In defence of the correlation coefficient

The relationship between calibrated range, residual standard deviation and correlation coefficient r is discussed. It is shown that in typical chemical calibration applications with appropriately distributed calibration points (particularly, with range comparable to mean and with approximately even or with ‘successive dilution’ spacing), the linear correlation coefficient has valid application as a routine criterion for acceptable fit if used with due care.

Linearity of calibration curves: use and misuse of the correlation coefficient

The correlation coefficient is commonly used to evaluate the degree of linear association between two variables. However, it can be shown that a correlation coefficient very close to one might also be obtained for a clear curved relationship. Other statistical tests, like the Lack-of-fit and Mandel’s fitting test thus appear more suitable for the validation of the linear calibration model. A number of cadmium calibration curves from atomic absorption spectroscopy were assessed for their linearity. All the investigated calibration curves were characterized by a high correlation coefficient (r >0.997) and low quality coefficient (QC <5%), but the straight-line model was systematically rejected at the 95% confidence level on the basis of the Lack-of-fit and Mandel’s fitting test. Furthermore, significantly different results were achieved between a linear regression model (LRM) and a quadratic regression (QRM) model in forecasting values for mid-scale calibration standards. The results obtained with the QRM did not differ significantly from the theoretically expected value, while those obtained with the LRM were systematically biased. It was concluded that a straight-line model with a high correlation coefficient, but with a lack-of-fit, yields significantly less accurate results than its curvilinear alternative. Received: 15 January 2002 Accepted: 18 April 2002     

Determination of thermal expansion coefficient of a monofilament polyamide fiber using digital image correlation

Coiled polymer actuators are a type of artificial muscles that are a promising development in the field of smart materials. The coefficient of thermal expansion of monofilament polyamide fibers is a crucial parameter for understanding the actuation of coiled fibers. The main purpose of this work is to develop a new methodology for estimating the coefficient of thermal expansion and the transition temperature of monofilament polymer fibers. In the experimental procedure, axial deformations of monofilament polyamide fiber samples were induced by temperature variations using a controlled thermal system. These deformations were determined from images of polyamide samples using the digital image correlation method. Two different approaches based on distinct temperature conditions were conducted. An alternative model with three parameters, including the coefficient of thermal expansion, was introduced to describe the thermal-mechanical behavior of monofilament polyamide fibers. Moreover, polyamide samples were also characterized using four conventional methodologies. Results indicated that the coefficient of thermal expansion changed of a modest negative value to a large negative value and this transition occurred around the glass transition temperature of the polyamide. The thermal expansion curves demonstrate good repeatability and all estimated parameters were in accordance with literature, indicating that the proposed approach can be suitable for the proposed study. This investigation may help in understanding of the intrinsic thermal-mechanical behavior of polymeric monofilaments employed as actuators.     

Comparing two K-category assignments by a K-category correlation coefficient

Predicted assignments of biological sequences are often evaluated by Matthews correlation coefficient. However, Matthews correlation coefficient applies only to cases where the assignments belong to two categories, and cases with more than two categories are often artificially forced into two categories by considering what belongs and what does not belong to one of the categories, leading to the loss of information. Here, an extended correlation coefficient that applies to K-categories is proposed, and this measure is shown to be highly applicable for evaluating prediction of RNA secondary structure in cases where some predicted pairs go into the category “unknown” due to lack of reliability in predicted pairs or unpaired residues. Hence, predicting base pairs of RNA secondary structure can be a three-category problem. The measure is further shown to be well in agreement with existing performance measures used for ranking protein secondary structure predictions. Server and software is available at http://rk.kvl.dk/     

An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient

After a measurement, a measured value and a measurement uncertainty are produced as a measurement result. By a repeated measurement, another measurement result is produced. Between the individual results of the two measurements, it is shown that there may be a significant correlation. A correlation coefficient can be determined when a GUM-compliant uncertainty budget for a measurement is available. Utilizing the correlations between the N individual results, an equation is derived to combine the N individual uncertainties of N measurements. Using the newly derived equation including the correlation coefficient, three measurement uncertainties of three measurement results are combined as an example. The combined uncertainty is compared with the uncertainty of a measurement which treats the three individual measurements as one process. Papers published in this section do not necessarily reflect the opinion of the editors, the editorial board, or the publisher.     

Electrolyte-UNIQUAC-NRF model for the correlation of the mean activity coefficient of electrolyte solutions

The new electrolyte-UNIQUAC-NRF excess Gibbs function is obtained for calculation of the activity coefficient of the binary electrolyte solutions. The excess Gibbs energy of the model consists of the Pitzer–Debye–Hückel equation, describing the long-range electrostatic contribution and the electrolyte-UNIQUAC-NRF model to account for the short-range contributions. With two adjustable parameters per electrolyte, the new model is applied to correlation of the mean activity coefficients of more than 130 binary aqueous electrolyte solutions at 25 °C. Also the binary parameters, obtaining from regression of mean activity data, are used for prediction of osmotic coefficient data for the same electrolytes. The results are compared with various local composition models such as the electrolyte-NRTL, electrolyte NRF-Wilson, electrolyte-NRTL-NRF, N-Wilson-NRF models. The comparison of the results with experiment demonstrates that the new model can correlate the experimental activity coefficient data and predict the osmotic coefficient data of binary electrolytes accurately.     

The stabilization effect of radiation crosslinking on positive temperature coefficient performances of carbon black-polymer composites

In this paper, we report the results of systematic study of effect of radiation crosslinking on PTC/NTC phenomena of carbon black(CB) filled vinyl polymers. We conclude that the effect of radiation crosslinking greatly improves the PTC intensity and electrical reproductivity in thermal cycling.     

Combining the least correlation design, wavelet packet transform and correlation coefficient test to reduce the size of calibration set for NIR quantitative analysis in multi-component systems

The paper focuses on solving a common and important problem of NIR quantitative analysis in multi-component systems: how to significantly reduce the size of the calibration set while not impairing the predictive precision. To cope with the problem orthogonal discrete wavelet packet transform (WPT), the least correlation design and correlation coefficient test (r-test) have been combined together. As three examples, a two-component carbon tetrachloride system with 21 calibration samples, a two-component aqueous system with 21 calibration samples, and a two-component aqueous system with 41 calibration samples have been treated with the proposed strategy, respectively. In comparison with some previous methods based on much more calibration samples, the results out of the strategy showed that the predictive ability was not obviously decreased for the first system while being clearly strengthened for the second one, and the predictive precision out of the third one was even satisfactory enough for most cases of quantitative analysis. In addition, all important factors and parameters related to our strategy are discussed in detail.     

The antiparallel electron correlation coefficient and Coulomb hole structure

Based on the charge density functional theory, a simple method is proposed to calculate the antiparallel electron correlation coefficient, correlation charge, and correlation energy. In contrast to the result stated in the literature, our analysis reveals that the Coulomb hole has a fine structure. To our knowledge, this is the first time it has been proven that the electrons obey the curve correlation law in an intermediate separation range but the electrons obey the linear correlation law in small and large separation ranges. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 1–6, 1999     

A nonelectrolyte local composition model and its application in the correlation of the mean activity coefficient of aqueous electrolyte solutions

The local composition models have been widely used for the correlation of activity coefficient of nonelectrolyte and electrolyte solutions. A new equation for the excess Gibbs energy function is developed based on the local composition expression of Wilson and the random reference state. This new function, the nonelectrolyte Wilson nonrandom factor (N-Wilson-NRF) model, is presented in the form of a molecular framework so that it can be used for both nonelectrolyte and electrolyte solutions. Without any particular assumptions for ionic solutions, the new function is used to described the short-range contribution of the excess Gibbs energy of electrolyte solutions. The long-range contribution is represented by Pitzer–Debye–Hückel model. With two adjustable parameters per electrolyte, the new model is applied to correlate the mean activity coefficients of more than 150 binary aqueous electrolyte solutions at 25 °C. The results are compared with various local composition models such as the electrolyte-NRTL, electrolyte NRF-Wilson and electrolyte-NRTL-NRF models. The comparison of the results with experiment demonstrates that the new model can correlate the experimental data accurately. Moreover, the model shows high precision of predictability for the osmotic coefficient of binary electrolyte solutions.     

The adhesion coefficient of fiber-reinforced polymers evaluated by dynamic measurements

A theoretical model, consisting of a series of infinite concentric cylinders surrounding a fiber in a composite material, was introduced in this paper to give a quantitative account of interface phenomena, already experimentally observed. A series of specimens, conveniently designed to represent the theoretical model, were subjected to dynamic modes of loading to measure the amount of adhesion between fibers and matrices by means of an adhesion coefficient developed in the theory. It was found that theoretical results for the adhesion between matrix and filler were compatible with the structural characteristics of the specimens tested.     

On the accuracy of correlation energy calculated by the correlation factor method: first- and second-row atoms

Summary The correlation energy of two- and four-isoelectronic series, a representative example for which the existing spin-density functionals fails, is calculated using the Colle and Salvetti method, considering mono- and multideterminantal wave functions. The results are in agreement with experimental data, and show the potentiality of this method when it is applied to wave functions including the most relevant configurational features. Also, results for the ionization energies and electron affinities of first- and second-row atoms are reported.     

Optical sensor-based measurements of thermal expansion coefficient in additive manufacturing

The applicability of fiber optic sensors for measuring the coefficient of thermal expansion (CTE) of thermoplastic parts fabricated through Additive Manufacturing is investigated. Prismatic specimens were fabricated via the Fused Deposition Modeling (FDM) technique and Fiber Bragg Grating (FBG) sensors were embedded within the structures during the building process. The specimens were manufactured featuring various raster orientations and were then subjected to thermal cycling. The wavelength measurements recorded from the optical sensors were employed for the calculation of the thermally induced strains in the specimens as a function of temperature. The obtained curves presented a linear behavior until the T_g, where the internal phase change is manifested with a peak and a subsequent decline of the curve. The CTE was calculated and the effect of raster orientation on the thermal expansion behavior was identified.     

Assessing the coefficient of variations of chemical data using bootstrap method

The coefficient of variation is frequently used in the comparison and precision of results with different scales. This work examines the comparison of the coefficient of variation without any assumptions about the underlying distribution. A family of tests based on the bootstrap method is proposed, and its properties are illustrated using Monte Carlo simulations. The proposed method is applied to chemical experiments with iid and non‐iid observations. Copyright © 2011 John Wiley & Sons, Ltd.     

连续环状色谱分离木糖—山梨糖混合溶液的研究──轴向扩散系数和总传质系数的辨识

本文提出了连续环状色谱分离过程的参数辨识方法，包括其轴向扩散系数的辨识模型，以惰性物ＣａＣｌ２作示踪剂，００１×７树脂为填料，采用扰动－应答法技术在环状色谱床中测定轴向扩散系数。考虑轴向扩散的作用，建立了线性吸附平衡体系的相平衡常数及总传质系数的辨识模型，并对木糖──山梨糖体系，采用扰动－应答技术测定相平衡常数及总传质系数。考察了操作因素对轴向扩散系数与总传质系数的影响规律。结果表明，流速增加，轴向扩散系数加大，环状色谱床中的流动明显地偏离理想活塞流；流速增加，总传质系数也相应增大。     

The temperature variation of the Harned coefficient in the system HCl−NaCl

The Harned coefficient measures the effect of an added electrolyte on the activity coefficient of another electrolyte. It is composite, depending partly on properties of solutions containing only one electrolyte and partly on the magnitude of a coefficient characteristic of interaction between two like charged ions in the mixed electrolyte solution. From the meager data available, the latter coefficient depends little upon temperature. The consequences of assuming temperature independence are discussed.Prepared for publication by R. G. Bates.     

The absorption coefficient spectrum of poly(methyl methacrylate) in the soft X-ray region

The total photoabsorption cross section for poly(methyl methacrylate) (PMMA) is measured experimentally, using synchrotron radiation, over the photon energy range 40 < E < 1000 eV. The experiments utilize nominally monochromatic radiation from a grazing incidence grating monochromator (the “Grasshopper” at the Canadian Synchrotron Radiation Facility) on the storage ring “Aladdin” at the University of Wisconsin. The raw experimental data are corrected for the spectral contamination of the incident radiation from the monochromator by utilizing a calculated photoabsorption spectrum for PMMA. The calculations are carried out using the “mixture” rule and reliable dipole oscillator strength distributions for smaller molecules related to PMMA.     

函数连接型网络应用于维生素B族四组分同时测定

报道了函数连接型网络（ＦＬＮ）用于维生素Ｂ族四组分同时测定。采用相关系数和标准偏差从原始紫外光谱数据中挑选１１个波长点供网络处理。在函数连接型网络中，非线性输入模式得到了增强，并使用了推广的δ学习规则。预测结果极行，其相关系数和标准偏差分别为０．９９９０４和０．２６８８５。     

A spark modulator for correlation chromatography

Summary A spark modulator has been tested as an injection system for correlation gas chromatography. An air/hydrocarbon vapour mixture was used as the carrier gas. Single negative sample injections were performed using a single spark period. The experiments resulted in negative sample peaks and a positive product peak. The intensity of all peaks showed a characteristic dependence on the spark frequency and power used. Ghostpeaks were absent when a correlation experiment was performed with a high concentration of a test compound in pure air. The signal to noise ratio enhancement however, was lower than the theoretical value. More experiments are necessary to expalain this phenomenon.     

Uncertainty due to the quantification step in analytical methods

The quantification step is an important source of uncertainty in analytical methods, but it is frequently misunderstood and disregarded. In this paper, it is shown how this uncertainty is closely related to the linear response range of a method, and to the Pearson correlation coefficient of the calibration line. So, if there is a need for a pre-fixed quantification uncertainty, the linear response range will be affected. Some practical cases are given showing the quantification uncertainty significance. The theoretical equation giving the value of the quantification uncertainty is deduced from which new conclusions can be taken out. Because of that, the quantification uncertainty can easily be calculated and the parameters that really affect its value are shown along the paper. Some final considerations about detection limits and two-point calibration lines are also given. The paper can also be considered a reflection on uncertainty owed to calibration and on their consequences on the analytical methodology.