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Posts Tagged ‘nonnegative matrix factorization’

Vibramycin Buy

Monday, January 11th, 2010

Vibramycin buy, In this continuing series, we explore the NMath Stats functions for performing cluster analysis. (For previous posts, see Part 1 - PCA , Part 2 - K-Means, Vibramycin doxycycline, Part 3 - Hierarchical, and Part 4 - NMF.) The sample data set we're using classifies 89 single malt scotch whiskies on a five-point scale (0-4) for 12 flavor characteristics. To visualize the data set and clusterings, we make use of the free Microsoft Chart Controls for .NET, which provide a basic set of charts, doxycycline vibramycin.

In this post, the last in the series, we'll look at how NMath provides a Monte Carlo method for performing multiple non-negative matrix factorization (NMF) clusterings using different random starting conditions, Vibramycin suspension, and combining the results.

NMF uses an iterative algorithm with random starting values for W and H. This, coupled with the fact that the factorization is not unique, means that if you cluster the columns of V multiple times, you may get different final clusterings, vibramycin buy. The consensus matrix is a way to average multiple clusterings, to produce a probability estimate that any pair of columns will be clustered together.

To compute the consensus matrix, the columns of V are clustered using NMF n times, vibramycin drug. Each clustering yields a connectivity matrix. Recall that the connectivity matrix is a symmetric matrix whose i, jth entry is 1 if columns i and j of V are clustered together, Order vibramycin, and 0 if they are not. The consensus matrix is also a symmetric matrix, whose i, jth entry is formed by taking the average of the i, jth entries of the n connectivity matrices.

Thus, vibramycin drug, each i, jth entry of the consensus matrix is a value between 0, when columns i and j are not clustered together on any of the runs, Vibramycin suspension, and 1, when columns i and j were clustered together on all runs. Vibramycin buy, The i, jth entry of a consensus matrix may be considered, in some sense, a "probability" that columns i and j belong to the same cluster.

NMF uses an iterative algorithm with random starting values for W and H. (See Part IV for more information on NMF.) This, coupled with the fact that the factorization is not unique, means that if you cluster the columns of V multiple times, vibramycin doxycycline, you may get different final clusterings. The consensus matrix is a way to average multiple clusterings, to produce a probability estimate that any pair of columns will be clustered together. Comprare vibramycin online, To compute the consensus matrix, the columns of V are clustered using NMF n times. Each clustering yields a connectivity matrix. Recall that the connectivity matrix is a symmetric matrix whose i, jth entry is 1 if columns i and j of V are clustered together, and 0 if they are not, vibramycin buy. The consensus matrix is also a symmetric matrix, whose i, vibramycin dose, jth entry is formed by taking the average of the i, jth entries of the n connectivity matrices. The i, Doxycycline vibramycin, jth entry of a consensus matrix may be considered a "probability" that columns i and j belong to the same cluster.

NMath Stats provides class NMFConsensusMatrix for computing a consensus matrix. NMFConsensusMatrix is parameterized on the NMF update algorithm to use. Vibramycin buy, Additional constructor parameters specify the matrix to factor, the order k of the NMF factorization (the number of columns in W), and the number of clustering runs. The consensus matrix is computed at construction time, so be aware that this may be an expensive operation, vibramycin uses.

For example, the following C# code creates a consensus matrix for 100 runs, clustering the scotch data (loaded into a dataframe in Part I) into four clusters:

int k = 4;
int numberOfRuns = 100;
NMFConsensusMatrix<NMFDivergenceUpdate> consensusMatrix =
new NMFConsensusMatrix<NMFDivergenceUpdate>(
data.ToDoubleMatrix().Transpose(), Vibramycin syrup, k,
numberOfRuns);

Console.WriteLine("{0} runs out of {1} converged.",
consensusMatrix.NumberOfConvergedRuns, numberOfRuns);


The output is:
100 runs out of 100 converged.

NMFConsensusMatrix provides a standard indexer for getting the element value at a specified row and column in the consensus matrix. For instance, vibramycin chlamydia, one of the goals of Young et al. was to identify single malts that are particularly good representatives of each cluster. This information could be used, for example, to purchase a representative sampling of scotches, vibramycin buy. As described in Part IV, Vibramycin doxycycline hyclate, they reported that these whiskies were the closest to each flavor profile:

  • Glendronach and Macallan

  • Tomatin and Speyburn

  • AnCnoc and Miltonduff

  • Ardbeg and Clynelish


The consensus matrix reveals, however, that the pairings are not equally strong:
Console.WriteLine("Probability that Glendronach is clustered with Macallan = {0}",
consensusMatrix[data.IndexOfKey("Glendronach"), data.IndexOfKey("Macallan")]);
Console.WriteLine("Probability that Tomatin is clustered with Speyburn = {0}", vibramycin medication,
consensusMatrix[data.IndexOfKey("Tomatin"), data.IndexOfKey("Speyburn")]);
Console.WriteLine("Probability that AnCnoc is clustered with Miltonduff = {0}",
consensusMatrix[data.IndexOfKey("AnCnoc"), Vibramycin tablets, data.IndexOfKey("Miltonduff")]);
Console.WriteLine("Probability that Ardbeg is clustered with Clynelish = {0}",
consensusMatrix[data.IndexOfKey("Ardbeg"), data.IndexOfKey("Clynelish")]);

The output is:
Probability that Glendronach is clustered with Macallan = 1
Probability that Tomatin is clustered with Speyburn = 0.4
Probability that AnCnoc is clustered with Miltonduff = 0.86
Probability that Ardbeg is clustered with Clynelish = 1

Thus, although Glendronach and Macallan are clustered together in all 100 runs, Tomatin and Speyburn are only clustered together 40% of the time, vibramycin hyclate.

A consensus matrix, C, can itself be used to cluster objects, Generic vibramycin, by perform a hierarchical cluster analysis using the distance function:

nmf_distance_function

For example, this C# code creates an hierarchical cluster analysis using this distance function, then cuts the tree at the level of four clusters, printing out the cluster members:

DoubleMatrix colNumbers = new DoubleMatrix(consensusMatrix.Order, 1, vibramycin dosage, 0, 1);
string[] names = data.StringRowKeys;

Distance.Function distance =
delegate(DoubleVector data1, DoubleVector data2)
{
int i = (int)data1[0];
int j = (int)data2[0];
return 1.0 - consensusMatrix[i, Vibramycin for cats, j];
};

ClusterAnalysis ca = new ClusterAnalysis(colNumbers, distance, Linkage.WardFunction);

int k = 4;
ClusterSet cs = ca.CutTree(k);
for (int clusterNumber = 0; clusterNumber < cs.NumberOfClusters; clusterNumber++)
{
int[] members = cs.Cluster(clusterNumber);
Console.Write("Objects in cluster {0}: ", clusterNumber);
for (int i = 0; i < members.Length; i++)
{
Console.Write("{0} ", names[members[i]]);
}
Console.WriteLine("\n");
}


The output is:
Objects in cluster 0:
Aberfeldy Auchroisk Balmenach Dailuaine Glendronach
Glendullan Glenfarclas Glenrothes Glenturret Macallan
Mortlach RoyalLochnagar Tomore

Objects in cluster 1:
Aberlour ArranIsleOf Belvenie BenNevis Benriach Benromach
Bladnoch BlairAthol Bowmore Craigallechie Dalmore
Dalwhinnie Deanston GlenElgin GlenGarioch GlenKeith
GlenOrd Glenkinchie Glenlivet Glenlossie Inchgower
Knochando Linkwood OldFettercairn RoyalBrackla
Speyburn Teaninich Tomatin Tomintoul Tullibardine

Objects in cluster 2:
AnCnoc Ardmore Auchentoshan Aultmore Benrinnes
Bunnahabhain Cardhu Craigganmore Dufftown Edradour
GlenGrant GlenMoray GlenSpey Glenallachie Glenfiddich
Glengoyne Glenmorangie Loch Lomond Longmorn
Mannochmore Miltonduff Scapa Speyside Strathisla
Strathmill Tamdhu Tamnavulin Tobermory

Objects in cluster 3:
Ardbeg Balblair Bruichladdich Caol Ila Clynelish
GlenDeveronMacduff GlenScotia Highland Park
Isle of Jura Lagavulin Laphroig Oban OldPulteney
Springbank Talisker


Once again using the cluster assignments to color the objects in the plane of the first two principal components, vibramycin uses, we can see the grouping represented by the consensus matrix (k=4).

nmf2

Well, this concludes are tour through the NMath clustering functionality. Vibramycin chlamydia, Techniques such as principal component analysis, k-means clustering, hierarchical cluster analysis, and non-negative matrix factorization can all be applied to data such as these to explore various clusterings. Vibramycin buy, Choosing among these approaches is ultimately a matter of domain knowledge and performance requirements. Is it appropriate to cluster based on distance in the original space, vibramycin hyclate, or should dimension reduction be applied. If dimension reduction is used, are negative component parameters meaningful. Doxycycline vibramycin, Are there sufficient computational resource available to construct a complete hierarchical cluster tree, or should a k-means approach be used. If an hierarchical cluster tree is computed, what distance and linkage function should be used. NMath provides a powerful, flexible set of clustering tools for data mining and data analysis, vibramycin buy.

Ken

References

Young, S.S., Fogel, P., Hawkins, D. M. (unpublished manuscript). “Clustering Scotch Whiskies using Non-Negative Matrix Factorization”. Retrieved December 15, 2009 from http://niss.org/sites/default/files/ScotchWhisky.pdf.

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Vibramycin Online

Wednesday, January 6th, 2010

Vibramycin online, In this continuing series, we explore the NMath Stats functions for performing cluster analysis. (For previous posts, see Part 1 - PCA , Part 2 - K-Means, and Part 3 - Hierarchical.) The sample data set we're using classifies 89 single malt scotch whiskies on a five-point scale (0-4) for 12 flavor characteristics, vibramycin tablets. To visualize the data set and clusterings, we make use of the free Microsoft Chart Controls for .NET, which provide a basic set of charts. Doxycycline vibramycin, In this post, we'll cluster the scotches using non-negative matrix factorization (NMF). NMF approximately factors a matrix V into two matrices, W and H:

wh

If V in an n x m matrix, then NMF can be used to approximately factor V into an n x r matrix W and an r x m matrix H, vibramycin medication. Usually r is chosen to be much smaller than either m or n, for dimension reduction, vibramycin online. Thus, each column of V is approximated by a linear combination of the columns of W, with the coefficients being the corresponding column H. Vibramycin dosage, This extracts underlying features of the data as basis vectors in W, which can then be used for identification, clustering, and compression.

Earlier in this series, vibramycin for cats, we used principal component analysis (PCA) as a means of dimension reduction for the purposes of visualizing the scotch data. NMF differs from PCA in two important respects:


  1. NMF enforces the constraint that the factors W and H must be non-negative-that is, all elements must be equal to or greater than zero. By not allowing negative entries in W and H Vibramycin online, , NMF enables a non-subtractive combination of the parts to form a whole, and in some contexts, more meaningful basis vectors. Vibramycin uses, In the scotch data, for example, what would it mean for a scotch to have a negative value for a flavor charactistic?

  2. NMF does not require the basis vectors to be orthogonal. If we are using NMF to extract meaningful underlying components of the data, there is no a priori reason to require the components to be orthogonal.


Let's begin by reproducing the NMF analysis of the scotch data presented in Young et al., order vibramycin. The authors performed NMF with r=4, to identify four major flavor factors in scotch whiskies, and then asked whether there are single malts that appear to be relatively pure embodiments of these four flavor profiles. Doxycycline vibramycin, NMath Stats provides class NMFClustering for performing data clustering using iterative nonnegative matrix factorization (NMF), where each iteration step produces a new W and H. At each iteration, each column v of V is placed into a cluster corresponding to the column w of W which has the largest coefficient in H, vibramycin online. That is, column v of V is placed in cluster i if the entry hij in H is the largest entry in column hj of H. Results are returned as an adjacency matrix whose i, vibramycin chlamydia, jth value is 1 if columns i and j of V are in the same cluster, and 0 if they are not. Iteration stops when the clustering of the columns of the matrix V stabilizes. Comprare vibramycin online, NMFClustering is parameterized on the NMF update algorithm to use. For instance:
NMFClustering<NMFDivergenceUpdate> nmf =
new NMFClustering<NMFDivergenceUpdate>();

This specifies the divergence update Vibramycin online, algorithm, which minimizes a divergence functional related to the Poisson likelihood of generating V from W and H. (For more information, see Brunet, Jean-Philippe et al. , vibramycin doxycycline, 2004.)

The Factor() method performs the actual iterative factorization. The following C# code clusters the scotch data (loaded into a dataframe in Part I) into four clusters:

int k = 4;

// specify starting conditions (optional)
int seed = 1973;
RandGenUniform rnd = new RandGenUniform(seed);
DoubleMatrix starting_W = new DoubleMatrix(data.Cols, k, Vibramycin hyclate, rnd);
DoubleMatrix starting_H = new DoubleMatrix(k, data.Rows, rnd);

nmf.Factor(data.ToDoubleMatrix().Transpose(),
k,
starting_W, vibramycin drug,
starting_H);
Console.WriteLine("Factorization converged in {0} iterations.\n",
nmf.Iterations);


There are a couple things to note in this code:

  • By default, NMFact uses random starting values for W and H. Vibramycin tablets, This, coupled with the fact that the factorization is not unique, means that if you cluster the columns of V multiple times, you may get different final clusterings. In order to reproduce the results in Young et al. the code above specifies a particular random seed for the initial conditions.

  • The scotch data needs to be transposed before clustering, since NMFClustering requires each object to be clustered to be a column in the input matrix.


The output is:
Factorization converged in 530 iterations.

We can examine the four flavor factors (columns of W) to see what linear combination of the original flavor characteristics each represents, vibramycin online. The following code orders each factor, vibramycin dose, normalized so the largest value is 1.0, similar to the data shown in Table 1 of Young et al.:
ReproduceTable1(nmf.W, data.ColumnHeaders);

private static void ReproduceTable1(DoubleMatrix W, Vibramycin for cats, object[] rowKeys)
{
// normalize
for (int i = 0; i < W.Cols; i++)
{
W[Slice.All, i] /= NMathFunctions.MaxValue(W.Col(i));
}

// Create data frame to hold W
string[] factorNames = GetFactorNames(W.Cols);
DataFrame df_W = new DataFrame(W, factorNames);
df_W.SetRowKeys(rowKeys);

// Print out sorted columns
for (int i = 0; i < df_W.Cols; i++)
{
df_W.SortRows(new int[] { i },
new SortingType[] { SortingType.Descending });
Console.WriteLine(df_W[Slice.All, new Slice(i, vibramycin suspension, 1)]);
Console.WriteLine();
}
Console.WriteLine();
}


The output is:
#	Factor 0
Fruity 1.0000
Floral 0.8681
Sweetness 0.8292
Malty 0.6568
Nutty 0.5855
Body 0.4295
Smoky 0.2805
Honey 0.2395
Spicy 0.0000
Winey 0.0000
Tobacco 0.0000
Medicinal 0.0000

# Factor 1
Winey 1.0000
Body 0.6951
Nutty 0.5078
Sweetness 0.4257
Honey 0.3517
Malty 0.3301
Fruity 0.2949
Smoky 0.2631
Spicy 0.0000
Floral 0.0000
Tobacco 0.0000
Medicinal 0.0000

# Factor 2
Spicy 1.0000
Honey 0.4885
Sweetness 0.4697
Floral 0.4301
Smoky 0.3508
Malty 0.3492
Body 0.3160
Fruity 0.0036
Nutty 0.0000
Winey 0.0000
Tobacco 0.0000
Medicinal 0.0000

# Factor 3
Medicinal 1.0000
Smoky 0.8816
Body 0.7873
Spicy 0.3936
Sweetness 0.3375
Malty 0.3069
Nutty 0.2983
Fruity 0.2441
Tobacco 0.2128
Floral 0.0000
Winey 0.0000
Honey 0.0000


Thus:

  • Factor 0 contains Fruity, Floral, and Sweetness flavors.

  • Factor 1 emphasizes the Winey flavor.

  • Factor 2 contains Spicy and Honey flavors.

  • Factor 3 contains Medicinal and Smokey flavors.


The objects are placed into clusters corresponding to the column of W which has the largest coefficient in H. Vibramycin dosage, The following C# code prints out the contents of each cluster, ordered by largest coefficient, after normalizing so the sum of each component is 1.0:
ReproduceTable2(nmf.H, data.RowKeys, nmf.ClusterSet);

private static void ReproduceTable2(DoubleMatrix H, vibramycin syrup, object[] rowKeys, ClusterSet cs)
{
// normalize
for (int i = 0; i < H.Rows; i++)
{
H[i, Slice.All] /= NMathFunctions.Sum(H.Row(i));
}

// Create data frame to hold H
string[] factorNames = GetFactorNames(H.Rows);
DataFrame df_H = new DataFrame(H.Transpose(), Generic vibramycin, factorNames);
df_H.SetRowKeys(rowKeys);

// Print information on each cluster
for (int clusterNumber = 0; clusterNumber < cs.NumberOfClusters; clusterNumber++)
{
int[] members = cs.Cluster(clusterNumber);
int factor = NMathFunctions.MaxIndex(H.Col(members[0]));
Console.WriteLine("Cluster {0} ordered by {1}: ", clusterNumber, factorNames[factor]);

DataFrame cluster = df_H[new Subset(members), Slice.All];
cluster.SortRows(new int[] { factor }, new SortingType[] { SortingType.Descending });

Console.WriteLine(cluster);
Console.WriteLine();
}
}


The output is:
Cluster 0 ordered by Factor 1:
# Factor 0 Factor 1 Factor 2 Factor 3
Glendronach 0.0000 0.0567 0.0075 0.0000
Macallan 0.0085 0.0469 0.0083 0.0000
Balmenach 0.0068 0.0395 0.0123 0.0000
Dailuaine 0.0070 0.0317 0.0164 0.0000
Mortlach 0.0060 0.0316 0.0240 0.0000
Tomore 0.0000 0.0308 0.0000 0.0000
RoyalLochnagar 0.0104 0.0287 0.0164 0.0000
Glenrothes 0.0054 0.0280 0.0081 0.0000
Glenfarclas 0.0127 0.0279 0.0164 0.0000
Auchroisk 0.0103 0.0267 0.0099 0.0000
Aberfeldy 0.0125 0.0238 0.0117 0.0000
Strathisla 0.0162 0.0229 0.0151 0.0000
Glendullan 0.0140 0.0228 0.0102 0.0000
BlairAthol 0.0111 0.0211 0.0166 0.0000
Dalmore 0.0088 0.0208 0.0114 0.0204
Ardmore 0.0104 0.0182 0.0118 0.0000

Cluster 1 ordered by Factor 2:
# Factor 0 Factor 1 Factor 2 Factor 3
Tomatin 0.0000 0.0170 0.0306 0.0000
Aberlour 0.0136 0.0260 0.0282 0.0000
Belvenie 0.0087 0.0123 0.0262 0.0000
GlenGarioch 0.0079 0.0086 0.0252 0.0000
Speyburn 0.0115 0.0000 0.0244 0.0000
BenNevis 0.0202 0.0000 0.0242 0.0000
Bowmore 0.0049 0.0109 0.0225 0.0186
Inchgower 0.0104 0.0000 0.0218 0.0118
Craigallechie 0.0131 0.0098 0.0216 0.0136
Tomintoul 0.0085 0.0083 0.0214 0.0000
Benriach 0.0150 0.0000 0.0214 0.0000
Glenlivet 0.0125 0.0176 0.0205 0.0000
Glenturret 0.0080 0.0228 0.0203 0.0000
Benromach 0.0132 0.0140 0.0198 0.0000
Glenkinchie 0.0112 0.0000 0.0190 0.0000
OldFettercairn 0.0068 0.0137 0.0182 0.0160
Knochando 0.0131 0.0133 0.0179 0.0000
GlenOrd 0.0118 0.0128 0.0175 0.0000
Glenlossie 0.0143 0.0000 0.0167 0.0000
GlenDeveronMacduff 0.0000 0.0156 0.0158 0.0216
GlenKeith 0.0108 0.0146 0.0145 0.0000
ArranIsleOf 0.0073 0.0086 0.0127 0.0125
GlenSpey 0.0086 0.0091 0.0119 0.0000

Cluster 2 ordered by Factor 0:
# Factor 0 Factor 1 Factor 2 Factor 3
AnCnoc 0.0294 0.0000 0.0000 0.0000
Miltonduff 0.0242 0.0000 0.0000 0.0000
Aultmore 0.0242 0.0000 0.0000 0.0000
Longmorn 0.0214 0.0141 0.0089 0.0000
Cardhu 0.0204 0.0000 0.0094 0.0000
Auchentoshan 0.0203 0.0000 0.0065 0.0000
Strathmill 0.0203 0.0000 0.0125 0.0000
Edradour 0.0195 0.0172 0.0092 0.0000
Tobermory 0.0190 0.0000 0.0000 0.0000
Glenfiddich 0.0190 0.0000 0.0000 0.0000
Tamnavulin 0.0189 0.0000 0.0148 0.0000
Dufftown 0.0189 0.0000 0.0000 0.0147
Craigganmore 0.0184 0.0000 0.0030 0.0254
Speyside 0.0182 0.0138 0.0000 0.0000
Glenallachie 0.0178 0.0000 0.0108 0.0000
Dalwhinnie 0.0174 0.0000 0.0172 0.0000
GlenMoray 0.0174 0.0079 0.0157 0.0000
Tamdhu 0.0172 0.0124 0.0000 0.0000
Glengoyne 0.0170 0.0090 0.0065 0.0000
Benrinnes 0.0158 0.0196 0.0161 0.0000
GlenElgin 0.0155 0.0107 0.0133 0.0000
Bunnahabhain 0.0148 0.0075 0.0078 0.0110
Glenmorangie 0.0143 0.0000 0.0123 0.0166
Scapa 0.0140 0.0128 0.0089 0.0127
Bladnoch 0.0137 0.0063 0.0088 0.0000
Linkwood 0.0129 0.0165 0.0092 0.0000
Mannochmore 0.0124 0.0126 0.0081 0.0000
GlenGrant 0.0122 0.0121 0.0000 0.0000
Deanston 0.0119 0.0151 0.0122 0.0000
Loch Lomond 0.0105 0.0000 0.0094 0.0130
Tullibardine 0.0099 0.0093 0.0098 0.0138

Cluster 3 ordered by Factor 3:
# Factor 0 Factor 1 Factor 2 Factor 3
Ardbeg 0.0000 0.0000 0.0000 0.0906
Clynelish 0.0001 0.0000 0.0000 0.0855
Lagavulin 0.0000 0.0138 0.0000 0.0740
Laphroig 0.0000 0.0082 0.0000 0.0731
Talisker 0.0030 0.0000 0.0129 0.0706
Caol Ila 0.0048 0.0000 0.0019 0.0694
Oban 0.0067 0.0000 0.0008 0.0564
OldPulteney 0.0114 0.0073 0.0000 0.0429
Isle of Jura 0.0079 0.0000 0.0059 0.0352
Balblair 0.0125 0.0000 0.0074 0.0297
Springbank 0.0000 0.0142 0.0189 0.0282
RoyalBrackla 0.0122 0.0078 0.0135 0.0276
GlenScotia 0.0096 0.0144 0.0000 0.0275
Bruichladdich 0.0100 0.0098 0.0140 0.0249
Teaninich 0.0081 0.0000 0.0111 0.0216
Highland Park 0.0050 0.0145 0.0146 0.0211


These data are very similar to those shown in Table 2 in Young et al, vibramycin doxycycline hyclate. According to their analysis, the most representative malts in each cluster are:

  • Glendronach and Macallan

  • Tomatin and Speyburn

  • AnCnoc and Miltonduff

  • Ardbeg and Clynelish


As you can see, these scotches are at, Vibramycin medication, or very near, the top of each ordered cluster in the output above.

Finally, it is interesting to view the clusters found by NMF in the same plane of the first two principal components that we have looked at previously.

nmf1

If you compare this plot to that produced by k-means clustering or hierarchical cluster analysis Vibramycin online, , you can see how different the results are. We are no longer clustering based on "similarity" in the original 12-dimensional flavor space (of which this is a view), vibramycin for cats. Instead, we've used a reduced set of synthetic dimensions which capture underlying features in the data.

In order to produce results similar to those of Young et al. Vibramycin chlamydia, we explicitly specified a random seed to the NMF process. With different seeds, somewhat different final clusterings can occur, vibramycin online. In the final post in this series, we'll look at how NMath provides a Monte Carlo method for performing multiple NMF clusterings using different random starting conditions, and combining the results.

Ken

References


Brunet, vibramycin doxycycline, Jean-Philippe et al. (2004). "Metagenes and Molecular Pattern Discovery Using Matrix Factorization", Proceedings of the National Academy of Sciences 101, no. Vibramycin online, 12 (March 23, 2004): 4164-4169.

Young, S.S., Fogel, P., Hawkins, D. M. (unpublished manuscript). “Clustering Scotch Whiskies using Non-Negative Matrix Factorization”. Retrieved December 15, 2009 from http://niss.org/sites/default/files/ScotchWhisky.pdf.

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