Complex TimeSeries Analysis

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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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Knowledge Management Group

Institute of Applied Informatics and Formal Description Methods2 20-07-2012

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

Binh LuongComplex Time Series Analysis


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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


Problems Analysis and ApproachesIntroduction

Conclusions and Future Works


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Evaluation Results

Time Series - Components

A TS is a combination of 4 components: trend, seasonal, cycle, error





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Evaluation Results

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




Kernel-Run Analysis: Detected seasonsSeason: 10.1 Weight: 0.67Season: 4.9 Weight: 0.47Detected periodicity = 10


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Evaluation Results

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


Problems Analysis and ApproachesIntroduction Conclusions and Future Works


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PROBLEMS ANALYSIS AND

APPROACHES




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Evaluation Results

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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Evaluation Results

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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01.01.04 01.02.04 01.03.04 01.04.04 01.05.04 01.06.04 01.07.04 01.08.04 01.09.04 01.10.04

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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Evaluation Results

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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Evaluation Results

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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Linear Splines Interpolation


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Evaluation Results

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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Evaluation Results

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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Evaluation Results

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



Linear Splines Interpolation

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Evaluation Results

Issue 3: Multiple Seasonal Patterns


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:


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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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Adjusted TS with 1 seasonalpattern p=9

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Original TS with 2 seasonalpatterns p1=9 and p2=15

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Adjusted TS with no seasonal pattern

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Adjusted TS with 1 seasonalpattern p=15


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EVALUATION RESULTS




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Evaluation Metrics

Root Mean Square Error (RMSE) [4,5]

1 ( )

=

Percentage Improvement

100%


Problems Analysis and ApproachesIntroduction Conclusions and Future WorksEvaluation Results


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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%



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Non-Integer Periodicity

Before (Exponential Smoothing) After (Exponential Smoothing)

Existing implementation vs. our approach

Binh Luong


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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%



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Real data: hourly utility demand from a company in USA

Existing implementation with ARIMA

Our approach with ARIMA

Binh Luong


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ARIMARMSEbefore 6535,4RMSEafter 1515,96Improvement 76,80%


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Real data: hourly utility demand from a company in USA

Existing implementation with Exponential Smoothing

Our approach with Exponential Smoothing

Binh Luong


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Exponential SmoothingRMSEbefore 1398,67RMSEafter 853,05Improvement 39,01%


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CONCLUSIONS AND

FUTURE WORKS


Binh Luong



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Knowledge Management GroupInstitute of Applied Informatics and Formal Description Methods

26 20-07-2012

Conclusions

Design solution approaches that can be combined with the

existing modelling techniques (ARIMA or ExponentialSmoothing) to analyse time series with:

Unstable seasonal pattern



Prototype implementation.

Evaluate our approaches and get a significant improvement offorecast accuracy compared to the existing implementation.

Binh Luong




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Future Works


Extend the algorithm to handle time series with numericaltime column


Distinguish between real non- integer periodicities and thosecaused by rounding error of the spectral analysis


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



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

Binh Luong




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Knowledge Management Group29 20-07-2012

Thank youfor your attention!

Binh Luong


Complex TimeSeries Analysis

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