Standardization. The last major technique for processing your tree-ring data.The last major...

StandardizationStandardization

• The last major technique for processing your The last major technique for processing your tree-ring data.tree-ring data.

• Despite all this measuring, you can use raw Despite all this measuring, you can use raw measurements only rarely, such as for age measurements only rarely, such as for age structure studies and growth rate studies.structure studies and growth rate studies.

• Remember that we’re after average growth Remember that we’re after average growth conditions, but can we really average all conditions, but can we really average all measurements from one year?measurements from one year?

• In most dendrochronological studies, you can In most dendrochronological studies, you can NOT use raw measurement data for your NOT use raw measurement data for your analyses. WHY NOT?analyses. WHY NOT?

• StandardizationStandardization

• You can not use raw measurements because…You can not use raw measurements because…

• Normal age-related trend exists in all Normal age-related trend exists in all tree-ring data = negative exponential or tree-ring data = negative exponential or negative slope.negative slope.

• Some trees simply grow faster/slower Some trees simply grow faster/slower despite living in the same location.despite living in the same location.

• Despite careful tree selection, you may Despite careful tree selection, you may collect a tree that has aberrant growth collect a tree that has aberrant growth patterns = disturbance.patterns = disturbance.

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



Notice different trends in growth rates among these different trees.

• You must first transform all your raw You must first transform all your raw measurement data to some common average. measurement data to some common average. But how?But how?

• Detrending! This is a common technique used Detrending! This is a common technique used in many fields when data need to be averaged in many fields when data need to be averaged but have different means or undesirable but have different means or undesirable trends.trends.

• Tree-ring data form a Tree-ring data form a time seriestime series. Most time . Most time series (like the stock market) have trends.series (like the stock market) have trends.

• All trends can be characterized by either a All trends can be characterized by either a straight line a simple curve, or a more straight line a simple curve, or a more complex curve.complex curve.

• That means that all trends in tree-ring time That means that all trends in tree-ring time series data can be mathematically modeled series data can be mathematically modeled with simple and complex equations.with simple and complex equations.


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

y = ax + y = ax + bb

or downward trending (negative or downward trending (negative slope)slope)


• Curves are mostly negative exponential…Curves are mostly negative exponential…

y = aey = ae -b -b


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

y = aey = ae –b –b + + kk


• Curves can also be a polynomial or smoothing Curves can also be a polynomial or smoothing spline.spline.


• Curves can also be a polynomial or modeled as Curves can also be a polynomial or modeled as a smoothing spline.a smoothing spline.

• Remember, all curves can be represented with Remember, all curves can be represented with a mathematical expression, some less complex a mathematical expression, some less complex and others more complex.and others more complex.

• Coefficients = the numbers before the x Coefficients = the numbers before the x variable (= years or age, doesn’t matter).variable (= years or age, doesn’t matter).

• y = ax + by = ax + b (1 (1 coefficient)coefficient)

• y = ax + bxy = ax + bx22 + c + c (2 (2 coefficients)coefficients)

• y = ax + bxy = ax + bx22 + cx + cx33 + d + d (3 (3 coefficients)coefficients)

• y = ax + bxy = ax + bx22 + cx + cx33 + dx + dx44 + e + e (4 (4 coefficients)coefficients)


• The smoothing splineThe smoothing spline


5 10 15

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Index

(y1

8)



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y18

Minimize the error terms!



5 10 15

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(y1

8)

Minimize the error terms!


• The spline function (g) at point (a,b) can be The spline function (g) at point (a,b) can be modeled as:modeled as:

• where g is any twice-differentiable function on where g is any twice-differentiable function on (a,b)(a,b)

• and and αα is the smoothing parameter is the smoothing parameter

• Alpha is very important. A large value means Alpha is very important. A large value means more data points are used in creating the more data points are used in creating the smoothing algorithm, causing a smoother line.smoothing algorithm, causing a smoother line.

• A small value means fewer data points are A small value means fewer data points are involved when creating the smoothing involved when creating the smoothing algorithm, resulting in a more flexible curve.algorithm, resulting in a more flexible curve.


b

a

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5 10 15

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y18

• Large value for alpha

• The cubic smoothing splineThe cubic smoothing spline


5 10 15

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y18

• Small value for alpha

Examples of Trend Fitting using Smoothing Examples of Trend Fitting using Smoothing SplinesSplines


• SO! What do all these lines and curves mean SO! What do all these lines and curves mean and, again, why are we interested in them?and, again, why are we interested in them?

• Remember, we need to remove the age-related Remember, we need to remove the age-related trend in tree growth series because, most trend in tree growth series because, most often, this represents noise.often, this represents noise.


More Examples More Examples of Trend of Trend FittingFitting

• Once we’re able to fit a line or curve to our Once we’re able to fit a line or curve to our tree-ring series, we will then have an equation.tree-ring series, we will then have an equation.

• We can use that equation to generate We can use that equation to generate predicted predicted valuesvalues of tree growth for each year via of tree growth for each year via regression analysis.regression analysis.

• How is this done? Simple…How is this done? Simple…


• For each x-value (the age of the tree or year), For each x-value (the age of the tree or year), we can generate a predicted y-value using the we can generate a predicted y-value using the equation itself:equation itself:

• y = ax + by = ax + b is the form of a straight lineis the form of a straight line

• BUT, in regression, we generate a predicted y-BUT, in regression, we generate a predicted y-value which occurs either on the line or curve value which occurs either on the line or curve itself.itself.•̂y = ax + b + ey = ax + b + e is the form of a is the form of a

regression lineregression line


Actual valuesPredicted values


• For each year, we now have:For each year, we now have:

• an actual value = measured ring widthan actual value = measured ring width

• a predicted value = from curve or linea predicted value = from curve or line

• To detrend the tree-ring time series, we To detrend the tree-ring time series, we conduct a data transformation for each year:conduct a data transformation for each year:

• I = A/PI = A/P

• Where I = INDEX, A = actual, and P = predictedWhere I = INDEX, A = actual, and P = predicted


• Note what happens in this simple Note what happens in this simple transformation: I = A/Ptransformation: I = A/P

• If the actual ring width is equal to the predicted If the actual ring width is equal to the predicted value, you obtain an index value of ?value, you obtain an index value of ?

• If the actual is greater than the predicted, you If the actual is greater than the predicted, you obtain an index value of ?obtain an index value of ?

• If the actual is less than the predicted, you If the actual is less than the predicted, you obtain an index value of ?obtain an index value of ?

• Another (simplistic) way to think of it: an index Another (simplistic) way to think of it: an index value of 0.50 means that growth during that value of 0.50 means that growth during that year was 50% of normal!year was 50% of normal!



We go from this …

… to this! Age trend now gone!


… to this!

From this …


From this …

… to this!

• Now, ALL series have a mean of 1.0.Now, ALL series have a mean of 1.0.

• Now, ALL series have been transformed to Now, ALL series have been transformed to dimensionless index values.dimensionless index values.

• Now, ALL series can be averaged together by Now, ALL series can be averaged together by year to develop a master tree-ring index year to develop a master tree-ring index chronology for a site.chronology for a site.

• Remember, this master chronology now Remember, this master chronology now represents the average growth conditions per represents the average growth conditions per year from ALL measured series!year from ALL measured series!



Index Series 1

Index Series 2

Index Series 3

Master Chronology!

+

+

Calculate Mean

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

Standardization. The last major technique for processing your tree-ring data.The last major...

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