Biotechnological innovations in the last decade have enabled systematic large-scale cancer genome studies. Cancer projects such as The Cancer Genome Atlas (TCGA) and The International Cancer Genome Consortium (ICGC) coordinate the generation of catalogues of tumors from over 50 different cancer types that are of clinical importance across the globe. These catalogues contain comprehensive data of more than 25,000 cancer genomes at a high, often at a single nucleotide, resolution level. The genomes of individual cancers are accompanied by the analyses of transcriptome, epigenome and sequence variation. Furthermore, where possible, measurements are provided for matched tumor and non-tumor tissue to distinguish inherited from oncogenic variation.

Diversity and abundance of data provided by the cancer projects challenge computer scientists of all kinds to develop innovative software, hardware and analytic solutions for data analysis. The hope is that with the computationally and statistically stronger approaches we will be able to reveal biological features that drive cancer development, define cancer types relevant for prognosis and, ultimately, enable the development of new cancer therapies.

Here, we analyze a transcriptome data set on breast cancer obtained from the ICGC. The analysis relies on a popular class of data mining techniques, called nonnegative matrix factorization, to find groups of patients with similar expression profiles and groups of genes with similar activation levels across patients.

In [2]:

```
%matplotlib inline
from collections import defaultdict, Counter
import urllib
import numpy as np
from matplotlib import pyplot as plt
import matplotlib.gridspec as gridspec
from sklearn import preprocessing
import scipy.cluster.hierarchy as sch
import nimfa
```

In [3]:

```
clinical_fname = 'clinical.BRCA-US.tsv.gz'
urllib.request.urlretrieve('https://dcc.icgc.org/api/v1/download?' \
'fn=/release_18/Projects/BRCA-US/' \
'clinical.BRCA-US.tsv.gz', clinical_fname);
expression_fname = 'protein_expression.BRCA-US.tsv.gz'
urllib.request.urlretrieve('https://dcc.icgc.org/api/v1/download?' \
'fn=/release_18/Projects/BRCA-US/' \
'protein_expression.BRCA-US.tsv.gz', expression_fname);
```

Once the download has completed, we have two new files on our local disk. The first file, `clinical.BRCA-US.tsv.gz`

, contains basic clinical information, such as patient and specimen identifiers and categorization of specimens into tumor and non-tumor types. The second file, `protein_expression.BRCA-US.tsv.gz`

, is a protein expression data set of breast cancer patients (`BRCA`

is a standard acronym used by the cancer projects to indicate data related to breast cancer). Each row in the expression file contains normalized expression value of a particular protein in a given specimen of a specific patient.

we read data columns that correspond to patient and specimen identifiers and expression measurements into two `numpy`

arrays, `C`

and `E`

.

In [12]:

```
C = np.genfromtxt(clinical_fname, delimiter='\t', dtype='object',
skip_header=1, usecols=(0, 17, 19))
```

In [13]:

```
E = np.genfromtxt(expression_fname, delimiter='\t', dtype='object',
skip_header=1, usecols=(0, 2, 7, 10))
```

In [14]:

```
donors = set(E[:, 0])
genes = set(E[:, 2])
print(len(donors))
print(len(genes))
```

In [15]:

```
spec2gene = defaultdict(set)
for spec_id, gene in E[:, 1:3]:
spec2gene[spec_id].add(gene)
donor2spec = defaultdict(set)
for donor, spec_id in E[:, :2]:
donor2spec[donor].add(spec_id)
# how many specimens are there per donor?
tmp = [len(v) for v in donor2spec.values()]
print(Counter(tmp))
# protein expression data is available only for tumor data (no matched normal tissue)
spec2spec_type = {C[idx, 1]: C[idx, 2] for idx, spec in enumerate(C[:, 1]) if spec in spec2gene}
print(set(spec2spec_type.values()))
```

In [16]:

```
data = np.zeros((len(donors), len(genes)))
donor2id = {donor.decode('UTF-8'): idx for idx, donor in enumerate(donors)}
id2donor = dict(zip(donor2id.values(), donor2id.keys()))
gene2id = {gene.decode('UTF-8'): idx for idx, gene in enumerate(genes)}
id2gene = dict(zip(gene2id.values(), gene2id.keys()))
for donor, gene, val in E[:, [0, 2, 3]]:
data[donor2id[donor.decode('UTF-8')], gene2id[gene.decode('UTF-8')]] = float(val)
```

In [17]:

```
# level of sparsity
print(np.sum(data != 0)/data.size)
# negative elements
print(np.sum(data < 0))
# remove negative elements
data = data - np.min(data)
# normalize data
data = preprocessing.Normalizer().fit_transform(data)
```

Nonnegative matrix factorization (NMF) is typically concerned with the decomposition of a nonnegative data matrix into two low-dimensional (latent) data matrices, such that their matrix product provides a good surrogate of the original data matrix.

There are a few important concepts to understand in order to fully appreciate the NMF. First, since not only input data matrix is nonnegative, but also the latent matrices contain solely nonnegative values, NMF is able to learn a parts-based data representation. For example, if we apply NMF to a database of facial images, then the latent components (vectors) of the matrices estimated by the NMF correspond to parts of faces. If we applied NMF to a text corpus, then the latent components would encode semantic features of text. Second, high generalization power of NMF is due to its ability to compress large input data matrix and represent it with latent matrices, which have substantially fewer dimensions. Technically, the dimensionality of latent matrices is known as factorization rank. In this sense, NMF might be a good example where less can be more.

Many variants of nonnegative matrix factorization have been developed in recent years. These augment the objective function with different types of regularization imposed on the latent matrices, provide various generalizations of the notion of a good matrix surrogate, and extend factorization of a single data matrix to simultaneous decomposition of multiple matrices.

We consider **Nimfa**, an open-source Python library for NMF, to factorize our matrix of protein expression measurements. Nimfa implements over a dozen of nonnegative matrix factorization algorithms, initialization techniques and provides quality scoring.

```
Zitnik, Marinka, and Blaz Zupan. Nimfa: A Python library for nonnegative matrix factorization. The Journal of Machine Learning Research 13, 1:849-853, 2012.
```

In our example, we run one of the most widely used NMF variants. This variant estimates the NMF model by iteratively improving current estimates of the latent matrices until a local minimum is attained, measured as Kullback-Leibler divergence between the input matrix and its estimated matrix surrogate.

In [19]:

```
rank = 10
nmf = nimfa.Nmf(data, rank=rank, seed='random_vcol', max_iter=100,
update='divergence', objective='div', n_run=50, track_factor=True)
nmf_fit = nmf()
```

Next, we use the estimated NMF model to cluster genes into groups. Whereas the latent components obtained from an image database correspond to parts of faces, the latent components of a protein expression data set represent subpopulations of patients and gene subsets. Conveniently, we refer to these latent components as *hidden patients* and *hidden genes*.

We now use the memberships of genes to hidden genes as a means to group genes together. Notice that above we ran the NMF algorithm 50 times (`n_run = 50`

). To obtain a stable gene clustering we examine gene-hidden gene memberships from all factorization runs and average gene membership indicators to obtain a *consensus clustering*.

Once we obtain the consensus clustering we can visualize it in a heatmap with the rows reordered to make the groups explicit.

In [66]:

```
def clean_axis(ax):
ax.get_xaxis().set_ticks([])
ax.get_yaxis().set_ticks([])
for sp in ax.spines.values():
sp.set_visible(False)
fig = plt.figure(figsize=(13.9, 10))
heatmapGS = gridspec.GridSpec(1, 2, wspace=.01, hspace=0., width_ratios=[0.25,1])
C = 1 - nmf_fit.fit.consensus()
Y = sch.linkage(C, method='average')
denAX = fig.add_subplot(heatmapGS[0,0])
denD = sch.dendrogram(Y, orientation='right', link_color_func=lambda k: 'black')
clean_axis(denAX)
heatmapAX = fig.add_subplot(heatmapGS[0,1])
D = C[denD['leaves'], :][:, denD['leaves']]
axi = heatmapAX.imshow(D, interpolation='nearest', aspect='equal', origin='lower', cmap='RdBu')
clean_axis(heatmapAX)
cb = fig.colorbar(axi, fraction=0.046, pad=0.04, aspect=10)
cb.set_label('Distance', fontsize=20)
```

In [67]:

```
rank = 10
nmf = nimfa.Nmf(data.T, rank=rank, seed='random_vcol', max_iter=200,
update='euclidean', objective='conn', conn_change=40,
n_run=50, track_factor=True)
nmf_fit = nmf()
```

In [69]:

```
def clean_axis(ax):
ax.get_xaxis().set_ticks([])
ax.get_yaxis().set_ticks([])
for sp in ax.spines.values():
sp.set_visible(False)
fig = plt.figure(figsize=(13.9, 10))
heatmapGS = gridspec.GridSpec(1, 2, wspace=.01, hspace=0., width_ratios=[0.25,1])
C = 1 - nmf_fit.fit.consensus()
Y = sch.linkage(C, method='average')
denAX = fig.add_subplot(heatmapGS[0,0])
denD = sch.dendrogram(Y, orientation='right', link_color_func=lambda k: 'black')
clean_axis(denAX)
heatmapAX = fig.add_subplot(heatmapGS[0,1])
D = C[denD['leaves'], :][:, denD['leaves']]
axi = heatmapAX.imshow(D, interpolation='nearest', aspect='equal', origin='lower', cmap='RdBu')
clean_axis(heatmapAX)
cb = fig.colorbar(axi, fraction=0.046, pad=0.04, aspect=10)
cb.set_label('Distance', fontsize=20)
```

In [27]:

```
rank_cands = range(3, 30, 5)
snmf = nimfa.Snmf(data, seed='random_vcol', max_iter=100)
summary = snmf.estimate_rank(rank_range=rank_cands, n_run=10, what='all')
```

In [28]:

```
summary[3].keys()
```

Out[28]:

In [29]:

```
rss = [summary[rank]['rss'] for rank in rank_cands]
coph = [summary[rank]['cophenetic'] for rank in rank_cands]
disp = [summary[rank]['dispersion'] for rank in rank_cands]
spar = [summary[rank]['sparseness'] for rank in rank_cands]
spar_w, spar_h = zip(*spar)
evar = [summary[rank]['evar'] for rank in rank_cands]
plt.plot(rank_cands, rss, 'o-', label='RSS', linewidth=2)
plt.plot(rank_cands, coph, 'o-', label='Cophenetic correlation', linewidth=2)
plt.plot(rank_cands, disp,'o-', label='Dispersion', linewidth=2)
plt.plot(rank_cands, spar_w, 'o-', label='Sparsity (Basis)', linewidth=2)
plt.plot(rank_cands, spar_h, 'o-', label='Sparsity (Mixture)', linewidth=2)
plt.plot(rank_cands, evar, 'o-', label='Explained variance', linewidth=2)
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), ncol=3, numpoints=1);
```

*hidden gene* corresponds to a latent component (a vector) in the low-dimensional matrices estimated by the NMF. We would like to know, which genes are most strongly associated with a particular hidden gene.

In [82]:

```
# SNMF/L: for sparse W (where ‘l’ denotes the sparseness imposed on the left factor)
# SNMF/R: for sparse H (where ‘r’ denotes the sparseness imposed on the right factor)
rank = 13
snmf = nimfa.Snmf(data.T, rank=rank, seed='random_vcol', version='l', max_iter=200)
snmf_fit = snmf()
```

In [91]:

```
W = snmf_fit.fit.basis()
c = 0
k = 10
topk = np.argsort(np.asarray(W[:, c]).flatten())[-k:]
val = W[topk, c]
plt.barh(np.arange(k) + .5, val, align="center")
labels = [id2gene[idx] for idx in topk]
plt.yticks(np.arange(k) + .5, labels)
plt.xlabel("Weight")
plt.ylabel("Gene");
```

In [92]:

```
print('\n'.join(labels))
```

A detailed explanation of gene scoring and selection is provided in

`Kim, Hyunsoo, and Haesun Park. Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis. Bioinformatics 23, 12:1495-1502, 2007.`

In [96]:

```
gs = snmf_fit.fit.score_features()
```

In [97]:

```
gss = snmf_fit.fit.select_features()
gene_set = [id2gene[idx] for idx in np.nonzero(gss)[0]]
```

In [98]:

```
print('\n'.join(gene_set))
```

*hidden patient* is a latent component (a vector) of the low-dimensional matrices estimated by the NMF. We would like to know, which patient are most strongly associated with a particular hidden patient.

In [88]:

```
# SNMF/L: for sparse W (where ‘l’ denotes the sparseness imposed on the left factor)
# SNMF/R: for sparse H (where ‘r’ denotes the sparseness imposed on the right factor)
rank = 13
snmf = nimfa.Snmf(data.T, rank=rank, seed='random_vcol', version='r', max_iter=200)
snmf_fit = snmf()
```

In [89]:

```
H = snmf_fit.fit.coef().T
c = 0
k = 30
topk = np.argsort(np.asarray(H[:, c]).flatten())[-k:]
val = H[topk, c]
plt.barh(np.arange(k) + .5, val, color="yellow", align="center")
labels = [id2donor[idx] for idx in topk]
plt.yticks(np.arange(k) + .5, labels)
plt.xlabel("Weight")
plt.ylabel("Donor");
```

In [ ]:

```
```