Wavelet Analysis

By Sam Luxenberg

23rd March 2017, Tel Aviv

INTRODUCTION TO WAVELETS

Applications:

▪ Data Compression

▪ Signal Analysis

▪ Smoothing/De-noising Data

▪ Modeling Abrupt Changes

▪ Pattern Recognition

▪ Solutions to Partial Differential Equations

WAVELETS AND FINANCIAL MARKETS

▪Markets are complex chaotic systems that have self-similarity properties.

▪Financial signals can be thought of as fractal signals having self-similarity properties which allows for analysis with wavelets.

SIGNAL PROCESSING AND ANALYSIS

▪ The conventional tool is Fourier Analysis which can represent a signal (e.g. audio signal, images, seismic signal, financial signal) as a sum of sinusoids (think cosine and sine curves).

▪ We may be interested in certain cyclic or frequency components present in a signal.

▪ The Fourier Transform allows us to study the signal not just as it relates to time, but also these frequency components.

REPRESENTATION OF SIGNALS

▪ In Fourier Analysis, a signal g(t) can be represented as:

Where are called the Fourier coefficients (aka Fourier

Transform of g(t))

▪ In Wavelet Analysis, a signal g(t) can be represented as:

Where are called the wavelet coefficients of g(t) with

respect to the basis

SIGNAL PROCESSING AND ANALYSIS

▪ Consider the Discrete Fourier Transform (DFT)

Let x0, x1, …, xN-1 be a signal sampled at N points in time.

Then the DFT is: where

▪Note that this is a linear transformation of the signal and can therefore be represented as a matrix.

▪ This sum can be thought of as the correlation between the signal xn and the different frequency components.

SIGNAL PROCESSING AND ANALYSIS

▪ Consider the Discrete Wavelet Transform (DWT)

Let x0, x1, …, x2n-1 be a signal sampled at 2n points in time.

Then via the DWT, x has its wavelet decomposition of the form:

where is a scaling function.

▪ Note that the DWT is also a linear (and orthogonal) transformation and can therefore be represented as a matrix.

GENERAL STRUCTURE OF WAVELET ANALYSIS

2 Steps

▪ Decomposition with the DWT

▪ Reconstruction/Synthesis with the Inverse DWT (IDWT)

ONE LEVEL OF WAVELET DECOMPOSITION

▪ The DWT of a signal x is calculated by passing it through a series of filters.

▪ Samples are simultaneously passed through a lower-pass and high-pass filter resulting in a convolution of the two.

▪ (approximation or scaling coefficients)

▪ (detail coefficients)

ADVANTAGES OVER FOURIER TRANSFORM

▪ Due to the Heisenberg Uncertainty Principle, we can study a signal with a Fourier Transform with information in time or with information in frequency BUT NOT BOTH.

▪ E.g. We may know some frequency component occurs often throughout the existence of a signal but we cannot know the timing this frequency occurs in the signal.

▪The two graphs below demonstrate the difference between examining a signal in the time domain versus the frequency domain.

ADVANTAGES OVER THE FOURIER TRANSFORM

ADVANTAGES OVER THE FOURIER TRANSFORM

▪ The Wavelet Transform allows us to get both time and scale (the analogue for frequency for wavelets) information simultaneously.

▪ J represents the “scale” or the “resolution level” at which we would want to examine the signal.

▪ K represents the “translation” or “shift” in time.

ADVANTAGES OVER THE FOURIER TRANSFORM

▪ The Fourier Transform does NOT represent abrupt changes efficiently

▪ To accurately analyze signals and images with abrupt changes, use the wavelet transform which is localized in time and frequency

WAVELET ANALYSIS AND TIME SERIES

“Time Series Forecasts Via Wavelets: An Application to Car Sales in the Spanish Market” by Miguel Ariño, Ph.D.

GENERAL OUTLINE

▪ We will first forecast the time series using conventional time series modeling tools such as Autoregressive Integrated Moving Average (ARIMA) and Seasonal ARIMA (SARIMA).

▪ We want to compare the accuracy of these forecasts to forecasts combining wavelet analysis and ARIMA models.

COMBINING WAVELET ANALYSIS AND ARIMA

▪ Decompose the time series into its long-term trend and seasonal component using wavelet decomposition and reconstruction.

▪ Using ARIMA models, forecast each component separately.

▪ Combine these forecasted components to get the forecast for the original time series.

DATA

▪ Monthly Car Sales in Spain from January 1974 to December 1994.

▪ 252 total observations

▪ We will use the first 240 observations to build the models and the last 12 to compare our forecasts with the actual number of sales during 1994.

ARIMA MODELS

▪ There are general rules of thumb to follow when identifying ARIMA models.

▪ Examine the Autocorrelation (correlation between each observation and the past observations) and Partial Autocorrelation (correlation not accounted for by lags in-between)

▪ Take differences (if necessary) to remove non-stationarity of the time series

▪ Two good models describe our time series:

▪ ARIMA(0, 1, 1)x(0, 1, 1)12

▪ ARIMA(2, 1, 0)x(0, 1, 1)12

ARIMA (0, 1, 1) X (0, 1, 1)12

▪ While both models give similar forecasts, the first model is simpler so we will use this one.

▪ No AR component, 1 non-seasonal difference, 1 moving average component

▪ No seasonal AR component, 1 seasonal difference, 1 seasonal moving average component with seasonal frequency being 12 months

▪ The out-of-sample root mean square error (RMSE) is 16,963.9

WAVELET MODEL – PREPARING TO USE DWT

▪ In order to use the DWT, we need the number of data points to be a power of 2

▪ Center the time series by subtracting its mean 60,603

▪ This centered or zero-mean series is called x = (xt)

▪ Daubechies of order 8 Wavelet basis

WAVELET MODEL - DWT

▪ Apply DWT to our time series x to obtain another series or vector called d which will represent our scaling and wavelet coefficients.

▪ The DWT can be represented as a matrix, so applying the DWT can be thought of as:

▪ For each level there will be associated coefficients

LONG-TERM TREND AND SEASONAL COMPONENT

▪ Scalogram is a graph of the amount of “energy” for each level of resolution to identify the two most dominant resolution levels to use as the long-term trend and seasonal components.

▪ For each resolution level and amount of shift/translation k

▪ Example for 2nd Resolution level:

SPLITTING THE COEFFICIENTS INTO 2 SETS

▪ There are two major peaks in the scalogram at levels 1 (7th decomposition) and 7 (1st decomposition).

▪ Existence of peaks at high or low levels indicates the existence of high or low frequency components.

▪ Take coefficients at levels around each of the major peaks and pad each of these 2 coefficient vectors with zeros.

RECONSTRUCTION OF THE COMPONENTS

▪ Now that we have the 2 separate sets of coefficients, let’s reconstruct the individual components using the IDWT.

BACK TO FORECASTING

▪ We have decomposed our time series into simpler and easier-to-forecast components.

▪ In order to forecast y we need to add back in the mean of the original time series that we subtracted before doing the wavelet analysis.

▪ Decomposition and Reconstruction on the boundary

▪ Delete first 36 and last 16 data points of each of our series x, y, and z.

▪ Left with car sales from January 1977 to December 1993

FORECASTING THE COMPONENTS

▪ Seasonality removed from the long-term trend y

▪ Best model for y: ARIMA(1,3,0)

▪ Seasonal model z

▪ Best model for z: ARIMA(0,1,1)12

▪ Forecasts for x

▪ x = y + z

COMPARISON OF THE MODELS

▪ Wavelet Model RMSE = 12,194

▪ SARIMA Model RMSE = 16,964

POTENTIAL APPLICATIONS FOR I KNOW FIRST

▪ Could provide more confidence to investment forecasts

▪ Overlay current forecasting systems on top of each of the decomposed wavelet components.

▪ Wavelet analysis does not have to be constrained to 1-dimensional problems

▪Could be used for n-dimensional problems which could include considering an entire portfolio of investments.

USEFUL RESOURCES

▪ Wavelet Toolbox User’s Guide

▪ http://web.mit.edu/1.130/WebDocs/wavelet_ug.pdf

▪The Discrete Wavelet Transform in S

▪ http://www.stat.ucla.edu/~cocteau/stat204/readings/nasonsilverman.pdf

▪Wavelet Scalograms and Their Applications in Economic Time Series

▪ https://www.ime.usp.br/~pam/amv.pdf

▪“Conceptual Wavelets in Digital Signal Processing: An In-Depth, Practical Approach for the Non-Mathematician” by D. Lee Fugal