15.5.2 Finding Y/X from X/Y Standard CurvesFitRef-FindVal-StandardCurve
The Find X/Y table allows you to obtain either a dependent variable value given an independent variable value, or an independent variable value given a dependent variable value, from the fit you performed on the data.
After fitting, there will a new worksheet named something like *FindXfromY* (or *FindYfromX*). Enter the Y (or X) values into the first column in the new sheet and the X (or Y) values will be calculated automatically.
Control over output options varies by fitting tool. In the Linear Fit tool, for instance, a check box turns on FindX/Y output and a second check box allows calculation of 95% LCL and UCL values at FindX/Y values. In the Nonlinear curve fit tool (NLFit), you can specify extra X/Y columns using a drop-down list. If there are multiple solutions, those values will be added in the extra columns. For example, in the model, there might be two x values for a given y value. So you might specify the Number of X Columns as 2 in the Find X from Y branch.
Generally, Find Y from X runs faster than Find X from Y. When finding Y from X, Origin uses the fit parameter values directly in the fitting model, to calculate the Y values. However, if you choose to find X from Y, Origin will not be able to derive an X~Y equation automatically. It has to calculate the approximate values by iteration. The algorithm used is illustrated below:
Origin first creates a uniform linear curve. For a given Y value, the range containing the Y value is found. For example, the Y value of a given point might fall into the range [yn, yn+1]. We know the X value should be within the range [xn, xn+1]. Then a new Y value y', which corresponds to X = (xn + xn+1) / 2, is computed. With the computed y', we can divide the range [yn, yn+1] into two subranges [yn, y'] and [y', yn+1]. Then y and y' are compared to see in which subrange y is located. These steps are repeated until the difference between y and y' is within some tolerance .
Note that the iteration method is used in Find X from Y. You can only find the X values for Y values within the source data range.
Algorithm for Calculating 95% Confidence Interval
When finding X from Y or Y from X, you can opt to calculate the 95% confidence interval.
- 1. Find X From Y
- Currently, Origin only supports calculating 95% confidence intervals for non-replicated data. For the specified Y value , the resulting X is .
- If is within the range of measurement, by interpolation, the standard deviation in is given by
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- where is the fitted slope, is number of points, is the sample variance, and and are the mean values of the X and Y data respectively.
- If is out of the range of measurement, by extrapolation, the standard deviation in is given by
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- The interpolated or extrapolated X with confidence interval is calculated as:
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- where is the critical t value for (one tail) and n-2 degrees of freedom.
- Note: If the weighting is specified, or there is a fixed parameter, the confidence interval for Find X From Y will not be calculated. If the Calculate 95% Confidence Interval box was checked, the result will be missing values.
- 2. Find Y From X
- The confidence band will be used to calculate the confidence interval of Y.
- Polynomial Fit and Nonlinear Fit
- 1. Find X From Y
- The confidence band will be interpolated at a given Y to calculate the confidence interval of X. If X or Y is out of the range of input data, the confidence interval (CI) of X is not calculated.
- 2. Find Y From X
- The confidence band will be used to calculate the confidence interval of Y.
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