1. Standardization

2. 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

3. 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

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

5. 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

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

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

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

9. 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 + bx 2 + c(2 coefficients) y = ax + bx 2 + cx 3 + d(3 coefficients) y = ax + bx 2 + cx 3 + dx 4 + e(4 coefficients) Standardization

10. Curves can also be a polynomial or smoothing spline. Standardization

11. The smoothing spline Standardization

12. The smoothing spline Standardization Minimize the error terms!

13. The smoothing spline Standardization Minimize the error terms!

14. 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

15. The smoothing spline Standardization Large value for alpha

16. The smoothing spline Standardization Small value for alpha

17. More Examples of Trend Fitting

18. Examples of Trend Fitting using Smoothing Splines Standardization

19. 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

20. 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

21. 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 + bis 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 + eis the form of a regression line Standardization

22. Actual values Predicted values Standardization

23. 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

24. 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

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

26. Standardization … to this! From this …

27. Standardization From this … … to this!

28. 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

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

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