Simulation for Time Series Analysis & Forecasts

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Liophant SimulationLiophant Simulation

Agostino Agostino Bruzzone &Bruzzone & Simone Simone Simeoni Simeoni

Liophant Liophant SimulationSimulation

Simulation for Time SeriesSimulation for Time SeriesAnalysis & ForecastsAnalysis & Forecasts

[email protected]@liophant.org

www.liophant.org

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Tutorial Goals

� To identify Critical Issues in Forecasting

� To define M&S Potential in Testing Forecasting

� To introduce Forecast Algorithms

� To describe methodologies for Best Fitting ofForecast Algorithms

� Examples and Exercise in Logistics Networks

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Forecasts

� Forecast is a statement of what is judged tohappen in the future in connection with historicaldata.

� Forecast is what you expect to happen in thefuture.

� Forecasts must be Credible

� People need to Trust Forecastsfor using these with success

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Precision vs. Robustness

In Forecasting the Robustness is the most critical aspect interm of performance evaluation.In this context, usually it is not important to have a highprecision in forecasts if their robustness is still weak.

Robustnessthe degree to which a system orcomponent can function correctly inthe presence of invalid inputs orstressful environment conditionsInstitute of Electrical and Electronics Engineers. IEEE Standard Computer Dictionary: A Compilation ofIEEE Standard Computer Glossaries. New York, NY: 1990.

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Liophant SimulationLiophant Simulation Predicting the Past andPredicting the Future

The forecast methodologies use historical data to makeassumption about what will happen, but it is really critical forthem to predict major scenario changes.

In this context it is critical to check if:The past can predict the future ?!

Often it is necessary the human support to analyze and notethe market scenario, because the assumption that was trueyesterday will be true tomorrow not ever works.

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Goodyear implemented a demand forecasting systemin mid-2000 but hasn’t shown significantimprovement in managing inventory and in the lastyear tire lost more money then the years ago.

Nike implemented a demand forecasting (i2) in June2000 and nine month later its executivesacknowledged that they would be taking a majorinventory write-off because the forecasts from theautomated system had been so inaccurate.

Forecasting CompanyExamples

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� Forecasting systems are only as good as the data putin.

� Software can't predict the future, particularly sudden,unexpected shifts in economic or market conditions.

� Forecasts sometime affect the System Evolution

“ ...I would say that if you looked at the split betweenpeople, science and process, people are half theequation. Algorithms are algorithms. That is not whatwill win the game for me." Sumatra Sengupta CIO for Scotts Co. 1.8 Billion USD revenues in 2003, Lawn & Garden Care products

Forecasting Systems Highlights

"Anyone who thinks you can do it with just mathematicsand statistics is only partly right. Human intelligence isalso required." Doug Richardson, CIO of Vicor,

150million USD revenues in 2003, managing 1.7trillionUSD transactions over internet

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Liophant SimulationLiophant Simulation Forecasting SystemsInvestments

In 2002 alone, companies spent $19 billion ondemand forecasting software and othersupply chain solutions. IDC

In a survey of 196 senior executives, 45percent said that supply chain technology ingeneral had failed to meet their expectations.

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Crystal Ball vs. Forecasts

� Crystal Ball don’t Exist

� Forecasts are not Point Values but ConfidenceRegions

� Reliability, Fidelity and Precision of Forecastsneed to be investigated during Analysis

� Sometime Reality is Unpredictable

� Predictability needs to be referred the specificProblem

� Good demand forecasting requires a combinationof accurate data and smart people.

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Advanced TechniquesAdvanced Techniques can be used in order to providesignificant improvements in forecasts performances;among the others�Artificial Neural Networks (ANN)�Fuzzy Logic�Knowledge Based System KBS�Genetic Algorithms�Data Fusion

However� New Forecasts Techniques don’t change the Real

System Predictabilityor

� New Forecasts Techniques can’t overpasstraditional methods and become a Crystal Ball

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Data Fusion & Forecasting� Most Promising advances in Forecasting are related

to merging information techniques devoted toimprove their reliability (Data Fusion, Fuzzy Logic)

Orders

Orders

Success Prob. No ?

Yes

Orders +Virtual Orders generated byForecast Fusion

Real Order Portfolio

Orders to be Simulated

Montecarlo Technique

ITEM iMetaProduct

Mix Component

ITEM jMix

Component

FrequencyDiagram

Random Number

ExtractedValue

CumulativeDiagram

MontecarloTechnique

Forecast

Quantity Estimation

Time Definition



Selection of theItem Mixage

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Simulation & Forecasts

Forecasting techniques are devoted to extract in the mostefficient way Knowledge from various Input for improvingForecasting Capabilities

Historical DataExpert EstimationPhenomena SymptomsBoundary Conditions

Modeling & Simulation can be used for testing ForecastingTechniques and for Measuring their Performances overcomplex scenarios

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Stochastic SimulationThe Stochastic Simulation allow to verify the robustness of thedifferent algorithm, we have to consider the following componentcombination on the time series to obtain the test scenario.

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Stochastic Punctual Variations Trend Components

The Simulator carries out the tests on different replications and itmakes an average of the obtained results on the different scenarios.

Trend hypothesis on DemandTime

SeriesStochastic DemandVariability

Time series

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Advanced Forecasting vs M&S

�Advanced techniques can improve theCapability to Predict a Real System respecttraditional methods

�Modeling & Simulation is still necessary tomeasure robustness and efficiency of theForecasting System

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Forecasting CostsForecasting Costs� Management that not pays attention to forecasting

assumes that what will happen in the future is thesame to what has happened in the past.

� An inadequate forecasting increase:

�Work

�Materials

�Costs (Expediting or Stock - Out ones)

� Increasing forecasting we increase cost ofcollecting and analyzing data.

� The optimal level cost of forecasting implies thebalance of the aspects above.

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Generic ModelsGeneric Models� ARIMA (Autoregressive Integrated Moving

Average) was introduced by Box and Jenkins in1970, who supposed to eliminate the trend andseasonal component in a time series bydifferencing.

� ARMA (Autoregressive Moving Average) modelsa time series that was eliminated trend andseasonal components by differencing.

� ARMAX (Autoregressive Integrated MovingAverage with exogenous input) introduced by Boxand Tiao in 1976 consider the influence ofexogenous variable on the output variable ofinterest.

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State Space MethodsState Space Methods� The State Space Method model separately each

term of a time series 0bservation as :

�Trend Term

�Seasonal Term

�Regression Term

�Disturbance Term

� The State Space Method puts together eachmodels of components in order to obtain a singlemodel for time series analysis.

� The State Space began in 1960 with Kalman.

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Forecasting in Logistics

In logistics and supply chain management Forecasting is used inorder to:

-Estimate the Customer Demand

-Estimates Market Evolution

-Support Inventory Management

-Support Production Planning

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Logistics & Forecasts

In logistics forecasting is really important to predict thefuture demand.In this scenario is really important to determinate where toget the data for the forecasts for the different actors involvedin supply chain.

As a study of Procter & Gamble the further away your data is from the point of sale, the more data accuracy decreased and forecasting errors increases .

In logistic forecasting is necessary use point of sale formationto directly from the retailer to avoid overforecasting demand.

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Supply Chain & ForecastsDemand Forecasts are a particularly important piece ofinformation within a supply chain.

• Each level of a supply chain makes decisions that haveramifications throughout the entire system.

• The quality of a given decision depends on what the decisionmaker knows.

• As a result, the dissemination of accurate information is criticalfor the supply chain to operate effectively.

• Credibility is a key factor in the exchange of information: Willand should the receiver of information trust the veracity of thereported information.

• While credibility is easily established in some cases, it is oftenmore elusive.

• This is especially true when the informed party has an incentiveto distort her message to influence the receiver’s actions.

Gérard P. Cachon, The Wharton School, University of Pennsylvania,Martin A. Lariviere, Kellogg Graduate School of Management, Northwestern University

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Example of CPFR

Collaborative Planning,Forecasting and Replenishment

CPFTRSyncra forP&G2004

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Applications for Forecasting

To Forecasts the Future Forecasting Algorithms are requested.Many different applications (Weather, Economics, Logistics, etc.)

There are several Forecast Algorithms types, many of them arecustomizable, but in the developing phase for all of them thesetting is a really critical issue.

First Order Exponential Smoothing in 1944 was Implemented by Brown and Itwas promoted and used in 1959/1962

Weather Market Logistics Power Medicine Military

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Algorithm Efficiency

The Forecast Algorithm effectiveness has to be measured onstatistic benefits in a medium long scenario not on the precisionon a single point

To use correctly a forecast algorithm is necessary to consider:

- Implementation Methodologies

-Algorithm Robustness

-Management and Procedure consolidation

-Parameter Proper Setting

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Performance/Process ControlConsidering the scenario stochastic nature and variability it isreally important to guarantee a Control System for thePerformance and for the Process.

The Control System has to verify and correct Forecast SystemSettings:- To develop correct time series analysis and synthesis (i.e. Representative

Samples, various mix, errors/ periods with not representativeconsumption)

- To estimate Costs, Benefits and Risks of different Policies (i.e.Warehouse Costs, Stock-Out Risks, Service Quality)

- To develop a forecasting system easy to manage and to maintain (i.e.fitting of forecast algorithm)

- To set an implementing plan in order to control the advancement

- To measure Performances and critic event

- To control the Start - Up and the operative management

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Algorithm’s Selection

In a case study in partnership with one of the most importantItalian Logistic Company that was implementing a new ERPSystem we used DICOSAP Simulator in order to to select theRobustness Algorithm and the parameter setting .

In the specific we used:

- A time series Analyzer in order to detect errors/ periods withnot representative consumption

- A simulator to select the best algorithm and the parametersettings

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DICOSAP Depots & Inventory Clustering Optimal Setting, Analysis & PlanningDepots & Inventory Clustering Optimal Setting, Analysis & Planning

� Fit the parameters of each forecastalgorithms

� Choose the best algorithms

� Define the best algorithm and fit theparameters for different clustering

� Simulate stochastically the time seriesto estimate robustness of algorithms

� test the results on n-replications

� Define the seasonal period on timeseries

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To analyze the effect ofpromotional policies on thecentral warehouse, we usemodel in order to collect allthe historical data about theDepartures and the Arrivalsflows and to analyze them.

Not representativeevent’s Detection

The Model consider inventory management hypothesis on differentscenarios, with days and/or week periods, cutting off the eventsrelated to values not compatible in a statistic way with the timeseries and evaluating the variance on stochastic factor withMontecarlo techniques.

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The affordability and the coherence of the inputsare important issues to carry out a good forecast.

The DICOSAP Simulator was used over a Retail Case Study for:

• Elaborating the consumption historical data of over 15,000 items in a 48months scenario

• Optimal Best Fitting for the parameters ha of 15 algorithms

• Optimal Algorithm Identification for each item for week/day scenarios

Total elaboratingtime: 3 days on 33computers in alocal LAN

Input Importance

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�Item

�Supplier

�Lead Time

�Safety Stock

�Stock -Out Number

�Rotation Index

�Overall

�Best Supply Policy

�Parameter Fitting of the selected algorithm

DICOSAP Simulator

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OVERALL: is the measure of the algorithm excellenceconsidering the warehouse fees and stock - out costs withappropriate weight.

In the case of critical products, the strategicimportance define that Stockout and WarehouseWeights Factors; usually Stockout is moreimportant then Fees.

Evaluation Criteria

∑∑=

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tOperazionsstHandlingCotevelInventoryLTf

feescoeffeesWarehouseF

1 0

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−⋅=Np

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For each item, the parameters of every algorithms arevisible in a window or in a file. In the output there aresafety stock level, and the coefficient and parametervalues.

Using automaticoptimizationmethodologies themodel select the mostrobust algorithm.

Parameter Fitting

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ALGOALGO: File with the best algorithm and its parameter fitting

SIMULSIMUL: File with simulation results in term ofconsumption, economic quantification , rotation index,stock -out for each simulation run

SUMMASUMMA: file with season period, the consumption with ashort/long scenario,quantification of fees, stockout andoverall

For each item simulator analyses all data.

DICOSAP Outputs

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•Moving Average

•Weighted MovingAverage

•Single Exp. Smoothing

•Double Exp. Smoothing

•Triple Exp. Smoothing

Forecast Algorithm Examples

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Moving Average AlgorithmThe Moving Average consider a time series before the period topredict and use the historical data to carry out the forecast forthe next period.

For each forecasting step the Moving Average consider the lasthistorical data and delete the oldest one, calculating the new ofaverage on the new time series.

This algorithm allow to smooth the irregular behavior on timeseries.

Order = number of periods

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Weighted Moving AverageThis algorithm use the same principle of the simple MovingAverage , The Weighted Moving Average introduce weightingfactors for each historical data.

Typically, the weights are decreasing for the most remote data.In this algorithm the more weight is assigned to more recentdata.

Otherwise it is possible use different criteria to attribute theweight (i.e. seasonal).

WMA(3) = Pt+1 = p1(X t) + p2(X t-1) + p3(X t-2)

Third Order Weighted Moving Average

pi = weight

Pt +1 = forecast

X t = time series

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Single Exp. SmoothingThe most fundamental aspect for this algorithm is the recent timeseries give a best support to make assumption on the futurebehavior. Using this algorithm it is possible reduce the number ofnecessary data for forecasting ad so reduce the database size.

0 ≤ α ≤ 1

High values of α means a major weightattributed to time series

P= Forecast

X=Time Series

Low αValue

High αValue

ttttt PXXPfP )1()(1 αα −+==+

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Double Exp. Smoothing

The Double Exponential Smoothing use two coefficients α e β,making forecasts in two steps: a Basic Value (St) and a Trend Value(Gt), in order to consider also trend effects.

S1 e G1 can be estimated on the time series or usingregression methodology. The α e β factors can be

optimized using algorithms.τ = interval time

))(1( 11 −− +−+= tttt GSXS αα

11 )1()( −− −+−= tttt GSSG ββ

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Triple Exp. SmoothingThis algorithm consider three components to carry out forecasting :

- a Basic Value St

- a Trend Value Gt

- a Seasonal Value Ct

The lack of seasonal effect is the most important critical issue ofDouble Exp. Smoothing

τ=Analysis Period

S = seasonal period

))(1( 11 −− +−+= tttt GSXS αα

11 )1()( −− −+−= tttt GSSG ββ

stt

tt C

S

XC −−+= )1( γγ

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Parameter OptimizationThere are several methodologies to calculate the forecast accuracy ofthe different algorithms. The most common used are:

Total Error Minimization

M.A.D. : Mean Absolute Deviation

M.S.E. : Mean Squared Error

The algorithm forecast accuracy is not measured on the minimizationof the difference between forecast and actual value but on therobustness.

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Algorithm Comparison

Each model has specific features; we have to pay attention aboutthe Robustness, that is the capability to suggest optimum resultseven if there are aleatory components to disturb the scenario.

It is necessary to guarantee a management and accurate settingsof the forecast algorithms parameters in order to represent thereality evolution.

Growing

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Forecast vs. Time SeriesMoving Average Examples

The Forecast inertia increase withthe increasing of the period number.

Moving Average

Periods

Con

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Time Series

2° Moving Average

4° Moving Average

8° Moving Average

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Forecast vs. Time SeriesExp. Smoothing Examples

α

With α increasing, last valuesinfluence is more sensible on forecast

Periods

Con

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Time Series

Single Exp. Sm. 0.2

Single Exp. Sm. 0.5

Single Exp. Sm. 0.8

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The different types of Exp.Smoothing allowto consider trend and seasonal behavior

Forecast vs. Time SeriesSingle vs. Double vs. Triple

Periods

Con

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Time Series

Single Exp. Sm.

Double Exp. Sm.

Triple Exp. Sm.

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Seasonal Period Analyses

A prior knowledge of a probable seasonal period in the timeseries allow to know better the studying scenario and to carryout strategic forecast on the real case study at the same time.

For some Forecast algorithms (i.e. Triple ExponentialSmoothing) the seasonal period of time series is an input data.

We has estimate the seasonal period converting the time seriesin the frequency range.

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Example from a Realistic Case

Typically real behaviors with strong stochasticcomponent and high values of Standard Deviationrespect the mean value

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Period Analysis

In order to identify a priori possible seasonal behaviors on thedemand it is useful to acquire knowledge related to theprocesses under analysis and to its characteristics.

Some predictive algorithms (i.e. Triple Exponential Smoothing)have parameters considering the periodic component in timeseries.

By a frequency analysis based on Fourier transform it ispossible to identify the impact of periodic components.

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Autocorrelation Analysis

Most Common Periods

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Autocorrelation analysis allows to identify the most significantperiodic component over a time series.

These figures are related to the case studyproposed about the demand of frozen goods.These figures consider the week withoutSundays (due to the delivery policies in use) itis evident that six days and multiple periodsare the most common and significant periodiccomponent.

In this figure the same representation isproposed considering weeks including all thedays; in this case the most significant periodis seven days and its multiples so is evidentthat consumption of these goods arecharacterized by week periodic behavior.

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Autocorrelation Example

Autocorrelazione ITEM

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Autocorrelation Analysis

• Autocorrelation highlights thepresence of periodic components

• In this case the behavior seemsincluding high randomcomponents (noise)

• Autocorrelation have is maxvalue in correspondence with 7days. However this value is just alittle bit higher than other timeshifts

• Zoom on a time windows it isevident that the week periodiccomponent is present but not toomuch significant

Real Consumption

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Stochastic and PeriodicComponents

• Define the periodic componentinfluence over the time series.

• Main Frequency in this casereproduces just qualitatively theoverall behavior.

• It is evident that in the proposed casethe combination of first fivefrequencies have very lowreproduction capability over thishistorical behavior.

• The overall periodic behavior can bereconstructed just by adding all thehigh frequency components (noise).

Real Demand and Main Frequency

Real Demand and First 5 Frequencies

Total Time Series Reconstruction with all the Frequencies

By Fourier Analysis it is possibleto:

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Time Series & Frequencies

0.001%

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Different PeriodicComponents Analysis

Using the Fourier Series Methodology we identify the frequency’sinfluence determined on the each time series behavior.

In our case, even if thesix and seven daysfrequencies are themost significancetogether the semesterand the annual one,their influence is lessimportant thenaleatory components inthe time series (ALTRO).

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The M.A.D. CalculationWe have choose the M.A.D. as optimization parameter because in acomplex scenario it allows to obtain number easier to manage.

In the case study proposed the M.A.D.calculation was made in twodifferent ways:

� Lead Time M.A.D.: the forecast is carried out on the Lead Timeand the delta among forecast and the time series are calculated.

� Future M.A.D.: the time orison is divided in two part, the firstpart is the total scenario less then the lead time and the other oneis the lead time. On the first part the algorithm is optimized and onthe second part the selected optimum one is tested.

The Future M.A.D. emphasize in a clear way the real performance ofthe selected algorithm.

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The different M.A.D.

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• Mean MAD in Lead Timeover Complete Time Series

• Parameters Optimized onMAD over All Data

• Tested and Measured overAll Data

• Mean MAD in Last LeadTime

• Parameters Optimized onMAD over “Old” Data

• Tested and Measured over“New” Data

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Complexity of the Frameworks

It is important to state the complexity of the framework to beanalyzed.

To identify the complexity of the problem we can have an hypothesisabout the case related to frozen goods over a regional area with about30 millions inhabitants involving about 1470 types of item from 90suppliers organized in categories 96 with 33 subcategories and 11product segments.

Demand is based on historical data about quantities delivered dailyover the supply chain.

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ExerciseExercise

� The proposed exercises is related to the Demandestimation of books

� In this Application case we need to identify howsimulation can support forecasting and whatresults we can obtain

� Scenario Characteristics:�Book Demand

�1242 Books Types

�6 Books Categories

�6 Books Subcategories

�6 Books Segments

�6 Books Suppliers

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Simulation SettingsSimulation Settings

� Simulation in this exercise will be used to:

�Reconstruct Scenarios from Historical Databased on stochastic components

�Estimate robustness of algorithms to demandobscillations

� As Simulationists we need to:

�Identify the Variability and Trend to be addedto historical data

�Identify the number of Simulation RunsRequired to estimate overall performances

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Trend & Trend & NoiseNoise� The analysis of the historical data plus the expert

knowledge in this case allows to identify the bestsetting:

�Trend Usually have to be estimated based onlast significant period trend analysis(i.e.regression) plus expert expectations

�Noise could be based on Standard deviationcollected over an analysis window related toa significant periodplus expertexpectation

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ReplicationsReplications

� The simulation requires N replications changingthe random seeds in order to provide stableperformance; this could be estimated by applyingMean Square pure Error Temporal EvolutionAnalysis. Mean Square pure Error of TF

0.00E+00

1.00E+11

2.00E+11

3.00E+11

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Replications

MS

pE

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E 1

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Periodic ComponentPeriodic ComponentIdentification Identification && Evaluation Evaluation

� In order to complete this part we will:

�Take a Look on Item Demand over time

�Proceed to Autocorrelate each Item Demand

�Identification of Most Significant Period

�Proceed to Fourier Analysis on Each Item

�Identification of Periodic Component Influences

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Best ClusteringBest Clustering

� Another important result is to use simulation inorder to group in the best ways items.

�Optimizing each grouping opportunity on bestalgorithms will provide an estimation of bestgrouping under different hypotheses

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Best AlgorithmBest Algorithm

� For each Group it is possible to use simulation tomeasure overall performances and robustness for:

�identification of best algorithm

�best fitting of parameters for best algorithm

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ConclusionsConclusions

� Simulation can improve performances of forecastingtechniques by:�Testing Hypotheses about the real System�Testing a priori their robustness� Identify best Management Policies� Identify best Algorithms and Fitting their Parameters�Evaluating the Operative Margins versus Predictability

Levels

� Simulation can’t provide:�Better Estimations on the Models than on real System�Crystal Ball on the Future�Correct Prediction in Random Systems

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Amico Vince, Guha R., Bruzzone A.G. (2000) "Critical Issues in Simulation", Proceedings of Summer Computer Simulation Conference, Vancouver, JulyBox, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994). "Time Series Analysis,Forecasting and Control" 3rd edition. San Francisco: Holden-DayBrockwell, P.J. and Davis, R.A. (1987). "Time Series: Theory and Methods". New York: Springer-Verlag.Bruzzone A., Mosca R., Orsoni A., Revetria R. (2001) "Forecasts Modelling in Industrial Applications Based on AI Techniques", International Journal of

Computing Anticipatory Systems (extended from Proeedings of CASYS2001, Liege Belgium August 13-18), Vol 11, pp. 245-258, ISSN1373-5411

Bruzzone A.G., Briano C., Brandolini M. (2000) "Forecast Modelling integrated with ERP Systems", Proceedings of HMS2000, Portofino, October 5-7Bruzzone A.G., Giribone P. (1998) "Decision-Support Systems and Simulation for Logistics: Moving Forward for a Distributed, Real-Time, Interactive

Simulation Environment", Proceedings of the Annual Simulation Symposium IEEE, Boston, 4-9 AprilBruzzone A.G., Giribone P. (1998), "Robust and Central Composite Design as Collaborative Techniques for Production Planning using Simulation",

Proceedings of Eurosim98, Helsinki, April 14-17Bruzzone A.G., Giribone P., Mosca R. (1996) "Simulation of Hazardous Material Fallout for Emergency Management During Accidents", Simulation, vol.

66, no.6, June, 343-355Bruzzone A.G., Giribone P., Revetria R., Solinas F, Schena F. (1998) "Artificial Neural Networks as a Support for the Forecasts in the Maintenance

Planning", Proceedings of Neurap 98, Marseilles, 11-13 MarchBruzzone A.G., Kerckhoffs (1996) "Simulation in Industry", Genoa, Italy, October, Vol. I & II, ISBN 1-56555-099-4Bruzzone A.G., Merkuryev Y.A., Mosca R. (1999) "Harbour Maritime & Industrial Logistics Modelling & Simulation", SCS Europe, Genoa, ISBN 1-

56555-175-3Bruzzone A.G., Mosca R. (1998) "Special Issue: Harbour and Maritime Simulation", Simulation, Vol.71, no.2, AugustBruzzone A.G., Mosca R. (1999) "Modelling & Simulation And Erp Systems For Supporting Logistics In Retail", Proceedings of ESS99, Erlangen,

OctoberBruzzone A.G., Mosca R., Revetria R. (2002) "Supply Chain Management Dynamic Negotiation using Web Integrated Logistics Designer (WILD II)",

Proceedings of MAS2002, Bergeggi October 3-5Bruzzone A.G., Revetria R. (1999) "Artificial Neural Networks as Support for Logistics in Super-Market Chains", Proceedings of HMS99, Genoa,

September 16-18Bruzzone A.G., Revetria R. (2000) "FRINE: Forecasts Robust Intelligent Evaluator", Marconi Communication Report, Genova, NovemberBruzzone A.G., Revetria R. (2003) "Simulation as Support for Contract Negotiation", Proceedings of ASTC2003, Orlando FL USA, AprilBruzzone A.G., Simeoni S., Briano C., Brandolini M. (2002) "Chaotic System Study In Process Plants By Using Simulation", Proceedings of AI2002,

Innsbruk, February 18-21Burman, J.P. (1980). "Seasonal adjustment by signal extraction", J.Royal Statistical Society A, 143,321-37Chatfield C.(2003) "The Analysis of Time Series: An Introduction",CRCde Jong P. and Shephard, N. (1995). "The simulation smoother for time series models", Biometrika, 82, 339-50

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Application Of Neural Models", Proceedings of 1st World Congress on Systems Simulation, Singapore, September 1-4Giribone P. & A.G.Bruzzone (1995) "Chaos Theory: a Coal Terminal Design Application", Proceedings of MIC95, Innsbruck, February 20-23Giribone P., A.G.Bruzzone (1994) "Analysis through Simulation of Unstable Variables based on Chaos Theory: Warehouse Management in a Port

Terminal", Proceedings MIC94, Grindelwald, February 21-23Giribone P., Bruzzone A.G. & Tenti M. (1996) "Local Area Service System (LASS): Simulation Based Power Plant Service Engineering & Management",

Proc. of XIII Simulators International Conference, New Orleans LA, April 8-11Granger, C.W.J. and Newbold, P. (1986). "Forecasting Economic Time Series" 2nd edition Orlando : Academic PressHamilton J.D. (2003) "Time Series Analysis", Princeton University pressHarvey A.C. (1989). "Forecasting, Structural Time Series Models and the Kalman Filter" Cambridge University PressKalman, R.E. (1960). "A new approach to linear filtering and prediction problems", J.Basic Engineering, Transactions ASMA, Series D, 82, 35-45Lambert D.M., Stock J.A., Ellram L.M. (1998) "Fundamentals of Logistics Management", McGraw Hill, NYCMakridakis, S., Wheelwright, S.C. and Hyndman, R.J. (1998). "Forecasting: Methods and Application" 3rd edition. New York: Wiley and SonsMerkuriev Y., Bruzzone A.G., Novitsky L (1998) "Modelling and Simulation within a Maritime Environment", SCS Europe, Ghent, Belgium, ISBN 1-

56555-132-XMills, T. C. (1993). "Time Series Techniques for Economists" 2nd edition. Cambridge: Cambridge University PressMonks J.G. (1996) “Operations Management” 2nd edition. New York: McGraw Hill.Mosca R., Bruzzone A.G. (1997) "Simulation as a Support for Customer Satisfaction-Oriented Planning", Proceedings of Simulators International XIV,

SMC'97, Atlanta, Georgia, April 6-10Naadimuthu G., Bronson R. (1997) “Operations Research” 2nd edition. New York: McGraw Hill.Reinsel C.G. (1997) “Elements of Multivariate Time Series Analysis ”. 2nd edition New York: SpringerSage, A.P. and Melsa, J.L.(1971). "Estimation Theory with Application to Communication and Control". New York: McGraw Hill.Shephard, N. (1993). "Fitting non-linear time series models, with aplications to stochastic variance models", J.Applied Econometrics, 8, S135-52Shumway, R.H. and Stoffer,D.S.(2000). "Time Series Analysis and Its Applications". New York: Springer-VerlagTaylor, S.J. (1986). "Modelling Financial Time Series". Chichester: John Wiley.Theil, H. and Wage, S. (1964). "Some observation on adaptive forecasting, Management Science", 10, 198-206


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