Constrained Curve Fitting

We survey techniques for constrained curve fitting, based upon Bayesian statistics, that offer significant advantages over conventional techniques used by lattice field theorists.Comment: Lattice2001(plenary); plenary talk given by G.P. Lepage at Lattice 2001 (Berlin); 9 pages, 5 figures (postscript specials

Bayesian curve fitting for lattice gauge theorists

A new method of extracting the low-lying energy spectrum from Monte Carlo estimates of Euclidean-space correlation functions which incorporates Bayesian inference is described and tested. The procedure fully exploits the information present in the correlation functions at small temporal separations and uses this information in a way consistent with fundamental probabilistic hypotheses. The computed errors on the best-fit energies include both statistical uncertainties and systematic errors associated with the treatment of contamination from higher-lying stationary states. Difficulties in performing the integrals needed to compute these error estimates are briefly discussed.Comment: 7 pages, 6 figures, 2 tables, uses espcrc2. Talk presented at the Workshop on Lattice Hadron Physics, Colonial Club Resort, Cairns, Australia, July 9-18, 200

A Critical Analysis of Dimensions and Curve Fitting Practice in Economics

When dealing with sustainability we are concerned with the biophysical as well as the monetary aspects of economic and ecological interactions. This multidimensional approach requires that special attention is given to dimensional issues in relation to curve fitting practice in economics. Unfortunately, many empirical and theoretical studies in economics, as well as in ecological economics, apply dimensional numbers in exponential or logarithmic functions. We show that it is an analytical error to put a dimensional unit x into exponential functions ( a x ) and logarithmic functions ( x a log ). Secondly, we investigate the conditions of data sets under which a particular logarithmic specification is superior to the usual regression specification. This analysis shows that logarithmic specification superiority in terms of least square norm is heavily dependent on the available data set. The last section deals with economists’ “curve fitting fetishism”. We propose that a distinction be made between curve fitting over past observations and the development of a theoretical or empirical law capable of maintaining its fitting power for any future observations. Finally we conclude this paper with several epistemological issues in relation to dimensions and curve fitting practice in economics.dimensions, logarithmic function, curve fitting, logarithmic specification

Fast B-spline Curve Fitting by L-BFGS

We propose a novel method for fitting planar B-spline curves to unorganized data points. In traditional methods, optimization of control points and foot points are performed in two very time-consuming steps in each iteration: 1) control points are updated by setting up and solving a linear system of equations; and 2) foot points are computed by projecting each data point onto a B-spline curve. Our method uses the L-BFGS optimization method to optimize control points and foot points simultaneously and therefore it does not need to perform either matrix computation or foot point projection in every iteration. As a result, our method is much faster than existing methods

Investigating bias in the application of curve fitting programs to atmospheric time series

The decomposition of an atmospheric time series into its constituent parts is an essential tool for identifying and isolating variations of interest from a data set, and is widely used to obtain information about sources, sinks and trends in climatically important gases. Such procedures involve fitting appropriate mathematical functions to the data. However, it has been demonstrated that the application of such curve fitting procedures can introduce bias, and thus influence the scientific interpretation of the data sets. We investigate the potential for bias associated with the application of three curve fitting programs, known as HPspline, CCGCRV and STL, using multi-year records of CO2, CH4 and O3 data from three atmospheric monitoring field stations. These three curve fitting programs are widely used within the greenhouse gas measurement community to analyse atmospheric time series, but have not previously been compared extensively. The programs were rigorously tested for their ability to accurately represent the salient features of atmospheric time series, their ability to cope with outliers and gaps in the data, and for sensitivity to the values used for the input parameters needed for each program. We find that the programs can produce significantly different curve fits, and these curve fits can be dependent on the input parameters selected. There are notable differences between the results produced by the three programs for many of the decomposed components of the time series, such as the representation of seasonal cycle characteristics and the long-term (multi-year) growth rate. The programs also vary significantly in their response to gaps and outliers in the time series. Overall, we found that none of the three programs were superior, and that each program had its strengths and weaknesses. Thus, we provide a list of recommendations on the appropriate use of these three curve fitting programs for certain types of data sets, and for certain types of analyses and applications. In addition, we recommend that sensitivity tests are performed in any study using curve fitting programs, to ensure that results are not unduly influenced by the input smoothing parameters chosen. Our findings also have implications for previous studies that have relied on a single curve fitting program to interpret atmospheric time series measurements. This is demonstrated by using two other curve fitting programs to replicate work in Piao et al. (2008) on zero-crossing analyses of atmospheric CO2 seasonal cycles to investigate terrestrial biosphere changes. We highlight the importance of using more than one program, to ensure results are consistent, reproducible, and free from bias