QUB | Archaeology and Palaeoecology | The 14Chrono Centre

Manual for psimpoll and pscomb

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Menu L: age-depth conversion

Menu Ld: age model

1 Linear interpolation between dates
2 Cubic spline interpolation
3 General linear line-fitting by weighted least-squares
4 General linear line-fitting by singular value decomposition
5 Curve-fitting by Bernshtein polynomial
6 Loess smoothing

Enter age model

Ld.1

Linear interpolation is a good standby, and surprisingly hard to beat, but confidence intervals are poor.

Ld.2

Cubic spline is a form of curve fitting, based on 4-term polynomials, that passes through all the given points. It can be interesting, but is highly susceptible to wild swings even short distances outside the range of given dates.

Ld.3

The age-depth model is a polynomial curve, fitted using `normal' equations, of the type
y = a + bx + cx² + dx³ etc.,
where y is estimated age, and x is sediment depth. Having selected one of these options, you may need to indicate how many terms you want in the polynomial (menu Le). A two-term model (i.e., y = a + bx) is identical to linear regression. The number should be in the range from 2 up to the number of age estimates available, inclusive, or else 0 to obtain the results of chi² from attempts to fit curves using first 2, then 3, up to the maximum number of terms in turn. You then need to choose which number of terms to use. In general, the idea is to use as few terms as possible, consistent with a low value for chi². To help evaluate chi², a measure of the goodness-of-fit is also given. This number is the probability that results as bad as yours would have been obtained if the selected model and number of terms is the right one. This value should, ideally, exceed 0.05, but often the best available value will be much less than this. You should notice that chi² values become lower with more terms used: a better fit with more terms. Occasionally, the goodness-of-fit will increase slightly as the number of terms increases. This is because the number of degrees of freedom decreases with increasing numbers of terms.

Ld.4

As for Ld.3, except that curves are fitted using singular value decomposition (SVD): see Press et al. (1992) for details. SVD should always give results at least as good as the normal equations: with some datasets it may do better. Try both.

Ld.5

The age-depth model is constructed using a Bernshtein polynomial, also known as a Bézier curve. This is a curve that passes smoothly through or near all the points, and is fitted by successive approximation. It is constrained so that it must pass through both endpoints (the highest and lowest ages). It will always fall within a polygon that bounds all the points. The curve uses all points for estimating an age for any given depth, so changing any point influences the entire curve. I find that it tends to fit close to most points, but will effectively bypass an outlier. The curve also tends to be rather stiff, but this has the advantage that it does not behave wildly in extrapolated portions (cf. the spline in menu Ld.2, for example). Newman and Sproull (1981) discuss the properties of these curves in more detail.

Gradients on this curve are obtained by numerical differentiation, using a central difference formula.

Ld.6

The age-depth model is constructed by local regression. This is an approach that fits curves to noisy data by a multivariate smoothing procedure: fitting a linear or quadratic function of the variables in a moving fashion that is analogous to how a moving average is computed for a time series. The method is described by Cleveland (1979) and Cleveland and Devlin (1988).

Graphical and data output is provided even where accumulation rates are negative. This may look strange, especially on the main pollen diagram.

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