EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

Yoo, S. S. et al. Increasing cortical activity in auditory areas through
neurofeedback functional magnetic resonance imaging. neuroreport 17,
1273–1278 (2006).
A.1 – Minimization of RMS Error
43- Appendix - A
Time series are sequences, discrete or continuous, of quantitative data of specific
moments in time. They may be simple such as a single numerical observation at each
moment of time and studied with respect to their distribution in time, or multiple in which
case they consist of a number of separate quantities tabulated according to a common
time base (e.g., a mixture of sine waves beginning at time = 0).
The statistics of a time series is the science of predicting an immediate or long
time future sequence based on a sample of past sequential quantitative data. In general,
the longer the sample of past quantitative moments of time then the greater the accuracy
of predicting future sequence(s).
The fine details of accuracy of prediction of the future based upon past samples is
generally governed by the relationship of 1 / sq rt. of N. To understand why this is the
case let us define a statistic of a time series based on the “signal” or “message” that is
transmitted and the “noise” or randomness that the signal is embedded in. This
relationship was described by the Nobel laureate Normbet Wiener (N. Wiener, Time
Series, MIT Press, Cambridge, Mass., 1949) in which a time series is a combination of a
signal + noise or the signal f(t) and the message g(t) + noise, where noise is defined as
f(t) – g(t). In other words noise is defined as the difference between the “message” and
the measured quantitative values or f(t) – g(t). For example, noise = 0 when f(t) – g(t) =
0.
Let us consider the output of an electrical circuit with input f(t). If the circuit has
the response A (t) to a unit-step function, then the output is given by:
+∞
∫
0
F( t)
= A'
( τ ) f ( t −τ
) dτ
+ A(0)
f ( t)
The goal is to have F(t) approximate as closely as possible the message g(f). That
is, we want to minimize [F(t) – g(t)]. As a criterion
The Ergotic goal of time series statistics is to minimize the difference between the
measured values f(t) and the “signal” g(t).