In order to deal with symmetry operations or condensed phase structures, several "fingerprint" frameworks have been developed [8, 28 40], that assign a unique vector of order parameters to each molecular or crystalline configuration: a metric can then be easily built by taking some norm of the difference between fingerprint vectors.
Similarly, we define the augmented backwards vector of order n ≥ 1 of x t as the 2n-vector x t b = [ x t + n − 1, x t + n − 1 ∗, x t + n − 2, x t + n − 2 ∗, …, x t, x t ∗ ] T. The following results establish the relation between the signals x t and their augmented forwards and backwards versions.
In addition, we write (0_{ntimes m}) to denote the (ntimes m) zero matrix, (I_{n}) to denote the (ntimes n) identity matrix, and (mathbf {e}_{n}inmathbb{R}^{n}) to denote the ones vector of order n, that is, (mathbf{e}_{n}=[1,1,ldots,1]^{maT}}m{T}}); if the order of (mathbf{e}_{n}) is clear from the context we simply write e for (mathbf {e}_{n}).
Let σ=[σ 1,σ 2,⋯,σ K ]T be the vector of ordered eigenvalues of (widehat {mathbf {Sigma }}_{K} = I_{K} + ^{mathbf {Omega }_{K}}!/_{N}), then all but one eigenvalue of (widehat {mathbf {Sigma }}_{K}) are still equal to 1 (eigenvalues of I K ) while σ 1 is given by sigma_{1} = 1 + frac{omega_{1}}{N}.
Denote the augmented forwards vector of order n ≥ 1 of x t as the 2n-vector x t = [ x t, x t ∗, x t − 1, x t − 1 ∗, …, x t − n + 1, x t − n + 1 ∗ ] T. and its correlation function by R ( t, s ) = E [ x t x s H ]. From now on, we assume that det {R t } ≠ 0 with R t :=R t,t).
Since U ′ U = I n, the model (1) becomes: d = Sb + ε ˜ (2)In (2), d and ε ˜ are vectors each of order n×1 and b is a vector of order p×1.
The sum, at every channel, was multiplied by a vector of ordered ranks (either [1 2 3] or [3 2 1]) to generate the non-parametric ordered-hypothesis test-statistic L (Page 1963) with 11° of freedom (12 subjects).
We compare results for polynomial regression vectors of orders p=0,1,2.
However, an straightforward solution which yield to singular values and singular vectors of order M is complicated.
For the traffic noise codebook, LP coefficient vectors of order 6 extracted from 2 min of nonstationary traffic noise were used.
In short, we therefore sorted the probe intensities of each of the n microarrays according to rank r, aligned these vectors of ordered intensity values into columns of a matrix and averaged each row to give the average quantiles for the reference distribution (calculated as the QPN procedure).
