FORECASTING USING NONLINEAR TECHNIQUES IN TIME SERIES ANALYSIS

EARLY FORECASTING Maltese Stone Age Hunter–Gatherer used Mnajdra to forecast seasons (among other purposes)

FORECASTING TODAY Data/Images acquired at EOS/OCS, CIF-US, Universidad de Sonora, Mexico. Observer(s): M. C.

APPLICATION AREAS § § § § MEDICAL MILITARY MANAGEMENT FINANCE ASTRONOMY DEMOGRAPY … Forecasting

TIME SERIES TECHNIQUES LINEAR VS NON LINEAR Forecasting using Non Linear Techniques in Time

LINEAR TECHNIQUES § Linear methods try to model closely underlying subsystems § Require identification

NON LINEAR TECHNIQUES § Non-linear techniques exploit measurement data and computer power: § Mimic

BASIC STEPS TO FORECASTING § § § § COLLECT DATA EXAMINE DATA PREPROCESS DATA

A PRACTICAL EXAMPLE § CREATE OWN DATA SET (3000 pts) WITH RANDOM NOISE §

DATA CREATION Created function X(t) vs t Where t = DISCRETE VALUES OF TIME

UNDERLYING SUBSYSTEMS SINE FUNCTION + COSINE FUNCTION Forecasting using Non Linear Techniques in Time

MEASUREABLE SIGNAL SUBSYSTEMS + NOISE Forecasting using Non Linear Techniques in Time Series Analysis

SEPARATE THE DATA Forecasting using Non Linear Techniques in Time Series Analysis – Michel

FUTURE SET (HIDDEN FROM SYSTEM) Forecasting using Non Linear Techniques in Time Series Analysis

EXAMINE DATA § § § § VISUAL INSPECTION STATIONARITY PHASE SPACE MAPPING AUTOCORRELATION LYAPUNOV

PHASE STATE Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri

PHASE SPACE MAP Forecasting using Non Linear Techniques in Time Series Analysis – Michel

AUTO CORRELATION SUM Forecasting using Non Linear Techniques in Time Series Analysis – Michel

MAX LYAPUNOV EXPONENT Forecasting using Non Linear Techniques in Time Series Analysis – Michel

TIME DELAY EMBEDDING THE ATTRACTOR DIMENSIONS = 100 TIME DELAY = 1 PREDICTOR POINT

PREPROCESSING DATA § § FILTERING NOISE REDUCTION TEMPORAL ABSTRACTIONS CATEGORIZE ETHERNET PACKETS BY SIZE

NON LINEAR NOISE REDUCTION Noise reduced by 8 % Forecasting using Non Linear Techniques

APPLY PREDICTION TECHNIQUE § Set initial parameters – Time delay, dimensions, distance, box size,

COMPARE ATTRACTOR ALONG TRAINING SET Forecasting using Non Linear Techniques in Time Series Analysis

FINDING A NEIGHBOUR Forecasting using Non Linear Techniques in Time Series Analysis – Michel

FIND ALL NEIGHBOURS OF SELECTED POINT ID 9, M=9 Err= 2 NEIGHBOURS FOUND 15

FIND PREDICTED SET FOR NEIGHBOUR Forecasting using Non Linear Techniques in Time Series Analysis

FINAL PREDICTION SETS OF ALL NEIGHBOURS AVERAGE of PREDICTION SETS Forecasting using Non Linear

FIRST PREDICTION ATTEMPT Forecasting using Non Linear Techniques in Time Series Analysis – Michel

NEED TO VARY PARAMETERS I Forecasting using Non Linear Techniques in Time Series Analysis

EXAMINE MORE CLOSELY I Forecasting using Non Linear Techniques in Time Series Analysis –

A BETTER ATTRACTOR Time Delay = 9, Dimensions = 9 Forecasting using Non Linear

A BETTER PREDICTION Delay=9, Dim=9, Err=2 neighb=15, rms = 1. 09 Forecasting using Non

CHANGE DELAY, DIMENSIONS Delay=1, Dim=20, Err=2 neighb=1, rms = 1. 37 Forecasting using Non

CHANGE DISTANCE Delay=1, Dim=20, Err=3 neighb=34, rms = 1. 09 Forecasting using Non Linear

PROCESSING CONSIDERATIONS § Multiple attempts at prediction, calculation of invariants, noise reduction, require increasing

THE END (AS FORECAST) Forecasting using Non Linear Techniques in Time Series Analysis –

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FORECASTING USING NON-LINEAR TECHNIQUES IN TIME SERIES ANALYSIS AN OVERVIEW OF RELATED TECHNIQUES AND MAIN ISSUES Michel Camilleri Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 1

EARLY FORECASTING Maltese Stone Age Hunter–Gatherer used Mnajdra to forecast seasons (among other purposes) Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 2

FORECASTING TODAY Data/Images acquired at EOS/OCS, CIF-US, Universidad de Sonora, Mexico. Observer(s): M. C. Marianna Lyubarets Non linear time series techniques are being used to to forecasting sun spot activity (among other uses) Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 3

APPLICATION AREAS § § § § MEDICAL MILITARY MANAGEMENT FINANCE ASTRONOMY DEMOGRAPY … Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 4

TIME SERIES TECHNIQUES LINEAR VS NON LINEAR Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 5

LINEAR TECHNIQUES § Linear methods try to model closely underlying subsystems § Require identification & measurement of several system features - seasons, trends, cycles, outliers Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 6

NON LINEAR TECHNIQUES § Non-linear techniques exploit measurement data and computer power: § Mimic dynamic system without having to understand exactly the underlying processes § Better results than Linear in certain areas Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 7

BASIC STEPS TO FORECASTING § § § § COLLECT DATA EXAMINE DATA PREPROCESS DATA OPTIMIZE PARAMETERS APPLY PREDICTION TECHNIQUES MEASURE PREDICTION ERROR REVIEW AND UPDATE Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 8

A PRACTICAL EXAMPLE § CREATE OWN DATA SET (3000 pts) WITH RANDOM NOISE § SEPARATE TRAINING SET, ATTRACTOR, FUTURE (HIDDEN SET) § EXAMINE DATA § PREPARE DATA § PREDICT § MEASURE SUCCESS OF PREDICTION § OPTIMISE PARAMETERS Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 9

DATA CREATION Created function X(t) vs t Where t = DISCRETE VALUES OF TIME (1. . 3000) And X(t) = A 1 * SINE (t * F 1) + A 2 * COS (t * F 2) * Random () * N Amplitude A 1 = 0. 1 , Frequency F 1 = 5 Amplitude A 2 = 0. 2 , Frequency F 2 = 0. 33 Noise factor N = 3 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 10

UNDERLYING SUBSYSTEMS SINE FUNCTION + COSINE FUNCTION Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 11

MEASUREABLE SIGNAL SUBSYSTEMS + NOISE Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 12

SEPARATE THE DATA Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 13

FUTURE SET (HIDDEN FROM SYSTEM) Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 14

EXAMINE DATA § § § § VISUAL INSPECTION STATIONARITY PHASE SPACE MAPPING AUTOCORRELATION LYAPUNOV EXPONENT DELAY SPACE EMBEDDING MINIMAL EMBEDDING DIMENSION Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 15

PHASE STATE Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 16

PHASE SPACE MAP Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 17

AUTO CORRELATION SUM Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 18

MAX LYAPUNOV EXPONENT Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 19

TIME DELAY EMBEDDING THE ATTRACTOR DIMENSIONS = 100 TIME DELAY = 1 PREDICTOR POINT Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 20

PREPROCESSING DATA § § FILTERING NOISE REDUCTION TEMPORAL ABSTRACTIONS CATEGORIZE ETHERNET PACKETS BY SIZE § CATEGORIZE ECG SIGNALS BY TYPE Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 21

NON LINEAR NOISE REDUCTION Noise reduced by 8 % Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 22

APPLY PREDICTION TECHNIQUE § Set initial parameters – Time delay, dimensions, distance, box size, number of future steps ahead § Choose measure of success and apply it to output (Various) § Find optimal set of parameters Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 23

COMPARE ATTRACTOR ALONG TRAINING SET Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 24

FINDING A NEIGHBOUR Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 25

FIND ALL NEIGHBOURS OF SELECTED POINT ID 9, M=9 Err= 2 NEIGHBOURS FOUND 15 Neigbour 1 2 Time point 2523 2711 Neighbor 9 10 Time point 1769 1770 3 4 5 6 7 8 447 1013 1768 1203 1392 1581 11 12 13 14 15 1956 1958 2145 2334 2335 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 26

FIND PREDICTED SET FOR NEIGHBOUR Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 27

FINAL PREDICTION SETS OF ALL NEIGHBOURS AVERAGE of PREDICTION SETS Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 28

FIRST PREDICTION ATTEMPT Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 29

NEED TO VARY PARAMETERS I Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 30

EXAMINE MORE CLOSELY I Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 31

A BETTER ATTRACTOR Time Delay = 9, Dimensions = 9 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 32

A BETTER PREDICTION Delay=9, Dim=9, Err=2 neighb=15, rms = 1. 09 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 33

CHANGE DELAY, DIMENSIONS Delay=1, Dim=20, Err=2 neighb=1, rms = 1. 37 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 34

CHANGE DISTANCE Delay=1, Dim=20, Err=3 neighb=34, rms = 1. 09 Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 35

PROCESSING CONSIDERATIONS § Multiple attempts at prediction, calculation of invariants, noise reduction, require increasing orders of operations § Each operation may require comparison of every point on attractor with respective points for each training point. § Number of operations to find neighbours can be reduced by comparing attractor only to points in same phase state e. g. Box or Tree assisted neighbour search in Phase space. Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 36

THE END (AS FORECAST) Forecasting using Non Linear Techniques in Time Series Analysis – Michel Camilleri – September 2004 37