Last update April 27 2015. (A new reference at the end of the document.)
What it's all about: CDendro is intended to help you crossdate wood samples towards each other!
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".
For a small update of this algorithm, please see the note at the end of this section!
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 strongly defend the Gleichläufigkeit values and pretend they are a good complement to
other values. Today I would partly agree  though only when "as a complement" is being stressed.
Gleichläufigkeit values are available within CDendro so you can build your own opinion of its usefulness.
You can turn on or off Gleichläufigkeit calculations in the panel "Options for normalization of ring widths and for matching".
Here is a citation from a paper by Douglas Keenan (see the section on Other sites):

Another statistical method used in treering matching relies on what I will call "gscores" 1).
The gscore 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 gscore method might be expected to be less reliable than the
tscore method [TTEST in CDendro]. Experience at Hohenheim, Germany, where gscores 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 gscores for matches known to be incorrect
[Schweingruber, 1989: p.77]. In the precomputer age, though, gscores had one
advantage: being easy to calculate. They are still sometimes used, perhaps out of habit.
1) The gscore 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 invention of the Gleichläufigkeit (GLK) value is sometimes described as a breakthrough in dendrochronological technology.
I have thus implemented mechanisms in CDendro to get some simple statistics on the quality of GLK based crossdating.
First, we will see how a group 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 P2Yrs method for crossdating (proportion of last two years growth) 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 a mean value
curve created from the ITRDB collection swed302.rwl.
To see a summary of this short investigation, scroll down to the end of this section!


Cropping the collection
To make sure that each block should have a right position to match we have to certify that there are at least two trees covering each yearly ring.
For that we will extract data from the inner part of the collection. First we will check the sample depth (number of rings per year) to find a suitable year interval for
getting a properly cropped collection.


This is a mean value sample created from our full collection.
The sample depth is shown as red numbers along the top of the diagram. As "Sum by stem" was checked
when this sumsample was created, these numbers represent the number of trees behind each year.

From the two diagrams above it seems that the interval 1810  1995 would be suitable for cropping the collection.



Diagram created with "Sum by stem" unchecked so that each radii is shown above.

The diagram shows the members of the cropped collection. Each 80 years long block (segment) will have a right position for matching towards the rest of the collection
even when we exclude other radii from the same tree, i.e. running with Sum by stem checked.


Before you click the "Test towards rest of collection" button, see that Sum by stem and With block checking are checked.


The result above shows that when using the P2Yrs normalization method for crossdating, then the best match for each block is also the right match!
If we only accept Tvalues above 6 we will be able to crossdate som 80% of our 80 years long blocks without any crossdating errors when we use the P2Yrs method!



Now select Gleichläufigkeit as the primary method to find the right match. Please notice also the block length settings already used above for running the example.



The result after again running "Test towards rest of collection"
shows that if we only accept GLKvalues >= 0.70 then we will have no errors though we will only be able to crossdate some 65% of the blocks. (The
others will have their best GLKvalue below 0.70!)  If we accept all GLKvalues >= 0.64, then we will be able to crossdate 93% of our blocks, though 10% of our crossdatings will be wrong!
That was using GLK to crossdate a tree towards a reference consisting of trees which had grown in the very same 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.



For this we will use a mean value .wid curve from the island of Nämdö. It is created from the ITRDB swed302.rwl collection using
the "Create Mean value sample" button with Heavy detrending selected and with a frame length of 25.
We open this sample and select it as the reference. Then we run "Test towards reference".



The table tells that when crossdating towards a nearby region and if we accept only GLKvalues above 0.72 then we will be able to
crossdate only 1/3rd or our blocks though some 10% of our datings will anyhow be wrong!

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 like P2Yrs 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 P2YrsL method in CDendro.
Note: If you try to reproduce the results above with your own software, you may get somewhat different percentage values.
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.


Crossdating with the P2YrsL normalization method towards a reference from not the same but a nearby region.
The table says that if we only accept Tvalues >= 6 then we will be able to crossdate some 40% of our 80 years long blocks probably without any crossdating errors!


This small investigation shows that
 if we use Gleichläufigkeit for crossdating towards the same
region, then it works reasonably well  though I would not recommend it because it is error prone.
 if we use Gleichläufigkeit towards a nearby region and even if we only accept GLKvalues as high as above 0.72 then we will
only be able to crossdate one third of a group of 80 years long blocks while some 10% of our datings will anyhow be wrong! This is because
there is often no clear dividing line between the right match and all the false matches. This is the real weakness with Gleichläufigkeit: It lacks good discrimination.
My conclusion: The Gleichläufigkeit method on its own should not be used for crossdating, though it might be
a useful complement to other methods!


An example of good and bad discrimination: Correlation coefficients and Gleichläufigkeit values from crossdating SNKB07B of swed308 towards swed302NamdoHD25.wid.

Some dendrochronologists want to see the Gleichläufigkeit (or "Trend") for the current overlapping position plotted along the ring width curve.
The horizontal "trend bar" is then plotted gray for a year when the two curves "agree", i.e. when they both either increase or decrease their growing.
In the right diagram above, a ring width has been removed from the curve (at the vertical line) which makes the trend bar go almost white on the right side of the vertical line = no match here!
 The overall GLK value for the left curve is 0.71, i.e. quite a high GLK value! Sample: SNKB07B.


P.S.
For a polemical text related to Gleichläufigkeit, see this by Douglas Keenan!

An update to the common definition of the Gleichläufigkeit algorithm
Allan Buras at the university of Greifswald has remarked that the common GLK value is erronously calculated when two successive
ring widths in both curves have their ring width values repeated, i.e. when the width increment (or decrement) from one year to the next happens to be exactly zero in both series.
In this case the tree growth is actually similar though that similarity does not give any "credit" to the mean GLK value.
This problem is taken care of from CDendro version 7.8.1. For practical usage this update has little relevance.
A concoct example demonstrating the problem is a series measured in millimeters but written to a file with only one decimal. This easily results in many successive ring widths
becoming the same. Crossdating that series towards itself will give correlation coefficients of 1.0 as expected but the corresponding Gleichläufigkeit value will be only 0.92
The updated GLK algorithm in CDendro 7.8.1 will produce GLK=1.00 as to be expected.
Reference:
Allan Buras, Martin Wilmking: "Correcting the calculation of Gleichläufigkeit", Dendrochronologia 34 (2015) 29–30.


LarsÅke Larsson, April 27 2015.
