Identification with Generalized Orthonormal Basis Functions - Statistical Analysis and Error Bounds

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations.

This paper is concerned with the computation of uncertainty bounds for the expansion of uncertain Volterra models into an orthonormal basis of functions, such as the Laguerre or Kautz bases. This problem has already been addressed in the context of linear systems by means of an approach in which the uncertainty bounds of the expansion coefficients have been estimated from a structured set of impulse responses describing a linear uncertain process. This approach is extended here towards nonlinear Volterra models through the computation of the uncertainty bounds of the expansion coefficients from a structured set of uncertain Volterra kernels. The proposed formulation assures that the resulting model is able to represent all the original uncertainties with minimum intervals for the expansion coefficients. An example is presented to illustrate the effectiveness of the proposed formulation.

On parameter estimation for biaxial mechanical behavior of arteries

In this paper, model sets for continuous-time linear time invariant systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalise the well known Laguerre and Kautz bases. It is shown that the obtained model sets are complete in all of the Hardy spaces Hp(Π), 1 ≤ p p (Π), 1 ≤ p p (Π), 1 ≤ p < ∞. The results in this paper have application in system identification, model reduction and control system synthesis.

A dynamic system with both input and output measurement errors is termed as an errors-in-variables (EIV) system. Employment of traditional identification methods for EIV systems will result in biased parameter estimates. The existing methods of EIV identification like Bias Eliminated Least Squares (BELS), and Subspace EIV methods may not always yield good parameter estimates. To address these open issues, this paper presents a maximum-likelihood (ML) approach using the Expectation Maximization (EM) algorithm. The efficacy of the proposed method is demonstrated with an experimental study.

A new method is proposed for the blind subspace-based identification of the coefficients of time-varying (TV) single-input multiple-output (SIMO) FIR channels. The TV channel coefficients are represented via a finite basis expansion model, i.e. linear combination of known basis functions. In contrast to earlier related works, the basis functions need not be limited to complex exponentials, and therefore do not necessitate the a priori estimation of frequency parameters. This considerably simplifies the implementation of the proposed method and provides added flexibility in applications. The merits of the proposed technique, including asymptotic consistency, are demonstrated by numerical simulations.

Parameter estimation for a laboratory water—gas-shift reactor using a nonlinear error-in-variables method

In order to improve classification accuracy, multiple classifier systems have provided better pattern classification over single classifier systems in different applications. The theoretical frameworks proposed in [5], [7] present important tools for estimating and minimizing the added error of linearly combined classifier systems. In this work, we present a theoretical model that estimates the added error using geometric mean rule which is a nonlinear combining rule. In the derivation, we assume classifiers' outputs are uncorrelated and have identical distributions for a given class case. We also show that by setting the number of classifiers to one (a single classifier system), the derived formula is modified and matches the results given in [5]. We validated our derivations with computer simulations and compared these with the analytical results. Due to the nonlinearity of the geometric mean, theoretical results show that the bias and variance errors are mixed together in their contribution to the added error. It was shown that the bias error dominated the contribution to the added error compared to the variance error. It is possible to minimize the variance error by increasing the ensemble size (number of classifiers) while the bias error is minimized under certain conditions. The proposed theoretical work can help in investigating the added error for other nonlinear arithmetic combining rules.

In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l/sub 2sup n/. To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, for example, the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

&amp;lt;p&amp;gt;The objective of this study is the combination of different types of basis functions applied separately to different kinds of gravity observations. We use two types of regional data sets: terrestrial gravity data and airborne gravity data, covering an area of about 500 km &amp;amp;#215; 800 km in Colorado, USA. These data are available within the &amp;amp;#8220;1 cm geoid experiment&amp;amp;#8221; (also known as the &amp;amp;#8220;Colorado Experiment&amp;amp;#8221;). We apply an approach for regional gravity modeling based on series expansions in terms of spherical radial basis functions (SRBF). Two types of basis functions covering the same spectral domain are used, one for the terrestrial data and another one for the airborne measurements. To be more specific, the non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The Cubic Polynomial (CuP) function is applied to the airborne data as a low-pass filter, and the smoothing features of this type of SRBF are used for filtering the high-frequency noise in the airborne data. In the parameter estimation procedure, these two modeling parts are combined to calculate the quasi-geoid.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;The performance of our regional quasi-geoid model is validated by comparing the results with the mean solution of independent computations delivered by fourteen institutions from all over the world. The comparison shows that the low-pass filtering of the airborne gravity data by the CuP function improves the model accuracy by 5% compared to that using the Shannon function. This result also makes evident the advantage of combining different SRBFs covering the same spectral domain for different types of observations.&amp;lt;/p&amp;gt;

We perform a reconstruction of the cosmological large-scale flows in the nearby Universe using two complementary observational sets. The first, the SFI++ sample of Tully–Fisher (TF) measurements of galaxies, provides a direct probe of the flows. The second, the whole sky distribution of galaxies in the 2MASS (Two Micron All Sky Survey) redshift survey (2MRS), yields a prediction of the flows given the cosmological density parameter, Ω, and a biasing relation between mass and galaxies. We aim at an unbiased comparison between the peculiar velocity fields extracted from the two data sets and its implication on the cosmological parameters and the biasing relation. We expand the fields in a set of orthonormal basis functions, each representing a plausible realization of a cosmological velocity field smoothed in such a way as to give a nearly constant error on the derived SFI++ velocities. The statistical analysis is done on the coefficients of the modal expansion of the fields by means of the basis functions. Our analysis completely avoids the strong error covariance in the smoothed TF velocities by the use of orthonormal basis functions and employs elaborate mock data sets to extensively calibrate the errors in 2MRS predicted velocities. We relate the 2MRS galaxy distribution to the mass density field by a linear bias factor, b, and include a luminosity-dependent, ∝Lα, galaxy weighting. We assess the agreement between the fields as a function of α and β=f(Ω)/b, where f is the growth factor of linear perturbations. The agreement is excellent with a reasonable χ2 per degree of freedom. For α= 0, we derive 0.28 < β < 0.37 and 0.24 < β < 0.43, respectively, at the 68.3 per cent and 95.4 per cent confidence levels (CLs). For β= 0.33, we get α < 0.25 and α < 0.5, respectively, at the 68.3 per cent and 95.4 per cent CLs. We set a constraint on the fluctuation normalization, finding σ8= 0.66 ± 0.10, which is only 1.5σ deviant from Wilkinson Microwave Anisotropy Probe (WMAP) results. It is remarkable that σ8 determined from this local cosmological test is close to the value derived from the cosmic microwave background, an indication of the precision of the standard model.

CO₂ emission is considerably dependent on energy consumption and on share of energy sources as well as on the extent of economic activities. Consequently, these factors must be considered for CO₂ emission prediction for seven middle eastern countries including Iran, Kuwait, United Arab Emirates, Turkey, Saudi Arabia, Iraq and Qatar. In order to propose a predictive model, a Multilayer Perceptron Artificial Neural Network (MLP ANN) is applied. Three transfer functions including logsig, tansig and radial basis functions are utilized in the hidden layer of the network. Moreover, various numbers of neurons are applied in the structure of the models. It is revealed that using MLP ANN makes it possible to accurately predict CO₂ emission of these countries. In addition, it is concluded that using logsig transfer function leads to the highest accuracy with minimum value of mean squared error (MSE) which is followed by the networks with radial basis and tansig transfer functions. The R-squared of the networks with logsig, radial basis and tansig transfer functions are 0.9998, 0.9997 and 0.9996, respectively. Finally, comparison of the proposed model with a similar study, considered five countries in the same region, reveals higher accuracy in term of MSE.

In a regression model, prediction is typically performed after model selection. The large variability in the model selection makes the prediction unstable. Thus, it is essential to reduce the variability in model selection and improve prediction accuracy. To achieve this goal, a parametric bootstrap smoothing can be applied. In this method, model selection is performed for each resampling from a parametric distribution, and these models are then averaged such that the distribution of the selected models is considered. Here, the prediction accuracy is highly dependent on the choice of a distribution for resampling. In particular, an experimental study shows that the choice of error variance significantly changes the distribution of the selected model and thus plays a key role in improving the prediction accuracy. We also observe that the true error variance does not always provide optimal prediction accuracy. Therefore, it is not always appropriate to use unbiased estimators of the true parameters or standard estimators of the parameters for the resampling distribution. In this study, we propose employing cross-validation to choose a suitable resampling distribution rather than unbiased estimators of parameters. Our proposed method was applied to several data sets. The results indicate that the proposed method provides a better prediction accuracy than the existing method.

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

Parameter Bounds for Models with Bounded Errors in all Variables