We consider a chaotic system with a double-scroll attractor proposed by Elwakil, composing with a second-order system, which has low-dimensional multiple invariant subspaces and multi-level on-off intermittency. This type of composite system always includes a skew-product structure and some invariant subspaces, which are associated with different levels of laminar phase. In order for the level of laminar phase be adjustable, we adopt a nonlinear function with saturation characteristic to tune the...

Let us consider the linear model covering the one-way classification as a special casse. In the paper the relationship between testing of some linear hypothesis and estimating of parameters in the linear model by common software packages is examined.

Planteamos el modelo lineal sobre variables dummy difusas asociadas a variables cuantitativas codificadas o cualitativas. En el modelo bicriterio aparecen las mismas dificultades conceptuales que se presentan en el caso multicriterio, pero su desarrollo resulta mucho más ágil y transparente. Resolvemos las ecuaciones normales, caracterizando el conjunto de soluciones y encontramos finalmente las funciones paramétricas estimables para este modelo.

Continuando el desarrollo del modelo lineal sobre dos variables dummy difusas, encontramos expresiones para las descomposiciones clásicas en sumas de cuadrados y estudiamos sus distribuciones bajo hipótesis de normalidad. Damos una interpretación geométrica del ajuste cuando las variables numéricas explicativas se ven sometidas a una codificación difusa de tipo semilineal y contrastamos en estos casos la mejora de este nuevo modelo respecto de la regresión sobre las variables originales. Concluimos...

Binary operations on commutative Jordan algebras are used to carry out the ANOVA of a two layer model. The treatments in the first layer nests those in the second layer, that being a sub-model for each treatment in the first layer. We present an application with data retried from agricultural experiments.

For estimating the variance components of a one-way random effect model recently Uhlig (1995, 1997) and Lischer (1996) proposed non-iterative estimators with high breakdown points. These estimators base on the high breakdown point scale estimators of Rousseeuw and Croux (1992, 1993), which they called Q-estimators. In this paper the asymptotic normal distribution of the new variance components estimators is derived so that the asymptotic efficiency of these estimators can be compared with that of...

In the paper necessary and sufficient conditions for the existence and an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components are presented for the mixed linear model $𝐭=𝐗\beta +ϵ$, $𝐄\left(𝐭\right)=𝐗\beta$, $\mathrm{𝐕𝐚𝐫}\left(𝐭\right)={\mathbf{0}}_{\mathbf{1}}{𝐔}_{\mathbf{1}}+{\mathbf{0}}_{\mathbf{2}}{𝐔}_{\mathbf{2}}+{\mathbf{0}}_{\mathbf{3}}{𝐔}_{\mathbf{3}}$, with three unknown variance components in the normal case. An application to some examples from the analysis of variance is given.

In the paper an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components is presented for the mixed linear model $𝐭=𝐗\beta +ϵ$, $𝐄\left(𝐭\right)=𝐗\beta$, $𝐃\left(𝐭\right)={\mathbf{0}}_{\mathbf{1}}{𝐔}_{\mathbf{1}}+{\mathbf{0}}_{\mathbf{2}}{𝐔}_{\mathbf{2}}$ with the unknown variance componets in the normal case. The matrices ${𝐔}_{\mathbf{1}}$, ${𝐔}_{\mathbf{2}}$ may be singular. Applications to two examples of the analysis of variance are given.

We study properties of Linear Genetic Programming (LGP) through several regression and classification benchmarks. In each problem, we decompose the results into bias and variance components, and explore the effect of varying certain key parameters on the overall error and its decomposed contributions. These parameters are the maximum program size, the initial population, and the function set used. We confirm and quantify several insights into the practical usage of GP, most notably that (a) the...

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the...

For any orthogonal multi-way classification, the sums of squares appearing in the analysis of variance may be expressed by the standard quadratic forms involving only squares of the marginal and total sums of observations. In this case the forms are independent and nonnegative definite. We characterize all two-way classifications preserving these properties for some and for all of the standard quadratic forms.

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

Statistical analysis of compositional data, multivariate observations carrying only relative information (proportions, percentages), should be performed only in orthonormal coordinates with respect to the Aitchison geometry on the simplex. In case of three-part compositions it is possible to decompose the covariance structure of the well-known principal components using variances of log-ratios of the original parts. They seem to be helpful for the interpretation of these special orthonormal coordinates....