Standardization

The last major technique for processing your tree-ring data.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.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?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?In most dendrochronological studies, you can NOT use raw measurement data for your 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 tree-ring data = negative exponential or negative slope.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.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.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.Therefore, you can NOT average all measurements together for a single year. StandardizationStandardization

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

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

Curves are mostly negative exponential…Curves are mostly negative exponential… y = ae -b StandardizationStandardization

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

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

Curves can also be a polynomial or modeled as a smoothing spline.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.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).Coefficients = the numbers before the x variable (= years or age, doesn’t matter). y = ax + b(1 coefficient)y = ax + b(1 coefficient) y = ax + bx 2 + c(2 coefficients)y = ax + bx 2 + c(2 coefficients) y = ax + bx 2 + cx 3 + d(3 coefficients)y = ax + bx 2 + cx 3 + d(3 coefficients) y = ax + bx 2 + cx 3 + dx 4 + e(4 coefficients)y = ax + bx 2 + cx 3 + dx 4 + e(4 coefficients) StandardizationStandardization

The smoothing splineThe smoothing spline StandardizationStandardization

The smoothing splineThe smoothing spline StandardizationStandardization Minimize the error terms!

The smoothing splineThe smoothing spline StandardizationStandardization Minimize the error terms!

The smoothing splineThe smoothing spline The spline function (g) at point (a,b) can be modeled as:The spline function (g) at point (a,b) can be modeled as: where g is any twice-differentiable function on (a,b)where g is any twice-differentiable function on (a,b) and α is the smoothing parameterand 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.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.A small value means fewer data points are involved when creating the smoothing algorithm, resulting in a more flexible curve. StandardizationStandardization

The smoothing splineThe smoothing spline StandardizationStandardization Large value for alpha

The cubic smoothing splineThe cubic smoothing spline StandardizationStandardization Small value for alpha

Examples of Trend Fitting using Smoothing Splines StandardizationStandardization

SO! What do all these lines and curves mean and, again, why are we interested in them?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.Remember, we need to remove the age-related trend in tree growth series because, most often, this represents noise. StandardizationStandardization

More Examples of Trend Fitting

Once we’re able to fit a line or curve to our tree-ring series, we will then have an equation.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.We can use that equation to generate predicted values of tree growth for each year via regression analysis. How is this done? Simple…How is this done? Simple… StandardizationStandardization

For each x-value (the age of the tree or year), we can generate a predicted y-value using the equation itself:For each x-value (the age of the tree or year), we can generate a predicted y-value using the equation itself: y = ax + bis the form of a straight liney = ax + bis the form of a straight line BUT, in regression, we generate a predicted y-value which occurs either on the line or curve itself.BUT, in regression, we generate a predicted y-value which occurs either on the line or curve itself. ^ y = ax + b + eis the form of a regression liney = ax + b + eis the form of a regression line StandardizationStandardization

Actual values Predicted values StandardizationStandardization

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

Note what happens in this simple transformation: I = A/PNote 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 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 greater than the predicted, you obtain an index value of ? If the actual is less 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!Another (simplistic) way to think of it: an index value of 0.50 means that growth during that year was 50% of normal! StandardizationStandardization

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

StandardizationStandardization … to this! From this …

StandardizationStandardization 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 dimensionless index values.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.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!Remember, this master chronology now represents the average growth conditions per year from ALL measured series! StandardizationStandardization

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).