announcements - carleton university
TRANSCRIPT
Announcements
I IBM Lecture on Watson Analytics will be next MondayMarch 07 in RB 3201http://carleton.ca/ims/rooms/river-building-3201/
I Schedule of project presentations. Enter yourpreferences to the file shared on Slack
I Details about Data Day 3.0I Register (free) and attend Data Day on Tuesday March 29
http://carleton.ca/cuids/cu-events/data-day-3-0-2/I Consider participating in Graduate Student Poster
Competition (prizes: 750$, 500$, 250$ for 1st, 2nd and 3rdplace, respectively)http://carleton.ca/cuids/cu-events/data-day-3-0-graduate-student-poster-competition/
I Volunteers wanted. Please email Kathryn Elliot([email protected]) if interested
Machine Learning
February 29, 2016
Naïve Bayes Classification
Naive Bayes classifiers are especially useful for problems:
I with many input variables,I categorical input variables with a very large number of possible values,I text classification.
Naive Bayes would be a good first attempt at solving thecategorization problem.
Naïve Bayes Classification
I Applicable for categorical response with categorical predictors.I Bayes theorem says that
P(Y = y |X1 = x1,X2 = x2) =P(Y = y)P(X1 = x ,X2 = x2|Y = y)
P(X1 = x1,X2 = x2)
I The denominator can be expanded by conditioning on Y
P(X1 = x1,X2 = x2) =∑
z
P(X1 = x1,X2 = x2|Y = z)P(Y = z)
I The Naïve Bayes method is to assume the Xj are mutually conditionallyindependent, i.e.
P(X1 = x1,X2 = x2|Y = z) = P(X1 = x1|Y = z)P(X2 = x2|Y = z)
I Now the probabilities on the right-hand side can be estimated bycounting from the data.
Example of Naïve Bayeslibrary(e1071)D <- mutate(Default, income=cut(income, 3), balance=cut(balance, 2))nb.D <- naiveBayes(default~., data=D, subset=train)
* * *A-priori probabilities:Y
No Yes0.96570645 0.03429355
Conditional probabilities:student
Y No YesNo 0.7073864 0.2926136Yes 0.6181818 0.3818182
balanceY (-2.65,1.33e+03] (1.33e+03,2.66e+03]No 0.86454029 0.13545971Yes 0.09090909 0.90909091
incomeY (699,2.5e+04] (2.5e+04,4.93e+04] (4.93e+04,7.36e+04]No 0.3242510 0.5497159 0.1260331Yes 0.3927273 0.4836364 0.1236364
Example of Naïve Bayes
D <- mutate(Default, income=cut(income, 10), balance=cut(balance, 10))nb.D <- naiveBayes(default~., data=D, subset=train)nb.pred <- predict(nb.D, subset(D, test))table(Actual=D$default[test], Predicted=nb.pred)
PredictedActual No Yes
No 1905 18Yes 40 18
Neural Networks
X1Input #1
X2Input #2
X3Input #3
X4Input #4
Z1
Z2
Z3
Z4
Z5
Y1 Output #1
Y2 Output #2
Y3 Output #3
Hiddenlayer
Inputlayer
Outputlayer
Neural Networks
Zm = σ(α0m + α1mX1 + · · ·αpmXp)
Yj = β0j + β1jZ1 + · · ·+ βMjZM
I The input neurons are attached to the predictorsX1, . . . ,Xp.
I They are activated by a function σ(v) = 11+e−v .
I The neurons in the hidden layer, Z1, . . . ,Zm are linearcombinations of the inputs.
I There may be zero, one, or multiple hidden layers, witheach layer being a linear combination of the previous one.
I The last layer is attached to the outputs.
Neural Networks Example> library(nnet)> nnet.fit <- nnet(default~., data=Default, subset=train, size=5)# weights: 26initial value 6553.347412iter 10 value 1136.024073iter 20 value 1135.901203final value 1135.901077converged
> summary(nnet.fit)a 3-5-1 network with 26 weightsoptions were - entropy fittingb->h1 i1->h1 i2->h1 i3->h1-0.10 -0.22 -0.37 -0.47b->h2 i1->h2 i2->h2 i3->h20.05 -0.46 -0.25 0.25b->h3 i1->h3 i2->h3 i3->h3-0.33 0.55 0.44 0.40b->h4 i1->h4 i2->h4 i3->h40.30 0.27 0.08 -0.28b->h5 i1->h5 i2->h5 i3->h5-0.04 0.01 -0.06 -0.07b->o h1->o h2->o h3->o h4->o h5->o
-22.19 -0.01 8.29 10.50 0.18 0.35
Neural Networks Example
> nnet.pred <- predict(nnet.fit, newdata=subset(Default, test),type="class")
> table(Actual=Default$default[test], Predicted=nnet.pred)Predicted
Actual NoNo 1939Yes 76
I The table is missing the "Yes" column because the neuralnetwork didn’t predict any positives.
I The neural network model is over-parametrized and thereis danger of over-fitting.
I The minimization is unstable and random initializationleads to different solution each time.
K-Means Clustering
I Pick a number of clusters, say K .I Start with a random assignment of each observation to one
of the K clusters.I For each cluster, compute the centroid as the mean of the
points in the cluster.I Reassign observations to clusters, with each observation
going to the cluster with the nearest centroid.I Repeat until convergence.
K-Means ClusteringExample with simulated data.
pts <- read.csv(’pts_2clusters.csv’, header=TRUE)qplot(x, y, data=pts, color=cl) + labs(color="Cluster")
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0.0 2.5x
yCluster
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2
Actual clusters
K-Means ClusteringSolve for two clusters.
km.out <- kmeans(pts, 2)qplot(x, y, data=mutate(pts, cl.1=factor(km.out$cluster)), color=cl.1)
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0.0 2.5x
yCluster
●●●●
1
2
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K-Means Clustering
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0.0 2.5x
y
Cluster●●●●●●●●
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3
4
Predicted clusters
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0.0 2.5x
y
Cluster●●●●●●●●●●
1
2
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4
5
Predicted clusters
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0
2
4
6
0.0 2.5x
y
Cluster●●●●●●●●●●●●●●●●●●●●
1
2
3
4
5
6
7
8
9
10
Predicted clusters
Hierarchical Clustering
I Here we don’t pick the number of clusters in advance, this is decided bythe algorithm.
I We need a distance or dissimilarity measureI Start with each point in its own cluster.I Compute all pairwise dissimilarities and merge the two most similar
clusters.I Repeat until some stopping criterion is reached.
I To compute dissimilarity between two clusters, A and B, one may lookat different possibilities.
I Take the dissimilarity of the two centroids.
Compute all pairwise dissimilarities between points in A andpoints in B.
I Complete linkage: take the maximumI Single linkage: take the minimum;I Average linkage: take the average.
Hierarchical ClusteringExample with the same simulated data.
library(grDevices)hc.out <- hclust(dist(pts[c(’x’,’y’)]), method="complete")plot(hc.out, xlab="", main="Complete linkage", sub="")
1273 1684
860
1074 1500
1592
663 743
613
1060
1065 1399
638
1737 1836
1579
582 1369
1383
586 1597
946 1120
1246
639 1482
728 1315
1745
536 1878
1807
543 617
1056
1342 1476
1204 1734
1611 1921
553 1219
504
538
868 2000
1292 1381
883
620 1319 816
1455 1740
1424 1731 604 1654
1233
1919 1983 1404 1651
1739 1900
802
1239 1679
948 1314
741 1152
1206 1528
777 1421
1212 1716
531
674 1426
1445
579
1137 1768 747 1285
773 808 903 1561
1505
884 1781 1201 1949
1609 1995
1681
1180
544 1160
1671
1214
1935
1008 1111
1131 1447
820
560 1667
955 1640
593
1930 1988
806 1857
947 1989 819 1720
801 1359
1312
1331 1498
1149 1959 771 1772 631 1075
1112 1465
1079
708 1785
1963
1485
832 1566
644 1127
1182 1322
810
1450 1813
537 1159 602 910
1194
670 1569
1437 1947 716 1590 588 1773
1076 1270
1594 1940
1657
624
770 1797 693 1296
625
1143 1329
1439
758 1529
1829
523
831 1370
754 1371
1621
790 909
999
519 943 1730
1126 1543
628 1258
1073
906 1588
156 490
5
119 451
210
418
128 157
237 316
298
8 87
41
63 106
16 391 29 373
345 363
289 409
45
252 422
162
267
124 309
484
238 473
291
183 279
366 476
283 349
56 416
654
341 467
28 141
241
257 357
72
64 102
1253 1601
290 672
1749
823 1166
666
1324
632 1001
1725
448 1479 525 1220
1546
1336 1858
760 1924
1052 1608
1393 1409
1580
1545 1855
1384
610 1632
530 1299
1560
1154 1196
1164
1457 1922
862 1715
637 1205
1095 1452 1821
1395 1633
738
1035 1338
578 1349
1438
1040 1788 891 1994
698 1089 1287
811 1225
1966
925 1174
720 1624
690 938 506
648 1612
1041 1568
575 1547 1981 836 1467
849
1271
709 1661
1352
1577 1707
1185 1774
798 866
1215
1531 1973
671 1330
608 872
1104
559 678
1026
1004 1009
1710
930
1019
1708 1869
1713
1509 1726
701
1692
841 1307
618 1235 980 1458 942 1962
576 813
815
1366
629 960
368
682
189 225
1117
14 209
2 1593
796
914 1913
1520 1990
751
859
114 115
326
1340 1915
1279 1908 1741 1876
146
634 1165
1078 1589
1157 1620
885
984 1425
334 457
215 331
93 338
20
277
26 231
107
282 383
222
57 398 145 214
268
103 296
195
297 449
381 389
65 421
15 464
94
58 113
9 311
159
232 458
247 470
262
53 327
186
59 312
204
295 406
71
402 427
286 378
300 339
187
66 351
98 480
303
324 414
126
137 206
97
333
433
342 495
61 174
69
413
438 471
280
245 475
13
269 356
125
4
321 393
211
423
194 273
172 248
287 499
233
80 380
253
424 151 465
226
205 213
372
386
478
67 396
1
197 306
88 192
23
455 491
121
188 403
37 292
212 294
400
135 271
442
164 270
352 477
288 429
384
482
11 149
36
487
447 493
123 472
242 385
7 48
40
24 463 440
236 494
336
203
60 367
17
30
360 453
278 443
6 95
91 101
62
249 375
317
138
229 425
256
158 445
3 250
160 202
766 1317
786 1902
1237
596 740
679 1658
763
1635 1891
1744
511 1996
185 1401
805 1656
591
1456 1662 1538 540 1211
1411
733 1835
569 1197
1617
577 1581
1527 1586
1622 1905
851 1881
1834 1892
1506
1570 1937
1217 1574 1495
1249 1603
1038
143 1304 1484 1628
1087
1378
557 1904
1262
633 887
1490
1086
1138 1473
717
1861 1875
1190
941
1311
661 1792
1121
1703 1758
1228
546 897
1607 1961
1483 1668
1584
1054 1832
686 1142
598
1362 1956 1042 1816
699
991 1874 1259 1557
505 1459
136
772 1195
1530
1345 1394 765
1514 1863
1021 1818
1953
1416 1899
1368 1970
1125
844
171 1764
590 1277
1105
774 1034
1753
1555 1736
725 830
1735
907 1039
1451
996 1860
1098 1526
1158
979 1625
1335 1729
1877
1114 1199
700 1986
1751 1911
1702
1695 1825
1918
1787 1979 501 627
833 1815
963
1181
1036 1791
724
1097
715 923
1223
1032
522
1278
1238 1969
616
565 1916
1606
847 1755 870 1554
502
1412 1511
1927
1468 1810
1102 1308
676
1028
874 1176
1099 1128
1386 1844
508
687 1805
1072
1630 1653
517 568
1673 1939
635
592 1082
1548
1155
895 1859
1365 1733
814 892
1882
583
1031 1433
818
1171
521 1508
563 803
1045 1191
1619 1705
683 1954
721
1048 1776 1852 614
1268 1290 1309 1718
876
1841 953 1346
1704
1449
1010 1475
1002
856 1343
987 1440
1058
1701
780 1690
719
558
1015 1276
727
1248 1374
1742
993 1133
1840
1122
917 1700
1573
873 1496
857 962 681
951 1723
549 894
1565
1263 1685
1499
567 1286
379 921
609 1706
730
750
919 1487
1320 1814
655 1794 1462 1202 1626
1284
1848 1975
1820
737 978 1325 1763 764 905
992
1175 1209
769 1851
176 1539
1512
571
1660
1272
861 1865
867
564
826 1846 852 1912
692
515 1254
630 1291 1067 1872
1524
1265 1649
533 1251
1743
1837 1909
789 1184
1643
1163 1193
967 974
534 587
619 762 1232 1634 1894
1431
1062 1591
965
882
541 1923
902 1747 1819 1697
589 998 935 1101 1069 1208
1146 1413
1013 1059 784 875
1493 1800
1297
1110
1427 1910
600
1809 1880 908 1080
804 824
1389
1728
797 972 794 1472
526 1669
865
800 1977 1187 1407
554
552 1156 1229 1873
518 1093
1321 1693
825
936 1068
1326
898 1914
603 920 601 950
1222
933 949
1945
927 975
1081
1677
748 1955
611 1540
1141 1883 1521 1765 1134 1648 1186 1944
1144 1477 622 843
723
1750 1917
520 714
1585
695 994
768
1224 1236
1811
1061 1504
1494
1732
1145 1348
645 1328 1752
1227 1992
718 1522
787 1207
791
1895 1267 1980 1446
1542
1116 1382
1153 1417 1436 606 1481
542 1088 580
922
1129 1480
1629 1889 1775 1826
1355
1027 1727
208
969 1760
1167
731 986
788
1946
657 1646
1150 1748 615 1759
1302 1709
1463
585 1327 939 1867
1537
626 1402
1350
1360
1063 1518 1161 1092 1896
677 1595
817
1377 1943
1598
734 809
1886
597
1790
503 1007 1448 1664 1906 1123 1670 1170 1838
812 958 1968
879 1310
1316 1893 1576 668 1294 1562 1637 1600 1817
736
753 1991
1779
848
1541
1169 1866
746 1418 532 1644
1403
650 1856
1071
1443 1754
1879
642 1242
829
970 1675
1564
864 1053
1151
713
1261 1842
1280 1604
694
1982
667 1415
1376
1933
562 761
1295 1843
878
934
81
1168 1655
646
1782 1887
1410
1135 1503
1552
255 795 1301
954 989 656
1614 1958
1057
1756
827 1388
1351 1683
572
1106 1533 330 1441 263 712 1005
1266 1831
1853
1616 1960
1596
834 1183
997 1414
1650
1084 1247 959 1770 869 1801 840 1203
911
1051 1920 1298 1363
1392 1976
662
752
855 1241 1488 1971
983
1022
940 1400
1221
776 1623
838 1783
1833
1868 1950
595 1189
1430 1571 1172 1862
594
652
1103
1532 1536
1795
945 1006
612
1563
705 1293
643
1639 828
1289 1762
1523
1016 1358
370 1050
1839
1559 1898
1406
1429 1804
890
881 1613
561 1113
1029 1077
1689
1136 1491
573 1513
1525 1780
664 1812
837 1888 1094 988
581 888
1965
782 1964
1587
1264 1420 739
880 913
1627 1806 1423 1974
1305
850 1178
704
1422 1618
1391
1244
1987
793 1341
1255
621 1274 1631 1784
27 1240
1453
1641 1778 703 1572
1850
1033 1108
785 1674
929
745 853
889 1647
1901
976
675 1132
673 781
1680
328 1231 912 1396
1519 1845
665 931
1049 1507
556
982
1188 1721
18
551 1243
1442 641 918
1014 1118 1044 1551
660 896
469 886 1636
555 732
707
1018 1398
702 1024
539 1712
584 792
545
858 1719
547 1179
329
1682 1828
1688
1556
1216 1460 514 1356
1344 1687
937
1055 1218 900 1375 570
1925 1941
1306 1803
854
1461
863 1003
1012 807 1354
1173 1599 1432
944
1337 1353
1471
710 1119
726
697 899 756 990
179
901 1047
981 1130
1474 1510 1550
1544
1558
742 1957
1615
696 1066
507 1903 1332 1984
1864
109 1162
1847
308 1642
1796 1870
928 1638
706
684
1367 1497
1252
651 1469
964
822 1724
1793 1798
711
1405
1011 1575 1025 1582
799 1318
434
387 1951 1210
535 775
1043 1952
529
1722
932 1300
649 744 924
524 952
1694
1333
1148
1124
251 1978
1390
510
1808 1972
647
1361 1993 1535 1691
516 1372
1281
1830
640 839
767 1313 1147 1652
1777
966 1454
926
527 1897 599 1738
1107 1997 1672 1699 1926 1234 1492
680 957
605
548 574 722
755 1230
217
392 971
877
224 1139
607
1757 1985
1645
1686
845
1046 1177
1890 1931
147
419 1932
1198
133 348
22 431
1696
1373
51 485
729 1466
275
426 685
1023 1226
54 977
411
21
904 1275
779
1967
139 1250 915 1435
512
301 1364
410
127 1283
528 1516
497 1583
1802
228 749
566 1256
1200
1444 1549
1771
1605 1907
961
75 1385
1085
1000
1486 1799
1659
636 893
1517
1676 1746
320
1663
10 488
181
1464 1827
313 995
166 350
417
266 428
104 394
47 77
415 956
96
82 498
100 105
116
325
219 307
173
1934
259 264
293
207 454
395
377
84 335
170
305
134 468
466
220 460
180
397 401
117
227 374
420 1037
258
347
52 500 42 234
12 353
1823
68 346
435
165 274
486
86 142
73 399
216 337
120 230
319
355
38 452
196
153
199 456
55
191 322
92
19 130
201
281 439
99 218
365
32 148 318 405
39
340 462
359 371
163
364
190 239
110 361
332
150 344
169 483 111
83 407
432
78 254
34
343 496
276
235 323
436
184
122 412
244
49 390
178 240
89 140
437
362 450
272 408 152
757
354 1109
1408 1534
44 689
1064
871
76 1083
1334
1936 1998
513 1766 509
835 968
1100
369
735 1192
1288
1339 778 1357
669
90 1090
1115 1257
1665 1849
821
167 691
1489 1786
1282 1761 1929
1884
1379 1666 1824
1260 1470
1323 1501
688 1419
1020 1303 759 1478
1140 1854
658
1515 1938 1380 1999
46 1678 284 304
168 265
33
261
221 260 154 1578 653 916
200 1767
1928 1948 1822 1567 1698
550 1871
43 985
1017 1070
1387
1030 1213
1269 1714
1502
1347 1942
1610 1885
783
1397 1711
129 388
112 1428
973
177 1096
161
131
1091 1245
492
1602 1769
314 1434
842
144 1553
35 155
481
118 175
299
108 474
193 489
358
459 461
74 223
382 446
85 441
302
70
246
243 310
25 132
479
79 623
198 444
50 430
31 404
376 659
1789
315 846
285
182 1717
02
46
8
Complete linkage
Hei
ght
Hierarchical Clustering
cl.1 <- cutree(hc.out, k=2)qplot(x, y, data=mutate(pts, cl.1=factor(cl.1)), color=cl.1)plot(hc.out, xlab="", main="Complete linkage", sub="")
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4
6
0.0 2.5x
yCluster
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1
2
Complete linkage
Hierarchical Clustering
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Association rules
I Data is a binary matrix with columns corresponding toproducts and rows corresponding to baskets.
I Entry (i , j) is TRUE if customer i purchased product j .I Apriori algorithm looks at most probable sets of products
and combines them
Association rules
I Association rule is a claim such as: A & B ⇒ C.I Support for the rule is the probability of all items being
together
Support(A & B & C) =Number of baskets with A, B and C
Total number of baskets
I Confidence of a rule is the conditional probability of theimplied item
Confidence(A & B ⇒ C) =Support(A & B & C)
Support(A & B)
I Lift of a rule is
Lift(A & B ⇒ C) =Confidence(A & B ⇒ C)
Support(C)
Association rules
I We start by computing the supports of all single items andsort them.
I Then prune at say 80% and compute the support of allrules with two items of the remaining ones.
I Sort and prune. Then proceed with rules with three items,not including pairs that have been pruned. And so on.
Association rules
> mb <- read.csv(’MarketBasket.csv’) # Simulated data> library(arules)> rules <- apriori(mb, parameter=list(supp=0.8, conf=0.8, target="rules"))
Parameter specification:confidence minval smax arem aval originalSupport support minlen maxlen target
0.8 0.1 1 none FALSE TRUE 0.8 1 10 rulesext
FALSE
Algorithmic control:filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
apriori - find association rules with the apriori algorithmversion 4.21 (2004.05.09) (c) 1996-2004 Christian Borgeltset item appearances ...[0 item(s)] done [0.00s].set transactions ...[5 item(s), 500 transaction(s)] done [0.00s].sorting and recoding items ... [3 item(s)] done [0.00s].creating transaction tree ... done [0.00s].checking subsets of size 1 2 3 done [0.00s].writing ... [12 rule(s)] done [0.00s].creating S4 object ... done [0.00s].
Association rules
> inspect(rules)lhs rhs support confidence lift
1 {} => {V4} 0.932 0.9320000 1.00000002 {} => {V1} 0.950 0.9500000 1.00000003 {} => {V3} 1.000 1.0000000 1.00000004 {V4} => {V1} 0.882 0.9463519 0.99615995 {V1} => {V4} 0.882 0.9284211 0.99615996 {V4} => {V3} 0.932 1.0000000 1.00000007 {V3} => {V4} 0.932 0.9320000 1.00000008 {V1} => {V3} 0.950 1.0000000 1.00000009 {V3} => {V1} 0.950 0.9500000 1.000000010 {V1,
V4} => {V3} 0.882 1.0000000 1.000000011 {V3,
V4} => {V1} 0.882 0.9463519 0.996159912 {V1,
V3} => {V4} 0.882 0.9284211 0.9961599