Statistics linearity

Coomans D, Massart DL, Broeckaert I (1981) Potential methods in pattern recognition. A combination of ALLOC and statistical linear discriminant analysis. Anal Chim Acta 133 215... 

This is the basis for our new test of linearity. It has all the advantages we described it gives an unambiguous determination of whether any nonlinearity is affecting the relationship between the test results and the analyte concentration. It provides a means of distinguishing between different types of nonlinearity, if they are present, since only those that have statistically significant coefficients are active. It also is more sensitive than any other statistical linearity test including the Durbin-Watson statistic. The tables... 

Data Processing and Statistics. Linear, power, and exponential lines were fitted to the dust, trash, and reflectance data by standard regression methods. 

Ifnon-linearity is ob ious you may abstain from a statistical linearity check... 

In recent years, new methods have been introduced into chemistry for classification problems, and they have often been applied to food analytical data. The statistical linear discriminant analysis is still the most widely used method, as was noted in the previous section. 

Fig. 21. Equiprobability ellipses and discriminant lines for statistical linear discriminant analysis (bivariate case)...

Prom statistical linear response theory, the infrared spectrum of a sample at temperature T is ... 

Note Classification includes only arithmetic portions of text. Excluded are chapters of statistics, linear transformations, square roots, and coordinate geometry. 1-S = problems having one of the five situations 2-S = problems having two or more of the five situations other = problems otherwise uncoded, including probabilities, range, and geometry problems. 

Kruskal J, Three-way arrays rank and uniqueness of trilinear decompositions with applications to arithmetic complexity and statistics, Linear Algebra and its Applications, 1977,18, 95-138. 

Clarke, J.U. (1998). Evaluation of censored data methods to allow statistical comparisons among very small samples with below detection limits observations. Environmental Science Technology. Vol. 32, pp. 177-183. ISSN 1520-5851 Cole, R.A. Phelps, K. (1979). Use of canonical variate analysis in the differentiation of swede cultivars by gas-liquid chromatography of volatile hydrolysis products. Journal of the Science of Food and Agriculture. Vol. 30, pp. 669-676. ISSN 1097-0010 Coomans, D. Broeckaert, L Fonckheer, M Massart, D.L. Blocks, P. (1978). The application of linear discriminant analysis in the diagnosis of thyroid diseases. Analytica Chimica Acta. Vol. 103, pp. 409-415. ISSN 0003-2670 Coomans, D. Massart, D.L. Kaufman, L. (1979) Optimization by statistical linear discriminant analysis in analytical chemistry. Analytica Chimica Acta. Vol. 112, pp. 97-122. ISSN 0003-2670... 

Again interesting ranking criteria might be different from user to user or from application to application. We have chosen the following six ranking criteria that we think are fundamental and common for scatterplots, and we have implemented them in HCE. The first three criteria are useful to reveal statistical (linear or quadratic) relationships between two dimensions (or variables), and the next three are useful to find scatterplots of interesting distributions. 

As transformation via inversion is non-Hnear, the distributions of raw error scores and derived performance win be quite different. This has no effect on ordinal analyses, such as non-parametric statistics, but wiU have some effect on Hnear analyses, such as parametric statistics, linear regression/correlation, etc., and may include improvements due to a possible greater normality of the distributions of derived performances. An alternative transformation which would retain a linear relationship with the error scores is ... 

Polystyrene - poly(methylmethacrylate) copolymer (statistical linear)... 

When the cyclic dimers reach their plateau value (x 1), the total monomer (meta + para) concentration is well above the critical concentration. Under this condition, p andp , coincide with the initial mole fractions of the two monomers (equal to 0.5), because the distribution of monomeric units within the linear fraction is always purely statistical, and when the total monomer concentration is well above the CC, the statistical linear fraction overwhelms the nonstatistical contribution of the cyclic fraction. Since the EM values of 12 and 14 (EM21 = 13.4 mM and EM23 = 0.30 mM) were known from previous one-monomer DL experiments (see Section 4.1.3), we can expect that the plateau values for the concentrations of homodimers 12 and 14 are J4 of their EM values, that is, 3.35 mM and 0.075 mM, respectively. The plateau value for concentration of homodimer 12 in the two-monomer system was indeed found to be 3.2 mM which is very close to the estimated value, whereas that of the homodimer 14 was too low to be detected, again in accordance with the estimated low value. Saturation value for the equilibrium concentration of heterodimer 13 was 11.4 mM which corresponds to EM22 = 11.4/0.5 = 22.8 mM. It should be stressed... 

Fig. 2.20. The number S N) of distinct sites visited during an AT-step random walk on a plane square lattice small diamonds) and on a percolating cluster constructed over the same lattice, at threshold small triangles). In the first case, the increase is (statistically) linear at each step, the probability of finding a new site is constant. In the second case, the discovery of new sites is much slower, going as There is a tendency to retrace the same path although, in contrast, the...

Fig. 26. Universal calibration curve for SEC. Polystsrrene (linear) O polystyrene ( comb ) + polystyrene ( star ) A polystyrene-poly(methylmethacrylate) copolymer (heterograft) x poly(methylmethacrylate) (linear) poly(vinylchloride) V polystyrene-poly(methylmethacrylate) copolymer (graft-comb) Bpolylphenyl siloxane) A polystyrene-poly(methylmethacrylate) copolymer (statistical-linear) n polybutadiene. From Ref 31.

Roberts, J.B. Spanos, P.D. 1990. Random Vibration and Statistical Linearization. Chichester John Wiley and Sons. 

In this paper we shall establish rather interesting results relating the method of statistical linearization and estimation of parameters for linear systems. In particular we establish that the coefficient estimators for linear models, when the observed data comes from a non-linear structure are, surprisingly, the "true" statistical linearization coefficients. The significance of this result is discussed, along with important questions that are generated. 

The method of statistical linearization has remained a surprisingly popular tool over the many years since it was first formulated by Booton [ 1 ], Caughey [ 2], and later developed by Kazakov [3 ], Sunahara [4], Spanos [5 ]. The more recent investigations of Wen [6], Casciati [7], Lin and Bruckner [8 ], as well as others, have been motivated by the desire to study the statistical properties of structures that possess hysteretic non-linear response characteristics. 

The concept of statistical linearization replaces a non-linear dynamical model with a linear model, whose, coefficients are explicitly formulated. However, since these coefficients must be evaluated in terms of statistics that are generated by the non-linear system and, therefore, unknown the usual approach is to evaluate these coefficients based upon assumed statistics. In the gaussian excitation case, the assumed statistics are gaussian. Thus, this lead to a further error in the approximation. 

The unexpected answer that we establish in this paper is that these parameter estimates will converge to the "true" statistical. linearization coefficients for the white noise excitation case. Even of more conceptual interest is that the specific form of the non-linearity does not have to be known. Thus, in order to construct the linear model, we only require the model order, as well as the observed response data from the true system. 

We shall review a few of the Important features of statistical linearization, as well as parameter estimation for equations that are driven by the gaussian white noise. 

The general method of statistical linearization replaces (2.1) with a linear model. 

We shall now establish the very interesting connection between (2.7)-(2.10) and the method of statistical linearization. 

In any case, once a class of models is selected, then the optimal choice is made based upon observations on the true system. Let us look at the statistical linearization problem from this point of view. 

But, most surprising is that the expectation terms in (3.8) are exactly the constants for statistical linearization given by (2.5). 

We have shown that the "true" statistical linearization coefficients for the linearized model of a non-linear structure excited by gausslan white noise can be obtained merely from observations on the response vector alone, without requiring any detailed knowledge of the specific form of the non-linearities that govern the structural behavior. 

Thus, for any laboratory model of a structure undergoing wide-band gaussian excitations on a shake table, the statistically equivalent linear model for a given vector of outputs can be obtained directly from data analysis. The question that still remains concerns the quality of the linearized model as compared to the true non- linear model of the structure. We would like to be able to say that because we have shown the statistical linearization coefficients to be the asymptotic maximum likelihood estimates of the coefficients of a linear model, then the linear model is in some sense a projection of the true non-linear model onto linear model space. 

Fatigue Life Prediction under Variable Cyclic Loading Based on Statistical Linear Cumulative Damage Rule for CFRP Laminates. Journal of Reinforced Plastics and Composites,... 

Roberts J, Spanos P (1990) Random vibration and statistical linearization. Wiley, Chichester... 

Qin J, Nishijima K, Faber MH (2012) Extrapolation method for system reliability assessment a new scheme. Adv Struct Eng 15(11) 1893-1909 Rackwitz R (1982) Response surfaces in structural reliability. Berichte zur Zuverlassigkeitstheorie der Bauwerke. Heft 67, Miinchen Roberts JB, Spanos P (2003) Random vibration and statistical linearization. Dover Publications, Mineola... 

Studies in mathematics must be beyond trigonometry and must include differential and integral calculus and differential equations. ABET encourages additional mathematics work in one or more subjects of probability and statistics, linear algebra, numerical analysis, and advanced calculus. 

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