Simple walkthrough¶

The heart of the DDM code (found in the ddm_calc.py file) is the computation of the image structure function. This is found by taking the average of the Fourier transforms of all image differences. By “image differences,” I mean the result of subtracting two images separated by a given lag time \(\Delta t\).

To describe the process mathematically, we find the difference between images separated by some lag time \(\Delta t\):

\[\Delta I = I(x,y;t) - I(x,y;t + \Delta t)\]

For a given \(\Delta t\) all such image differences are calculated. We then Fourier transform each \(\Delta I\) and average all of the same \(\Delta t\).

This results in the image structure function \(D(q_x,q_y,\Delta t)\).

Importing the necessary modules¶

Modules for plotting¶

We use matplotlib for creating figures and plots. Note that we use:

%matplotlib inline

This sets the backend of matplotlib to inline which means the plots are included in the notebook. If you want the plots to also be interactive (e.g., having the ability to zoom, scroll, and save) then use:

%matplotlib notebook

[1]:

%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt

Modules for numerical work¶

Here, we import numpy and xarray.

Note that xarray might not have come with your Anaconda Python distribution (or whichever other distribution you installed). If that is the case, you’ll need to install this package.

[2]:

import numpy as np #numerical python used for working with arrays, mathematical operations
import xarray as xr #package for labeling and adding metadata to multi-dimensional arrays

Import the PyDDM package¶

Make sure you append to sys.path the directory containing the PyDDM code.

[3]:

import sys
sys.path.append("../../PyDDM") #must point to the PyDDM folder
import ddm_analysis_and_fitting as ddm

Initiazing DDM_Fit class and fitting our data to a model¶

Next, we initialize the DDM_Fit class which will help us fit the calculated DDM matrix to a model of our choosing.

When the DDM_Fit class is initialized, it will display a table of the parameters, with the initial guesses and bounds, that were in the parameters of the yaml file. It will also indicate if, for the chosen model, initial guesses for parameters are missing.

Note: The intial guesses and bounds for the parameters are given in the yaml file. But you may realize that you need to adjust the bounds. You could change the yaml file and reload it. But you can also change the initial guesses and bounds for the parameters as shown in this code snippet:

ddm_fit.set_parameter_initial_guess('Tau', 0.5)
ddm_fit.set_parameter_bounds('Tau', [0.001, 1000])

Also, note that to initialize the DDM_Fit class, we need to pass it the YAML file which contains the metadata and parameters. Below, we use the fact that the path to the YAML file is stored in the variable ddm_calc.data_yaml. But we could also use the path to the YAML file just like we did when initializing the DDM_Analysis class. That is, this would work as well:

ddm_fit = ddm.DDM_Fit("../../Examples/example_data_silica_beads.yml")

[6]:

ddm_fit = ddm.DDM_Fit(ddm_calc.data_yaml)

	Initial guess	Minimum	Maximum
Amplitude	100.0	1.000	1000000.0
Tau	1.0	0.001	10.0
Background	25000.0	0.000	10000000.0
StretchingExp	1.0	0.500	1.1

Loading file C:/Users/rmcgorty/Documents/GitHub/PyDDM/Examples/images_nobin_40x_128x128_8bit_ddmmatrix.nc ...

Below, the fit is performed and saved into an xarray Dataset. We also have a table of the parameters for each q. Since this table may be long (especially if you have a large image size), you might want to not display it. If that’s the case, then run this command with the display_table parmeter set to False, i.e.:

fit01 = ddm_fit.fit(name_fit = 'fit01', display_table=False)

[7]:

#Note that the method `fit` has many optional parameters, but here we
# allow those optional parameter to take their default values

fit01 = ddm_fit.fit(name_fit = 'fit01')

In function 'get_tau_vs_q_fit', using new tau...
Fit is saved in fittings dictionary with key 'fit01'.

	q	Amplitude	Tau	Background	StretchingExp
0	0.000000	1.000000	10.000000	0.000000	1.100000
1	0.202840	5009.160959	10.000000	125.265537	0.500000
2	0.405681	8482.808495	10.000000	0.000000	0.525256
3	0.608521	10442.590957	3.868067	110.613248	0.710762
4	0.811362	16595.852548	2.556070	83.269636	0.811578
5	1.014202	18774.162108	1.424364	101.316228	0.909133
6	1.217043	20126.206549	1.035536	37.943122	0.911751
7	1.419883	20269.935277	0.742219	30.165661	0.952913
8	1.622723	19408.933976	0.552100	0.000000	0.955882
9	1.825564	21442.103307	0.465083	0.000000	0.977973
10	2.028404	23388.120235	0.396199	0.000000	0.968451
11	2.231245	27490.759703	0.321218	0.000000	1.008676
12	2.434085	33464.835149	0.290591	0.000000	1.014272
13	2.636926	39525.290291	0.248451	0.000000	1.040113
14	2.839766	41793.934306	0.213868	0.000000	1.030231
15	3.042607	42078.124470	0.192926	0.000000	1.030781
16	3.245447	40768.461703	0.172733	0.000000	1.045874
17	3.448287	35721.127412	0.155423	0.000000	1.043916
18	3.651128	28431.332255	0.142240	0.000000	1.043399
19	3.853968	20619.257623	0.128071	0.000000	1.054034
20	4.056809	14370.154840	0.121052	0.000000	1.035008
21	4.259649	9293.652992	0.111708	0.000000	1.025896
22	4.462490	5662.007475	0.102243	0.000000	1.008149
23	4.665330	3487.994425	0.097775	0.000000	0.971809
24	4.868170	2077.865135	0.090379	0.000000	0.913350
25	5.071011	1223.115625	0.089518	83.952238	0.911770
26	5.273851	876.927434	0.083446	84.578516	0.860156
27	5.476692	690.124583	0.081228	107.731737	0.859201
28	5.679532	559.962532	0.074982	89.841197	0.757985
29	5.882373	417.404173	0.077395	113.007189	0.732307
30	6.085213	339.585926	0.074093	121.429878	0.702539
31	6.288053	328.928162	0.063312	104.741958	0.648010
32	6.490894	273.021363	0.065949	123.344201	0.657741
33	6.693734	210.366431	0.069732	138.341989	0.630490
34	6.896575	168.823091	0.078894	151.432022	0.624047
35	7.099415	120.146768	0.096006	168.399637	0.643459
36	7.302256	90.471468	0.114569	179.212366	0.696849
37	7.505096	73.280303	0.143625	183.974447	0.707400
38	7.707937	63.315036	0.145241	186.298290	0.743781
39	7.910777	54.490768	0.146684	188.785276	0.771679
40	8.113617	48.703834	0.161850	189.738060	0.784964
41	8.316458	46.274244	0.157325	189.710146	0.805431
42	8.519298	42.999737	0.150650	188.305785	0.739396
43	8.722139	41.307921	0.160369	188.695165	0.715729
44	8.924979	37.804395	0.155632	188.636134	0.761223
45	9.127820	35.459070	0.163358	188.442035	0.690826
46	9.330660	31.309845	0.158481	191.127615	0.879080
47	9.533500	27.495879	0.159557	190.705808	0.826357
48	9.736341	29.578351	0.164955	189.489710	0.747781
49	9.939181	27.750185	0.173154	190.573335	0.817289
50	10.142022	26.022042	0.169798	190.445764	0.694703
51	10.344862	25.817798	0.149565	188.926426	0.677988
52	10.547703	20.964509	0.171498	192.258742	0.865291
53	10.750543	22.212659	0.186064	191.315356	0.803059
54	10.953383	20.296560	0.175117	191.111831	0.815027
55	11.156224	17.733487	0.177052	191.985542	0.975116
56	11.359064	20.012534	0.174850	190.372072	0.764394
57	11.561905	17.740871	0.167563	190.855691	0.796393
58	11.764745	19.164617	0.177914	190.699793	0.827180
59	11.967586	17.974918	0.176198	190.556119	0.771980
60	12.170426	15.801139	0.197665	191.322769	0.965799
61	12.373267	15.561198	0.167777	190.627761	0.941686
62	12.576107	14.880392	0.194683	191.407404	0.907700
63	12.778947	15.539401	0.183432	190.803455	0.990262

Trying a different mathematical model¶

Above, we fit the DDM matrix to a model where the intermediate scattering function (ISF or \(f(q,\Delta t)\)) is an exponential. That is, we fit our data to:

\[D(q,\Delta t) = A(q) \left[ 1 - \exp \left(-\Delta t / \tau(q) \right) ^{s(q)} \right] + B(q)\]

where \(A\) is the amplitude, \(B\) is the background, \(\tau\) is the decay time, and \(s\) is the stretching exponent.

An alternative is to use other information to determine \(A(q)\) and \(B\) and then fit only the ISF. We will try this below.

[9]:

ddm_fit.reload_fit_model_by_name("ISF - Single Exponential")

	Initial guess	Minimum	Maximum
Tau	1.0	0.001	10.0
StretchingExp	1.0	0.500	1.1

[10]:

fit02 = ddm_fit.fit(name_fit = 'fit02', display_table=False)

In function 'get_tau_vs_q_fit', using new tau...
Fit is saved in fittings dictionary with key 'fit02'.

[11]:

ddm.fit_report(fit02, q_indices=[3,6,9,22], forced_qs=[4,16], use_new_tau=True, show=True)

In function 'get_tau_vs_q_fit', using new tau...
In hf.plot_one_tau_vs_q function, using new tau...

Interactive with matplotlib¶

Using the Browse_DDM_Fits class as shown below, one can interactively inspect the fits (to either the DDM matrix or the ISF) by selecting the appropriate point on the \(\tau\) vs \(q\) plot or by pressing ‘N’ or ‘P’ for next or previous.

[12]:

%matplotlib notebook
fig, (ax, ax2) = plt.subplots(2, 1, figsize=(10,10/1.618))
browser = ddm.Browse_DDM_Fits(fig, ax, ax2, fit02)

fig.canvas.mpl_connect('pick_event', browser.on_pick)
fig.canvas.mpl_connect('key_press_event', browser.on_press)

ax.set_title('Decay time vs wavevector')
ax.set_xlabel("q")
ax.set_ylabel("tau (s)")

Click on a point in the tau vs q plot to see a fit.
Or press 'N' or 'P' to display next or previous fit.

[12]:

Text(0, 0.5, 'tau (s)')

[ ]:

Saving the results¶

We have a few options for saving the results of the fit. We can save the information contained in the xarray Dataset fit01 as an Excel file. Or we can save it as a netCDF file using the xarray function `to_netcdf <https://xarray.pydata.org/en/stable/generated/xarray.Dataset.to_netcdf.html>`__. If we have the fit results as a netCDF file, then we can reload it easily with teh xarray function `open_dataset <https://xarray.pydata.org/en/stable/generated/xarray.open_dataset.html>`__.

[13]:

ddm.save_fit_results_to_excel(fit01)

[14]:

fit01.to_netcdf("fit_of_ddmmatrix.nc")

PyDDM v0.1.0 documentation

Simple walkthrough

Contents