Wavelet Multi-scale Analysis of High Frequency FX Rates

Wavelet Multi-scale Wavelet Multi-scale Analysis of High Analysis of High

Frequency FX RatesFrequency FX RatesSaif Ahmad

Department of Computing

University of Surrey, Guildford, UK

August 27, 2004

Intelligent Data Engineering and Automated Learning - IDEAL 2004

5th International Conference, Exeter, UKSeries: Lecture Notes in Computer Science , Vol. 3177

http://www.springer.com/east/home/computer/database?SGWID=5-153-69-173621324-0

Talk OutlineTalk Outline Describing Time Series Data Financial Time Series Data Characteristics Wavelet Multiscale Analysis Our Time Series Analysis Approach - Algorithms - Prototype System - Case Study - Conclusions Questions

What Is a Time Series?What Is a Time Series?

A chronologically arranged sequence of data on a particular variable

Obtained at regular time interval Assumes that factors influencing past and

present will continue

U.S. Retail SalesU.S. Retail SalesQuarterly DataQuarterly Data

200

250

300

350

400

450

83 84 85 86 87

Year

Sal

es (B

illio

ns)

Time Series ComponentsTime Series Components

Trend

Seasonal Cyclical

Irregular

TS Data

Trend ComponentTrend Component

Indicates the very long-term behavior of the time series

Typically as a straight line or an exponential curve

This is useful in seeing the overall picture

Cyclical ComponentCyclical Component

A non-seasonal component which varies in a recognizable period

Peak Contraction Trough Expansion

Due to interactions of economic factors The cyclic variation is especially difficult to

forecast beyond the immediate future more of a local phenomenon

Time

Seasonal ComponentSeasonal Component

Regular pattern of up and down fluctuations within a fixed time

Due to weather, customs etc. Periods of fluctuations more regular, hence more

profitable for forecasting

Time

Irregular ComponentIrregular Component

Random, unsystematic, “residual” fluctuations

Due to random variation or unforeseen events

Short duration and non-repeating A forecast, even in the best situation, can be

no closer (on average) than the typical size of the irregular variation

Time Series Data Broken-Down*Time Series Data Broken-Down*

Trend

Seasonal Index

Cyclic Behavior

Irregular

TS Data

*For illustration purposes only.

Financial Time Series Financial Time Series Data CharacteristicsData Characteristics

Evolve in a nonlinearnonlinear fashion over time

Exhibit quite complicated patterns, like trends, abrupt changes, and volatility clustering, which appear, disappear, and re-appear over time nonstationary nonstationary

There may be purely local changes in time domain, global changes in frequency domain, and there may be changes in the variance parameters


305

345

385

425

465

505

545

585

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351

0

0.02

0.04

0.06

0.08

0.1

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351

IBM

Pri

ce

sIB

M V

ola

tili

ty

Nonstationary

Time Varying Volatility

The nonlinearities and nostationarities do contain certain regularities or patterns

Therefore, an analysis of nonlinear time series data would involve quantitatively capturing such regularities or patterns effectively


Having said that…

How and Why?How and Why?

Wavelet Multiscale AnalysisWavelet Multiscale AnalysisOverview Wavelets are mathematical functions that cut up

data into different frequency components and then study each component with a resolution matched to its scale

Wavelets are treated as a ‘lens’ that enables the researcher to explore relationships that were previously unobservable

Provides a unique decomposition (deconstruction) of a time series in ways that are potentially revealing

Signal

Wavelet

C = C1

Step I:Step I: Take a wavelet and compare it to a section at the start of the original signal. Calculate C to measure closeness (correlation) of wavelet with signal

Wavelet Multiscale AnalysisWavelet Multiscale Analysis

Signal

Wavelet

C = C2

Step II:Step II: Keep shifting the wavelet to the right and repeating Step I until whole signal is covered


Signal


Wavelet

C = C3

Step III:Step III: Scale (stretch) the wavelet and repeat Steps I & II

Step IV:Step IV: Repeat Steps I to III for all scales


Discrete Convolution:Discrete Convolution: The original signal is convolved with a set of high or low pass filters corresponding to the prototype wavelet

iitxiwtx*w

Xt Original Signal

W High or low pass filters

Filter Bank Approach


Filter Bank Approach

H (f)

G (f) G* (f)

2 2 H* (f)

2 2

Xt

D1

A1

H: Bank of High Pass filters

G: Bank of Low Pass filters

H (f) – high-pass decomposition filter

H* (f) – high-pass reconstruction filter

G (f) – low-pass decomposition filter

G* (f) – low-pass reconstruction filter

Up arrow with 2 – upsampling by 2

Down arrow with 2 – downsampling by 2

Xt

A1 D1A1

A2 D2A2

A3 D3

L

Level 1

Xt = A1 + D1

Level 2

Level 3

L

L

H

H

L Xt = A2 + D1+ D2

Xt = A3 + D1+ D2 + D3

Level N

Fr e

qu

enc

yF

req

uen

cy

Xt = AN + D1+ D2 + … DN

Iteration gives scaling effect

at each level

Mallat’s Pyramidal Filtering ApproachMallat’s Pyramidal Filtering Approach

Wavelet Multiscale AnalysisWavelet Multiscale AnalysisW

avel

et D

eco

mp

osi

tio

ns

Wav

elet

Dec

om

po

siti

on

s

Fo

uri

er P

ow

er S

pec

tru

m

Fo

uri

er P

ow

er S

pec

tru

m

-1.1E+02

-6.1E+01

-1.1E+01

3.9E+01

8.9E+01

Le

ve

l -

1

-1.5E+02

-5.0E+01

5.0E+01

1.5E+02

2.5E+02

Le

ve

l -

2

-1.5E+02

-1.0E+02

-5.0E+01

0.0E+00

5.0E+01

Le

ve

l -

3

3.8E+03

4.0E+034.2E+03

4.4E+034.6E+03

4.8E+03

5.0E+035.2E+03

FT

SE

10

0

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

FF

T (

1)

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

FF

T (

2)

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

FF

T (

3)

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

FF

T (

FT

SE

)

Analyzing High-frequency Analyzing High-frequency Financial Data: Our ApproachFinancial Data: Our Approach

Tick Data

Preprocessing TransformationKnowledge Discovery Forecast

Data Compression

Multiscale Analysis

PredictionSummarization

AggregateAggregate the movement in the

dataset over a certain

period of time

Use the DWT to deconstructdeconstruct

the series

Describe market dynamics at

different scales (time horizons)

with chief featureschief features

Use the extracted

‘chief features’ to predictpredict

CycleCycle

TrendTrend

Turning PointsTurning Points

Variance ChangeVariance Change


I. CompressCompress the tick data to get Open (O), High (H), Low (L) and Close (C) value for a given compression period (for example, one minute or five minutes).

II. Calculate the level L of the DWT needed based on number of samples N in C of Step I,

L = floor [log (N)/log (2)].

III. Perform a level-L DWTlevel-L DWT on C based on results of Step I and Step II to get,

Di, i = 1, . . ., L, and AL.

III-1. Compute trendtrend by performing linear regression on AL.

III-2. Extract cyclecycle (seasonality) by performing a Fourier power spectrum analysis on each Di and choosing the Di with maximum power as DS.

III-3. Extract turning pointsturning points by choosing extremas of each Di.

IV. Locate a single variance changevariance change in the series by using the NCSS index on C.

V. Generate a graphical and verbal summarysummary for results of Steps III-1 to III-3 and IV.

Generalized Algorithm: Generalized Algorithm: SummarizationSummarization


I. SummarizeSummarize the tick data using the time series summarization algorithm.

II. For a N-step ahead forecast, extend the seasonalextend the seasonal component DS

symmetrically N points to the right to get DS, forecast.

III. For a N-step ahead forecast, extend the trend componentextend the trend component AN linearly N

points to the right to get AN, forecast.

IV. Add the results of Steps II and III to get an aggregateaggregate N-step ahead forecastforecast,

Forecast = DS, forecast + AN, forecast.

Generalized Algorithm: Generalized Algorithm: PredictionPrediction


Raw Signal

VolatilityVolatility

DWTDWT

Statistic

NCSS

Statistic

NCSS

DWTDWT

FFTFFT

Detect Turning

Points and Trends

Detect Turning

Points and Trends

Detect Inherent Cycles

Detect Inherent Cycles

Detect Variance Change

Detect Variance Change

Su

mm

arization

Pred

iction

A prototype systemprototype system has been implemented that automatically extracts “chief features” from a time series and give a prediction based on the extracted features, namely trend and seasonality

Analyzing High-frequency Analyzing High-frequency Financial Data: Our ApproachFinancial Data: Our ApproachA Case StudyA Case Study

Consider the five minutes compressed tick data for the £/$ exchange rate on March 18, 2004

1.82

1.82

1.83

1.83

1.84

0 25 50 75 100 125 150 175 200 225 250 275

0.0

0.2

0.4

0.6

0.8

1.0

0 25 50 75 100 125 150 175 200 225 250 275

Feature Phrases Details

Trend

1st Phase

2nd Phase

Turning Points

Downturns 108, 132, 164, and 178

Upturns 5, 12, 20 36, 68, and 201

Variance Change

Location 164

CyclePeriod 42

Peaks at 21, 54, 117, 181, 215, and 278

260 < t 1.81, + t 5-6.36eTrend1x

288 < t < 261 1.83, + t 6-3.65eTrend2x

Input DataInput Data

Sys

tem O

utp

ut

Sys

tem O

utp

ut

Analyzing High-frequency Analyzing High-frequency Financial Data: Our ApproachFinancial Data: Our ApproachA Case StudyA Case Study

For predictionprediction, we use the ‘chief features’ of the previous day (March 18, 2004), information about the dominant cycle and trend (summarization), to reproduce the elements of the series for the following day (March 19, 2004):

1.82

1.82

1.83

1.83

1.83

0 25 50 75 100 125 150 175 200 225 250

System OutputSystem Output

ActualMarch 19, 2004

Predicted (seasonal + trend)March 19, 2004

Root Means Square Error = 0.0000381

Correlation = + 62.4 %

Analyzing High-frequency Analyzing High-frequency Financial Data: Our ApproachFinancial Data: Our ApproachConclusionsConclusions

I. We have presentedpresented a time series summarization, annotation, and prediction framework based on the multiscale wavelet analysis to deal with nonstationary, volatile and high frequency financial data

II.II. Multiscale analysisMultiscale analysis can effectively deconstructdeconstruct the total series into its constituent time scales: specific forecasting techniques can be applied to each timescale series to gain efficiency in forecastefficiency in forecast

III.III. ResultsResults of experiments performed on Intraday exchange data show promiseshow promise for summarizing and predicting highly volatile time series

IV.IV. Continuously evolvingContinuously evolving and randomly shocked economic systemseconomic systems demand for a more rigorousmore rigorous and extended analysisanalysis, which is being planned

V. Successful analysis of agentsagents operatingoperating on several scalesseveral scales simultaneously and of modeling these componentsmodeling these components could result in more exact forecastsexact forecasts

Questions / CommentsQuestions / Comments

Wavelet Multi-scale Analysis of High Frequency FX Rates

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