Some of the mathematics used for crossdating is outlined in the section "Dendrochronology, curve matching and mathematics"
There you will find a description of "Normalization" as a way to prepare growth data to make finding the "right match" more successful.
After the normalization has been done, comparison of two curves are based on calculating correlation coefficient values out of all possible overlapping positions for the two curves. Such calculations can be quickly done with modern computers.
There are other methods for scoring how well two samples match at a certain overlapping position.
One of them is the "Gleichläufigkeit".
The Gleichläufigkeit score is based on counting how well the two trees have followed each other in growth over the years when comparing at a certain overlapping position.
Both trees growing more than the previous year gives a plus point. Both trees growing less than the previous year also gives a plus point.
One tree growing more, the other growing less gives a minus point.
The sum of these points divided by the number of years compared, gives the "Gleichläufigkeit".
As a few CDendro users asked for this type of score, it has been made available in CDendro. Though you have to turn on a setting to make it displayed in tables. In the diagram below, please find an example.
As you can see in the example above an incorrect dating may easily have a higher Gleichläufigkeit value than
the correct dating.
Typically this occurs when comparing a log from one area to a reference curve from - not the same but - a neighbour area.
In discussions, I have met people who defend the Gleichläufigkeit values and pretend they are a good complement to
other values.
Gleichläufigkeit values are available within CDendro so you can build your own opinion of its usefulness
- or unusefulness!
You can turn on Gleichläufigkeit calculations in the panel "Options for normalization of ring widths and for matching".
If you do not use it, turn it off!!!
Here is a citation from a paper by Douglas Keenan (see the section on Other sites):
-
Another statistical method used in tree-ring matching relies on what I will call "g-scores" 1).
The g-score is the proportion (or percentage) of years in which the two trees
ring widths increased or decreased together (i.e. increased or decreased from the prior
year). This method thus ignores the size of the increase or decrease. Because it ignores
so much information, the g-score method might be expected to be less reliable than the
t-score method [TTEST in CDendro]. Experience at Hohenheim, Germany, where g-scores were previously
used, seems to support this: matches were thrice found to be in error, each time after
strong assertions of reliability [Baillie, 1995: ch.2; Spurk et al., 1998]. Early trials in
Ireland also indicated problems, and the method was abandoned there [Baillie, 1982:
p.81–82,95]. Other testing found very high g-scores for matches known to be incorrect
[Schweingruber, 1989: p.77]. In the pre-computer age, though, g-scores had one
advantage: being easy to calculate. They are still sometimes used, perhaps out of habit.
1) The g-score is commonly called "Gleichläufigkeit" [Schweingruber, 1989] or "trend".
An exact description on calculating Gleichläufigkeit is found in the reference "BECKER AND GERMAN DENDROCHONOLOGY", see the section on Other sites.
More comments on Gleichläufigkeit
The examples from CDendro shown below, are taken from a development version not yet published.
I have thus implemented mechanisms in CDendro to get some statistics on the quality of GLK based crossdating.
First, we will see how a lot of 80 years long segments (blocks) taken out of samples (members) of the ITRDB collection swed308.rwl (Saltsjöbaden) crossdate
to the rest of that collection.
We will then first use the CDendro standard method for crossdating and then we will use the GLK method.
After that we will do the same crossdating with the reference taken from an island some 20 miles East of Saltsjöbaden. For that we will use the ITRDB collection swed302.rwl.
Open the swed308 collection with the command shown above!
Click the button Test towards rest of collection!
It shows that all blocks that have something else to crossdate against ("the rest") have found the best match at the right place!
i.e. how our blocks crossdate towards the rest of our collection when we use the greatest GLK value to select the best match.
Click OK.
Again click the button Test towards rest of collection!
That was using GLK to crossdate a tree towards a reference consisting of trees which had grown in the very neighbourhood of the first tree.
We will now continue by looking at how GLK works when we try to crossdate trees from one area towards the reference of a more distant but anyhow neighbour area.
A sum-sample will show up in a new window. It will automatically be selected as the Reference!
Click on the top of the Saltsjöbaden collection window to make it visible!
The small box "With block checking" is already checked, so just click the button "Test towards reference"
The table shows that we will then only be able to crossdate some 37% of our blocks and that we will have an error rate of 32% among these crossdatings!
This is what happens when we use GLK values to crossdate samples towards a reference which is not from the very same area as the samples to be crossdated!
The problem gets worse with a lower correlation coefficient value between references from the two areas.
The correlation coefficient between the Saltsjöbaden and the Nämdö collection is at 0.63 measured with the standard methods of CDendro.
When correlation coefficient is at 0.7 for two references from nearby areas, the problem with the GLK values is not as worse as shown above.
Longer blocks also lessens the problem.
To complete the comparisons, we will look at the result of the standard methods in CDendro.
This easily occurs because the sorting order for best match is undefined when there are more than one best match with exactly the same GLK value. This often occurs because of the construction of the Gleichläufigkeit algorithm.
Then again click the button "Test towards reference"!
Saltsjöbaden 2nd March 2008 (updated 8 April 2008)
Lars-Åke Larsson
For a polemical text related to this matter, see this one by Douglas Keenan!