Data Fitting in the Chemical Sciences: By the Method of Least Squares
Data Fitting in the Chemical Sciences Peter Gans, School of Chemistry, The University of Leeds, Leeds, UK Data fitting is a technique of central importance in modern experimental science. It is the means by which data is tested against a model of the experimental system, be it a theoretical or empirical model. In this book an all-round approach is adopted in which the first stage of data-fitting is seen as data collection, the second is numerical processing and the third a critical evaluation of the 'goodness' of fit in both statistical and common sense terms. Each stage is considered in detail: the sources and nature of experimental errors; the theory of least-squares fitting; probability theory; hypothesis testing, and the application of scientific criteria. The theory is complemented by three chapters on a wide range of applications. The emphasis of this book is on methodology: why certain procedures are preferred rather than how any one procedure is implemented. The author aims to assist people in extracting from their data its full information content, i.e. to use their data, not abuse it.
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Linear Least Squares
Nonlinear Least Squares
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approximation B-splines band calculated value codomain coin mass columns components concentration consider constraints contour convergence convolution coefficients convolution function correlation coefficient covariance curve data points decomposition diagonal distribution function equally spaced equilibrium constants example expectation value experimental data experimental error exponential expressions factor fitting follows formula Fourier transform Gaussian given half-width hypothesis independent variable inverse Jacobian least-squares method linear combination Lorentzian Marquardt parameter maximum mean measurements minimization minimum multiple noise non-linear normal distribution normal equations matrix objective function observations obtained orthogonal polynomials parameter values plot Poisson distribution probability distribution quadratic quantities random errors random variable reduced refinement rows sample second derivative Section shift vector shown in Figure smoothing function solution spectra spectrum spline function squared residuals standard deviation statistical straight line sum of squares symmetrical systematic error Taylor series variance variance-covariance matrix weight matrix zero