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

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EARLY FORECASTING

• Maltese Stone Age HunterGatherer used Mnajdra
  to forecast seasons
• (among other purposes)

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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)
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APPLICATION AREAS

• MEDICAL
• MILITARY
• MANAGEMENT
• FINANCE
• ASTRONOMY
• DEMOGRAPY

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TIME SERIES TECHNIQUES

• LINEAR
• VS
• NON LINEAR

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LINEAR TECHNIQUES

• Linear methods try to model closely underlying
  subsystems
• Require identification measurement of several
  system features - seasons, trends, cycles,
  outliers

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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

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BASIC STEPS TO FORECASTING

• COLLECT DATA
• EXAMINE DATA
• PREPROCESS DATA
• OPTIMIZE PARAMETERS
• APPLY PREDICTION TECHNIQUES
• MEASURE PREDICTION ERROR
• REVIEW AND UPDATE

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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

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DATA CREATION

• Created function X(t) vs t
• Where t DISCRETE VALUES OF TIME (1..3000)
• And X(t) A1 SINE (t F1) A2 COS (t
  F2) Random () N
• Amplitude A1 0.1 , Frequency F1 5
• Amplitude A2 0.2 , Frequency F2 0.33
• Noise factor N 3

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UNDERLYING SUBSYSTEMS
SINE FUNCTION COSINE FUNCTION
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MEASUREABLE SIGNALSUBSYSTEMS NOISE
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SEPARATE THE DATA
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FUTURE SET (HIDDEN FROM SYSTEM)
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EXAMINE DATA

• VISUAL INSPECTION
• STATIONARITY
• PHASE SPACE MAPPING
• AUTOCORRELATION
• LYAPUNOV EXPONENT
• DELAY SPACE EMBEDDING
• MINIMAL EMBEDDING DIMENSION

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PHASE STATE
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PHASE SPACE MAP
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AUTO CORRELATION SUM
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MAX LYAPUNOV EXPONENT
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TIME DELAY EMBEDDING THE ATTRACTOR
DIMENSIONS 100
TIME DELAY 1
PREDICTOR POINT
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PREPROCESSING DATA

• FILTERING
• NOISE REDUCTION
• TEMPORAL ABSTRACTIONS
• CATEGORIZE ETHERNET PACKETS BY SIZE
• CATEGORIZE ECG SIGNALS BY TYPE

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NON LINEAR NOISE REDUCTION
Noise reduced by 8
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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

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COMPARE ATTRACTOR ALONG TRAINING SET
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FINDING A NEIGHBOUR
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FIND ALL NEIGHBOURS OF SELECTED POINT
ID 9, M9 Err 2 NEIGHBOURS FOUND 15
Neigbour Time point Neighbor Time point
1 2523 9 1769
2 2711 10 1770
3 447 11 1956
4 1013 12 1958
5 1768 13 2145
6 1203 14 2334
7 1392 15 2335
8 1581
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FIND PREDICTED SET FOR NEIGHBOUR
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FINAL PREDICTION
PREDICTION SETS OF ALL NEIGHBOURS
AVERAGE of PREDICTION SETS
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FIRST PREDICTION ATTEMPT
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NEED TO VARY PARAMETERS
I
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EXAMINE MORE CLOSELY
I
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A BETTER ATTRACTOR
Time Delay 9, Dimensions 9
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A BETTER PREDICTION
Delay9,Dim9,Err2 neighb15,rms 1.09
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CHANGE DELAY, DIMENSIONS
Delay1,Dim20,Err2 neighb1,rms 1.37
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CHANGE DISTANCE
Delay1,Dim20,Err3 neighb34,rms 1.09
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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.

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THE END

• (AS FORECAST)