In a linear autoregressive model of order R, a time series y_{n}
is modelled as a linear combination of R earlier values in the time series,
with the addition of a correction term x_{n}:
y_{n}^{model} =
x_{n} 
a_{j} y_{nj} .
The autoregressive coefficients a_{j} (j = 1, ... R) are fit by
minimizing the meansquared difference between the modelled time series
y_{n}^{model} and the observed time series
y_{n}. The minimization process results in a system of
linear equations for the coefficients a_{n}, known as the
YuleWalker equations
[Yule, G.U. (1927) On a method of investigating periodicities
in disturbed series with special reference to Wolfer's sunspot numbers.
Phil. Trans. Roy. Soc. Lond. A 226, 267298].
Conceptually, the time series y_{n}
is considered to be the output of a discrete linear feedback circuit
driven by a noise x_{n},
in which delay loops of lag j have feedback strength a_{j}.
For Gaussian signals, an autoregressive model often provides
a concise description of the time series y_{n}, and calculation of
the coefficients a_{j} provides an indirect but highly efficient
method of spectral estimation.
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introduction
nonlinear modelling
NLAR fingerprints
significance testing
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In a full nonlinear autoregressive model,
quadratic (or higherorder) terms are added to the linear
autoregressive model. A constant term is also added, to counteract
any net offset due to the quadratic terms:
y_{n}^{model} =
x_{n}  a_{0}

a_{j} y_{nj}

b_{j,k} y_{nj}y_{nk}.
The autoregressive coefficients a_{j} (j = 0, ... R) and
b_{j,k} (j, k = 1, ...., R)
are fit by
minimizing the meansquared difference between the modelled time series
y_{n}^{model} and the observed time series
y_{n}. The minimization process also results in a system of
linear equations, which are generalizations of the YuleWalker equations for
the linear autoregressive model.
Conceptually, the time series y_{n} is considered to be the output of a circuit with nonlinear feedback, driven by a noise x_{n}. In principle, the coefficients b_{j,k} describe dynamical features that are not evident in the power spectrum or related measures.
Although the equations for the
autoregressive coefficients a_{j} and
b_{j,k} are linear, the estimates of these parameters are often
unstable, essentially because a large number of them
must be estimated. This is the motivatation for the
NLAR fingerprint.
introduction
nonlinear modelling
NLAR fingerprints
significance testing
references
To create the nonlinear autoregressive fingerprint,
only a single term of the full quadratic model is retained,
along with the constant term:
y_{n}^{model} =
x_{n}  a_{0}

a_{j} y_{nj}
 b_{u,v} y_{nu}y_{nv}.
The autoregressive coefficients a_{j} (j = 0, ... R) and
the single coefficient b_{u,v}
are fit by
minimizing the meansquared difference between the modelled time series
y_{n}^{model} and the observed time series
y_{n}. This involves estimation of only R+2 parameters
(compared with (R+1)(R+2)/2 equations for the
full quadratic autoregressive model),
and substantially more reliable values for the parameters.
However, like the full quadratic autoregressive model, it provides
a characterization of the nonlinear dynamics of the time series.
The fitting procedure is performed sequentially for all pairs of
lags u and v (u,v = 1, ...., R). The "NLAR fingerprint" consists
(see example)
of a contour map of the residuals
y_{n}^{model}  y_{n}^{2},
parametric in the choice of lags u and v.
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introduction
nonlinear modelling
NLAR fingerprints
significance testing
references
Introduction of an additional term into a linear or nonlinear autoregressive model always improves the fit (in the meansquared sense). Akaike [Akaike, H. (1974) A new look at statistical model identification. IEEE Trans. Auto. Control AC19,716723] showed that, for a linear autoregressive model, a significant improvement in the fit is associated with a reduction in the residual variance of at least 2V/N, where V is the variance without the candidate additional term, and N is the number of data points.
We showed that the same criterion, a reduction in residual variance by at least 2V/N, is a criterion for the significance of a single nonlinear term as well.