Tutorial 7 - Missing data

In this tutorial we will build a simple Bayesian network (shown below) using data that is incomplete, i.e. certain values in the data are missing (unobserved). We will then show how predictions can be performed with missing data. Finally we will demonstrate how to fill-in missing values using the Bayesian network.

Missing data network

To demonstrate learning with missing data, we will use data sampled from the multivariate Gaussian distribution tabulated below. One thousand samples were taken, following which 5% of the data was randomly set to missing.

X Y Z
Mean 6.5 2.9 5.5
Covariance (X) 0.4 0.1 0.3
Covariance (Y) 0.1 0.11 0.08
Covariance (Z) 0.3 0.08 0.3

The data in this example is very simplistic, but allows us to keep the example simple. Typically a more complex network would be used, and the network might include latent variables as used in the mixture model tutorial, allowing us to model more complex and hidden patterns (automatic feature engineering).

All the examples in this tutorial use continuous variables, however the same techniques can be used with discrete variables, and Dynamic Bayesian networks (time series).

The following concepts will be covered:

  • Continuous variables
  • Nodes with multiple variables
  • Creating a Data Connection
  • Parameter learning
  • Batch queries

Bayes Server must be installed, before starting this tutorial. An evaluation version can be downloaded from the Downloads page

Companion video (No Audio)

Create the model structure

  • Click New on the File tab to create a new empty network.

  • To add a new node, click Node on the Network tab, Editing group, to create a new node. This will launch the New node window.

  • Enter Multivariate Gaussian in the Name text box.

  • This node will contain three continuous variables, so in the Variable section, click the Multiple tab.

  • Click the Add Continuous toolbar button three times, to add three new continuous variables to the node.

  • Rename the new variables to the following, by clicking on the name of each new variable, and typing the new name.

    • X
    • Y
    • Z

    The New Node window should look like this:

    Missing data - new node

  • Click the OK button to create the new node.

    The network structure is now complete.

Learning with missing data

In this section we will learn the parameters of the multivariate Gaussian with data that contains missing values.

For convenience, we will use Microsoft Excel as the data source, however another database can be substituted.

Although Microsoft Excel is a convenient way of storing data, in practice we recommend using a database as the data source.

Adding a data connection

Note: You can skip this step, and instead use the pre-installed Tutorial data connection (Walkthrough Data in earlier versions).

  • Select the data (including the header) in the data section and copy it to the clipboard (Ctrl+C).

    If you use a database to store your data, missing data is usually denoted by (null) values, and columns will have an option to allow or disallow (null) values.

  • Open Microsoft Excel and paste the data into a new Microsoft Excel spreadsheet (Ctrl+V).

  • Save the new spreadsheet.

  • In Bayes Server, click the Data Connections button on the Data tab, Data Sources group. This will launch the Data connection manager.

  • Click the New button on the toolbar. This will launch the Data connection editor.

  • In the list of data providers, select the appropriate Excel Driver for the version of Microsoft Excel you are using.

  • Next to the File Name text box, click the Ellipsis (...) button, and select the Microsoft Excel spreadsheet created in an earlier step.

  • Click the Test Connection button, to ensure the new data connection is working.

  • Click OK to add the new Data Connection.

Parameter learning

  • Click the Parameter Learning button, on the Data tab. This will launch the Data tables window.

  • In the Data Connection drop down, select the new Data Connection created in an earlier step, or the Tutorial data connection if you skipped that step. This should enable the Data drop down.

  • In the Data drop down, select the worksheet that contains the data. (If the data is on the first worksheet, select Sheet1$). If you are using the pre-installed Tutorial data connection, select Tutorial 7 - Missing data.

  • Click the OK button. This will launch the Data map window.

  • In the Data map window, ensure that variable X has automatically been mapped to column X, variable Y has automatically been mapped to column Y, and variable Z has automatically been mapped to column Z.

    The window should look like this:

    Missing data - data map

  • Click the OK button. This will launch the Parameter learning wizard.

  • Click Next in the wizard, accepting all the default settings, until you reach the Run page. Click the Run button to start learning.

    Since we are learning with missing data, the learning procedure will require a number of iterations.

  • When learning has completed, click the Finish button on the wizard. This will launch the Candidate Networks window.

  • Click the OK button in the Candidate Networks window.

  • The distribution for the node Multivariate Gaussian has now been learned. Select the node, and click the Distribution button on the Network tab, Editing group. This will launch the Distribution editor window.

    The window should look like this:

    Missing data - distribution editor

  • Now compare the mean, variance and covariance parameters in the Distribution editor window, with those from the original sampling distribution tabulated in the introduction. The values are sufficiently close to indicate that we have successfully approximated the original distribution even though 5% of the data was discarded.

  • Click OK to close the Distribution Editor window.

Perform predictions with missing data

In this section we will predict the variable Z, using values for X and Y which might be missing. This demonstrates that we can still predict Z, even if have incomplete information. We will re-use the data we setup in the previous section.

Since we are predicting a continuous variable, the task is known as regression.

Batch query - predictions

  • Click the Batch query button, on the Data tab. This will launch the Data tables window.

  • In the Data Connection drop down, select the new Data Connection created in an earlier step, or the Tutorial data connection if you skipped that step. This should enable the Data drop down.

  • In the Data drop down, select the worksheet that contains the data. (If the data is on the first worksheet, select Sheet1$). If you are using the pre-installed Tutorial data connection, select Tutorial 7 - Missing data.

  • Click the OK button. This will launch the Data map window.

  • In the Data map window, ensure that variable X has automatically been mapped to column X, and variable Y has automatically been mapped to column Y.

Because we are predicting Z, we do not want the Z variable to be mapped.

  • Click the Un-map column button at the end of the Z row.

In order to test how well our model can predict Z, we want to have access to the Z data column, but we do not want to map it to the variable we are predicting.

  • Click on the Information tab, and click the check box next to Z.

Another way of performing the same prediction, would be to leave the default mappings (including Z) and use the Retract evidence feature which assumes the variable you are predicting is missing, even if it mapped to non missing data.

The data map windows should look like this:

Missing data - prediction data map

Missing data - prediction data map information

  • Click the OK button. This will launch the Batch query window.

  • In the query pane on the left hand side, ensure the following queries/information columns are checked.

    • Predict(Z)
    • Variance(Z)
    • X
    • Y
    • Z
  • Click the Start button on the Batch Query tab, Batch Query group. This outputs the predictions to the window.

    Instead of outputting to the window, you can also output the predictions to a database. This is useful if you are working with large datasets.

    The window should look like this:

    Missing data - batch query

    Note that we get a prediction for Z, even if the value for X or Y is missing, based on the information available. In fact, we still get a prediction if both X and Y are missing (see case 997).

    Along with the predicted value for Z, we also get a variance indicating the spread for the prediction.

  • Close the Batch query window.

Filling in missing data

In this section we will demonstrate how a Bayesian network can be used to fill-in missing values.

This is different from the prediction performed above, firstly because we are also predicting X and Y, but also because if any values are known, we retain their values.

Batch query - fill-in missing values

  • Click the Batch query button, on the Data tab. This will launch the Data tables window.

  • In the Data Connection drop down, select the new Data Connection created in an earlier step, or the Tutorial data connection if you skipped that step. This should enable the Data drop down.

  • In the Data drop down, select the worksheet that contains the data. (If the data is on the first worksheet, select Sheet1$). If you are using the pre-installed Tutorial data connection, select Tutorial 7 - Missing data.

  • Click the OK button. This will launch the Data map window.

  • In the Data map window, ensure that variable X has automatically been mapped to column X, and variable Y has automatically been mapped to column Y and variable Z has automatically been mapped to column Z.

    Unlike the previous section we include the mapping for Z.

    The data map window should look like this:

    Missing data - fill missing data map

  • Click the OK button. This will launch the Batch query window.

  • On the Batch query tab, Batch query group, ensure that the Retract option is turned OFF. This ensures that existing values will be output, and missing values will be replaced by predicted values.

  • In the query pane on the left hand side, ensure the following queries/information columns are checked.

    • Predict(X)
    • Predict(Y)
    • Predict(Z)
    • X
    • Y
    • Z

    It is the columns Predict(X), Predict(Y) and Predict(Z) that will contain our original data with missing values filled in. Although not required, we have included the columns X, Y and Z for comparison.

  • Click the Start button on the Batch Query tab, Batch Query group. This outputs the predictions to the window.

    Instead of outputting to the window, you can also output the predictions to a database. This is useful if you are working with large datasets.

    The window should look like this:

    Missing data - batch query fill missing

  • Note that the original data is retained, however missing values are replaced with predicted values.

Data

Case X Y Z
0 7.361322367 3.643201047 6.588605476
1 5.922611866 5.177020763
2 6.734592687 2.540409254 5.735911197
3 6.214891612 3.098860437 5.212103279
4 7.245333106 3.737858898 6.246807317
5 6.187227835 2.809222957 5.154439521
6 6.127841245 5.257808701
7 5.420549855 2.789983186 3.824171095
8 7.636637234 2.822159507 6.0035082
9 6.865330437 2.707942279 5.560468267
10 7.026775618 3.18926625 5.544403824
11 8.241347584 3.667843081 7.136554568
12 6.862291101 3.384921933
13 6.382521691 2.616473832 5.686374713
14 7.741580895 3.649089818 6.209658017
15 7.35690619 5.858972228
16 6.283815662 3.102976165 5.383477518
17 8.1692888 3.393410808 7.410928557
18 6.415122356 3.37686942 5.360733995
19 7.319021079 2.888678131 6.32623456
20 5.487696 4.761963138
21 5.537592284 2.641392062
22 5.549138903 2.687142009 4.946209321
23 2.915170336 5.926632454
24 6.814135318 2.860260755 5.840769372
25 5.969226118 2.557137533 4.962035067
26 7.328855174 3.424848988 5.699110923
27 5.992640112 2.821734388 5.39441374
28 3.30077682 5.597015287
29 6.125132829 2.994691325 5.021193115
30 2.91047831 5.123655817
31 6.437989569 3.124747556 5.841610558
32 6.534085647 2.855246834 5.120398352
33 3.019425824 5.72764605
34 6.563657499 3.07146835 5.695404153
35 7.14056442 2.941180993 6.042606067
36 6.30789806 2.458248557 5.30204915
37 6.879490735 3.475622606 5.742896246
38 6.407409625 3.127032583 5.514673104
39 5.88065558 3.351495179 5.148263324
40 2.994561306 5.174832252
41 6.353335182 2.854178844 5.315722038
42 6.976010328 3.3518604
43 7.46693531 3.648246109 6.615507994
44 6.244280149 2.869164132 5.08913155
45 6.085534082 2.64720529 5.200212027
46 5.9713148 2.913353784 5.037540821
47 2.977520081 5.937204306
48 6.057136532 4.951382998
49 6.882216192 3.240281267 5.743295966
50 5.640954857 3.015062939 4.75385882
51 7.321360904 6.305422628
52 6.455554432 2.818199177 5.850607894
53 6.268045882 3.038089256 5.129929448
54 6.345488741 3.136579476 5.695596935
55 7.060588272 2.988998422 5.69256609
56 6.501012554 2.830530268 5.321441873
57 6.382532128 3.120994255 5.660898431
58 5.662831617 2.697172784
59 7.048640482 2.828555842 5.645351624
60 6.805833875 2.490609499 6.097848922
61 6.607422216 2.982227352 5.509822749
62 7.186044038 2.971533988 6.067915739
63 6.358017453 2.893218043 5.4341287
64 7.518403921 3.318224851 6.461504966
65 6.020192143 2.139412397 5.488469128
66 5.825310702 2.167001882 5.199221927
67 6.853488349 3.20644835 6.055716077
68 5.802990108 2.431395262 4.951872716
69 6.347060776 2.59097972 5.275174484
70 2.867355123 4.53991452
71 7.551006666 2.930814592 6.252212258
72 6.956474922 2.946300113 5.684037963
73 6.576540302 3.241121899 5.396611449
74 3.421800591
75 6.086830191 3.043787569 5.348294414
76 6.240607142 2.210123178 4.897220562
77 6.605607336 3.346278478 5.750291587
78 3.030177481 5.266824964
79 5.490461783 2.249058077 4.380099178
80 6.31461542 2.949349575 5.689228517
81 5.931297807 2.868347476 4.939817685
82 2.69959019 4.296612051
83 6.689073094 2.683420306 5.945552455
84 7.00757263 2.971770331 6.04460193
85 6.149398359 2.540029993 5.302082457
86 6.465218078 2.803324551 6.113201536
87 6.637031623 2.910439685 5.285953636
88 7.077141159 3.927992996 5.687541369
89 6.28687294 3.026804266 5.062739577
90 6.028051908 2.363302372 5.27109251
91 6.402809092 2.904332673 5.016257518
92 6.775156922 2.97920289 5.969680585
93 5.987869675 2.821823643 5.289889639
94 6.333704604 3.052481508 5.122340327
95 6.308542175 2.913524776 5.510700359
96 6.032685882 2.355387057 5.052397241
97 6.570714516 5.602559121
98 5.878432738 2.604579845 4.96776982
99 6.556585753 2.686323245 5.595305408
100 6.487414818 3.036945011 5.447483183
101 6.184605404 2.978941104 5.5573754
102 7.630436437 6.267985493
103 7.492649134 3.57768332 5.952459046
104 6.059613772 2.723297138 5.309575852
105 7.016809185 3.303013279 5.675482494
106 5.353811728 2.199801987 4.635987666
107 5.735329006 3.001137479 4.411810395
108 6.614689885 2.734556349
109 6.134413542 3.20116548 5.425713252
110 6.676398064 2.90098181 5.717644583
111 5.907583177 2.726105676 5.259474915
112 4.895193617 2.674765559 4.1694444
113 7.218781954 3.057266354 6.624032652
114 2.862572558 5.714951145
115 6.753359476 2.873792528
116 7.363700426 2.568008828 6.053455074
117 7.100027953 2.874574307 6.117414783
118 6.007840116 2.341920886 5.123433543
119 6.659391721 2.952358938 5.852424552
120 6.502406223 2.662795828
121 6.684169599 3.336343763 5.30299905
122 6.96896762 3.23246908 6.090332869
123 5.916009156 2.752365768 5.126880456
124 6.61792677 5.177048925
125 7.194942641 2.63576156 6.175994666
126 3.931668721
127 6.918439527 3.052689784 5.982199991
128 7.550337227 2.476160292 6.817891974
129 6.144706782 2.521898648 5.320753668
130 6.65632187 2.611218013 5.797190702
131 6.527684824 2.847870383 4.971596694
132 5.895315478 2.832152111 5.078776905
133 6.806538025 2.877300436 6.061073965
134 3.089530712 5.295367399
135 6.804589737 3.049129925 5.55281585
136 5.622716653 2.537849724 4.554709924
137 6.708489801 2.606386902 5.728295962
138 5.8207204 2.598661969 5.194597464
139 5.708823368 2.871966811 4.895632787
140 7.074762068 3.124964175 5.971105588
141 5.613388619 2.243767581 4.813146675
142 7.904074888 3.597291299 6.245908958
143 6.836083109 2.791231488 5.971413994
144 6.230064831 3.212994667 4.650919116
145 6.407247247 3.026649985 5.3961284
146 6.406918863 2.334885183 5.424689734
147 6.896141251 6.223953765
148 7.084181012 3.265680002 6.13185279
149 7.296952683 3.276935815 5.5306946
150 6.682895844 2.700447218 5.297744203
151 7.518296677 3.182496741 5.904263233
152 6.91625288 2.85600896 5.760381359
153 6.633209915 3.201560441 5.639482636
154 7.0211199 3.073996556 6.261321843
155 9.042261327 3.396359008 7.467300747
156 7.340466132 3.28017227 6.072604745
157 7.116118021 2.924143917 5.847241518
158 6.738063099 3.032361028 5.860911661
159 5.661035287 2.598830681 5.098418561
160 5.467996357 2.130209122 4.979502813
161 6.810160147 3.158106994 5.890262741
162 6.458575035 5.275777978
163 6.684370778 2.706043352 6.024105122
164 4.716045252 2.190078374 3.953523841
165 5.663167389 2.269116781 4.938256185
166 6.773706218 2.941131718 5.379892249
167 5.765264396 2.703574481 5.06304299
168 6.682023872 3.171587498 5.721538225
169 6.647114191 3.043006859 5.819118688
170 5.857453521 3.103367816 5.044418451
171 6.362953926 2.779090487 5.496660196
172 2.750106448 5.620231942
173 6.620380813 2.815381747 5.656687971
174 6.562366738 2.845120077 4.904661668
175 5.911450816 2.69524639 4.907616019
176 6.382663563 3.10714041 5.351039152
177 6.320558523 2.558464132 5.51372535
178 6.833375657 3.409966744 5.473128053
179 6.529581221 3.163521942
180 6.604001786 3.338420733 5.522778394
181 6.727290032 3.164081889 5.604317953
182 7.727182523 3.271305037 6.775289327
183 6.585840552
184 6.236682989 5.144884782
185 6.750598465 3.293780293 6.022768974
186 6.267825001 2.726435819 5.13251046
187 5.084060905 2.459250609 4.643122219
188 7.255458803 3.514863884 6.723481947
189 6.257109069 3.069849909 5.499189305
190 5.29045878 2.763953807 4.758097645
191 5.248249298 2.765725421 4.322178467
192 6.197662722 3.124649836 4.902952645
193 6.010748546 3.151295502 5.247275268
194 6.345150647 2.461619527 5.291160011
195 6.382340529 2.515344238 5.124481293
196 4.742123649 1.58560423 3.727960242
197 8.050102687 3.388625637 6.34535555
198 5.367090269 2.683282031 4.477070129
199 5.789875651 3.060283635 4.721495016
200 7.060934829 2.888908381 5.777440892
201 5.579312157 2.638520322
202 5.407984306 2.804468106 3.970304972
203 7.148962252 2.996374199 6.323515097
204 6.21996113 2.783326812 5.365056987
205 6.940566544 2.671973316 5.953801851
206 2.851085241 6.24884169
207 6.104653012 2.832020146 5.904489021
208 6.519754562 2.673512687 5.212443068
209 6.24329454 2.948045361 5.158280301
210 6.726815926 2.680989145 5.805426901
211 5.864914459 2.231366834 4.897888097
212 7.684424618 2.628395105 6.331617828
213 6.301504366 2.806705235 5.035330369
214 5.934760174 2.731122755 5.167197478
215 2.374489734 5.155584496
216 6.262896227 2.893371788 4.889274143
217 6.13108405 5.255268995
218 7.20345026 2.720879613 5.688009374
219 7.105877838 2.385246479 6.080502904
220 7.647816162 3.639451525
221 5.864234272 3.114445292 5.485023223
222 6.919252083 3.05592096 5.964954805
223 7.763399308 3.2864443 6.69982554
224 5.788338016 2.685650525 4.46595344
225 5.524163548 2.426012627 4.857926829
226 5.48352112 3.265825448 4.957074521
227 6.623673254 2.995940275 5.426464583
228 6.007779669 3.034766575 5.021032533
229 6.136415972 2.905689718
230 7.520810679 2.978274163 5.980608183
231 7.007319537 2.887988637 5.845245019
232 6.853985014 3.266344686 5.778224245
233 6.696603305 2.960287983 5.487629921
234 2.543641755 5.869672408
235 5.460273853 3.006902483 5.080712726
236 6.965434924 2.758561011 6.060851212
237 7.29459662 3.646288437 5.593804793
238 5.494701858 3.048535373 4.340565358
239 7.388649717 3.339175555 6.533795452
240 7.061732471 3.481626588 5.369566725
241 6.827873459 3.336083303 5.720575492
242 5.882931848 2.972155881 5.14349811
243 5.944861653 2.395395069 5.576154307
244 6.897629245 3.077573654 5.362000638
245 7.31903224 2.858639672 6.02236564
246 6.214176594 2.285488554 4.951820611
247 6.502005371 3.036749653 5.438227763
248 5.732930279 2.609417703 4.857265443
249 6.374070178 3.034794344 5.357797353
250 7.062750527 3.708045937 5.949497907
251 6.968881831 3.17161348 6.051975841
252 6.383283577 2.904838668 5.084764712
253 7.489233312 3.0227553
254 5.785144535 2.833847428 5.179988734
255 6.239538136 3.045525977 5.676288516
256 5.910886334 2.7305909 4.722666065
257 6.831709152 2.857182027 6.160585597
258 6.661868375
259 7.209293632 3.174941953 5.959259482
260 7.478288425 2.884518033 6.248677989
261 6.764856901 3.059109185 6.106891685
262 6.292423183 2.860118661 5.387718189
263 6.152288318 2.757391083 5.228163787
264 5.490373884 2.513561973 4.549994791
265 6.796748537 3.663243423 5.321266648
266 7.224875183 3.260390795 6.022155241
267 6.489760406 2.546732962 5.204169778
268 6.384950486 3.09994258 5.330099251
269 5.972655349 2.285614567 5.176702807
270 6.024848086 2.636999519 5.22957755
271 7.497851908 3.178935022 5.635675904
272 6.71320192 3.208327813 5.738962687
273 5.795397655 2.92554303 5.286875123
274 3.212513829 5.821696699
275 6.39816874 3.101349167 5.0804893
276 7.363001272 3.010460414 6.187855708
277 6.952919819 3.18807165 6.187265414
278 7.136279958 2.776992939 5.68236817
279 6.330816734 2.428827377
280 6.987624078 3.577918526 5.616478142
281 5.955293575 2.805577637 5.040326988
282 5.339612485 2.428563209 4.628874963
283 5.588760622 2.702812167 4.909561004
284 6.301962163 2.138631065 5.0626364
285 6.603624567 3.157228949 5.487790498
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