StandardizationStandardization

• 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.

• Standardization

• 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.

• Standardization

• Standardization

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.

• Standardization

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

y = ax + b

or downward trending (negative slope)

• Standardization

• Curves are mostly negative exponential…

y = ae -b

• Standardization

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

y = ae –b + k

• Standardization

• 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)

• Standardization

• Curves can also be a polynomial or smoothing spline.

• Standardization

• The smoothing spline

• Standardization

5 10 15

24

68

10

Index

(y1

8)

• The smoothing spline

• Standardization

5 10 15

24

68

10

Index

y18

Minimize the error terms!

• The smoothing spline

• Standardization

5 10 15

24

68

10

Index

(y1

8)

Minimize the error terms!

• 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.

• Standardization

b

a

n

iii dttgtgYgS 2

1

2 )('')()(

• The smoothing spline

• Standardization

5 10 15

24

68

10

Index

y18

• Large value for alpha

• The smoothing spline

• Standardization

5 10 15

24

68

10

Index

y18

• Small value for alpha

More Examples of Trend Fitting

Examples of Trend Fitting using Smoothing Splines

• Standardization

• 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.

• Plus, each tree may be doing its own thing due to local microsite conditions and local disturbances.

• Remember that each tree must contribute equally to the final data set, necessitating that each raw measurement series must be transformed.

• The final data set will be a master tree-ring chronology using all measurement series, developed by averaging all yearly values from each measurement series to enhance the signal.

• Standardization

• Once we’re able to fit a line or curve to our tree-ring series, we will then have an equation, some very simple but others very complex. It doesn’t matter!

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

• In regression analysis, the raw measurements (y-values) are “regressed” or modeled as a function of tree age (x-values).

• This essentially says that tree growth (y) is a function of age (x): y = f(x).

• So, how is this done? Simple…

• Standardization

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

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

• BUT, using regression analysis, for each actual measurement value, we generate a predicted measurement value which occurs either on the line or curve itself.

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

• Standardization

Actual valuesPredicted values

• Standardization

• 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

• Standardization

• 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!

• Standardization

• Standardization

We go from this …

… to this! Age trend now gone!

• Standardization

… to this!

From this …

• Standardization

From this …

… to this!

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

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

• Now, ALL measurement 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!

• Standardization

• Standardization

Index Series 1

Index Series 2

Index Series 3

Master Chronology!

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Calculate Mean

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