**Contents**

1. Daily exchange rate values from 1999 January 1 to date.

2. Filtered data.

3. Wavelet transform decomposition.

4. "Patterns from the past".

5. Scatter plots of signal smooths.

6. Nowcast.

7. One day ahead forecast.

This web page is recreated each day at 06:00 UT.

**1. Daily exchange rate values from 1999 January
1 to date.** This is
the data we are analyzing. There is an additional value each day. The
abscissa on the plot is the day number from 1999 January 1. The ordinate
is the number of Euros corresponding to 1 US dollar.

**2. Filtered data.** This is
based on optimizing the separation of signal and noise
at the different resolution levels of an à trous wavelet transform.
Gaussian noise is assumed and the optimization uses an entropy
characterization of signal and noise. (On this signal we did not find
a non-stationary noise model to be particularly useful.)

**3. Wavelet transform decomposition.**
The Haar à trous wavelet
transform used is a new method. (i) It is redundant, which is best
for finding patterns in the data at various resolution scales. (ii)
The all-important final data values are rigorously respected, without
any "wrap-around" or other signal boundary "fix-ups".

First we display the wavelet transform of the input data. From the upper left, we have resolution scales 1, 2, ... 7.

**Important remark:** we have deleted the first
600 values from the result of the wavelet transform, to show the
interesting part of the time series more clearly. Therefore the wavelet
transform was carried out on a far longer time series; and properties like
zero mean of the wavelet coefficients at each level will not hold -
evidently - for the final values of the time series displayed here. Note
also a perceptible "drift to the right", or increasing "delay" of
phenomena with resolution level: this is due to the smoothing which is
progressively happening at each level.

Next we do the same for the entropy-filtered signal. We carry out the
wavelet transform. Then we remove the first 600 values in order to display
the wavelet transform result more clearly for the remaining values.

**4. Patterns from the past.**
The following shows the sum of wavelet
coefficients at scales 3, 4, 5 and 6. (We have omitted scales 1 and 2
on the grounds of being too "noisy". We have also omitted scale 7, and
of course the final smoothed version of the data.) This figure is based
on all values in the time series studied. The values are necessarily of
zero mean since each of the wavelet scales is thus.

**5. Scatter plots of signal smooths.**
The following show scatter plots of

- Final smoothed version of the signal + wavelet scale 7 plotted against the final smoothed version of the signal + wavelet scale 7 + wavelet scale 6.
- Final smoothed version of the signal + wavelet scale 7 + wavelet scale 6 plotted against the final smoothed version of the signal + wavelet scale 7 + wavelet scale 6 + wavelet scale 5.

**6. Nowcast.**
This is based on the Haar à trous wavelet
transform. It tells us whether there is a significant move up
or down, at each resolution level. It tells us the importance
of the movement (SNR = signal-to-noise ratio). It also tells
us the duration of an upward movement, or a downward movement.

The algorithm for this nowcast program is as follows. We are characterizing the last value (i.e. time-point: now) in all cases.

- If the last value is significant and the penultimate value
is not significant, then we have a
**"NEW Upward transition"**or a**"NEW Downward transition"**. - If the last value is significant and the penultimate value
is also significant, then we have a continuing
**"Negative**(resp.**Positive**)**significant structure"**. We indicate how long this significant structure has lasted so far. - If the last value is not significant, but the penultimate value is significant, then we indicate the end of a significant structure, and we also indicate how long this significant structure has lasted.
- Finally, if the last value is not significant, and the
penultimate structure is not significant, then we indicate
**"No detection"**.

SCALE 1 No detection, SNR = 0.440819 SCALE 2 Negative significant structure [744,745] SNR = 3.84382 SCALE 3 NEW Downward transition: SNR = 3.75459 SCALE 4 Positive significant structure [738,745] SNR = 22.2902 SCALE 5 Positive significant structure [744,745] SNR = 9.94855 SCALE 6 Negative significant structure [641,745] SNR = 22.5916 SCALE 7 Negative significant structure [645,745] SNR = 143.121 SCALE 8 Negative significant structure [655,745] SNR = 206.863

**7. Forecast.**
This is based on an AR(5) model of the entropy-filtered data stream.
The standard deviation of the prediction error was found to be 0.00143783.
An almost equally good result is obtained from the MAR (Multiresolution
AR) model, stationarity assumed, and up to AR(1) at each resolution
level. Our one-step ahead forecast of the number of EUR corresponding to
1 USD is:

For further information, including documentation, see http://www.multiresolution.com.

Programs used: for the entropy-based filtering, mw1d_filter; for the wavelet transforms, a version of mr1d_trans; for the nowcast, mr1d_nowcast; for the forecast, mr1d_fcast.

Contact: fmurtagh @ multiresolution.com