Righteous Pathway Factor Analysis

pip install righteous-fa==1.2.1


A Statistical Learning library for Humans

This toolkit currently offers enrichment analysis, hierarchical enrichment analysis, novel PLS regression, shape alignment, connectivity clustering, clustering and hierarchical clustering as well as factor analysis methods. A novel graph construction and graph searching class can be found in src/impetuous/convert.py (GraphNode). It was developed and invented by the author as a faster alternative for hierarchical DAG construction and searching.

The fine grained data can be studied via a statistical tests that relates it to observables in a coarse grained journal file. The final p values can then be rank corrected by employing a novel q-value method.

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Visit the active code via : https://github.com/richardtjornhammar/impetuous

Pip installation with :

pip install impetuous-gfa

Version controlled installation of the Impetuous library

The Impetuous library

In order to run these code snippets we recommend that you download the nix package manager. Nix package manager links from Oktober 2020:


$ curl -L https://nixos.org/nix/install | sh

If you cannot install it using your Wintendo then please consider installing Windows Subsystem for Linux first:


In order to run the code in this notebook you must enter a sensible working environment. Don't worry! We have created one for you. It's version controlled against python3.7 (and python3.8) and you can get the file here:


Since you have installed Nix as well as WSL, or use a Linux (NixOS) or bsd like system, you should be able to execute the following command in a termnial:

$ nix-shell impetuous-shell.nix

Now you should be able to start your jupyter notebook locally:

$ jupyter-notebook impetuous.ipynb

and that's it.

Test installation

You can download and run that python file to verify the installation. If it isn't working then there is an error with the package

Example 0:

After installing impetuous-gfa version >=0.66.5 you should be able to execute the code

if __name__=='__main__':
    import impetuous as imp
    import impetuous.hierarchical as imphi
    import impetuous.clustering as impcl
    import impetuous.fit as impfi
    import impetuous.pathways as imppa
    import impetuous.visualisation as impvi
    import impetuous.optimisation as impop
    import impetuous.convert as impco
    import impetuous.probabilistic as imppr
    import impetuous.quantification as impqu
    import impetuous.spectral as impsp
    import impetuous.reducer as impre
    import impetuous.special as impspec

You can execute it easily when you are in the impetuous environment. Just write

$ wget https://gist.githubusercontent.com/richardtjornhammar/34e163cba547d6c856d902244edc2039/raw/2a069b062df486b8d081c8cfedbbb30321e44f36/example0.py
$ python3 example0.py

And if it doesn't work then contact me and I'll try and get back within 24h

Usage example 1: Elaborate informatics

code: https://gitlab.com/stochasticdynamics/eplsmta-experiments docs: https://arxiv.org/pdf/2001.06544.pdf

Usage example 2: Simple regression code

Now while in a good environment: In your Jupyter notebook or just in a dedicated file.py you can write the following:

import pandas as pd
import numpy as np

import impetuous.quantification as impq

analyte_df = pd.read_csv( 'analytes.csv' , '\t' , index_col=0 )
journal_df = pd.read_csv( 'journal.csv'  , '\t' , index_col=0 )

formula = 'S ~ C(industry) : C(block) + C(industry) + C(block)'

res_dfs 	= impq.run_rpls_regression ( analyte_df , journal_df , formula , owner_by = 'angle' )
results_lookup	= impq.assign_quality_measures( journal_df , res_dfs , formula )

print ( results_lookup )
print ( res_dfs )

Example 3: Novel NLP sequence alignment

Finding a word in a text is a simple and trivial problem in computer science. However matching a sequence of characters to a larger text segment is not. In this example you will be shown how to employ the impetuous text fitting procedure. The strength of the fit is conveyed via the returned score, higher being a stronger match between the two texts. This becomes costly for large texts and we thus break the text into segments and words. If there is a strong word to word match then the entire segment score is calculated. The off and main diagonal power terms refer to how to evaluate a string shift. Fortinbras and Faortinbraaks are probably the same word eventhough the latter has two character shifts in it. In this example both "requests" and "BeautifulSoup" are employed to parse internet text.

import numpy as np
import pandas as pd

import impetuous.fit as impf    # THE IMPETUOUS FIT MODULE
                                # CONTAINS SCORE ALIGNMENT ROUTINE

import requests                 # FOR MAKING URL REQUESTS
from bs4 import BeautifulSoup   # FOR PARSING URL REQUEST CONTENT

if __name__ == '__main__' :

    url_       = 'http://shakespeare.mit.edu/hamlet/full.html'

    response   = requests.get( url_ )
    bs_content = BeautifulSoup ( response.content , features="html.parser")

    name = 'fortinbras'
    score_co = 500
    S , S2 , N = 0 , 0 , 0
    for btext in bs_content.find_all('blockquote'):

        theTextSection = btext.get_text()
        theText        = theTextSection.split('\n')

        for segment in theText:
            pieces = segment.split(' ')
            if len(pieces)>1 :
                for piece in pieces :
                    if len(piece)>1 :
                        score = impf.score_alignment( [ name , piece ],
                                    main_diagonal_power = 3.5, shift_allowance=2,
                                    off_diagonal_power = [1.5,0.5] )
                        S    += score
                        S2   += score*score
                        N    += 1
                        if score > score_co :
                            print ( "" )
                            print ( score,name,piece )
                            print ( theTextSection )
                            print ( impf.score_alignment( [ name , theTextSection ],
                                        main_diagonal_power = 3.5, shift_allowance=2,
                                        off_diagonal_power = [1.5,0.5] ) )
                            print ( "" )

    print ( S/N )
    print ( S2/N-S*S/N/N )

Example 4: Diabetes analysis

Here we show how to use a novel multifactor method on a diabetes data set to deduce important transcripts with respect to being diabetic. The data was obtained from the Broad Insitute and contains gene expressions from a microarray hgu133a platform. We choose to employ the Diabetes_collapsed_symbols.gct file since it has already been collapsed down to useful transcripts. We have entered an impetuous-gfa ( version >= 0.50.0 ) environment and set up the a diabetes.py file with the following code content:

import pandas as pd
import numpy as np

if __name__ == '__main__' :
    analyte_df = pd.read_csv('../data/Diabetes_collapsed_symbols.gct','\t', index_col=0, header=2).iloc[:,1:]

In order to illustrate the use of low value supression we use the reducer module. A tanh based soft max function is employed by the confred function to supress values lower than the median of the entire sample series for each sample.

    from impetuous.reducer import get_procentile,confred
    for i_ in range(len(analyte_df.columns.values)):
        vals   = analyte_df.iloc[:,i_].values
        eta    = get_procentile( vals,50 )
        varpi  = get_procentile( vals,66 ) - get_procentile( vals,33 )
        analyte_df.iloc[:,i_] = confred(vals,eta,varpi)

    print ( analyte_df )

The data now contain samples along the columns and gene transcript symbols along the rows where the original values have been quenched with low value supression. The table have the following appearance

NAME NGT_mm12_10591 ... DM2_mm81_10199
215538_at 16.826041 ... 31.764484
LDLR 19.261185 ... 30.004612

We proceed to write a journal data frame by adding the following lines to our code

    journal_df = pd.DataFrame([ v.split('_')[0] for v in analyte_df.columns] , columns=['Status'] , index = analyte_df.columns.values ).T
    print ( journal_df )

which will produce the following journal table :

NGT_mm12_10591 ... DM2_mm81_10199
Status NGT ... DM2

Now we check if there are aggregation tendencies among these two groups prior to the multifactor analysis. We could use the hierarchical clustering algorithm, but refrain from it and instead use the associations method together with the connectivity clustering algorithm. The associations can be thought of as a type of ranked correlations similar to spearman correlations. If two samples are strongly associated with each other they will be close to 1 (or -1 if they are anti associated). Since they are all humans, with many transcript features, the values will be close to 1. After recasting the associations into distances we can determine if two samples are connected at a given distance by using the connectivity routine. All connected points are then grouped into technical clusters, or batches, and added to the journal.

    from impetuous.quantification import associations
    ranked_similarity_df = associations ( analyte_df .T )
    sample_distances = ( 1 - ranked_similarity_df ) * 2.

    from impetuous.clustering import connectivity
    cluster_ids = [ 'B'+str(c[0]) for c in connectivity( sample_distances.values , 5.0E-2 )[1] ]
    print ( cluster_ids )

    journal_df .loc['Batches'] = cluster_ids

which will produce a cluster list containing 13 batches with members whom are Normal Glucose Tolerant or have Diabetes Mellitus 2. We write down the formula for deducing which genes are best at recreating the diabetic state and batch identities by writing:

    formula = 'f~C(Status)+C(Batches)'

The multifactor method calculates how to produce an encoded version of the journal data frame given an analyte data set. It does this by forming the psuedo inverse matrix that best describes the inverse of the analyte frame and then calculates the dot product of the inverse with the encoded journal data frame. This yields the coefficient frame needed to solve for the numerical encoding frame. The method has many nice statistical properties that we will not discuss further here. The first thing that the multifactor method does is to create the encoded data frame. The encoded data frame for this problem can be obtained with the following code snippet

    encoded_df = create_encoding_data_frame ( journal_df , formula ).T
    print ( encoded_df )

and it will look something like this

NGT_mm12_10591 ... DM2_mm81_10199
B10 0.0 ... 0.0
B5 0.0 ... 0.0
B12 0.0 ... 1.0
B2 0.0 ... 0.0
B11 1.0 ... 0.0
B8 0.0 ... 0.0
B1 0.0 ... 0.0
B7 0.0 ... 0.0
B4 0.0 ... 0.0
B0 0.0 ... 0.0
B6 0.0 ... 0.0
B9 0.0 ... 0.0
B3 0.0 ... 0.0
NGT 1.0 ... 0.0
DM2 0.0 ... 1.0

This encoded dataframe can be used to calculate statistical parameters or solve other linear equations. Take the fast calculation of the mean gene expressions across all groups as an example

    print ( pd .DataFrame ( np.dot( encoded_df,analyte_df.T ) ,
                          columns = analyte_df .index ,
                          index   = encoded_df .index ) .apply ( lambda x:x/np.sum(encoded_df,1) ) )

which will immediately calculate the mean values of all transcripts across all different groups.

The multifactor_evaluation calculates the coefficients that best recreates the encoded journal by employing the psudo inverse of the analyte frame utlizing Singular Value Decomposition. The beta coefficients are then evaluated using a normal distribution assumption to obtain p values and rank corrected q values are also returned. The full function can be called with the following code

    from impetuous.quantification import multifactor_evaluation
    multifactor_results = multifactor_evaluation ( analyte_df , journal_df , formula )

    print ( multifactor_results.sort_values('DM2,q').iloc[:25,:].index.values  )

which tells us that the genes

['MYH2' 'RPL39' 'HBG1 /// HBG2' 'DDN' 'UBC' 'RPS18' 'ACTC' 'HBA2' 'GAPD'
 'ANKRD2' 'NEB' 'MYL2' 'MT1H' 'KPNA4' 'CA3' 'RPLP2' 'MRLC2 /// MRCL3'
 '211074_at' 'SLC25A16' 'KBTBD10' 'HSPA2' 'LDHB' 'COX7B' 'COX7A1' 'APOD']

have something to do with the altered metabolism in Type 2 Diabetics. We could now proceed to use the hierarchical enrichment routine to understand what that something is, but first we save the data


Example 5: Understanding what it means

If you have a well curated .gmt file that contains analyte ids as unique sets that belong to different groups then you can check whether or not a specific group seems significant with respect to all of the significant and insignificant analytes that you just calculated. One can derive such a hierarchy or rely on already curated information. Since we are dealing with genes and biologist generally have strong opinions about these things we go to a directed acyclic knowledge graph called Reactome and translate that information into a set of files that we can use to build our own knowledge hierarchy. After downloading that .zip file (and unzipping) you will be able to execute the following code

import pandas as pd
import numpy as np

if __name__=='__main__':
    import impetuous.pathways as impw

which will blurt out code you can use as inspiration to generate the Reactome knowledge hierarchy. So now we do that

    paths = impw.Reactome( './Ensembl2Reactome_All_Levels_v71.txt' )

but we also need to translate the gene ids into the correct format so we employ BioMart. To obtain the conversion text file we select Human genes GRCh38.p13 and choose attributes Gene stable ID, Gene name and Gene Synonym and save the file as biomart.txt.

    biomart_dictionary = {}
    with open('biomart.txt','r') as input:
        for line in input :
            lsp = line.split('\n')[0].split('\t')
            biomart_dictionary[lsp[0]] = [ n for n in lsp[1:] if len(n)>0 ]
    paths.add_pathway_synonyms( synonym_dict=biomart_dictionary )

    paths .make_gmt_pathway_file( './reactome_v71.gmt' )

Now we are almost ready to conduct the hierarchical pathway enrichment, to see what cellular processes are significant with respect to our gene discoveries, but we still need to build the Directed Acyclic Graph (DAG) from the parent child file and the pathway definitions.

    import impetuous.hierarchical as imph
    dag_df , tree = imph.create_dag_representation_df ( pathway_file = './reactome_v71.gmt',
                                                        pcfile = './NewestReactomeNodeRelations.txt' )

We will use it in the HierarchicalEnrichment routine later in order not to double count genes that have already contributed at lower levels of the hierarchy. Now where did we store those gene results...

    quantified_analyte_df = pd.read_csv('multifactor_dm2.csv','\t',index_col=0)
    a_very_significant_cutoff = 1E-10

    enrichment_results = imph.HierarchicalEnrichment ( quantified_analyte_df , dag_df , 
                                  ancestors_id_label = 'DAG,ancestors' , dag_level_label = 'DAG,level' ,
                                  threshold = a_very_significant_cutoff ,
                                  p_label = 'DM2,q' )

lets see what came out on top!

    print( enrichment_results.sort_values('Hierarchical,p').loc[:,['description','Hierarchical,p']].iloc[0,:] )

which will report that

description Striated Muscle Contraction
Hierarchical,p 6.55459e-05
Name: R-HSA-390522

is affected or perhaps needs to be compensated for... now perhaps you thought this exercise was a tad tedious? Well you are correct. It is and you could just as well have copied the gene transcripts into String-db and gotten similar results out. But, then you wouldn't have gotten to use the hierarchical enrichment method I invented!

Example 6: Absolute and relative coordinates

In this example, we will use the SVD based distance geometry method to go between absolute coordinates, relative coordinate distances and back to ordered absolute coordinates. Absolute coordinates are float values describing the position of something in space. If you have several of these then the same information can be conveyed via the pairwise distance graph. Going from absolute coordinates to pairwise distances is simple and only requires you to calculate all the pairwise distances between your absolute coordinates. Going back to mutually orthogonal ordered coordinates from the pariwise distances is trickier, but a solved problem. The distance geometry can be obtained with SVD and it is implemented in the impetuous.clustering module under the name distance_matrix_to_absolute_coordinates. We start by defining coordinates afterwhich we can calculate the pair distance matrix and transforming it back by using the code below

import pandas as pd
import numpy as np

coordinates = np.array([[-23.7100 ,  24.1000 ,  85.4400],
  [-22.5600 ,  23.7600 ,  85.6500],
  [-21.5500 ,  24.6200 ,  85.3800],
  [-22.2600 ,  22.4200 ,  86.1900],
  [-23.2900 ,  21.5300 ,  86.4800],
  [-20.9300 ,  22.0300 ,  86.4300],
  [-20.7100 ,  20.7600 ,  86.9400],
  [-21.7900 ,  19.9300 ,  87.1900],
  [-23.0300 ,  20.3300 ,  86.9600],
  [-24.1300 ,  19.4200 ,  87.2500],
  [-23.7400 ,  18.0500 ,  87.0000],
  [-24.4900 ,  19.4600 ,  88.7500],
  [-23.3700 ,  19.8900 ,  89.5200],
  [-24.8500 ,  18.0000 ,  89.0900],
  [-23.9600 ,  17.4800 ,  90.0800],
  [-24.6600 ,  17.2400 ,  87.7500],
  [-24.0800 ,  15.8500 ,  88.0100],
  [-23.9600 ,  15.1600 ,  86.7600],
  [-23.3400 ,  13.7100 ,  87.1000],
  [-21.9600 ,  13.8700 ,  87.6300],
  [-24.1800 ,  13.0300 ,  88.1100],
  [-23.2900 ,  12.8200 ,  85.7600],
  [-23.1900 ,  11.2800 ,  86.2200],
  [-21.8100 ,  11.0000 ,  86.7000],
  [-24.1500 ,  11.0300 ,  87.3200],
  [-23.5300 ,  10.3200 ,  84.9800],
  [-23.5400 ,   8.9800 ,  85.4800],
  [-23.8600 ,   8.0100 ,  84.3400],
  [-23.9800 ,   6.5760 ,  84.8900],
  [-23.2800 ,   6.4460 ,  86.1300],
  [-23.3000 ,   5.7330 ,  83.7800],
  [-22.7300 ,   4.5360 ,  84.3100],
  [-22.2000 ,   6.7130 ,  83.3000],
  [-22.7900 ,   8.0170 ,  83.3800],
  [-21.8100 ,   6.4120 ,  81.9200],
  [-20.8500 ,   5.5220 ,  81.5200],
  [-20.8300 ,   5.5670 ,  80.1200],
  [-21.7700 ,   6.4720 ,  79.7400],
  [-22.3400 ,   6.9680 ,  80.8000],
  [-20.0100 ,   4.6970 ,  82.1500],
  [-19.1800 ,   3.9390 ,  81.4700] ]);

if __name__=='__main__':

    import impetuous.clustering as impc

    distance_matrix = impc.absolute_coordinates_to_distance_matrix( coordinates )
    ordered_coordinates = impc.distance_matrix_to_absolute_coordinates( distance_matrix , n_dimensions=3 )

    print ( pd.DataFrame(ordered_coordinates).T )

You will notice that the largest variation is now aligned with the X axis, the second most variation aligned with the Y axis and the third most, aligned with the Z axis while the graph topology remained unchanged.

Example 7: Retrieval and analysis of obesity data

In this example, we will show an analysis similar to the one conducted in Example 4. The only difference here is that we will model all of the data present in the journal. This includes the simultaneous analysis of categorical and number range descriptors present in the journal. We use an impetuous shell and download the required python file and execute it in the shell. Now you are done! Was that too fast? ok, so what is this about?

You will see that the python code downloads a data directory (if you're using GNU/Linux), extracts it, curates it and performs the analysis. The directory contains sample data with information about both the platform and the sample properties. In our case a sample can come from any of 6 different platforms and belong to either lean or obese females or males. We collect the information and skip all but the GPL8300 platform data. Now we have a journal that describes how well the sample was collected (with integer value ranges) and the sample categories as well as gene transcripts belonging to the samples. We can see that the common property for all samples are that they all are dealing with obesity, adipocyte, inflammation and gene expression. The journal now has the form

GSM47229 GSM47230 GSM47231 GSM47232 ... GSM47334 GSM47335 GSM47336 GSM47337
C(Array) HG_U95Av2 HG_U95Av2 HG_U95Av2 HG_U95Av2 ... HG_U95Av2 HG_U95Av2 HG_U95Av2 HG_U95Av2
C(Types) lean-female lean-female lean-female lean-female ... obese-male obese-male obese-male obese-male
C(Type0) lean lean lean lean ... obese obese obese obese
C(Type1) female female female female ... male male male male
C(Platform) GPL8300 GPL8300 GPL8300 GPL8300 ... GPL8300 GPL8300 GPL8300 GPL8300
Marginal 355 340 330 362 ... 357 345 377 343
Present 5045 5165 5581 4881 ... 4355 4911 5140 5672
Absent 7225 7120 6714 7382 ... 7913 7369 7108 6610
NoCall 0 0 0 0 ... 0 0 0 0

Since we put extra effort into denoting all categoricals with C( ) we can solve the problem for the entire journal in one go with

formula = 'f~'+'+'.join(journal_df.index.values)

which becomes


and the final analysis of the data becomes exceptionally simple, again by writing

    from impetuous.quantification import multifactor_evaluation
    multifactor_results = multifactor_evaluation ( analyte_df , journal_df , formula )

Now we can see which transcripts are sensitive to the numerical quality measures as well as the categorical instances that we might be interested in. Take for example the genes that seem to regulate obesity

np.array([['HSPA1A','HSPA1B', 'HSPA1L', 'IGFBP7', 'TMSB10', 'TMSB4X', 'RPLP2',
        'SNORA52', 'COL3A1', 'CXCL12', 'FLNA', 'AGPAT2', 'GPD1', 'ACTB',
        'ACTG1', 'RARRES2', 'COL6A2', 'HSPB6', 'CLU', 'TAGLN', 'HLA-DRA',
        'PFKFB3', 'MAOB', 'DPT', 'NQO1', 'S100A4', 'LIPE', 'CCND1',
        'FASN', 'COL6A1', 'NOTCH3', 'PFKFB3'],
       ['ECM2', 'C1S', 'GLUL', 'ENPP2', 'PALLD', 'MAOA', 'B2M', 'SPARC',
        'HTRA1', 'CCL2', 'ACTB', 'AKR1C1', 'AKR1C2', 'LOC101930400',
        'EIF4A2', 'MIR1248', 'SNORA4', 'SNORA63', 'SNORA81', 'SNORD2',
        'PTPLB', 'GAPDH', 'CCL2', 'SAT1', 'IGFBP5', 'AES', 'PEA15',
        'ADH1B', 'PRKAR2B', 'PGM1', 'GAPDH','S100A10']], dtype=object)

which account for the top 64 obesity transcripts. We note that some of these are shared with diabetics. If we study which ones describes the Marginal or Absent genes we can see that there are some that we might want to exclude for technical reasons. We will leave that excercise for the curious reader.

Example 8: Latent grouping assumptions. Building a Parent-Child list

So you are sitting on a large amount of groupings that you have a significance test for. Testing what you are interested in per analyte symbol/id. Since you will conduct a large amount of tests there is also a large risk that you will technically test the same thing over and over again. In order to remove this effect from your group testing you could employ my HierarchicalEnrichment routine, but then you would also need a relationship file describing how to build a group DAG Hierarchy. This can be done with a relationship file that contains a parent id, a tab delimiter and a child id on each line. The routine that I demonstrate here uses a divide-and-conquer type approach to construct that information, which means that a subgroup, or child, is only assigned if it is fully contained within the parents definition. You can create redundant assignments by setting bSingleDescent=False, but it is not the recommended default setting.

Construction of the downward node relationships can be done with my build_pclist_word_hierarchy routine. Let us assume that you are sitting on the following data:

    portfolios = { 'PORT001' : ['Anders EQT' ,['AAPL','GOOG','IBM','HOUSE001','OTLY','GOLD','BANANAS'] ],
                   'PORT002' : ['Anna EQT'   ,['AAPL','AMZN','HOUSE001','CAR','BOAT','URANIUM','PLUTONIUM','BOOKS'] ],
                   'PORT003' : ['Donald EQT' ,['EGO','GOLF','PIES','HOUSE100','HOUSE101','HOUSE202'] ] ,
                   'PORT004' : ['BOB EQT'    ,['AAPL','GOOG'] ],
                   'PORT005' : ['ROB EQT'    ,['AMZN','BOOKS'] ],
                   'PORT006' : ['KIM EQT'    ,['URANIUM','PLUTONIUM'] ],
                   'PORT007' : ['LIN EQT'    ,['GOOG'] ] }

Then you might have noticed that some of the portfolios seem to contain the others completely. In order to derive the direct downward relationship you can issue the following commands (after installing impetuous version>=0.64.1

    import impetuous.hierarchical as imph
    pclist = imph.build_pclist_word_hierarchy ( ledger = portfolios , group_id_prefix='PORT' , root_name='PORT000')

which will return the list you need. You can now save it as a node relationship file and use that in my DAG construction routine.

Lets instead assume that you want to read the analyte groupings from a file, then you could issue :

    import os
    os.system('wget https://gist.githubusercontent.com/richardtjornhammar/6780e6d99e701fcc83994cc7a5f77759/raw/c37eaeeebc4cecff200bebf3b10dfa57984dbb84/new_compartment_genes.gmt')
    filename = 'new_compartment_genes.gmt'
    pcl , pcd = imph.build_pclist_word_hierarchy ( filename = filename , bReturnList=True )

If there are latent assumptions for some groupings then you can read them out by checking what the definitions refers to (here we already know that there is one for the mitochondrion definition):

    for item in pcl :
        if  'mito' in pcd[item[1]][0] or 'mela' in pcd[item[1]][0] :
            print ( pcd[item[0]][0] , ' -> ' , pcd[item[1]][0] )

which will tell you that

full cell  ->  melanosome membrane
full cell  ->  mitochondrial inner membrane
full cell  ->  mitochondrial matrix
melanosome membrane   ->  mitochondrion
full cell  ->  mitochondrial outer membrane
full cell  ->  mitochondrial intermembrane space

the definition for the mitochondrion is fully contained within the melanosome membrane definition and so testing that group should try and account for the mitochondrion. This can be done with the HierarchicalEnrichment routine exemplified above. We know that the melanosome membrane is associated with sight and that being diabetic is associated with mitochondrial dysfunction, but also that diabetic retinopathy affects diabetics. We see here that there is a knowledge based genetic connection relating these two spatially distinct regions of the cell.

Example 9: Impetuous deterministic DBSCAN (search for dbscan)

DBSCAN is a clustering algorithm that can be seen as a way of rejecting points, from any cluster, that are positioned in low dense regions of a point cloud. This introduces holes and may result in a larger segment, that would otherwise be connected via a non dense link to become disconnected and form two segments, or clusters. The rejection criterion is simple. The central concern is to evaluate a distance matrix with an applied cutoff this turns the distances into true or false values depending on if a pair distance between point i and j is within the distance cutoff. This new binary Neighbour matrix tells you wether or not two points are neighbours (including itself). The DBSCAN criterion states that a point is not part of any cluster if it has fewer than minPts neighbors. Once you've calculated the distance matrix you can immediately evaluate the number of neighbors each point has and the rejection criterion, via . If the rejection vector R value of a point is True then all the pairwise distances in the distance matrix of that point is set to a value larger than epsilon. This ensures that a distance matrix search will reject those points as neighbours of any other for the choosen epsilon. By tracing out all points that are neighbors and assessing the connectivity (search for connectivity) you can find all the clusters.

In this example we do exactly this for two gaussian point clouds. The dbscan search is just a single line dbscan ( data_frame = point_cloud_df , eps=0.45 , minPts=4 ), while the last lines are there to plot the results ( has graph revision dates )

The radial distribution function is a useful tool for visualizing whether or not there are radial clustering tendencies at any average distance between the group of interest and any other constituents of the system. This structure assessment method is usually used for analysis of particle systems, i.e. see liquid structure. It is implemented in the clustering module and is demonstrated here. If there is a significant density close to r=0 then you cannot separate the group from the mean background. This also means that any significance test between those groups will tell you that the grouping is insignificant. The resulting plot has revision dates. Since the radial distribution function calculates the spherically symmetric distribution of points surrounding an analyte, or analyte group, of interest it is effectively analogous to segmenting the distance matrix and leaving out any self interaction distances that may or may not be present.

The functions select_from_distance_matrix uses boolean indexing to select rows and columns (it is symmetric) in the distance matrix and the exclusive_pdist function calculates all pairs between the points in the two separate groups.

Example 10: Householder decomposition

In this example we will compare the decompostion of square and rectangular matrices before and after Householder decomposition. We recall that the Householder decomposition is a way of factorising matrices into orthogonal components and a tridiagonal matrix. The routine is implemented in the impetuous.reducer module under the name Householder_reduction. Now, why is any of that important? The Householder matrices are deterministically determinable and consitutes an unambigous decomposition of your data. The factors are easy to use to further solve what different types of operations will do to your original matrix. One can, for instance, use it to calculate the ambigous SVD decomposition or calculate eigenvalues for rectangular matrices.

Let us assume that you have a running environment and a set of matrices that you like

import numpy as np
import pandas as pd

if __name__=='__main__' :
    from impetuous.reducer import ASVD, Householder_reduction

    df = lambda x:pd.DataFrame(x)
    if True :
        B = np.array( [ [  4 ,  1 , -2 ,  2 ] ,
                   [  1 ,  2 ,  0 ,  1 ] ,
                   [ -2 ,  0 ,  3 , -2 ] ,
                   [  2 ,  1 , -2 , -1 ] ] )

    if True :
        A = np.array([ [ 22 , 10 ,  2 ,   3 ,  7 ] ,
                    [ 14 ,  7 , 10 ,   0 ,  8 ] ,
                    [ -1 , 13 , -1 , -11 ,  3 ] ,
                    [ -3 , -2 , 13 ,  -2 ,  4 ] ,
                    [  9 ,  8 ,  1 ,  -2 ,  4 ] ,
                    [  9 ,  1 , -7 ,   5 , -1 ] ,
                    [  2 , -6 ,  6 ,   5 ,  1 ] ,
                    [  4 ,  5 ,  0 ,  -2 ,  2 ] ] )

you might notice that the eigenvalues and the singular values of the square matrix B look similar

    print ( "FOR A SQUARE MATRIX:" )
    print ( "SVD DIAGONAL MATRIX ",df(np.linalg.svd(B)[1]) )
    print ( "SORTED ABSOLUTE EIGENVALUES ", df(sorted(np.abs(np.linalg.eig(B)[0]))[::-1]) )

but that the eigenvalues for the Householder reduction of the matrix B and the matrix B are the same

    HB = Householder_reduction ( B )[1]
    print ( np.linalg.eig( B)[0]  )
    print ( np.linalg.eig(HB)[0]  )

We readily note that this is also true for the singular values of the matrix B and the matrix HB. For the rectangular matrix A the eigenvalues are not defined when using numpy. The SVD decomposition is defined and we use it to check if the singular values are the same for the Householder reduction of the matrix A and the matrix A.

    HOUSEH = Householder_reduction ( A )[1]
    print ( "SVD ORIGINAL  : " , df(np.linalg.svd(A)[1]) )
    print ( "SVD HOUSEHOLD : " , df(np.linalg.svd(HOUSEH)[1]) )

and lo and behold.

    n = np.min(np.shape(HOUSEH))
    print ( "SVD SKEW   H : " , df(np.linalg.svd(HOUSEH)[1]) )
    print ( "SVD SQUARE H : " , df(np.linalg.svd(HOUSEH[:n,:n])[1]) )
    print ( "SVD ORIGINAL : " , df(np.linalg.svd(A)[1]) )
    print ( "EIGENVALUES  : " , np.linalg.eig(HOUSEH[:n,:n])[0] )

They are. So we feel confident that using the eigenvalues from the square part of the Householder matrix (the rest is zero anyway) to calculate the eigenvalues of the rectangular matrix is ok. But wait, why are they complex valued now? :^D

We can also reconstruct the original data by multiplying together the factors of either decomposition

    F,Z,GT = Householder_reduction ( A )
    U,S,VT = ASVD(A)

    print ( np.dot( np.dot(F,Z),GT ) )
    print ( np.dot( np.dot(U,S),VT ) )
    print ( A )

Thats all for now folks!

Example 11: The traveling salesman

This classic problem can be solved by first constructing a distance matrix for all the sites/cities that the person must visit. By performing hierarchical clustering and storing the clusters as nodes in a hierarchical DAG one can obtain the solution by doing a breadth first search on that hierarchy. It is demonstrated here and uses the newly developed NodeGraph functionalities as well as the numerical hierarchical clustering routines (in impetuous.hierarchical and impetuous.convert respectively). This section will be further elaborated.


These examples were meant as illustrations of some of the codes implemented in the impetuous-gfa package.

The impetuous visualisation codes requires Bokeh and are still being migrated to work with the latest Bokeh versions. For an example of the dynamic triplot routine (you can click on the lefthand and bottom scatter points) you can view it here ( with revision dates or download it here ).

Some of the algorithms rely on the SVD implementation in Numpy. A switch is planned for the future.

Manually updated code backups for this library :

GitLab: https://gitlab.com/richardtjornhammar/impetuous

CSDN: https://codechina.csdn.net/m0_52121311/impetuous

Bitbucket: https://bitbucket.org/richardtjornhammar/impetuous