Wavelet Multi Resolution Analysis of High Frequency Fx Rates 1203290417290522 5

8/6/2019 Wavelet Multi Resolution Analysis of High Frequency Fx Rates 1203290417290522 5

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Wavelet Multi-resolutionWavelet Multi-resolution

Analysis of HighAnalysis of HighFrequency FX RatesFrequency FX Rates

Department of ComputingUniversity of Surrey, Guildford, UK

August 27, 2004

Intelligent Data Engineering and Automated

Learning - IDEAL 2004

5th International Conference, Exeter, UK


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

q A chronologically arranged sequence of

data on a particular variable

q Obtained at regular time interval

q Assumes that factors influencing past and

present will continue


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U.S. Retail SalesU.S. Retail Sales

Quarterly DataQuarterly Data

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

8 3 8 4 8 5 8 6 8 7

Yea

S

ales(Billions)


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Time Series ComponentsTime Series Components

Trend

Seasonal Cyclical

Irregular

TS Data


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Trend ComponentTrend Component

q Indicates the very long-term behavior of the

time series

q Typically as a straight line or an

exponential curve

q This is useful in seeing the overall picture


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Cyclical ComponentCyclical Component

q A non-seasonal component which varies in arecognizable period

q Peak

q Contractionq Trough

q Expansion

q Due to interactions of economic factors

q The cyclic variation is especially difficult toforecast beyond the immediate future more of alocal phenomenon

Time


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Seasonal ComponentSeasonal Component

q Regular pattern of up and down fluctuationswithin a fixed time

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

profitable for forecasting

Time


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Irregular ComponentIrregular Component

q Random, unsystematic, residualfluctuations

q Due to random variation or unforeseenevents

q Short duration and non-repeating

q

A forecast, even in the best situation, can beno closer (on average) than the typical sizeof the irregular variation


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

Trend

Seasonal Index

Cyclic Behavior

Irregular

TS Data

*For illustration purposes only.


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Financial Time SeriesFinancial Time Series

Data CharacteristicsData Characteristicsq Evolve in a nonlinearnonlinear fashion over time

q Exhibit quite complicated patterns, like trends,abrupt changes, and volatility clustering, whichappear, disappear, and re-appear over timenonstationarynonstationary

q There may be purely local changes in time domain,global changes in frequency domain, and there may

be changes in the variance parameters


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Data CharacteristicsData Characteristics

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

IBMPrices

IBMVolati

lity

Nonstationary

Time VaryingVolatility


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q The nonlinearities and nostationarities do contain

certain regularities or patterns

q Therefore, an analysis of nonlinear time series

data would involve quantitatively capturing such

regularities or patterns effectively


Data CharacteristicsData CharacteristicsHaving said that

How and Why?How and Why?


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Wavelet Multiscale AnalysisWavelet Multiscale Analysis

Overviewq Wavelets are mathematical functions that cut up

data into different frequency components and thenstudy each component with a resolution matchedto its scale

q Wavelets are treated as a lens that enables theresearcher to explore relationships that were

previously unobservable

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

revealing


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Signal

Wavelet

C = C2

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



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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


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

==

i

itxiwtx*w

Xt Original Signal

W High or low pass filters

Filter Bank Approach


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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

Fre

que

nc

y

Fre

que

nc

y

Xt = AN + D1+ D2 + DN

Iteration gives scaling effect

at each level

Mallats Pyramidal Filtering ApproachMallats Pyramidal Filtering Approach


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Analyzing High-frequencyAnalyzing High-frequency

Financial Data: Our ApproachFinancial Data: Our Approach

Tick Data

Preprocessing TransformationKnowledge

Discovery Forecast

Data

Compression

Multiscale

AnalysisPredictionSummarization

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)

withchief featureschief features

Use the

extracted

chief features

to predictpredict

CycleCycle

TrendTrend

Turning PointsTurning Points

Variance ChangeVariance Change


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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 fiveminutes).

II. Calculate the level L of the DWT needed based on number of samples N in Cof 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 A

L.

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


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

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

N points to the right to get DS, forecast

.

III. For a N-step ahead forecast, extend the trend componentextend the trend componentANlinearlyN 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


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Raw Signal

VolatilityVolatility

DWTDWT

Statistic

NCSS

Statistic

NCSS

DWTDWT

FFTFFT

Detect

Turning

Points and

Trends

Detect

Turning

Points and

Trends

DetectInherent

Cycles

DetectInherent

Cycles

Detect

Variance

Change

Detect

Variance

Change

Su

mma

riz

ation

Pre

dic

tion

A prototype systemprototype system has been implemented that automaticallyextracts chief features from a time series and give a predictionbased on the extracted features, namely trend and seasonality


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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 . 8 2

1 . 8 2

1 . 8 3

1 . 8 3

1 . 8 4

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5

Feature Phrases Details

Trend 1st Phase

2nd Phase

TurningPoints

Downturns 108, 132, 164, and 178

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

VarianceChange

Location 164

Cycle Period 42

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

260


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Financial Data: Our ApproachFinancial Data: Our ApproachA Case StudyA Case Study

Forpredictionprediction, we use the chief features of the previous day (March 18, 2004), information about thedominant cycle and trend (summarization), to reproduce the elements of the series for the followingday (March 19, 2004):

1 . 8 2

1 . 8 2

1 . 8 3

1 . 8 3

1 . 8 3

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0

SystemOutputSystemOutput

Actual

March 19, 2004

Predicted

(seasonal + trend)

March 19, 2004

Root Means Square Error = 0.0000381

Correlation = + 62.4 %


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Wavelet Multi Resolution Analysis of High Frequency Fx Rates 1203290417290522 5