1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
|
package de.lmu.ifi.dbs.elki.utilities.scaling.outlier;
/*
This file is part of ELKI:
Environment for Developing KDD-Applications Supported by Index-Structures
Copyright (C) 2011
Ludwig-Maximilians-Universität München
Lehr- und Forschungseinheit für Datenbanksysteme
ELKI Development Team
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Affero General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Affero General Public License for more details.
You should have received a copy of the GNU Affero General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
import de.lmu.ifi.dbs.elki.database.ids.ArrayDBIDs;
import de.lmu.ifi.dbs.elki.database.ids.DBID;
import de.lmu.ifi.dbs.elki.database.ids.DBIDUtil;
import de.lmu.ifi.dbs.elki.logging.Logging;
import de.lmu.ifi.dbs.elki.math.MathUtil;
import de.lmu.ifi.dbs.elki.math.MeanVariance;
import de.lmu.ifi.dbs.elki.result.outlier.OutlierResult;
import de.lmu.ifi.dbs.elki.utilities.documentation.Reference;
/**
* Tries to fit a mixture model (exponential for inliers and gaussian for
* outliers) to the outlier score distribution.
*
* @author Erich Schubert
*/
@Reference(authors = "J. Gao, P.-N. Tan", title = "Converting Output Scores from Outlier Detection Algorithms into Probability Estimates", booktitle = "Proc. Sixth International Conference on Data Mining, 2006. ICDM'06.", url = "http://dx.doi.org/10.1109/ICDM.2006.43")
public class MixtureModelOutlierScalingFunction implements OutlierScalingFunction {
/**
* The logger for this class.
*/
private static final Logging logger = Logging.getLogger(MixtureModelOutlierScalingFunction.class);
/**
* Parameter mu of the gaussian distribution (outliers)
*/
protected double mu;
/**
* Parameter sigma of the gaussian distribution (outliers)
*/
protected double sigma;
/**
* Parameter lambda of the exponential distribution (inliers)
*/
protected double lambda;
/**
* Mixing parameter alpha
*/
protected double alpha;
/**
* Precomputed static value
*/
public final static double ONEBYSQRT2PI = 1.0 / MathUtil.SQRTTWOPI;
/**
* Convergence parameter
*/
private static final double DELTA = 0.0001;
/**
* Compute p_i (Gaussian distribution, outliers)
*
* @param f value
* @param mu Mu parameter
* @param sigma Sigma parameter
* @return probability
*/
protected static double calcP_i(double f, double mu, double sigma) {
final double fmu = f - mu;
return ONEBYSQRT2PI / sigma * Math.exp(fmu * fmu / (-2 * sigma * sigma));
}
/**
* Compute q_i (Exponential distribution, inliers)
*
* @param f value
* @param lambda Lambda parameter
* @return probability
*/
protected static double calcQ_i(double f, double lambda) {
return lambda * Math.exp(-lambda * f);
}
/**
* Compute the a posterior probability for the given parameters.
*
* @param f value
* @param alpha Alpha (mixing) parameter
* @param mu Mu (for gaussian)
* @param sigma Sigma (for gaussian)
* @param lambda Lambda (for exponential)
* @return Probability
*/
protected static double calcPosterior(double f, double alpha, double mu, double sigma, double lambda) {
final double pi = calcP_i(f, mu, sigma);
final double qi = calcQ_i(f, lambda);
final double val = (alpha * pi) / (alpha * pi + (1.0 - alpha) * qi);
return val;
}
@Override
public void prepare(OutlierResult or) {
// Initial parameters - are these defaults sounds?
MeanVariance mv = new MeanVariance();
for(DBID id : or.getScores().iterDBIDs()) {
double val = or.getScores().get(id);
if(!Double.isNaN(val) && !Double.isInfinite(val)) {
mv.put(val);
}
}
double curMu = mv.getMean() * 2;
if(curMu == 0) {
curMu = Double.MIN_NORMAL;
}
double curSigma = Math.max(mv.getSampleStddev(), Double.MIN_NORMAL);
double curLambda = Math.min(1.0 / curMu, Double.MAX_VALUE);
double curAlpha = 0.05;
ArrayDBIDs ids = DBIDUtil.ensureArray(or.getScores().getDBIDs());
// TODO: stop condition!
int iter = 0;
// logger.debugFine("iter #-1 mu = " + curMu + " sigma = " + curSigma +
// " lambda = " + curLambda + " alpha = " + curAlpha);
while(true) {
// E and M-Steps
// Sum of weights
double tisum = 0.0;
// Weighted sum
double wsum = 0.0;
// Weighted deviation from previous mean
double sqsum = 0.0;
for(int i = 0; i < ids.size(); i++) {
double val = or.getScores().get(ids.get(i));
// E-Step
double ti = calcPosterior(val, curAlpha, curMu, curSigma, curLambda);
// M-Step
tisum += ti;
wsum += ti * val;
sqsum += ti * val * val; // (val - curMu) * (val - curMu);
}
if(tisum <= 0.0 || wsum <= 0.0) {
logger.warning("MixtureModel Outlier Scaling converged to extreme.");
break;
}
double newMu = wsum / tisum;
double newSigma = Math.max(Math.sqrt(sqsum / tisum - newMu * newMu), Double.MIN_NORMAL);
double newLambda = Math.min(tisum / wsum, Double.MAX_VALUE);
double newAlpha = tisum / ids.size();
// converged?
{
boolean changed = false;
if(Math.abs(newMu - curMu) > DELTA) {
changed = true;
}
if(Math.abs(newSigma - curSigma) > DELTA) {
changed = true;
}
if(Math.abs(newLambda - curLambda) > DELTA) {
changed = true;
}
if(Math.abs(newAlpha - curAlpha) > DELTA) {
changed = true;
}
if(!changed) {
break;
}
}
if(newSigma <= 0.0 || newAlpha <= 0.0) {
logger.warning("MixtureModel Outlier Scaling converged to extreme.");
break;
}
// logger.debugFine("iter #"+iter+" mu = " + newMu + " sigma = " +
// newSigma + " lambda = " + newLambda + " alpha = " + newAlpha);
curMu = newMu;
curSigma = newSigma;
curLambda = newLambda;
curAlpha = newAlpha;
iter++;
if(iter > 100) {
logger.warning("Max iterations met in mixture model fitting.");
break;
}
}
mu = curMu;
sigma = curSigma;
lambda = curLambda;
alpha = curAlpha;
// logger.debugFine("mu = " + mu + " sigma = " + sigma + " lambda = " +
// lambda + " alpha = " + alpha);
}
@Override
public double getMax() {
return 1.0;
}
@Override
public double getMin() {
return 0.0;
}
@Override
public double getScaled(double value) {
final double val = 1.0 - calcPosterior(value, alpha, mu, sigma, lambda);
// Work around issues with unstable convergence.
if(Double.isNaN(val)) {
return 0.0;
}
return val;
}
}
|
