Errors

Error Tab

Error analysis is a very complex problem and different features in the data will affect the errors in the resulting distance distribution differently. Not only does the result depend on the S/N, modulation depth, and duration of the DEER trace, but also on the quality of the background determination, the strength of regularization, and even the presence of other peaks, their width, shape, and amplitude.

Additional errors can occur due to sample and instrumental artifacts such as orientation selection, excitation bandwidth corrections, and pulse overlap. These are not included here, but should be kept in mind.

Model based fitting will directly give estimated errors for all fitting parameters based on the covariance matrix according to NIST certified procedures. These values are shown in the results table after model based fitting has succeeded. Model-based error analysis will repeatedly generate results using randomly generated parameter sets that match the error and covariance of the fitting result (involves Cholesky factorization).

Error estimation for the model free analysis is more difficult. One way would be to repeatedly introduce small variations to the data and then statistically observe the magnitude of the variations in the different parts of the result.

The following perturbations can be introduced. The values are typically entered as dimensionless relative values as described, but the actual values will be displayed too. Each setting can be enabled or disabled by clicking the green LED while retaining the input setting for future use. If the LED is off, the actual value will be zero.

    1. Additional Noise: This adds normal distributed noise to the data. The input is in multiples of the actual noise in the data, so a Value of 1 (default) would add extra noise with the same standard deviation as contained in the original data.
    2. Background noise: This takes the points defining the background (2 or 3 points, resp, depending on the background model) and adds a small random vertical shift to each point to modify the background. As with the "Data Noise" above, the random displacements are normal distributed with a standard deviation in multiples of the noise in the original data. The default is 0.1. If "Background=none", this option cannot be used.
    3. Jitter: This recalculates the kernel where all time points are individually randomly shifted by a small amount scaled as a small multiple of the point spacing. This introduces random numerical noise and thus gauges errors due to limitations of the numerical regularization procedure. The statistical probability is even over the given range (not Gaussian).
    4. The t0 error: Randomly varies the position of the zero time point in multiples of the point spacing. For example if the point spacing is 16ns and the value is 0.1, the standard deviation of the shifted positions will be 1.6ns. The default is zero.
    5. Dimensionality error: This is only available if the background has variable dimension. The units are absolute, e.g. for a value of 0.1 (the default) the dimensionality is randomly varied by that standard deviation.
    6. Regularization error: This show the sensitivity of the results to the regularization parameter (alpha or smoothness) and allows judging how influential these parameters really are.

Whenever new data is loaded in this tab (Load Current Fit) it will run in the background and other data can be analyzed in the other tabs if desired. The code repeats as fast as the computer allows but it will temporarily pause whenever interactive computations occur in order not to slow down foreground operations and spoil the user experience.

Initially, it will show the results of repeated Tikhonov regularization, because that is much faster. You will see the result of repeated recalculation after the input data is spoiled based on the parameters described above. At this point, the two green cursors should be moved inwards to limit calculation to a smaller range and speed up the fitting, however make sure to leave sufficient baseline on each side to avoid bunching of probabilities at the cutoff point. Once the range is selected, switch to "fitting". Averaging of successive tries is enabled by default and resets whenever a parameter is changed. To observe single-shots, uncheck "Accumulate'.

Optionally, the code can stop after N accumulation. It can also be paused manually if desired. Current statistics are displayed (loop time, # of averages since reset, approximate normalized chisquared).

The result is shown as a normalized 2D histogram. Narrow white lines indicate little variability in the result, while shaded areas indicate sensitivity to small variations in the inputs and are thus less reliable (Click on picture below for a larger view).

Here is the result for a noisy grid data containing equally spaced Gaussian peaks from 25..65A. While the first three peaks are well defined, the longer distances get progressively less reliable as expected.

During each accumulation, a running tally of mean and standard deviation is maintained for each point is the distance axis. The graph in the lower left shows this mean distance distribution as well as shaded areas corresponding to mean+SD and mean-SD. In other words, the vertical width of the shaded area is 2xSD for each distance. The visible graph is limited by the pixel resolution, but the saved file will have all distance points (200 by default).

(Note: this is image is from an older version. The current version has significantly more features)

It is important to understand that these results should not be interpreted as real errors, because they strongly depend on the settings. It is a tool to gauge the propagation of small variation in the data (noise) and assumptions (zero position, background) across the analysis machinery.

Pressing "save" will output the analyzed data.

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