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1. Standardization

2. 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. • 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?

3. 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.

4. Standardization Notice different trends in growth rates among these different trees.

5. Standardization • 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.

6. or downward trending (negative slope) • Standardization • Straight lines can be either horizontal (zero slope), upward trending (positive slope), y = ax + b

7. Standardization • Curves are mostly negative exponential… y = ae -b

8. Standardization • …. but negative exponentials must be modified to account for the mean. y = ae –b + k

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

10. 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)

11. Standardization • The smoothing spline

12. Standardization • The smoothing spline Minimize the error terms!

13. Standardization • The smoothing spline Minimize the error terms!

14. Standardization • 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.

15. Standardization • The smoothing spline • Large value for alpha

16. Standardization • The cubic smoothing spline • Small value for alpha

17. Standardization Examples of Trend Fitting using Smoothing Splines

18. 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.

19. More Examples of Trend Fitting

20. Standardization • 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…

21. ^ • 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.

22. Predicted values • Standardization Actual values

23. 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

24. 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!

25. … to this! Age trend now gone! • Standardization We go from this …

26. … to this! • Standardization From this …

27. … to this! • Standardization From this …

28. Standardization • 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!

29. Master Chronology! • Standardization Index Series 1 + Index Series 2 + Index Series 3 Calculate Mean

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