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Standardization. Standardization. The last major technique for processing your tree-ring data. Despite all this measuring, you can use raw measurements only rarely, such as for age structure studies and growth rate studies.

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- The last major technique for processing your tree-ring data.
- Despite all this measuring, you can use raw measurements only rarely, such as for age structure studies and growth rate studies.
- Remember that we’re after average growth conditions, but can we really average all measurements from one year?
- In most dendrochronological studies, you can NOT use raw measurement data for your analyses. WHY NOT?

- You can not use raw measurements because…
- Normal age-related trend exists in all tree-ring data = negative exponential or negative slope.
- Some trees simply grow faster/slower despite living in the same location.
- Despite careful tree selection, you may collect a tree that has aberrant growth patterns = disturbance.

- Therefore, you can NOT average all measurements together for a single year.

Notice different trends in growth rates among these different trees.

- You must first transform all your raw measurement data to some common average. But how?
- Detrending! This is a common technique used in many fields when data need to be averaged but have different means or undesirable trends.
- Tree-ring data form a time series. Most time series (like the stock market) have trends.
- All trends can be characterized by either a straight line a simple curve, or a more complex curve.
- That means that all trends in tree-ring time series data can be mathematically modeled with simple and complex equations.

or downward trending (negative slope)

- Standardization

- Straight lines can be either horizontal (zero slope), upward trending (positive slope),

y = ax + b

- …. but negative exponentials must be modified to account for the mean.

y = ae –b + k

- Curves can also be a polynomial or smoothing spline.

- Curves can also be a polynomial or modeled as a smoothing spline.
- Remember, all curves can be represented with a mathematical expression, some less complex and others more complex.
- Coefficients = the numbers before the x variable (= years or age, doesn’t matter).
- y = ax + b (1 coefficient)
- y = ax + bx2 + c (2 coefficients)
- y = ax + bx2 + cx3 + d (3 coefficients)
- y = ax + bx2 + cx3 + dx4 + e (4 coefficients)

- The smoothing spline

- The smoothing spline
- The spline function (g) at point (a,b) can be modeled as:
- where g is any twice-differentiable function on (a,b)
- and α is the smoothing parameter
- Alpha is very important. A large value means more data points are used in creating the smoothing algorithm, causing a smoother line.
- A small value means fewer data points are involved when creating the smoothing algorithm, resulting in a more flexible curve.

Examples of Trend Fitting using Smoothing Splines

- SO! What do all these lines and curves mean and, again, why are we interested in them?
- Remember, we need to remove the age-related trend in tree growth series because, most often, this represents noise.

- Once we’re able to fit a line or curve to our tree-ring series, we will then have an equation.
- We can use that equation to generate predicted values of tree growth for each year via regression analysis.
- How is this done? Simple…

- y = ax + b + e is the form of a regression line

- Standardization

- For each x-value (the age of the tree or year), we can generate a predicted y-value using the equation itself:
- y = ax + b is the form of a straight line
- BUT, in regression, we generate a predicted y-value which occurs either on the line or curve itself.

- For each year, we now have:
- an actual value = measured ring width
- a predicted value = from curve or line

- To detrend the tree-ring time series, we conduct a data transformation for each year:
- I = A/P

- Where I = INDEX, A = actual, and P = predicted

- Note what happens in this simple transformation: I = A/P
- If the actual ring width is equal to the predicted value, you obtain an index value of ?
- If the actual is greater than the predicted, you obtain an index value of ?
- If the actual is less than the predicted, you obtain an index value of ?
- Another (simplistic) way to think of it: an index value of 0.50 means that growth during that year was 50% of normal!

- Now, ALL series have a mean of 1.0.
- Now, ALL series have been transformed to dimensionless index values.
- Now, ALL series can be averaged together by year to develop a master tree-ring index chronology for a site.
- Remember, this master chronology now represents the average growth conditions per year from ALL measured series!

This one curve represents information from hundreds of trees (El Malpais National Monument, NM).

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