Curve analysis with uncertainties package

[nkanazawa1989]

Summary

This is followup of #481 . This PR offloads error computation currently manually coded (sometime with error) to uncertainties package.

Details and comments

Usage of FitVal object is removed from experiment library / framework module and replaced with UFloat object. Both FitVal and UFloat are numerical object consisting of the nominal value and standard error. UFloat supports numerical operation with error propagation while keeping parameter correlation, so this will give us more precise estimate of error values that is computed based on fit values.

We need minor code update for many analysis modules, because of property name difference:

FitVal.value -> UFloat.nominal_value
FitVal.stddev -> UFloat.std_dev

However we can remove code that manually computes error propagation (without considering parameter correlation).

Effect of parameter correlation

In current Experiments curve fit code, pcov (covariance matrix) returned from curve_fit is just ignored. So parameter values are (written with ufloat object in uncertainties package, but same implementation with current code)

fit_params = [ufloat(n, e) for n, e in zip(popt, np.sqrt(np.diag(pcov)))]

on the other hand, uncertainties provides a convenient function that generates parameter set with covariance matrix.

fit_params = correlated_values(nom_values=popt, covariance_mat=pcov)

I tested the difference with T1 experiment with not enough scan range (so that error is amplified).

Without correlation

With correlation

# param_keys
['amp', 'base', 'tau']

# popt
array([9.62219381e-01, 3.90678859e-02, 2.36014395e-05])

# perr := np.sqrt(np.diag(pcov)), i.e. diagonal part. no correlation
array([3.51308248e-02, 3.79792306e-02, 1.80316371e-06])

# pcov
array([[ 1.23417485e-03, -1.31405326e-03,  5.94960145e-08],
       [-1.31405326e-03,  1.44242196e-03, -6.68205862e-08],
       [ 5.94960145e-08, -6.68205862e-08,  3.25139937e-12]])

Note that non-diagonal part (correlation) is not negligible in such configuration, and seems like we have been overestimating the confidence interval.

(note)
This PR doesn't update data representation in ExperimentData. This means once analysis complete the outcome is converted into FitVal anyways and we cannot compute error propagation anymore. Same for data loaded from the database. To support error propagation with loaded entries, we need to replace FitVal with UFloat in db service module. But this change will add more and more files to review, so will be done (if we want) in follow-up PR.

TODO

• update tutorials

[eggerdj]

Looks good. Minor suggestions.

[nkanazawa1989]

I also added 3-sigma interval according to https://en.wikipedia.org/wiki/68–95–99.7_rule

In the empirical sciences, the so-called three-sigma rule of thumb expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty

CI with parameter correlation nicely explains relation between sampling error and fit error.

[eggerdj]

Almost good. One minor naming concern.

[nkanazawa1989]

Closed since this is merged into #564 for review with full functionality