A-Function-for-Finding-Critical-Points-for-the-Relationship-Between-Wood-Temperature-and-Winter-Dormant-Season-Sap-Flow
There are situations where data points, when plotted, cluster around the horizontal axis except when they're near a particular value along that axis. The 'Example Plot 1.jpeg' figure in this repository illustrates this idea. In these scenarios, the value of the predictor variable at which the response variable is most variable is unknown. This value, which I'll call the 'critical value', should be identified and likely has some significance in the system being studied. Here, I propose a method to estimate these critical values.
To estimate a critical value, I propose to split the plane of the plot up into four quadrants. The quadrants will come together at a single point on the horizontal axis somewhere between the minimum and maximum values of the predictor variable. These quadrants will be delineated by two lines that cross on the horizontal axis and that have slopes of equal magnitude but of different signs (in other words, the lines that delineate the quadrants are reflections of each other about the vertical line that passes through the point where the lines delineating the quadrant cross).
To find the critical value, two major processes will happen. First, the lines delineating the quadrants will start at the leftmost data point and together incrementally move rightward along the horizontal axis until they reach the rightmost data point. Second, at each position along the horizontal axis where these two lines cross, the angles of these lines will vary incrementally from 0 to pi / 2 radians. The 'Example Plot 2.jpeg' figure in this repository illustrates this idea. For each combination of horizontal axis intercept and angle, a least-squares value will be calculated. This least-squares value will be the sum of the squared residuals from the topmost and bottommost quadrants taken together (these residuals are the difference between the data points in these two quadrants and the vertical line that passes through the point where the lines delineating the quadrants cross) and from the leftmost and rightmost quadrants taken together (these residuals are the difference between the data points in these two quadrants and the horizontal axis). The critical point will be the value of the predictor variable that generates the minimum least-squares value.
An even better and more descriptive figure that helps to explain this process is included in this repository - it's called 'Example Plot 2.jpeg'.
Actually, a third minor process will also happen concurrently. The vertical position of this critical point will be allowed to vary as well. Although, in theory, sap flow will approach 0 at extreme temperatures, in practice, sap flow values are typically off by a small amount due to probe misalignment (details can be found in Burgess et al., 2001). Therefore, in order to determine the empirical value for no sap flow, this critical point will be allowed to vary vertically, and the vertical position of this critical point will be subtracted from all sap flow values to reset the baseline to 0.
This function takes ten arguments. The first two are required.
The Predictor_Variable argument specifies which variable will be treated as the variable on the horizontal axis. If the Data_Frame argument is used to specify a data frame containing the predictor variable column, the Predictor_Variable argument should not be written as a character string (it should not be surrounded by quotation marks). If the Data_Frame argument is not used, the Predictor_Variable argument should be written as a character string, surrounded by quotation marks. This argument must be numeric and of the same length as the Response_Variable argument.
The Response_Variable argument specifies which variable will be treated as the variable on the vertical axis. If the Data_Frame argument is used to specify a data frame containing the response variable column, the Response_Variable argument should not be written as a character string (it should not be surrounded by quotation marks). If the Data_Frame argument is not used, the Response_Variable argument should be written as a character string, surrounded by quotation marks. This argument must be numeric and of the same length as the Predictor_Variable argument.
The Data_Frame argument is optional and can be used to identify an object of class 'data.frame' that contains both the predictor and response variables.
The Horizontal_Variable_Minimum_Value argument specifies the minimum possible value for the critical point's horizontal axis (predictor variable) value. Since it may not be computationally efficient to search for the critical value across the entire range of the predictor variable, a subset of predictor variable values can be specified.
The Horizontal_Variable_Maximum_Value argument specifies the maximum possible value for the critical point's horizontal axis (predictor variable) value. Since it may not be computationally efficient to search for the critical value across the entire range of the predictor variable, a subset of predictor variable values can be specified.
The Vertical_Variable_Minimum_Value argument specifies the minimum possible value for the critical point's vertical axis (response variable) value. Since it may not be computationally efficient to search for the critical value across the entire range of the response variable, a subset of response variable values can be specified.
The Vertical_Variable_Maximum_Value argument specifies the maximum possible value for the critical point's vertical axis (response variable) value. Since it may not be computationally efficient to search for the critical value across the entire range of the response variable, a subset of response variable values can be specified.
The Slope_Interval argument specifies, in radians, how gradually slopes (angles) should increase from 0 to pi / 2. Smaller Slope_Interval values will cause the function to take longer to run but will lead to more accurate critical values. This argument must be numeric, of length 1, and between 0 and pi / 2.
The Horizontal_Axis_Intercept_Interval argument specifies, in whatever units the predictor variable is in, how gradually the quadrants shift from right to left. Smaller Horizontal_Axis_Intercept_Interval values will cause the function to take longer to run but will lead to more accurate critical values. This argument must be numeric, of length 1, and between 0 and the domain of the predictor variable.
The Vertical_Axis_Intercept_Interval argument specifies, in whatever units the response variable is in, how gradually the quadrants shift from top to bottom. Smaller Vertical_Axis_Intercept_Interval values will cause the function to take longer to run but will lead to more accurate critical values. This argument must be numeric, of length 1, and between 0 and the domain of the response variable.