Least-square hyperbola

Given m pairs of experimental data ut and t , (i = 1. m), and a general hyperbola equation 

Then we lump together the m products m,P into a vector y, c — urrjv rj, urXl, and raj into a 3-vector X of unknowns and form the ith row of the Amx3 matrix with 1, r, and m . Then apply the usual method. 

Barling and Goldstein (1989) have measured Pb and Sr isotope compositions in recent lavas from Heard Island (Southern Indian Ocean). They obtained the data listed in Table 5.6. Find the parameters of a least-square mixing hyperbola fitting the observations. 

We should nevertheless be aware that this method does not guarantee that all points will fall onto the same branch of a hyperbola, o 

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Figure 5.5 Least-square mixing hyperbola for the isotopic data on Heard Island of Barling and Goldstein (1989). Data from Table 5.10. The 87Sr/86Sr value of the MORB source ( 0.7025) lies below the horizontal asymptote. Asthenosphere and oceanic lithosphere are unlikely source components of Heard Island basalts.

In this and subsequent spreadsheet exercises, we will use the method of least-squares to fit data to a function rather than to repeat measurements. This is based on several assumptions (1) that, except for the effect of random fluctuations, the experimental data can indeed be described by a particular function (say, a straight line, a hyperbola, a circle, etc.), that (2) the random fluctuations are predominantly in the dependent parameter, which we will here call y, so that random fluctuations in the independent parameterxcanbeneglected, and (3) that those random fluctuations can be described by a single Gaussian distribution. 

The hyperbohc saturation function of the form ax i [x + b) often arises in biophysical and biochemical appHcations. It is more obvious that this represents a hyperbola, if it is written in a double-reciprocal form, as in a Lineweaver-Burk transformation of the MichaeHs-Menten enzyme kinetics that employs this type of function. Other contexts where this function appears are monomolecular photochemical kinetics and visual physiology. In the present context of action spectroscopy, x would stand for the fluence or, in some cases, the fluence rate. When x = b, the function is at the half-maximum level that is often chosen to be the criterion response. So, when one performs least-squares fits using such a function, the parameter b is the estimate of the fluence needed for the criterion response, and the effectiveness (action spectrum ordinate) is just h -... 

Kjjj-values were computer-calculated using least-square fit methods fitted to a hyperbola as described by Henderson (1978). 

Data shown represent a typical experiment, whereas Km-values indicated were determined from about 15 experiments with different preparations each calculated by computerized least-square fit method fitted to a hyperbola with a standard deviation of 0.89. Inset Original hyperbolic curve of oxygen uptake vs. added nucleotides. The symbol ( ) indicates the rate obtained with gramicidin D present (10 g/ml). Rates of samples including gramicidin D were found identical to those without gramicidin in the light as well as in the dark. 

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