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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?


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  • 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.


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Notice different trends in growth rates among these different trees.


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  • 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.


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or downward trending (negative slope)

  • Standardization

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

y = ax + b


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  • Curves are mostly negative exponential…

y = ae -b


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

y = ae –b + k


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  • Curves can also be a polynomial or smoothing spline.


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  • 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)


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  • The smoothing spline


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  • The smoothing spline

Minimize the error terms!


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  • The smoothing spline

Minimize the error terms!


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  • 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.


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  • The smoothing spline

  • Large value for alpha


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  • The cubic smoothing spline

  • Small value for alpha


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Examples of Trend Fitting using Smoothing Splines


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  • 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.



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  • 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…


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^

  • 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.


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Predicted values

  • Standardization

Actual values


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  • 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


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  • 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!


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… to this! Age trend now gone!

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We go from this …


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… to this!

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From this …


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… to this!

  • Standardization

From this …


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  • 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!


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Master Chronology!

  • Standardization

Index Series 1

+

Index Series 2

+

Index Series 3

Calculate Mean



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