Help required: circular fuzzy, time series, statistical nature.

Barry Bayliss (B.A.Bayliss@open.ac.uk)
Thu, 23 Sep 1999 12:03:49 +0200 (MET DST)

Hello,

I am a research student looking at the signals generated by the brain
while processing information. The object is to be able to locate is
both the spatial and temporal components where differences in neural
processing occur.

There are several aspects which I believe that fuzzy logic may by able
to help.

1. The generation of a better estimate of the signal generated by
a specific experimental condition (including when the required
information may not be present at all times).

2. The generation of statistical methods that can handle both the
noise and variations in the timing of signals in the system.

3. The modification of statistical tests, mainly circular statistics,
to make better use of the information we have available.

Although I am interested in finding out where the signal differences occur,
I am not primarily interested in trying to model the time series.

I have applied several standard statistical tests, however, they
generally tend to ingore the error in measurement.

Any help on the above would be appreciated.

I would also appreciate any references on circular fuzzy logic, or fuzzy
geometry.

To help give a better idea of the problem, I have tried to put together
an introduction to the problem (following). If this is not clear, or
other information is required, let me know.

Barry.
B.A.Bayliss@open.ac.uk

INTRODUCTION
------------

Given two time series sets,

F(t) = { f_1(t), f_2(t), ... , f_n(t) }
and
G(t) = { g_1(t), g_2(t), ... , g_m(t) }

Each time series (ideally) is composed of three components

1. synchronous signals
2. non-synchronous signals
3. noise

Note: 1. It is possible that the events that I'm trying to extract are not at
exactly at the same point in each time series.

2. It is possible that the signal we are interested in does not appear
in all the time series (due to the nature of the generation of the signals
measured).

At each time point, we can generate a set of vectors (paired data).
Each component of the vector pair has normally distributed noise, ie

2
1 { - (x - \mu) }
a = s_a + --------------- exp | ------------- |
/-----| | 2 |
v 2 \pi \sigma { 2 \sigma }

where \mu and \sigma can be estimated.

By calculating the mean of the set at a fixed time point, we have the following
relationship with regards to the noise.

10 * | s_a | ~ | noise |

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