series.
Formal description for each problem:
Input:
- A time series with an above-mentioned issue
- Forecast horizon: a point of time in the future in which forecasts should bemade
Output:
- Forecasting results: a list of pairs of
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Issue 1: Unstable Seasonal Pattern
In some cases, the seasonal pattern in a TS is not stable, i.e. the length of
the periodicity varies over timeExample: monthly seasonal pattern in daily time series
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ARIMA1/8 31/8 30/9
Kernel-Run Analysis: Detected seasonsSeason: 30.428 Weight: 0.377501Season: 10.1519 Weight: 0.184886Season: 15.2192 Weight: 0.164569Detected periodicity = 30
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Approach for Unstable Seasonal Pattern (1)
Problem: the length of each period varies over time (e.g.monthly seasonal pattern with 29, 30 or 31 days / month)
Approach:
1. Transform each period in the original TS into newones based on a unique mean period length.
2. Apply ARIMA or Exponential Smoothing forforecasting.
3. At the end retransform the forecasting results basedon their real period lengths.
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Approach for Unstable Seasonal Pattern (2)
Illustration:Transformation of a month from 31 days into 30 days
All periods have a stable period length
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Problems Analysis and ApproachesIntroduction Conclusions and Future Works
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Linear Splines Interpolation
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Issue 2: Non Integer Periodicity
When the spectral analysis finds a seasonal pattern whose length is
not an integer
its length will be rounded up.
This problem causes inaccurate forecasted values.
To illustrate the problem we can use a trigonometrical function:
sin(2
). +
For p=7.5 with Exponential Smoothing:
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Kernel-Run Analysis: Detected seasonsSeason: 7.58903 Weight: 0.99993Detected periodicity = 8
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Approach for Non-Integer Periodicity (1)
Problem:
- Although the TS has a non-integer periodicity ExponentialSmoothing can not realize that and just use the roundedperiodicity found by FFT for further analyzing.
- ARIMA is not affected by this problem.
Approach:
1. Transform the original TS to a new one that has aninteger periodicity.
2. Apply Exponential Smoothing for the new TS.3. Retransform the forecasting results using the non-integer
periodicity at the beginning.
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Approach for Non-Integer Periodicity (2)
Illustration:Transformation a TS with p=7.5 into p=8
The new TS has now an integer periodicity p=8
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Problems Analysis and ApproachesIntroduction Conclusions and Future Works
Linear Splines Interpolation
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Evaluation Results
Issue 3: Multiple Seasonal Patterns
Binh LuongComplex Time Series Analysis
Some TS contain multiple seasonal patterns of different lengths
To illustrate the problem we can use a trigonometrical function:
sin 2 + sin2 . +
For p1=9 and p2=15 with Exponential Smoothing:
Problems Analysis and ApproachesIntroduction Conclusions and Future Works
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Kernel-Run Analysis: Detected seasonsSeason: 8.91999 Weight: 0.599654Season: 15.0681 Weight: 0.400238Detected periodicity = 9
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Evaluation Results
Approach for Multiple Seasonal Patterns (1)
Problem:
Exponential Smoothing can only handle one seasonalpattern. ARIMA provides quite good forecasts which still canbe improved.
Approach:1. Remove all seasonal patterns iteratively one after another
until there are no seasonal patterns in the TS.
2. Apply ARIMA or Exponential Smoothing for the
deseasonalized TS.3. Add all existing seasonal patterns into the forecasting
results.
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Evaluation Results
Approach for Multiple Seasonal Patterns (2)
Removing seasonal patterns iteratively:
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EVALUATION RESULTS
Complex Time Series Analysis
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Evaluation Metrics
Root Mean Square Error (RMSE) [4,5]
1 ( )
=
Percentage Improvement
100%
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Unstable Seasonal Pattern
ARIMA Exponential Smoothing
Existing implementation vs. our approach
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ARIMA Exponential SmoothingRMSEbefore 827,13 909,09RMSEafter 48,83 36,53Improvement 94,10% 95,98%
Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
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Non-Integer Periodicity
Before (Exponential Smoothing) After (Exponential Smoothing)
Existing implementation vs. our approach
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Complex Time Series Analysis
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Exponential Smoothingp= 3,5 p=7,5 p=18,5 p=30,5
RMSEbefore 34,36 23,74 15,13 7,88RMSEafter 5,2 1,21 0,21 0,08Improvement 41,69% 94,89% 98,58% 98,98%
Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
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Multiple Seasonal Patterns
Real data: hourly utility demand from a company in USA
Existing implementation with ARIMA
Our approach with ARIMA
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Complex Time Series Analysis
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Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
ARIMARMSEbefore 6535,4RMSEafter 1515,96Improvement 76,80%
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Multiple Seasonal Patterns
Real data: hourly utility demand from a company in USA
Existing implementation with Exponential Smoothing
Our approach with Exponential Smoothing
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Complex Time Series Analysis
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Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
Exponential SmoothingRMSEbefore 1398,67RMSEafter 853,05Improvement 39,01%
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CONCLUSIONS AND
FUTURE WORKS
Complex Time Series Analysis
Binh Luong
Complex Time Series Analysis
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Conclusions
Design solution approaches that can be combined with the
existing modelling techniques (ARIMA or ExponentialSmoothing) to analyse time series with:
Unstable seasonal pattern
Non-Integer Periodicity
Multiple Seasonal Patterns
Prototype implementation.
Evaluate our approaches and get a significant improvement offorecast accuracy compared to the existing implementation.
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Complex Time Series Analysis
Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
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Future Works
Unstable Seasonal Pattern
Extend the algorithm to handle time series with numericaltime column
Non-Integer Periodicity
Distinguish between real non- integer periodicities and thosecaused by rounding error of the spectral analysis
Multiple Seasonal Patterns
Specify a reasonable threshold to filter out the seasonalpatterns that are also results of spectral analysis but do notpresent real seasonal variation in the time series
Binh Luong
Complex Time Series Analysis
Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results
Time Column
real date numeric
Real non-integer vs. Rounding error
found periodicity weight real periodicityYYN ?
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References
[1] G.E.P Box and G.M. Jenkins, Time series analysis, forecasting
and control., Holden-Day, San Francisco, 1970
[2] E.S.Gardner, Jr, Exponential smoothing: the state of art, Journal of
Forecasting 4 (1985)
[3] E.S.Gardner, Jr, Exponential smoothing: the state of art part II,
International Journal of Forecasting 22 (2006)
[4] B. Abraham and J. Ledolter, Statistical methods for forecasting, John
Wiley & Sons, New York, 1983
[5] W. Reinmuth, W. Mendenhall, and R. J. Beaver, Statistics for
management and economics, Duxbury Press, Belmonth, California,
1993
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Thank youfor your attention!
Binh Luong
Complex Time Series Analysis
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