Some Data Processing and Analysis with Python

The following problems appeared as assignments in the edX course Analytics for Computing (by Gatech). The descriptions are taken from the assignments.

1. Association rule mining

First we shall implement the basic pairwise association rule mining algorithm.

Problem definition

Let’s say we have a fragment of text in some language. We wish to know whether there are association rules among the letters that appear in a word. In this problem:

  • Words are “receipts”
  • Letters within a word are “items”

We want to know whether there are association rules of the form, a⟹ b , where a and b are letters, for a given language (by calculating for each rule its confidencecon(⟹ b), which is an estimate of the conditional probability of b given a, or Pr[b|a].

Sample text input

Let’s carry out this analysis on a “dummy” text fragment, which graphic designers refer to as the lorem ipsum:

latin_text= """
Sed ut perspiciatis, unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam
rem aperiam eaque ipsa, quae ab illo inventore veritatis et quasi architecto beatae vitae dicta
sunt, explicabo. Nemo enim ipsam voluptatem, quia voluptas sit, aspernatur aut odit aut fugit, sed
quia consequuntur magni dolores eos, qui ratione voluptatem sequi nesciunt, neque porro quisquam est,
qui dolorem ipsum, quia dolor sit amet consectetur adipisci[ng] velit, sed quia non numquam [do] eius
modi tempora inci[di]dunt, ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi
consequatur? Quis autem vel eum iure reprehenderit, qui in ea voluptate velit esse, quam nihil molestiae
consequatur, vel illum, qui dolorem eum fugiat, quo voluptas nulla pariatur?

At vero eos et accusamus et iusto odio dignissimos ducimus, qui blanditiis praesentium voluptatum
deleniti atque corrupti, quos dolores et quas molestias excepturi sint, obcaecati cupiditate non
provident, similique sunt in culpa, qui officia deserunt mollitia animi, id est laborum et dolorum
fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est
eligendi optio, cumque nihil impedit, quo minus id, quod maxime placeat, facere possimus, omnis voluptas
assumenda est, omnis dolor repellendus. Temporibus autem quibusdam et aut officiis debitis aut rerum
necessitatibus saepe eveniet, ut et voluptates repudiandae sint et molestiae non recusandae. Itaque
earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur
aut perferendis doloribus asperiores repellat.
"""


Data cleaning

Like most data in the real world, this dataset is noisy. It has both uppercase and lowercase letters, words have repeated letters, and there are all sorts of non-alphabetic characters. For our analysis, we should keep all the letters and spaces (so we can identify distinct words), but we should ignore case and ignore repetition within a word.

For example, the eighth word of this text is “error.” As an itemset, it consists of the three unique letters{e,o,r}. That is, to treat the word as a set, meaning we only keep the unique letters. This itemset has six possible itempairs{e,o}{e,r}, and {o,r}.

  • We need to start by “cleaning up” (normalizing) the input, with all characters converted to lowercase and all non-alphabetic, non-space characters removed.
  • Next, we need to convert each word into an itemset like the following examples:

consequatur –> {‘a’, ‘e’, ‘c’, ‘s’, ‘n’, ‘o’, ‘u’, ‘t’, ‘q’, ‘r’}
voluptatem –> {‘l’, ‘a’, ‘e’, ‘o’, ‘p’, ‘u’, ‘m’, ‘t’, ‘v’}

Implementing the basic algorithm

The followed algorithm is implemented:

FindAssocRules--crop-small.png

 

First all item-pairs within an itemset are enumerated and a table that tracks the counts of those item-pairs is updated in-place.

  • Now, given tables of item-paircounts and individual item counts, as well as a confidence threshold, the rules that meet the threshold are returned.
  • The returned rules should be in the form of a dictionary whose key is the tuple(a,bcorresponding to the rule a⇒ b, and whose value is the confidence of the rule, conf(⇒ b).
  • The following functions were implemented to compute the association rules.
    from collections import defaultdict
    from itertools import combinations 
    
    def update_pair_counts (pair_counts, itemset):
        """
        Updates a dictionary of pair counts for all pairs of items 
        in a given itemset.
        """
        assert type (pair_counts) is defaultdict
    
        for item in list(combinations(itemset, 2)):
            pair_counts[item] += 1
            pair_counts[item[::-1]] += 1
        return pair_counts
    def update_item_counts(item_counts, itemset):
    
        for item in itemset:
            item_counts[item] += 1
        return item_counts
    def filter_rules_by_conf (pair_counts, item_counts, threshold):
        rules = {} # (item_a, item_b) -> conf (item_a => item_b)
    
        for (item_a, item_b) in pair_counts:
            conf = pair_counts[(item_a, item_b)] / float(item_counts[item_a])
            if conf >= threshold:
                rules[(item_a, item_b)] = conf
                                                         
        return rules
    def find_assoc_rules(receipts, threshold):
    
        pair_counts = defaultdict(int)
        item_counts = defaultdict(int)
        for receipt in receipts:
            update_pair_counts(pair_counts, receipt)
            update_item_counts(item_counts, receipt)
        return filter_rules_by_conf(pair_counts, item_counts, threshold)

 

  • For the Latin string, latin_text, the function find_assoc_rules() was used to compute the rules whose confidence is at least 0.75, with the following rules obtained as result.

    conf(q => u) = 1.000
    conf(x => e) = 1.000
    conf(x => i) = 0.833
    conf(h => i) = 0.833
    conf(v => t) = 0.818
    conf(r => e) = 0.800
    conf(v => e) = 0.773
    conf(f => i) = 0.750
    conf(b => i) = 0.750
    conf(g => i) = 0.750

 

  • Now, let’s look at rules common to the above Latin textandEnglish text obtained by a translation of the lorem ipsum text, as shown below:
    english_text = """
    But I must explain to you how all this mistaken idea of denouncing of a pleasure and praising pain was
    born and I will give you a complete account of the system, and expound the actual teachings of the great
    explorer of the truth, the master-builder of human happiness. No one rejects, dislikes, or avoids
    pleasure itself, because it is pleasure, but because those who do not know how to pursue pleasure
    rationally encounter consequences that are extremely painful. Nor again is there anyone who loves or
    pursues or desires to obtain pain of itself, because it is pain, but occasionally circumstances occur in
    which toil and pain can procure him some great pleasure. To take a trivial example, which of us
    ever undertakes laborious physical exercise, except to obtain some advantage from it? But who has any
    right to find fault with a man who chooses to enjoy a pleasure that has no annoying consequences, or
    one who avoids a pain that produces no resultant pleasure?
    
    On the other hand, we denounce with righteous indignation and dislike men who are so beguiled and
    demoralized by the charms of pleasure of the moment, so blinded by desire, that they cannot foresee the
    pain and trouble that are bound to ensue; and equal blame belongs to those who fail in their duty
    through weakness of will, which is the same as saying through shrinking from toil and pain. These
    cases are perfectly simple and easy to distinguish. In a free hour, when our power of choice is
    untrammeled and when nothing prevents our being able to do what we like best, every pleasure is to
    be welcomed and every pain avoided. But in certain circumstances and owing to the claims of duty or
    the obligations of business it will frequently occur that pleasures have to be repudiated and
    annoyances accepted. The wise man therefore always holds in these matters to this principle of
    selection: he rejects pleasures to secure other greater pleasures, or else he endures pains to
    avoid worse pains.
    """

 

  • Again, for the English string, english_text, the function find_assoc_rules()  was used to compute the rules whose confidence is at least 0.75, with the following rules obtained as result.

    conf(z => a) = 1.000
    conf(j => e) = 1.000
    conf(z => o) = 1.000
    conf(x => e) = 1.000
    conf(q => e) = 1.000
    conf(q => u) = 1.000
    conf(z => m) = 1.000
    conf(z => r) = 1.000
    conf(z => l) = 1.000
    conf(z => e) = 1.000
    conf(z => d) = 1.000
    conf(z => i) = 1.000
    conf(k => e) = 0.778
    conf(q => n) = 0.750

 

  • Let’s consider any rules with a confidence of at least 0.75 to be a “high-confidence rule“.  The common_high_conf_rules are all the high-confidence rules appearing in both the Latin text and the English text. The rules shown below are all such rules:High-confidence rules common to _lorem ipsum_ in Latin and English:

    q => u
    x => e

 

  • The following table and the figure show the high confidence rules for  the latin and the english texts.
    index rule confidence
    0 z=> o 1.000000 English
    1 z=> l 1.000000 English
    2 z=> m 1.000000 English
    3 q=> u 1.000000 English
    4 q=> e 1.000000 English
    5 x=> e 1.000000 English
    6 z=> e 1.000000 English
    7 j=> e 1.000000 English
    8 z=> a 1.000000 English
    9 z=> d 1.000000 English
    10 q=> u 1.000000 Latin
    11 z=> i 1.000000 English
    12 x=> e 1.000000 Latin
    13 z=> r 1.000000 English
    14 x=> i 0.833333 Latin
    15 h=> i 0.833333 Latin
    16 v=> t 0.818182 Latin
    17 r=> e 0.800000 Latin
    18 k=> e 0.777778 English
    19 v=> e 0.772727 Latin
    20 g=> i 0.750000 Latin
    21 q=> n 0.750000 English
    22 f=> i 0.750000 Latin
    23 b=> i 0.750000 Latin

    eng_lat.png

Putting it all together: Actual baskets!

Let’s take a look at some real data  from this link. First few lines of the transaction data is shown below:

citrus fruit,semi-finished bread,margarine,ready soups
tropical fruit,yogurt,coffee
whole milk
pip fruit,yogurt,cream cheese ,meat spreads
other vegetables,whole milk,condensed milk,long life bakery product
whole milk,butter,yogurt,rice,abrasive cleaner
rolls/buns
other vegetables,UHT-milk,rolls/buns,bottled beer,liquor (appetizer)
pot plants
whole milk,cereals
tropical fruit,other vegetables,white bread,bottled water,chocolate
citrus fruit,tropical fruit,whole milk,butter,curd,yogurt,flour,bottled water,dishes
beef
frankfurter,rolls/buns,soda
chicken,tropical fruit
butter,sugar,fruit/vegetable juice,newspapers
fruit/vegetable juice
packaged fruit/vegetables
chocolate
specialty bar
other vegetables
butter milk,pastry
whole milk
tropical fruit,cream cheese ,processed cheese,detergent,newspapers
tropical fruit,root vegetables,other vegetables,frozen dessert,rolls/buns,flour,sweet spreads,salty snack,waffles,candy,bathroom cleaner
bottled water,canned beer
yogurt
sausage,rolls/buns,soda,chocolate
other vegetables
brown bread,soda,fruit/vegetable juice,canned beer,newspapers,shopping bags
yogurt,beverages,bottled water,specialty bar

  • Our task is to mine this dataset for pairwise association rules to produce a final dictionary, basket_rules, that meet these conditions:
  1. The keys are pairs (a,b), where a and b are item names.
  2. The values are the corresponding confidence scores, conf(⇒ b).
  3. Only include rules ⇒ b where item a occurs at least MIN_COUNT times and conf(⇒ b) is at least THRESHOLD.

The result is shown below:

Found 19 rules whose confidence exceeds 0.5.
Here they are:

conf(honey => whole milk) = 0.733
conf(frozen fruits => other vegetables) = 0.667
conf(cereals => whole milk) = 0.643
conf(rice => whole milk) = 0.613
conf(rubbing alcohol => whole milk) = 0.600
conf(cocoa drinks => whole milk) = 0.591
conf(pudding powder => whole milk) = 0.565
conf(jam => whole milk) = 0.547
conf(cream => sausage) = 0.538
conf(cream => other vegetables) = 0.538
conf(baking powder => whole milk) = 0.523
conf(tidbits => rolls/buns) = 0.522
conf(rice => other vegetables) = 0.520
conf(cooking chocolate => whole milk) = 0.520
conf(frozen fruits => whipped/sour cream) = 0.500
conf(specialty cheese => other vegetables) = 0.500
conf(ready soups => rolls/buns) = 0.500
conf(rubbing alcohol => butter) = 0.500
conf(rubbing alcohol => citrus fruit) = 0.500

2. Simple string processing with Regex

Phone numbers

  • Write a function to parse US phone numbers written in the canonical “(404) 555-1212” format, i.e., a three-digit area code enclosed in parentheses followed by a seven-digit local number in three-hyphen-four digit format.
  • It should also ignore all leading and trailing spaces, as well as any spaces that appear between the area code and local numbers.
  • However, it should not accept any spaces in the area code (e.g., in ‘(404)’) nor should it in the local number.
  • It should return a triple of strings, (area_code, first_three, last_four).
  • If the input is not a valid phone number, it should raise a ValueError.
The following function implements the regex parser.
import re 
def parse_phone(s):
    pattern = re.compile("\s*\((\d{3})\)\s*(\d{3})-(\d{4})\s*")
    m = pattern.match(s)
    if not m:
        raise ValueError('not a valid phone number!')
    return m.groups()

#print(parse_phone1('(404) 201-2121'))    
#print(parse_phone1('404-201-2121'))  
  • Implement an enhanced phone number parser that can handle any of these patterns.
    • (404) 555-1212
    • (404) 5551212
    • 404-555-1212
    • 404-5551212
    • 404555-1212
    • 4045551212
  • As before, it should not be sensitive to leading or trailing spaces. Also, for the patterns in which the area code is enclosed in parentheses, it should not be sensitive to the number of spaces separating the area code from the remainder of the number.

The following function implements the enhanced regex parser.

import re 
def parse_phone2 (s):
    pattern = re.compile("\s*\((\d{3})\)\s*(\d{3})-?(\d{4})\s*")
    m = pattern.match(s)
    if not m:
        pattern2 = re.compile("\s*(\d{3})-?(\d{3})-?(\d{4})\s*")
        m = pattern2.match(s)
        if not m:
            raise ValueError('not a valid phone number!')
    return m.groups()

3. Tidy data and the Pandas

“Tidying data,” is all about cleaning up tabular data for analysis purposes.

Definition: Tidy datasets. More specifically, Wickham defines a tidy data set as one that can be organized into a 2-D table such that

  1. each column represents a variable;
  2. each row represents an observation;
  3. each entry of the table represents a single value, which may come from either categorical (discrete) or continuous spaces.

Definition: Tibbles. if a table is tidy, we will call it a tidy table, or tibble, for short.

Apply functions to data frames

Given the following pandas DataFrame (first few rows are shown in the next table),compute the prevalence, which is the ratio of cases to the population, using the apply() function, without modifying the original DataFrame.

country year cases population
0 Afghanistan ’99 745 19987071
1 Brazil ’99 37737 172006362
2 China ’99 212258 1272915272
3 Afghanistan ’00 2666 20595360
4 Brazil ’00 80488 174504898
5 China ’00 213766 1280428583

The next function does exactly the same thing.

def calc_prevalence(G):
    assert 'cases' in G.columns and 'population' in G.columns
    H = G.copy()
    H['prevalence'] = H.apply(lambda row: row['cases'] / row['population'], axis=1)
    return H

Tibbles and Bits

Now let’s start creating and manipulating tibbles.

Write a function, canonicalize_tibble(X), that, given a tibble X, returns a new copy Y of X in canonical order. We say Y is in canonical order if it has the following properties.

  1. The variables appear in sorted order by name, ascending from left to right.
  2. The rows appear in lexicographically sorted order by variable, ascending from top to bottom.
  3. The row labels (Y.index) go from 0 to n-1, where n is the number of observations.

The following code exactly does the same:

def canonicalize_tibble(X):
    # Enforce Property 1:
    var_names = sorted(X.columns)
    Y = X[var_names].copy()
    Y = Y.sort_values(by=var_names, ascending=True)
    Y.reset_index(drop=True, inplace=True)
    return Y

Basic tidying transformations: Implementing Melting and Casting

Given a data set and a target set of variables, there are at least two common issues that require tidying.

Melting

First, values often appear as columns. Table 4a is an example. To tidy up, we want to turn columns into rows:

tidy-9

Because this operation takes columns into rows, making a “fat” table more tall and skinny,  it is sometimes called melting.

To melt the table, we need to do the following.

  1. Extract the column values into a new variable. In this case, columns "1999"  and "2000" of table4 need to become the values of the variable, "year".
  2. Convert the values associated with the column values into a new variable as well. In this case, the values formerly in columns "1999" and "2000"become the values of the "cases" variable.

In the context of a melt, let’s also refer to "year" as the new key variable and
"cases" as the new value variable.

Implement the melt operation as a function,

def melt(df, col_vals, key, value):
        ...

It should take the following arguments:

  • df: the input data frame, e.g., table4 in the example above;
  • col_vals: a list of the column names that will serve as values;
  • key: name of the new variable, e.g., year in the example above;
  • value: name of the column to hold the values.

The next function implements the melt operation:

def melt(df, col_vals, key, value):
    assert type(df) is pd.DataFrame
    df2 = pd.DataFrame()
    for col in col_vals:
        df1 = pd.DataFrame(df[col].tolist(), columns=[value]) #, index=df.country)
        df1[key] = col
        other_cols = list(set(df.columns.tolist()) - set(col_vals))
        for col1 in other_cols:
            df1[col1] = df[col1]
        df2 = df2.append(df1, ignore_index=True)
    df2 = df2[other_cols + [key, value]]    
    return df2

with the following output

=== table4a ===
country 1999 2000
0 Afghanistan 745 2666
1 Brazil 37737 80488
2 China 212258 213766
=== melt(table4a) ===
country year cases
0 Afghanistan 1999 745
1 Brazil 1999 37737
2 China 1999 212258
3 Afghanistan 2000 2666
4 Brazil 2000 80488
5 China 2000 213766
=== table4b ===
country 1999 2000
0 Afghanistan 19987071 20595360
1 Brazil 172006362 174504898
2 China 1272915272 1280428583
=== melt(table4b) ===
country year population
0 Afghanistan 1999 19987071
1 Brazil 1999 172006362
2 China 1999 1272915272
3 Afghanistan 2000 20595360
4 Brazil 2000 174504898
5 China 2000 1280428583

Casting

The second most common issue is that an observation might be split across multiple rows. Table 2 is an example. To tidy up, we want to merge rows:

tidy-8.png

Because this operation is the moral opposite of melting, and “rebuilds” observations from parts, it is sometimes called casting.

Melting and casting are Wickham’s terms from his original paper on tidying data. In his more recent writing, on which this tutorial is based, he refers to the same operation as gathering. Again, this term comes from Wickham’s original paper, whereas his more recent summaries use the term spreading.

The signature of a cast is similar to that of melt. However, we only need to know the key, which is column of the input table containing new variable names, and the value, which is the column containing corresponding values.

Implement a function to cast a data frame into a tibble, given a key column containing new variable names and a value column containing the corresponding cells.

Observe that we are asking your cast() to accept an optional parameter, join_how, that may take the values 'outer' or 'inner' (with 'outer' as the default).

The following function implements the casting operation:

def cast(df, key, value, join_how='outer'):
    """Casts the input data frame into a tibble,
    given the key column and value column.
    """
    assert type(df) is pd.DataFrame
    assert key in df.columns and value in df.columns
    assert join_how in ['outer', 'inner']
    
    fixed_vars = df.columns.difference([key, value])
    tibble = pd.DataFrame(columns=fixed_vars) # empty frame
    
    fixed_vars = fixed_vars.tolist()
    #tibble[fixed_vars] = df[fixed_vars]
    cols = []
    for k,df1 in df.groupby(df[key]):
        #tibble = pd.concat([tibble.reset_index(drop=True), df1[value]], axis=1)
        #print(df1[fixed_vars+[value]].head())
        tibble = tibble.merge(df1[fixed_vars+[value]], on=fixed_vars, how=join_how)
        cols.append(str(k)) #list(set(df1[key]))[0])
    tibble.columns = fixed_vars + cols
 
    return tibble

with the following output:

=== table2 ===
country year type count
0 Afghanistan 1999 cases 745
1 Afghanistan 1999 population 19987071
2 Afghanistan 2000 cases 2666
3 Afghanistan 2000 population 20595360
4 Brazil 1999 cases 37737
5 Brazil 1999 population 172006362
6 Brazil 2000 cases 80488
7 Brazil 2000 population 174504898
8 China 1999 cases 212258
9 China 1999 population 1272915272
10 China 2000 cases 213766
11 China 2000 population 1280428583
=== tibble2 = cast (table2, "type", "count") ===
country year cases population
0 Afghanistan 1999 745 19987071
1 Afghanistan 2000 2666 20595360
2 Brazil 1999 37737 172006362
3 Brazil 2000 80488 174504898
4 China 1999 212258 1272915272
5 China 2000 213766 1280428583

Separating variables

Consider the following table.

 

country year rate
0 Afghanistan 1999 745/19987071
1 Afghanistan 2000 2666/20595360
2 Brazil 1999 37737/172006362
3 Brazil 2000 80488/174504898
4 China 1999 212258/1272915272
5 China 2000 213766/1280428583

In this table, the rate variable combines what had previously been the cases
andpopulation data. This example is an instance in which we might want to separate a column into two variables.

Write a function that takes a data frame (df) and separates an existing column (key) into new variables (given by the list of new variable names, into).  How will the separation happen? The caller should provide a function, splitter(x), that given a value returns a list containing the components.

 

The following code implements the function:

import re

def default_splitter(text):
    """Searches the given spring for all integer and floating-point
    values, returning them as a list _of strings_.
    
    E.g., the call
    
      default_splitter('Give me $10.52 in exchange for 91 kitten stickers.')
      
    will return ['10.52', '91'].
    """
    #fields = re.findall('(\d+\.?\d+)', text)
    fields = list(re.match('(\d+)/(\d+)', text).groups())
    return fields

def separate(df, key, into, splitter=default_splitter):
    """Given a data frame, separates one of its columns, the key,
    into new variables.
    """
    assert type(df) is pd.DataFrame
    assert key in df.columns    
    return (df.merge(df[key].apply(lambda s: pd.Series({into[i]:splitter(s)[i] for i in range(len(into))})), 
    left_index=True, right_index=True)).drop(key, axis=1)

with the following output:

=== table3 ===
country year rate
0 Afghanistan 1999 745/19987071
1 Afghanistan 2000 2666/20595360
2 Brazil 1999 37737/172006362
3 Brazil 2000 80488/174504898
4 China 1999 212258/1272915272
5 China 2000 213766/1280428583
=== tibble3 = separate (table3, ...) ===
country year cases population
0 Afghanistan 1999 745 19987071
1 Afghanistan 2000 2666 20595360
2 Brazil 1999 37737 172006362
3 Brazil 2000 80488 174504898
4 China 1999 212258 1272915272
5 China 2000 213766 1280428583
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Some Computational Photography: Image Quilting (Texture Synthesis) with Dynamic Programming and Texture Transfer (Drawing with Textures) in Python

The following problems appeared as a programming assignment in the Computation Photography course (CS445) at UIUC. The description of the problem is taken from the assignment itself. In this assignment, a python implementation of the problems will be described instead of matlab, as expected in the course.

 

The Problems

  • The goal of this assignment is to implement the image quilting algorithm for
    texture synthesis and transfer, described in this SIGGRAPH 2001 paper by Efros
    and Freeman.
  • Texture synthesis is the creation of a larger texture image from a small sample.
  • Texture transfer is giving an object the appearance of having the
    same texture as a sample while preserving its basic shape.
  • For texture synthesis, the main idea is to sample patches and lay them down in overlapping patterns, such that the overlapping regions are similar.
  • The overlapping regions may not match exactly, which will result in noticeable
    edges artifact. To fix this, we need to compute a path along pixels with similar intensities through the overlapping region and use it to select which overlapping patch from which to draw each pixel.
  • Texture transfer is achieved by encouraging sampled patches to have similar appearance to a given target image, as well as matching overlapping regions of already sampled patches.
  • In this project, we need to apply important techniques such as template matching, finding seams, and masking. These techniques are also useful for image stitching, image completion, image retargeting and blending.

 

Randomly Sampled Texture

First let’s randomly samples square patches of size patchsize from sample in order to create an output image of size outsize. Start from the upper-left corner, and tile samples until the image is full. If the patches don’t fit evenly into the output image, we can leave black borders at the edges. This is the simplest but least effective method. Save a result from a sample image to compare to the next two methods.

 

Overlapping Patches

Let’s start by sampling a random patch for the upper-left corner. Then sample new patches to overlap with existing ones. For example, the second patch along the top row will overlap by patchsize pixels in the vertical direction and overlap pixels in the horizontal direction. Patches in the first column will overlap by patchsize pixels in the horizontal direction and overlap pixels in the vertical direction. Other patches will have two overlapping regions (on the top and left) which should both be taken into account. Once the cost of each patch has been computed, randomly choose on patch whose cost is
less than a threshold determined by some tolerance value.

As described in the paper, the size of the block is the only parameter controlled by the user and it depends on the properties of a given texture; the block must be big enough to capture the relevant structures in the texture, but small enough so that the interaction between these structures is left up to the algorithm. The overlap size is taken to be one-sixth of the block size (B/6) in general.

 

Seam Finding

Next we need to find the min-cost contiguous path from the left to right side of the patch according to the cost. The cost of a path through each pixel is the square differences (summed over RGB for color images) of the output image and the newly
sampled patch. Use dynamic programming to find the min-cost path.

The following figure describes the algorithm to be implemented for image quilting.

f20.png

Texture Transfer

The final task is to implement texture transfer, based on the quilt implementation for creating a texture sample that is guided by a pair of sample/target correspondence images (section 3 of the paper). The main difference between this function and
quilt function is that there is an additional cost term based on the difference between
the sampled source patch and the target patch at the location to be filled.

Image quilting (texture synthesis) results

The following figures and animations show the results of the outputs obtained with the quilting algorithm. The input texture images are mostly taken from the paper .

Input sample Texture
q2.png

100 sampled blocks of a fixed size (e.g. 50×50) from the input sample
samples_q2.png

The next animation shows how the large output texture gets created (100 times larger than the input sample texture) with the quilting algorithm.

quilt.gif

Output Texture (10×10 times larger than the input) created with texture synthesis (quilting)
quilt_q2.png

 

Input Texture

q6.png

Output Texture (25 times larger than the input) created with texture synthesis (quilting) with the minimum cost seams (showed as red lines) computed with dynamic programming

quilt_boundary_q6

Output Texture (25 times larger than the input) created with quiltingquilt_q6

 

Input Texture

q1

Output Texture (25 times larger than the input) created with quilting

quilt_q1_.png

Input Texture

q3

Output Texture (25 times larger than the input) created with quilting

quilt_q3.png

Input Texture

q4

Output Texture (12 times larger than the input) created with quilting

quilt_q4.png

Input Texture

q5

Output Texture (25 times larger than the input) created with quilting

quilt_q5.png

Input Texture

q7

Output Texture (25 times larger than the input) created with quilting

quilt_q7_.png

Input Texture

q9

Output Texture (36 times larger than the input) created with quilting

quilt_q9.png

Input Texture

q11

Output Texture (9 times larger than the input) created with quilting

quilt_q11_.png

Input Texture

q12

Output Texture (25 times larger than the input) created with quilting

quilt_q12.png

Input Texture

q13

Output Texture (9 times larger than the input) created with quilting along with the min-cost seams (shown as red lines) computed with dynamic programming 

quilt_boundary_q13

Output Texture (9 times larger than the input) created with quilting

quilt_q13

 

Texture transfer results

The following figures and animations show how the texture from an input image can be transferred to the target image using the above-mentioned simple modification of the quilting algorithm. Again, some of the images are taken from the paper.

Input Texture (milk)

q14.png

Target Image

qt1.png

Output Image after Texture Transfer

text_tran_q14

 

Input Texture (milk)

q14.png

Target Image

mona

Output Image after Texture Transfer

text_tran_q14_

 

The following figures show the output image obtained when a few textures were transferred to my image.

Target Image (me)

me1.png

 

Input Texture (fire)

fire.png

Output Image after Texture Transfer  (with small block size)

text_tran_fire

 

 

Input Texture (cloud)

cloud1.png

Output Image after Texture Transfer  (with small block size)

text_tran_cloud1

 

Input Texture (toast)
toast.png

Output Image after Texture Transfer  (with small block size)text_tran_toast

Now applying some gradient mixing such as Poisson blending  on the original image and the the one after texture transfer with the target toast image we get the following two images respectively.

pe_toast1

pe_toast

Input Texture
q7.png

Output Image after Texture Transfer  (with small block size)

text_tran_q7

 

Input Texture

draw.png

Output Image after Texture Transfer  (with small block size)text_tran_draw

 

 

Drawing with Textures

 

The following figures show the output image obtained when a few textures were transferred to another image of mine.

 

Target Image (me)

me6.png

 

 

Input Texture

epii

Output Images after Texture Transfer  (with 2 different patch sizes)

text_tran_012_epiitext_tran_024_epii

 

Input Texture

ds

Output Images after Texture Transfer  (with 2 different patch sizes)

text_tran_012_dstext_tran_024_ds

 

Input Texture

alphabet.png

Output Image after Texture Transfer

text_tran_024_alphabet.png

 

Input Texture

rabi

Output Images after Texture Transfer  (with 2 different patch sizes)

text_tran_012_rabitext_tran_024_rabi

 

Input Texture

notes

Output Images after Texture Transfer  (with 2 different patch sizes)

text_tran_012_notestext_tran_024_notes

 

Input Texture

india

Output Images after Texture Transfer (with 2 different patch sizes)

text_tran_012_india.png

text_tran_024_india.png

 

 

Input Texture

bangla2.png

Output Image after Texture Transfer

text_tran_024_bangla2.png

 

 

Input Texture

r1

Output Image after Texture Transfer

text_tran_012_r1

 

 

Input Texture

ens

Output Image after Texture Transfer

text_tran_012_ens

 

Input Texture

rabi2

Output Image after Texture Transfer

text_tran_012_rabi2

 

 

Input Texture

 

toast

Output Image after Texture Transfer

text_tran_012_toast_12.png

 

 

Input Texture

q14

Output Image after Texture Transfer  (with 2 different patch sizes)

text_tran_012_q14_12text_tran_024_q14

 

Target Image (from The Mummy 1999)
mm.png

Input Texture (sand)

msky.png

Output Image after Texture Transfer

text_tran_012_msky_1.png

 

 

The following animation shows how the milk texture is being transformed to the target image of mine with the quilting algorithm with modified code.

Input Texture

q14

Target Image (me)

me4

Output Image after Texture Transfer  (with small block size)

ttmea

 

The next figures and animations show the output image obtained after milk texture gets transferred to the target image of mine, for different block size of the samples (shown in red). As can be seen from the following outputs, the target image gets more and more prominent as the sampling block size gets smaller and smaller.

text_tran_036_q14

text_tran_804_q14

ttme

Feature Detection with Harris Corner Detector and Matching images with Feature Descriptors in Python

The following problem appeared in a project in this Computer Vision Course (CS4670/5670, Spring 2015) at Cornell. In this article, a python implementation is going to be described. The description of the problem is taken (with some modifications) from the project description. The same problem appeared in this assignment problem as well. The images used for testing the algorithms implemented are mostly taken from these assignments / projects.

The Problem

In this project, we need to implement the problem of detect discriminating features in an image and find the best matching features in other images.  The features should be reasonably invariant to translation, rotation, and illumination.  The slides presented in class can be used as reference.

Description

The project has three parts:  feature detection, feature description, and feature matching.

1. Feature detection

In this step, we need to identify points of interest in the image using the Harris corner detection method.  The steps are as follows: 

  • For each point in the image, consider a window of pixels around that point.
  • Compute the Harris matrix H for (the window around) that point, defined asharriseq-structuretensor.pngwhere the summation is over all pixels p in the window.   is the x derivative of the image at point p, the notation is similar for the y derivative.
  •  The weights  are chosen to be circularly symmetric,  a 9×9 Gaussian kernel with 0.5 sigma is chosen to achieve this.
  • Note that H is a 2×2 matrix.  To find interest points, first we need to compute the corner strength function

  • Once c is computed for every point in the image, we need to choose points where c is above a threshold.
  • We also want c to be a local maximum in a 9×9 neighborhood (with non-maximum suppression).
  • In addition to computing the feature locations, we need to compute a canonical orientation for each feature, and then store this orientation (in degrees) in each feature element.
  • To compute the canonical orientation at each pixel, we need to compute the gradient of the blurred image and use the angle of the gradient as orientation.

2. Feature description

  • Now that the points of interest are identified,  the next step is to come up with a descriptor for the feature centered at each interest point.  This descriptor will be the representation to be used to compare features in different images to see if they match.
  • In this article, we shall implement a simple descriptor, a 8×8 square window without orientation. This should be very easy to implement and should work well when the images we’re comparing are related by a translation. We also normalize the window to have zero mean and unit variance, in order to obtain illumination invariance.
  • In order to obtain rotational invariance MOPS descriptor, by taking care of the orientation that is not discussed in this article for the time being.

3. Feature matching

  • Now that the features in the image are detected and described, the next step is to write code to match them, i.e., given a feature in one image, find the best matching feature in one or more other images.
  • The simplest approach is the following:  write a procedure that compares two features and outputs a distance between them.  For example, we simply sum the absolute value of differences between the descriptor elements.
  • We then use this distance to compute the best match between a feature in one image and the set of features in another image by finding the one with the smallest distance. The distance used here is the Manhattan distance.

 

The following theory and math for the Harris Corner Detection will be used that’s taken from this youtube video.

f19.png

The following figure shows the structure of the python code to implement the algorithm.

hcode.png

Feature Detection (with Harris Corner Detection): Results on a few images

  • The threshold to be used for the Harris Corner Detection is varied (as shown in the following animations in red, with the value of the threshold being 10^x, where x is shown (the common logarithm of the threshold is displayed).
  • The corner strength function with kappa=0.04 is used instead of the minimum eigenvalue (since it’s faster to compute).
  • As can be seen from the following animations, lesser and lesser corner features are detected when the threshold is increased.
  • The direction and magnitude of the feature is shown by the bounding (green) square’s angle with the horizontal and the size of the square respectively, computed from the gradient matrices.

Input Imageflower.jpg

Harris Corner Features Detected for different threshold values (log10)
hcflw

Input Image
liberty3

The following figure shows the result of thresholding on

  1. the Harris corner strength R values and
  2. the minimum eigenvalue for the Harris matrix respectively,

for each pixel, before applying non-maximum suppression (computing the local maximum).

hc1_liberty1.jpg

The next animation shows the features detected after applying non-maximum suppression, with different threshold values.

Harris Corner Features Detected for different threshold values (log10)
hclib

Input Image
features_small.png

Harris Corner Features Detected for different threshold values (log10)hcfea

Input Image
chess

Harris Corner Features Detected for different threshold values (log10)

hcchs

Input Image
yosemite1.jpg

Harris Corner Features Detected for different threshold values (log10)
hcyos

Computing Feature descriptors

  • In this article, we shall implement a very simple descriptor, a 8×8 square window without orientation. This is expected to work well when the images being compared are related by a translation.
  • We also normalize the window to have zero mean and unit variance, in order to obtain illumination invariance.
  • In order to obtain rotational invariance MOPS descriptor, by taking care of the orientation that is not discussed in this article for the time being.

 

Matching Images with Detected Features: Results on a few images

  • First the Harris corner features and the simple descriptors are computed for each of the images to be compared.
  • Next, distance between each pair of corner feature descriptors is computed, by simply computing the sum the absolute value of differences between the descriptor elements.
  • This distance is then used to compute the best match between a feature in one image and the set of features in another image by finding the one with the smallest distance.
  • The following examples show how the matching works with simple feature descriptors around the Harris corners for images obtained using mutual translations.

Input images (one is a translation of the other) 

liberty2           liberty1
Harris Corner Features Detected for the images
hc_0.000500liberty1hc_0.000500liberty2

Matched Features with minimum sum of absolute distancemhclib.gif

 

Input images
me        me3

Harris Corner Features Detected for the images

hc_0.000010me        hc_0.000010me3

Matched Features with minimum sum of absolute distance
mhcme.gif

The following example shows the input images to be compared being created more complex transformations (not only translation) and as expected, the simple feature descriptor does not work well in this case, as shown. We need some feature descriptors like SIFT to obtain robustness against rotation and scaling too.

Input images
trees_002  trees_003

Harris Corner Features Detected for the images
hc_0.000050trees_002   hc_0.000050trees_003

Matched Features with minimum sum of absolute distance
match_hc_403_trees_002.jpg

Seam Carving: Using Dynamic Programming to implement Content-Aware Image Resizing in Python

The following problem appeared as an assignment in the Algorithm Course (COS 226) at Princeton University taught by Prof. Sedgewick.  The following description of the problem is taken from the assignment itself.

The Seam Carving Problem

  • Seam-carving is a content-aware image resizing technique where the image is reduced in size by one pixel of height (or width) at a time.
  • vertical seam in an image is a path of pixels connected from the top to the bottom with one pixel in each row.
  • horizontal seam is a path of pixels connected from the left to the right with one pixel in each column.
  • Unlike standard content-agnostic resizing techniques (such as cropping and scaling), seam carving preserves the most interesting features (aspect ratio, set of objects present, etc.) of the image.
  • In this assignment, a data type needs to be created that resizes W-by-H image using the seam-carving technique.
  • Finding and removing a seam involves three parts:
    1. Energy calculation.The first step is to calculate the energy of a pixel, which is a measure of its importance—the higher the energy, the less likely that the pixel will be included as part of a seam (as you will see in the next step).In this assignment, we shall use the dual-gradient energy function, which is described below.Computing the energy of a pixel. With the dual-gradient energy function, the energy of  pixel (x,y) is  given by the following:f17.png
    2. Seam identification.The next step is to find a vertical seam of minimum total energy. (Finding a horizontal seam is analogous.) This is similar to the classic shortest path problem in an edge-weighted digraph, but there are three important differences:
      • The weights are on the vertices instead of the edges.
      • The goal is to find the shortest path from any of the W pixels in the top row to any of the W pixels in the bottom row.
      • The digraph is acyclic, where there is a downward edge from pixel (xy) to pixels (x − 1, y + 1), (xy + 1), and (x + 1, y + 1). assuming that the coordinates are in the prescribed ranges.
      • Also, Seams cannot wrap around the image (e.g., a vertical seam cannot cross from the leftmost column of the image to the rightmost column).
      • As described in the paper, the optimal seam can be found using dynamic programming. The first step is to traverse the image from the second row to the last row and compute the cumulative minimum energy M for all possible connected seams for each pixel (i, j):f18
    3. Seam removal.The final step is remove from the image all of the pixels along the vertical or horizontal seam.

Results with a few images

The following image is the original 507×284 pixel input image taken from the same assignment. The next few images and animations show the outputs of the seam carving algorithm implementation. The shape (the transpose of the image shape is reported) and size of the image (in terms of the memory taken by python np array as float, to store the image) is also reported for each iteration of the algorithm. As can be seen, the algorithm resizes the image by removing  the minimum energy vertical (and horizontal seams) iteratively one at a time, by keeping the main objects as is.

Input Original Image

HJoceanSmall.png

Removing the Vertical Seams

energy_000HJoceanSmall
vseam

After Removing 200 Vertical Seams

seam_199_vHJoceanSmall

energy

Removing the Vertical and the Horizontal Seams in alternating manner

vhseam
After Removing 100 Vertical Seams and 100 Horizontal Seams

seam_099_hHJoceanSmall

seam_099_vHJoceanSmall

The following is the original 1024×576 image of the Dakshineswar Temple, Kolkata along with the removed vertical seams with content-aware resizing.

temple

energy__temple
vseamtemple (2).gif

Output image after removing 500 Vertical Seams

seam_499_vtemple
The next animation again shows how the content-aware resizing preserves the objects in the original image.

vseam_sr

energy_000vr

The next image is the original dolphin 239×200 image taken from the paper, along with the removed vertical seams with content-aware resizing.

dolphin

energy_000dolphin.jpg

vseamdolphin

           After removing 112 Vertical Seams

seam_113_vdolphin

The next animation shows how the seams are deleted from the image in the reverse order of deletion.

vseamidolphin

 

The next image is the original 750×498 bird image taken from the paper, along with the removed vertical seams with content-aware resizing.

bird2.png

energy_000bird2.png

vseambird
         After Removing 297 Vertical Seams

seam_299_vbird2

 

The next image is the original 450×299 sea image taken from the paper, along with the removed vertical seams with content-aware resizing.

sea2.png

vseamsea2

 

The next image is the original 628×413 cycle image taken from the paper, along with the removed vertical seams with content-aware resizing.

cycle.png

energy_000cycle

vseamcycle

After Removing 99 Vertical Seams

seam_199_vcycle

 

The next image is the original 697×706 Fuji image again taken from the paper, along with the removed vertical seams with content-aware resizing.

Fuji

 

vseamfuji

               After Removing 282 Vertical Seams

seam_280_vFuji

 

Object Removal

The same technique can be applied with mask to remove objects from an image. For example. consider the following image of the shoes taken from the same paper.

shoes

Let’s use a black mask to remove a shoe that we don’t want, as shown in the next figure.

masked_shoes.jpg
Finally the vertical seams can be forced to go through the masked object, as shown in the next animation,  in order to remove the masked object completely just by using context-aware resizing.

or_shoes.gif

Output after removing the shoe with content-aware image resize algorithm

seam_115_vshoes.jpg

Measuring Semantic Relatedness using the Distance and the Shortest Common Ancestor and Outcast Detection with Wordnet Digraph in Python

The following problem appeared as an assignment in the Algorithm Course (COS 226) at Princeton University taught by Prof. Sedgewick.  The description of the problem is taken from the assignment itself. However, in the assignment, the implementation is supposed to be in java, in this article a python implementation will be described instead. Instead of using nltk, this implementation is going to be from scratch.

 

The Problem

 

  • WordNet is a semantic lexicon for the English language that computational linguists and cognitive scientists use extensively. For example, WordNet was a key component in IBM’s Jeopardy-playing Watson computer system.
  • WordNet groups words into sets of synonyms called synsets. For example, { AND circuitAND gate } is a synset that represent a logical gate that fires only when all of its inputs fire.
  • WordNet also describes semantic relationships between synsets. One such relationship is the is-a relationship, which connects a hyponym (more specific synset) to a hypernym (more general synset). For example, the synset gatelogic gate } is a hypernym of { AND circuitAND gate } because an AND gate is a kind of logic gate.
  • The WordNet digraph. The first task is to build the WordNet digraph: each vertex v is an integer that represents a synset, and each directed edge v→w represents that w is a hypernym of v.
  • The WordNet digraph is a rooted DAG: it is acyclic and has one vertex—the root— that is an ancestor of every other vertex.
  • However, it is not necessarily a tree because a synset can have more than one hypernym. A small subgraph of the WordNet digraph appears below.

 

wordnet-event.png

 

The WordNet input file formats

 

The following two data files will be used to create the WordNet digraph. The files are in comma-separated values (CSV) format: each line contains a sequence of fields, separated by commas.

  • List of synsets: The file synsets.txt contains all noun synsets in WordNet, one per line. Line i of the file (counting from 0) contains the information for synset i.
    • The first field is the synset id, which is always the integer i;
    • the second field is the synonym set (or synset); and
    • the third field is its dictionary definition (or gloss), which is not relevant to this assignment.

      wordnet-synsets.png

      For example, line 36 means that the synset { AND_circuitAND_gate } has an id number of 36 and its gloss is a circuit in a computer that fires only when all of its inputs fire. The individual nouns that constitute a synset are separated by spaces. If a noun contains more than one word, the underscore character connects the words (and not the space character).
  • List of hypernyms: The file hypernyms.txt contains the hypernym relationships. Line i of the file (counting from 0) contains the hypernyms of synset i.
    • The first field is the synset id, which is always the integer i;
    • subsequent fields are the id numbers of the synset’s hypernyms.

      wordnet-hypernyms.png

      For example, line 36 means that synset 36 (AND_circuit AND_Gate) has 42338 (gate logic_gate) as its only hypernym. Line 34 means that synset 34 (AIDS acquired_immune_deficiency_syndrome) has two hypernyms: 47569 (immunodeficiency) and 56099 (infectious_disease).

 

 

The WordNet data type 

 

Implement an immutable data type WordNet with the following API:

wn.png

 

  • The Wordnet Digraph contains 76066 nodes and 84087 edges, it’s very difficult to visualize the entire graph at once, hence small subgraphs will be displayed as and when required relevant to the context of the examples later.

 

  • The sca() and the distance() between two nodes v and w are implemented using bfs (bread first search) starting from the two nodes separately and combining the distances computed.

 

Performance requirements 

  • The data type must use space linear in the input size (size of synsets and hypernyms files).
  • The constructor must take time linearithmic (or better) in the input size.
  • The method isNoun() must run in time logarithmic (or better) in the number of nouns.
  • The methods distance() and sca() must make exactly one call to the length() and ancestor() methods in ShortestCommonAncestor, respectively.


The Shortest Common Ancestor
 

 

  • An ancestral path between two vertices v and w in a rooted DAG is a directed path from v to a common ancestor x, together with a directed path from w to the same ancestor x.
  • shortest ancestral path is an ancestral path of minimum total length.
  • We refer to the common ancestor in a shortest ancestral path as a shortest common ancestor.
  • Note that a shortest common ancestor always exists because the root is an ancestor of every vertex. Note also that an ancestral path is a path, but not a directed path.

wordnet-sca.png

  • The following animation shows how the shortest common ancestor node 1 for the nodes 3 and 10  for the following rooted DAG is found at distance 4 with bfs, along with the ancestral path 3-1-5-9-10.sca2.gif
  • The following animation shows how the shortest common ancestor node 5 for the nodes 8 and 11  for the following rooted DAG is found at distance 3 with bfs, along with the ancestral path 8-5-9-11.sca1.gif
  • The following animation shows how the shortest common ancestor node 0 for the nodes 2 and for the following rooted DAG is found at distance 4 with bfs, along with the ancestral path 6-3-1-0-2.sca3.gif
      
  • We generalize the notion of shortest common ancestor to subsets of vertices. A shortest ancestral path of two subsets of vertices A and B is a shortest ancestral path over all pairs of vertices v and w, with v in A and w in B.
  • The figure (digraph25.txt) below shows an example in which, for two subsets, red and blue, we have computed several (but not all) ancestral paths, including the shortest one.wordnet-sca-set.png

     

  • The following animation shows how the shortest common ancestor node 3 for the set of nodes {13, 23, 24} and {6, 16, 17}  for the following rooted DAG is found at associated length (distance) with bfs, along with the ancestral path 13-7-3-9-16.sca4Shortest common ancestor data type

     

    Implement an immutable data type ShortestCommonAncestor with the following API:

    sca.png

 

Basic performance requirements 

The data type must use space proportional to E + V, where E and V are the number of edges and vertices in the digraph, respectively. All methods and the constructor must take time proportional to EV (or better).

 

Measuring the semantic relatedness of two nouns

Semantic relatedness refers to the degree to which two concepts are related. Measuring semantic relatedness is a challenging problem. For example, let’s consider George W. Bushand John F. Kennedy (two U.S. presidents) to be more closely related than George W. Bush and chimpanzee (two primates). It might not be clear whether George W. Bush and Eric Arthur Blair are more related than two arbitrary people. However, both George W. Bush and Eric Arthur Blair (a.k.a. George Orwell) are famous communicators and, therefore, closely related.

Let’s define the semantic relatedness of two WordNet nouns x and y as follows:

  • A = set of synsets in which x appears
  • B = set of synsets in which y appears
  • distance(x, y) = length of shortest ancestral path of subsets A and B
  • sca(x, y) = a shortest common ancestor of subsets A and B

This is the notion of distance that we need to use to implement the distance() and sca() methods in the WordNet data type.

wordnet-distance.png

 

Finding semantic relatedness for some example nouns with the shortest common ancestor and the distance method implemented

 

apple and potato (distance 5 in the Wordnet Digraph, as shown below)

dag_apple_potato.png

As can be seen, the noun entity is the root of the Wordnet DAG.

beer and diaper (distance 13 in the Wordnet Digraph)

dag_beer_diaper.png

 

beer and milk (distance 4 in the Wordnet Digraph, with SCA as drink synset), as expected since they are more semantically closer to each other.

dag_beer_milk.png

bread and butter (distance 3 in the Wordnet Digraph, as shown below)

dag_bread_butter.png

cancer and AIDS (distance 6 in the Wordnet Digraph, with SCA as disease as shown below, bfs computed distances and the target distance between the nouns are also shown)

dag_cancer_AIDS.png

 

car and vehicle (distance 2 in the Wordnet Digraph, with SCA as vehicle as shown below)

dag_car_vehicle.png
cat and dog (distance 4 in the Wordnet Digraph, with SCA as carnivore as shown below)

dag_cat_dog.png

cat and milk (distance 7 in the Wordnet Digraph, with SCA as substance as shown below, here cat is identified as Arabian tea)

dag_cat_milk.png

 

Einstein and Newton (distance 2 in the Wordnet Digraph, with SCA as physicist as shown below)

dag_Einstein_Newton

Leibnitz and Newton (distance 2 in the Wordnet Digraph, with SCA as mathematician)

dag_Leibnitz_Newton

Gandhi and Mandela (distance 2 in the Wordnet Digraph, with SCA as national_leader synset)dag_Gandhi_Mandela

laptop and internet (distance 11 in the Wordnet Digraph, with SCA as instrumentation synset)dag_laptop_internet

school and office (distance 5 in the Wordnet Digraph, with SCA as construction synset as shown below)

dag_school_office

bed and table (distance 3 in the Wordnet Digraph, with SCA as furniture synset as shown below)
dag_table_bed

Tagore and Einstein (distance 4 in the Wordnet Digraph, with SCA as intellectual synset as shown below)

dag_Tagore_Einstein

Tagore and Gandhi (distance 8 in the Wordnet Digraph, with SCA as person synset as shown below)

dag_Tagore_Gandhi

Tagore and Shelley (distance 2 in the Wordnet Digraph, with SCA as author as shown below)
dag_Tagore_Shelley

text and mining (distance 12 in the Wordnet Digraph, with SCA as abstraction synset as shown below)
dag_text_mining

milk and water (distance 3 in the Wordnet Digraph, with SCA as food, as shown below)dag_water_milk

Outcast detection

 

Given a list of WordNet nouns x1x2, …, xn, which noun is the least related to the others? To identify an outcast, compute the sum of the distances between each noun and every other one:

di   =   distance(xix1)   +   distance(xix2)   +   …   +   distance(xixn)

and return a noun xt for which dt is maximum. Note that distance(xixi) = 0, so it will not contribute to the sum.

Implement an immutable data type Outcast with the following API:

outc.png

 

Examples

As expected, potato is the outcast  in the list of the nouns shown below (a noun with maximum distance from the rest of the nouns, all of which except potato are fruits, but potato is not). It can be seen from the Wordnet Distance heatmap from the next plot, as well as the sum of distance plot from the plot following the next one.
outcast_apple_pear_peach_banana_lime_lemon_blueberry_strawberry_mango_watermelon_potato

dag_apple_bananadag_strawberry_bananadag_strawberry_blueberry

dag_apple_potatodag_strawberry_potato

Again, as expected, table is the outcast  in the list of the nouns shown below (a noun with maximum distance from the rest of the nouns, all of which except table are mammals, but table is not). It can be seen from the Wordnet Distance heatmap from the next plot, as well as the sum of distance plot from the plot following the next one.

outcast_horse_zebra_cat_bear_table

dag_cat_beardag_cat_tabledag_horse_zebra

Finally, as expected, bed is the outcast  in the list of the nouns shown below (a noun with maximum distance from the rest of the nouns, all of which except bed are drinks, but bed is not). It can be seen from the Wordnet Distance heatmap from the next plot, as well as the sum of distance plot from the plot following the next one.

outcast_water_soda_bed_orange_juice_milk_apple_juice_tea_coffee

dag_bed_tea

dag_orange_juice_tea

Some more Variational Image Processing: Diffusion, TV Denoising, TV Image Inpainting in Python

In this article, a few variational image processing techniques will  be described along with application of those techniques with some images, most of the problems are taken from the assignments from this course.

 

Some preliminaries: The Calculus of Images – Computing Curvature and TV

 

  • Let’s first compute the Euclidian Curvature of a few images.  The curvature measures the rate at which the unit gradient vector is changing and is given byf11.png
  • The following couple of images are used to compute the curvature. As can be seen from the below figures, the curvature is zero in flat regions and along straight edges and non-zero along the rounded edges of the circles, as expected.cur_testcur_source
  • Now, let’s compute the total variation (TV), which is given by the following.f12.png
  • First we need to approximate the partial derivatives using a forward difference.
  • Let’s compute the TV for the grayscale image Cameraman. Now let’s add more and more Salt & Pepper noise (by increasing the probability threshold p) to the image and see how the norm of the gradient matrix along with the TV value changes from the following figure.cam
    tv_Cameraman256.png

The Heat Equation and Diffusion

Let’s implement the isotropic and anisotropic diffusion by solving PDEs numerically.
The following figure shows the math.

f13.png

The following shows the isotropic diffusion output with Δt = 0.1, with gradient descent.  As can be seen, the results are same as applying gaussian blur kernel on the image.

iso_cam.gif

The following shows the anisotropic diffusion output with Δt = 0.1, with gradient descent, with a = 5, 20, 100 respectively.  As can be seen, unlike isotropic diffusion, the anisotropic diffusion preserves the edges.

aniso_cam_5

aniso_cam_20

aniso_cam_100

Creating Cartoon / flat-texture Images with Anisotropic Diffusion

As can be seen from the following figures and animations, we can create cartoons from the real images with anisotropic diffusion, by applying the diffusion on each channel, this time on color images (the barbara image and my image).

Original image

barbara

Cartoon Image with anisotropic diffusion (a=5)

aniso_barbara_020.0_5

aniso_bar_col_10

Original Image

me2

Cartoon Image with anisotropic diffusion (a=5)

aniso_me2_020.0_5

 

 

me_col_7.5

Total Variation Denoising

The following math is going to be used for TV denoising, the Euler-Lagrange equation is used to solve the minimum of the functional, as shown in the following figures with proof.

f14.pngf9

f10.png

  • First a noisy grayscale image is prepared by adding Gaussian noise to the cameraman image.

 

Original Cameraman
Cameraman256

Noisy Cameraman

noisy_cam

 

  • Let’s first denoise this image with linear TV denoising. The next animations show the results obtained , using the fidelity weight λ=1. As can be seen, even with the fidelity term, this model blurs the edges.tvld_cam.gif
  • Now let’s denoise this image with Nonlinear TV denoising. The next animations show the results obtained , using the fidelity weight λ=0.01 and λ=1 respectively.tvn_cam_0.01.gif
    tvn_cam_1

 

Image Inpainting

Inpainting is the process of restoring damaged or missing parts of an image. Suppose we have a binary mask D that specifies the location of the damaged pixels in the input image f as shown below:

f15.png

The following theory is going to be used for TV inpainting.

f16.png

Damaged image  

tampered_cmask1_Cameraman256

cam_in0

Damaged image  

tampered_cmask2_Cameraman256cam_in1

Damaged image  

tampered_text_Cameraman256cam_in2cam_in2

Damaged image
tampered_lena2

lena_in.gif

tv_inpaint_gd_tampered_lena2.png

Some Variational Image Processing: Poisson Image Editing and its applications in Python

Poisson Image Editing

The goal of Poisson image editing is to perform seamless blending of an object or a texture from a source image (captured by a mask image) to a target image. We want to create a photomontage by pasting an image region onto a new background using Poisson image editing. This idea is from the P´erez et al’s SIGGRAPH 2003 paper Poisson Image Editing.

The following figures describe the basic theory. As can be seen, the problem is first expressed in the continuous domain as a constrained variational optimization problem (Euler-Lagrange equation is used to find a solution) and then can be solved using a discrete Poisson solver.

f7.png

 

f8.png

 

As described in the paper and also in this assignment from this MIT course on Computational Photography, the main task of Poisson image editing is to solve a huge linear system Ax = b (where I is the new unknown image and S and T are the known images).

 

Seamless Cloning

The following images are taken from an assignment  from the same MIT course, where the Poisson image editing had to be used to blend the source inside the mask inside the target image. The next few figures show the result obtained.

Source Imagebear

Mask Imagemask

Target Imagewaterpool

Output Gray-Scale Image with Poisson Image Editing
pe_waterpool

The next animation shows the solution gray-scale images obtained at different iterations using Conjugate Gradient method when solving the linear system of equations.

peditmit1

Output Color Image with Poisson Image Editingpe_waterpool_color1

The next animation shows the solution color images obtained at different iterations using Conjugate Gradient method to solve the linear system of equations, applying the discrete Poisson solver on each channel.

pbear.gif

 

 

The following images are taken from this UNC course on computational photography. Again, the same technique is used to blend the octopus from the source image to the target image.

Source Imagesource

Mask Imagemask

Target Imagetarget

Output Imagepe_target_color

The next animation shows the solution color images obtained at different iterations using Conjugate Gradient method to solve the linear system of equations, applying the discrete Poisson solver on each channel.

poct.gif

 

Again, Poisson gradient domain seamless cloning was used to blend the penguins’ image  inside the following target image with appropriate mask.

Source Image
peng1

Target Image                                                                                                            trekking

Output Imagepe_trekking.jpg

The next animation again shows the solution color images obtained at different iterations using Conjugate Gradient method to solve the linear system of equations, applying the discrete Poisson solver on each channel.

pe-trekk

 

The next figures show how a source bird image is blended into the target cloud image with seamless cloning.

Source Image
bird1

Target Imagecloud

Output gray-scale imagepe_cloud

The next animation shows the solution gray-scale images obtained at the first few iterations using Conjugate Gradient method when solving the linear system of equations.

pedit2

 

Finally, the next figures show how the idol of the Goddess Durga is blended into the target image of the city of kolkata with seamless cloning. As can be seen, since the source mask had its own texture and there is a lots of variations in the background texture, the seamless cloning does not work that well.

Source Image
madurga

Target Image
kol

Output Image
pe_kol_color

The next animation shows the solution gray-scale images obtained at the first few iterations using Conjugate Gradient method when solving the linear system of equations.

pedit

The next animation again shows the solution color images obtained at different iterations using Conjugate Gradient method while solving the linear system of equations, applying the discrete Poisson solver on each channel.

pe-madurga

 

Feature Cloning: Inserting objects

The next figures are taken from the same paper, here the eyes, nose and lips from the  source face image is going to be inserted into the target monalisa face.

  Source image          Target image         Output image
face mona pe_mona

The next animation again shows the solution color images obtained at different iterations using Conjugate Gradient method while solving the linear system of equations, applying the discrete Poisson solver on each channel.

pe-mona

 

Texture Swapping: Feature exchange with Seamless Cloning

As discussed in the paper, seamless cloning allows the user to replace easily certain features of one object by alternative features. The next figure shows how the texture of the other fruit was transferred to the orange, the images being taken from the same paper.

Source Image        Target Image        Mask Image
sfruit   dfruit    fmask

Output Image
pe_dfruit

The next animation again shows the solution color images obtained at different iterations using Conjugate Gradient method while solving the linear system of equations, applying the discrete Poisson solver on each channel.

pfruit

 

Gradient Mixing: Inserting objects with holes

Again, the next figures are taken from the same paper, this time the source objects contain holes. As can be seen from the following results, the seamless cloning does not work well in this case for inserting the object with holes into the target, the target texture is lost inside the mask after blending.

Source Image                                            Target Image
srch            dsth

Output Image with Poisson Seamless Cloning
pe_tran1

The next animation again shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for seamless cloning, applying the discrete Poisson solver on each channel.

pe-tran1

Using the mixing gradients method the blending result obtained is far better, as shown below, it preserves the target texture.

Output Image with Poisson Mixing Gradients
pe_tran

The next animation again shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for mixing gradients, applying the discrete Poisson solver on each channel.

pe-tran

 

Mixing Gradients: Inserting transparent objects

The next example shows the insertion of a rainbow into a target image, the images are taken from the paper again. As can be seen, the seamless cloning wrongly places the rainbow in front of the coconut tree in the target image. Using gradient mixing, the stronger gradient is used as the image gradient and this solves the issue.

 

    Source Image                              Target Image                         Mask Imagerainbow    sky1      rmask

Output Image with Seamless Cloning
pe_sky
Output Image with Mixing Gradients
pe_rainbow

The next animation again shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for mixing gradients, applying the discrete Poisson solver on each channel.

pe-rainbow

 

The next few figures show the results obtained using mixing gradients on another set of images, the seamless cloning does not work well in this case, but mixing gradient works just fine.

 

Source Image
liberty1

Target Image
vic

Output Image with mixing gradients
pe_vic

The next animation again shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for mixing gradients, applying the discrete Poisson solver on each channel.

pe-vic

Texture Flattening: Creating Flat-texture Cartoon-like images

As illustrated in the paper, by retaining only the gradients at edge locations, before integrating with the Poisson solver, one washes out the texture of the selected region, giving its contents a flat aspect.

The following figures show the output cartoon-like image obtained using texture flattening, using the canny-edge detector to generate the  mask.

Source Image
pe_face1

Mask Image created with Canny edge detector
mface1

Output cartoon obtained with texture flattening from the source with the mask
pe_cart1

The next animation shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for texture flattening, applying the discrete Poisson solver on each channel.

pe-cart1

 

Again, the next figures show the output cartoon-like image obtained using texture flattening, using the canny-edge detector to generate the  mask on my image.

Source Image
me2

Output image obtained with texture flattening
pe-cart-me

The next animation again shows the solution color images obtained at the first few iterations using Conjugate Gradient method while solving the linear system of equations for texture flattening, applying the discrete Poisson solver on each channel.

pe-cart-me

 

Some more Social Network Analysis with Python: Centrality, PageRank/HITS, Random Network Generation Models, Link Prediction

In this article, some more social networking concepts will be illustrated with a few problems. The problems appeared in the programming assignments in the
coursera course Applied Social Network Analysis in Python.  The descriptions of the problems are taken from the assignments. The analysis is done using NetworkX.

The following theory is going to be used to solve the assignment problems.

 

f01.png

1. Measures of  Centrality

In this assignment, we explore measures of centrality on the following two networks:

  1. A friendship network
  2. A political blog network.

 

  • First let’s do some analysis with different centrality measures using the friendship network, which is a network of friendships at a university department. Each node corresponds to a person, and an edge indicates friendship. The following figure visualizes the network, with the size of the nodes proportional to the degree of the nodes.

 

f33.png

Degree Centrality

The next figure shows the distribution of the degree centrality of the nodes in the friendship network graph.

f34.png
The following figure visualizes the network, with the size of the nodes again
proportional to the degree centrality of the nodes.
f35.png

Closeness Centrality

The next figure shows the distribution of the closeness centrality of the nodes in the friendship network graph.

f36.png

 

The following figure again visualizes the network, with the size of the nodes being
proportional to the closeness centrality of the nodes.

f37.png

 

 

Betweenness Centrality

The next figure shows the distribution of the betweenness centrality of the nodes in the friendship network graph.

 

f38.png

 

The following figure again visualizes the network, with the size of the nodes being
proportional to the betweenness centrality of the nodes.

 

f39.png

 

  • Now, let’s consider another network, which is a directed network of political blogs, where nodes correspond to a blog and edges correspond to links between blogs.
  • This network will be used to compute  the following:
    • PageRank of the nodes using random walk with damping factor.
    • Authority and Hub Score of the nodes using the HITS.
  • The blog network looks like the following:
    source value
    tsrightdominion.blogspot.com Blogarama 1
    rightrainbow.com Blogarama 1
    gregpalast.com LabeledManually 0
    younglibs.com Blogarama 0
    blotts.org/polilog Blogarama 0
    marylandpolitics.blogspot.com BlogCatalog 1
    blogitics.com eTalkingHead 0
    thesakeofargument.com Blogarama 1
    joebrent.blogspot.com Blogarama 0
    thesiliconmind.blogspot.com Blogarama 0
    40ozblog.blogspot.com Blogarama,BlogCatalog 0
    progressivetrail.org/blog.shtml Blogarama 0
    randomjottings.net eTalkingHead 1
    sonsoftherepublic.com Blogarama 1
    rightvoices.com CampaignLine 1
    84rules.blog-city.com eTalkingHead 1
    blogs.salon.com/0002874 LeftyDirectory 0
    alvintostig.typepad.com eTalkingHead 0
    notgeniuses.com CampaignLine 0
    democratreport.blogspot.com BlogCatalog 0
  • First let’s visualize the network, next figure shows the visualization. The network has nodes (blogs) from 47 different unique sources, each node belonging to a source is colored with a unique color. The gray lines denote the edges (links) between the nodes (blogs).

f40.png

  • The next figure shows the same network graph, but without the node labels (blog urls).

f41.png

Page-Rank and HITS

  • Now, let’s apply the Scaled Page Rank Algorithm to this network.,with damping value 0.85. The following figure visualizes the graph with the node size proportional to the page rank of the node.f42.png
  • The next animations show how the page rank of the nodes in the network changes with the first 25 iterations of the power-iteration algorithm.

 

anipranipr1

  • The top 5 nodes with the highest page rank values after 25 iterations of the power-iteration page-rank algorithm are shown below, along with their ranks scores.
    • (u’dailykos.com’, 0.020416972184975967)
    • (u’atrios.blogspot.com’, 0.019232277371918939)
    • (u’instapundit.com’, 0.015715941717833914)
    • (u’talkingpointsmemo.com’, 0.0152621016868163)
    • (u’washingtonmonthly.com’, 0.013848910355057181)

 

  • The top 10 nodes with the highest page rank values after the convergence of the  page-rank algorithm are shown below, along with their ranks scores.
    • (u’dailykos.com’, 0.01790144388519838)
    • (u’atrios.blogspot.com’, 0.015178631721614688)
    • (u’instapundit.com’, 0.01262709066072975)
    • (u’blogsforbush.com’, 0.012508582138399093)
    • (u’talkingpointsmemo.com’, 0.012393033204751035)
    • (u’michellemalkin.com’, 0.010918873519905312)
    • (u’drudgereport.com’, 0.010712415519898428)
    • (u’washingtonmonthly.com’, 0.010512012551452737)
    • (u’powerlineblog.com’, 0.008939228332543162)
    • (u’andrewsullivan.com’, 0.00860773822610682)

 

  • The following figure shows the distribution of the page-ranks of the nodes after the convergence of the algorithm.f43.png

 

  • Next, let’s apply the HITS Algorithm to the network to find the hub and authority scores of node.
  • The following couple of figures visualizes the network graph where the node size is proportional to the hub score and the authority score for the node respectively, once the HITS algorithm converges.f44f45
  • Next couple of figures show the distribution of the hub-scores and the authority-scores for the nodes once the HITS converges.

 

  • f46f47

 

 

2. Random Graph Identification

 

Given a list containing 5 networkx graphs., with each of these graphs being generated by one of three possible algorithms:

  • Preferential Attachment ('PA')
  • Small World with low probability of rewiring ('SW_L')
  • Small World with high probability of rewiring ('SW_H')

Anaylze each of the 5 graphs and determine which of the three algorithms generated the graph.

The following figures visualize all the graphs along with their degree-distributions. Since the Preferential Attachment model generates graphs with the node degrees following  the power-law distribution (since rich gets richer), the graphs with this pattern in their degree distributions are most likely generated by this model.

On the other hand, the Small World model generates graphs with the node degrees not following the power-law distribution, instead the distribution shows fat tails. If this pattern is seen, we can identify the network as to be generated with this model.
f23f24f25f26f27f28f29f30f31f32

 

3. Prediction using Machine Learning models with the graph-centrality and local clustering features

 

For this assignment we need to work with a company’s email network where each node corresponds to a person at the company, and each edge indicates that at least one email has been sent between two people.

The network also contains the node attributes Department and ManagmentSalary.

Department indicates the department in the company which the person belongs to, and ManagmentSalary indicates whether that person is receiving a managment position salary. The email-network graph has

  • Number of nodes: 1005
  • Number of edges: 16706
  • Average degree: 33.2458

The following figure visualizes the email network graph.

f48.png

Salary Prediction

Using network G, identify the people in the network with missing values for the node attribute ManagementSalary and predict whether or not these individuals are receiving a managment position salary.

To accomplish this, we shall need to

  • Create a matrix of node features
  • Train a (sklearn) classifier on nodes that have ManagementSalary data and
  • Predict a probability of the node receiving a managment salary for nodes where ManagementSalary is missing.

The predictions will need to be given as the probability that the corresponding employee is receiving a managment position salary.

The evaluation metric for this assignment is the Area Under the ROC Curve (AUC).

A model needs to achieve an AUC of 0.75 on the test dataset.

Using the trained classifier, return a series with the data being the probability of receiving managment salary, and the index being the node id (from the test dataset).

The next table shows first few rows of the dataset with the degree and clustering features computed. The dataset contains a few ManagementSalary values are missing (NAN), the corresponding tuples form the test dataset, for which we need to predict the missing ManagementSalary values. The rest will be the training dataset.

Now,  a few classifiers will be trained on the training dataset , they are:

  • RandomForest
  • SVM
  • GradientBoosting

with 3 input features for each node:

  • Department
  • Clustering (local clustering coefficient for the node)
  • Degree

in order to predict the output variable as the indicator receiving  managment salary.

 

 Index Department ManagementSalary clustering degree
0 1 0.0 0.276423 44
1 1 NaN 0.265306 52
2 21 NaN 0.297803 95
3 21 1.0 0.384910 71
4 21 1.0 0.318691 96
5 25 NaN 0.107002 171
6 25 1.0 0.155183 115
7 14 0.0 0.287785 72
8 14 NaN 0.447059 37
9 14 0.0 0.425320 40

 

Typical 70-30 validation is used for model selection. The next 3 tables show the first few rows of the train, validation and the test datasets respectively.

 

Department clustering degree ManagementSalary
421 14 0.227755 52 0
972 15 0.000000 2 0
322 17 0.578462 28 0
431 37 0.426877 25 0
506 14 0.282514 63 0
634 21 0.000000 3 0
130 0 0.342857 37 0
140 17 0.394062 41 0
338 13 0.350820 63 0
117 6 0.274510 20 0
114 10 0.279137 142 1
869 7 0.733333 6 0
593 5 0.368177 42 0
925 10 0.794118 19 1
566 14 0.450216 22 0
572 4 0.391304 26 0
16 34 0.284709 74 0
58 21 0.294256 126 1
57 21 0.415385 67 1
207 4 0.505291 30 0

 

 Index Department clustering degree ManagementSalary
952 15 0.533333 8 0
859 32 0.388106 74 1
490 6 0.451220 43 0
98 16 0.525692 25 0
273 17 0.502463 31 0
784 28 0.000000 3 0
750 20 0.000000 1 0
328 8 0.432749 21 0
411 28 0.208364 106 1
908 5 0.566154 28 0
679 29 0.424837 20 0
905 1 0.821429 10 0
453 6 0.427419 34 1
956 14 0.485714 15 0
816 13 0.476190 23 0
127 6 0.341270 28 0
699 14 0.452899 26 0
711 21 0.000000 2 0
123 13 0.365419 36 0
243 19 0.334118 53 0
Department clustering degree
1 1 0.265306 52
2 21 0.297803 95
5 25 0.107002 171
8 14 0.447059 37
14 4 0.215784 80
18 1 0.301188 56
27 11 0.368852 63
30 11 0.402797 68
31 11 0.412234 50
34 11 0.637931 31

The following table shows the first few predicted probabilities  by  the RandomForest classifier on the test dataset.

 Index 0 1
0 1.0 0.0
1 0.2 0.8
2 0.0 1.0
3 1.0 0.0
4 0.5 0.5
5 1.0 0.0
6 0.7 0.3
7 0.7 0.3
8 0.3 0.7
9 0.7 0.3
10 0.9 0.1
11 0.8 0.2
12 1.0 0.0
13 0.6 0.4
14 0.7 0.3
15 0.5 0.5
16 0.0 1.0
17 0.2 0.8
18 1.0 0.0
19 1.0 0.0

 

The next figure shows the ROC curve to compare the performances (AUC) of the classifiers on the validation dataset.

As can be seen, the GradientBoosting  classifier performs the best (has the highest AUC on the validation dataset).

 

f49.png

 

The following figure shows the Decision Surface for the Salary Prediction learnt by the RandomForest Classifier.

f50.png

 

4. New Connections Prediction (Link Prediction with ML models)

For the last part of this assignment, we shall predict future connections between employees of the network. The future connections information has been loaded into the variable future_connections. The index is a tuple indicating a pair of nodes that currently do not have a connection, and the FutureConnectioncolumn indicates if an edge between those two nodes will exist in the future, where a value of 1.0 indicates a future connection. The next table shows first few rows of the dataset.

 Index Future Connection
(6, 840) 0.0
(4, 197) 0.0
(620, 979) 0.0
(519, 872) 0.0
(382, 423) 0.0
(97, 226) 1.0
(349, 905) 0.0
(429, 860) 0.0
(309, 989) 0.0
(468, 880) 0.0

 

Using network G and future_connections, identify the edges  in future_connections
with missing values and predict whether or not these edges will have a future connection.

To accomplish this, we shall need to

  1. Create a matrix of features for the edges found in future_connections
  2. Train a (sklearn) classifier on those edges in future_connections that have Future Connection data
  3. Predict a probability of the edge being a future connection for those edges in future_connections where Future Connection is missing.

The predictions will need to be given as the probability of the corresponding edge being a future connection.

The evaluation metric for this assignment is again the Area Under the ROC Curve (AUC).

Using the trained classifier, return a series with the data being the probability of the edge being a future connection, and the index being the edge as represented by a tuple of nodes.

Now,  a couple of classifiers will be trained on the training dataset , they are:

  • RandomForest
  • GradientBoosting

with 2 input features for each edge:

  • Preferential attachment
  • Common Neighbors

in order to predict the output variable Future Connection.

The next table shows first few rows of the dataset with the preferential attachment and Common Neighbors  features computed.

 Index Future Connection preferential attachment Common Neighbors
(6, 840) 0.0 2070 9
(4, 197) 0.0 3552 2
(620, 979) 0.0 28 0
(519, 872) 0.0 299 2
(382, 423) 0.0 205 0
(97, 226) 1.0 1575 4
(349, 905) 0.0 240 0
(429, 860) 0.0 816 0
(309, 989) 0.0 184 0
(468, 880) 0.0 672 1
(228, 715) 0.0 110 0
(397, 488) 0.0 418 0
(253, 570) 0.0 882 0
(435, 791) 0.0 96 1
(711, 737) 0.0 6 0
(263, 884) 0.0 192 0
(342, 473) 1.0 8140 12
(523, 941) 0.0 19 0
(157, 189) 0.0 6004 5
(542, 757) 0.0 90 0

 

Again, typical 70-30 validation is used for model selection. The next 3 tables show the first few rows of the train, validation and the test datasets respectively.

 

preferential attachment Common Neighbors Future Connection
(7, 793) 360 0 0
(171, 311) 1620 1 0
(548, 651) 684 2 0
(364, 988) 18 0 0
(217, 981) 648 0 0
(73, 398) 124 0 0
(284, 837) 132 1 0
(748, 771) 272 4 0
(79, 838) 88 0 0
(207, 716) 90 1 1
(270, 928) 15 0 0
(201, 762) 57 0 0
(593, 620) 168 1 0
(494, 533) 18212 40 1
(70, 995) 18 0 0
(867, 997) 12 0 0
(437, 752) 205 0 0
(442, 650) 28 0 0
(341, 900) 351 0 0
(471, 684) 28 0 0

 

 Index preferential attachment Common Neighbors Future Connection
(225, 382) 150 0 0
(219, 444) 594 0 0
(911, 999) 3 0 0
(363, 668) 57 0 0
(161, 612) 2408 4 0
(98, 519) 575 0 0
(59, 623) 636 0 0
(373, 408) 2576 6 0
(948, 981) 27 0 0
(690, 759) 44 0 0
(54, 618) 765 0 0
(149, 865) 330 0 0
(562, 1001) 320 1 1
(84, 195) 4884 10 1
(421, 766) 260 0 0
(599, 632) 70 0 0
(814, 893) 10 0 0
(386, 704) 24 0 0
(294, 709) 75 0 0
(164, 840) 864 3 0

 

preferential attachment Common Neighbors
(107, 348) 884 2
(542, 751) 126 0
(20, 426) 4440 10
(50, 989) 68 0
(942, 986) 6 0
(324, 857) 76 0
(13, 710) 3600 6
(19, 271) 5040 6
(319, 878) 48 0
(659, 707) 120 0

The next figure shows the ROC curve to compare the performances (AUC) of the classifiers on the validation dataset.

As can be seen, the GradientBoosting  classifier again performs the best (has the highest AUC on the validation dataset).

f51.png

 

The following figure shows the Decision Boundary for the Link Prediction learnt by the RandomForest Classifier.

f52.png

Some Social Network Analysis with Python

The following problems appeared in the programming assignments in the coursera course Applied Social Network Analysis in Python.  The descriptions of the problems are taken from the assignments. The analysis is done using NetworkX.

The following theory is going to be used to solve the assignment problems.

f0.png

1. Creating and Manipulating Graphs

  • Eight employees at a small company were asked to choose 3 movies that they would most enjoy watching for the upcoming company movie night. These choices are stored in a text file Employee_Movie_Choices , the following figure shows the content of the file.
     Index Employee Movie
    0 Andy Anaconda
    1 Andy Mean Girls
    2 Andy The Matrix
    3 Claude Anaconda
    4 Claude Monty Python and the Holy Grail
    5 Claude Snakes on a Plane
    6 Frida The Matrix
    7 Frida The Shawshank Redemption
    8 Frida The Social Network
    9 Georgia Anaconda
    10 Georgia Monty Python and the Holy Grail
    11 Georgia Snakes on a Plane
    12 Joan Forrest Gump
    13 Joan Kung Fu Panda
    14 Joan Mean Girls
    15 Lee Forrest Gump
    16 Lee Kung Fu Panda
    17 Lee Mean Girls
    18 Pablo The Dark Knight
    19 Pablo The Matrix
    20 Pablo The Shawshank Redemption
    21 Vincent The Godfather
    22 Vincent The Shawshank Redemption
    23 Vincent The Social Network
  • A second text file, Employee_Relationships, has data on the relationships between different coworkers.  The relationship score has value of -100 (Enemies) to +100 (Best Friends). A value of zero means the two employees haven’t interacted or are indifferent. The next figure shows the content of this file.

 

 Index Employee1 Employee2 Score
0 Andy Claude 0
1 Andy Frida 20
2 Andy Georgia -10
3 Andy Joan 30
4 Andy Lee -10
5 Andy Pablo -10
6 Andy Vincent 20
7 Claude Frida 0
8 Claude Georgia 90
9 Claude Joan 0
10 Claude Lee 0
11 Claude Pablo 10
12 Claude Vincent 0
13 Frida Georgia 0
14 Frida Joan 0
15 Frida Lee 0
16 Frida Pablo 50
17 Frida Vincent 60
18 Georgia Joan 0
19 Georgia Lee 10
20 Georgia Pablo 0
21 Georgia Vincent 0
22 Joan Lee 70
23 Joan Pablo 0
24 Joan Vincent 10
25 Lee Pablo 0
26 Lee Vincent 0
27 Pablo Vincent -20
0 Claude Andy 0
1 Frida Andy 20
2 Georgia Andy -10
3 Joan Andy 30
4 Lee Andy -10
5 Pablo Andy -10
6 Vincent Andy 20
7 Frida Claude 0
8 Georgia Claude 90
9 Joan Claude 0
10 Lee Claude 0
11 Pablo Claude 10
12 Vincent Claude 0
13 Georgia Frida 0
14 Joan Frida 0
15 Lee Frida 0
16 Pablo Frida 50
17 Vincent Frida 60
18 Joan Georgia 0
19 Lee Georgia 10
20 Pablo Georgia 0
21 Vincent Georgia 0
22 Lee Joan 70
23 Pablo Joan 0
24 Vincent Joan 10
25 Pablo Lee 0
26 Vincent Lee 0
27 Vincent Pablo -20
  • First, let’s load the bipartite graph from Employee_Movie_Choices file, the following figure visualizes the graph. The blue nodes represent the employees and the red nodes represent the movies.

f1.png

 

  • The following figure shows yet another visualization of the same graph, this time with a different layout.f2.png
  • Now, let’s find a weighted projection of the graph which tells us how many movies different pairs of employees have in common. We need to compute an L-bipartite projection for this, the projected graph is shown in the next figure.f3.png
  • The following figure shows the same projected graph with another layout and with weights. For example, Lee and Joan has the weight 3 for their connecting edges, since they share 3 movies in common as their movie-choices.f4.png

 

  • Next, let’s load the graph from Employee_Relationships  file, the following figure visualizes the graph. The nodes represent the employees and the edge colors and widths (weights) represent the relationships. The green edges denote friendship, the red edges enmity and blue edges neutral relations. Also, the thicker an edge is, the more powerful is a +ve or a -ve relation.f5.png
  • Suppose we like to find out if people that have a high relationship score also like the same types of movies.
  • Let’s find the Pearson correlation between employee relationship scores and the number of movies they have in common. If two employees have no movies in common it should be treated as a 0, not a missing value, and should be included in the correlation calculation.
  • The following data frame is created from the graphs and will be used to compute the correlation.
     Index Employee1 Employee2 Relationship Score Weight in Projected Graph
    0 Andy Claude 0 1.0
    1 Andy Frida 20 1.0
    2 Andy Georgia -10 1.0
    3 Andy Joan 30 1.0
    4 Andy Lee -10 1.0
    5 Andy Pablo -10 1.0
    6 Andy Vincent 20 0.0
    7 Claude Frida 0 0.0
    8 Claude Georgia 90 3.0
    9 Claude Joan 0 0.0
    10 Claude Lee 0 0.0
    11 Claude Pablo 10 0.0
    12 Claude Vincent 0 0.0
    13 Frida Georgia 0 0.0
    14 Frida Joan 0 0.0
    15 Frida Lee 0 0.0
    16 Frida Pablo 50 2.0
    17 Frida Vincent 60 2.0
    18 Georgia Joan 0 0.0
    19 Georgia Lee 10 0.0
    20 Georgia Pablo 0 0.0
    21 Georgia Vincent 0 0.0
    22 Joan Lee 70 3.0
    23 Joan Pablo 0 0.0
    24 Joan Vincent 10 0.0
    25 Lee Pablo 0 0.0
    26 Lee Vincent 0 0.0
    27 Pablo Vincent -20 1.0
    28 Claude Andy 0 1.0
    29 Frida Andy 20 1.0
    30 Georgia Andy -10 1.0
    31 Joan Andy 30 1.0
    32 Lee Andy -10 1.0
    33 Pablo Andy -10 1.0
    34 Vincent Andy 20 0.0
    35 Frida Claude 0 0.0
    36 Georgia Claude 90 3.0
    37 Joan Claude 0 0.0
    38 Lee Claude 0 0.0
    39 Pablo Claude 10 0.0
    40 Vincent Claude 0 0.0
    41 Georgia Frida 0 0.0
    42 Joan Frida 0 0.0
    43 Lee Frida 0 0.0
    44 Pablo Frida 50 2.0
    45 Vincent Frida 60 2.0
    46 Joan Georgia 0 0.0
    47 Lee Georgia 10 0.0
    48 Pablo Georgia 0 0.0
    49 Vincent Georgia 0 0.0
    50 Lee Joan 70 3.0
    51 Pablo Joan 0 0.0
    52 Vincent Joan 10 0.0
    53 Pablo Lee 0 0.0
    54 Vincent Lee 0 0.0
    55 Vincent Pablo -20 1.0
  • The correlation score is 0.788396222173 which is a pretty strong correlation. The following figure shows the association between the two variables with a fitted regression line.

    f6.png 

 

2. Network Connectivity

 

  • In this assignment we shall go through the process of importing and analyzing an internal email communication network between employees of a mid-sized manufacturing company.

    Each node represents an employee and each directed edge between two nodes represents an individual email. The left node represents the sender and the right node represents the recipient, as shown in the next figure.

    f7.png

  •  First let’s load the email-network as a directed multigraph and visualize the graph in the next figure. The graph contains 167 nodes (employees) and 82927 edges (emails sent). The size of a node in the figure is proportional to the out-degree of the node.f8.png

 

  • The next couple of figures visualize the same network with different layouts.f9.png

    f10.png

     

  • Assume that information in this company can only be exchanged through email.When an employee sends an email to another employee, a communication channel has been created, allowing the sender to provide information to the receiver, but not vice versa.Based on the emails sent in the data, is it possible for information to go from every employee to every other employee?

    This will only be possible if the graph is strongly connected, but it’s not. The following figure shows 42 strongly-connected components (SCC) of the graph. Each SCC is shown using a distinct color.

     
    f11.png

    As can be seen from the following histogram, only one SCC contains 126 nodes, each of the remaining 41 SCCs contains exactly one node.

    f12.png

  • Now assume that a communication channel established by an email allows information to be exchanged both ways.Based on the emails sent in the data, is it possible for information to go from every employee to every other employee?This is possible since the graph is weakly connected. 
  • The following figure shows the subgraph  induced by the largest SCC with 126 nodes.f13.png
  • The next figure shows the distribution of the (shortest-path) distances between the node-pairs in the largest SCC.f14.png

    As can be seen from above, inside the largest SCC, all  the nodes are reachable from one another with at most 3 hops, the average distance between any node pairs belonging to the SCC being 1.6461587301587302.

  • Diameter of the largest SCC: The largest possible distance between two employees, which is 3.
  • Find the set of nodes in the subgraph with eccentricity equal to the diameter: these are exactly the nodes that are on the periphery. As can be seen from the next figure (the largest component is shown along with few other nodes from some other components in the graph, all the nodes and edges in the graph are not shown to avoid over-crowding), there are exactly 3 nodes on the periphery of the SCC, namely the node 97, 129 and 134.Each of the following 3 shortest paths shown in the next figure
    • 97->14->113->133
    • 129->1->38->132 and
    • 134->1->38->132

    has length equal to the diameter of this SCC.

    f15.png

 

  • Center of the largest SCC: The set of node(s) in the subgraph induced by the largest SCC with eccentricity equal to the radius (which is 1). There is exactly one such node (namely, node 38), as shown in the next figure, all the nodes belonging to the largest SCC are distance-1 reachable from the center node 38 (again, the largest component is shown along with few other nodes from some other components in the graph, all the nodes and edges in the graph are not shown to avoid over-crowding).f16.png
  • The following figure shows the distribution of eccentricity in the largest SCC.f17.png
  • Which node in the largest SCC has the most shortest paths to other nodes whose distance equal the diameter ?  How many of these paths are there?As can be seen from the following figure, the desired node is 97 and there are 63 such shortest paths that have length equal to the diameter of the SCC,  5 of such paths (each with length 3) are shown in the next figure, they are:
    • 97->14->113->133
    • 97->14->113->130
    • 97->14->113->136
    • 97->14->45->131
    • 97->14->45->132
      .f18.png
  • Suppose we want to prevent communication from flowing to the node 97 (the node that has the most shortest paths to other nodes whose distance equal the diameter), from any node in the center of the largest component, what is the smallest number of nodes we would need to remove from the graph (we’re not allowed to remove the node 97 or the center node(s))?

    As obvious, the minimum number of nodes required to be removed exactly equals to the size of the min-cut with the source node (center node 38) to the target node (node 97), shown in red in the next figure. The size of the min-cut is 5 and hence 5 nodes (shown in pale-golden-red color) need to be removed, the corresponding node numbers are: 14, 32, 37, 45 and 46. As done in the earlier figures, all the nodes and edges in the email-net graph are not shown to avoid over-crowding.

    emails_comp5.png

    The min-cut is separately shown in the following figure from source node 38 to target node 97.

    emails_comp4

  • Construct an undirected graph from the subgraph induced by the largest component on the email-net directed multi-graph. 

    The next figure shows the undirected graph constructed. As before, the node size is proportional to the degree of the node.f19.png

  • What is the transitivity and average clustering coefficient of the undirected graph?The transitivity and average clustering coefficient of the undirected graph are 0.570111160700385 and 0.6975272437231418 respectively.

    The following figure shows the distribution of the local clustering coefficients.

    f20.png

    The following figure shows the undirected graph, this time the node size being proportional to the local clustering coefficient of the node.

    f21.png

    The next figure shows the degree distribution of the undirected graph.

    f22.png

    Since there are more nodes with lower degrees than with higher degrees and the transitivity weights the nodes with higher degree more, the undirected graph has  lower transitivity and higher average clustering coefficient.

Implementing kd-tree for fast range-search, nearest-neighbor search and k-nearest-neighbor search algorithms in 2D (with applications in simulating the flocking boids: modeling the motion of a flock of birds and in learning a kNN classifier: a supervised ML model for binary classification) in Java and python

The following problem appeared as an assignment in the coursera course Algorithms-I by Prof. Robert Sedgewick from the Princeton University few years back (and also in the course cos226 offered at Princeton). The problem definition and the description is taken from the course website and lectures.  The original assignment was to be done in java, where in this article both the java and a corresponding python implementation will also be described.

  • Use a 2d-tree to support
    • efficient range search (find all of the points contained in a query rectangle)
    • nearest-neighbor search (find a closest point to a query point).

    2d-trees have numerous applications, ranging from classifying astronomical objects to computer animation to speeding up neural networks to mining data to image retrieval. The figure below describes the problem:

    kdtree-ops.png

    2d-tree implementation: A 2d-tree is a generalization of a BST to two-dimensional keys. The idea is to build a BST with points in the nodes, using the x– and y-coordinates of the points as keys in strictly alternating sequence, starting with the x-coordinates, as shown in the next figure.

    kdtree3

    • Search and insert. The algorithms for search and insert are similar to those for BSTs, but at the root we use the x-coordinate (if the point to be inserted has a smaller x-coordinate than the point at the root, go left; otherwise go right); then at the next level, we use the y-coordinate (if the point to be inserted has a smaller y-coordinate than the point in the node, go left; otherwise go right); then at the next level the x-coordinate, and so forth.
    • The prime advantage of a 2d-tree over a BST is that it supports efficient implementation of range search and nearest-neighbor search. Each node corresponds to an axis-aligned rectangle, which encloses all of the points in its subtree. The root corresponds to the entire plane [(−∞, −∞), (+∞, +∞ )]; the left and right children of the root correspond to the two rectangles split by the x-coordinate of the point at the root; and so forth.
      • Range search: To find all points contained in a given query rectangle, start at the root and recursively search for points in both subtrees using the following pruning rule: if the query rectangle does not intersect the rectangle corresponding to a node, there is no need to explore that node (or its subtrees). That is, search a subtree only if it might contain a point contained in the query rectangle.
      • Nearest-neighbor search: To find a closest point to a given query point, start at the root and recursively search in both subtrees using the following pruning rule: if the closest point discovered so far is closer than the distance between the query point and the rectangle corresponding to a node, there is no need to explore that node (or its subtrees). That is, search a node only if it might contain a point that is closer than the best one found so far. The effectiveness of the pruning rule depends on quickly finding a nearby point. To do this, organize the recursive method so that when there are two possible subtrees to go down, you choose first the subtree that is on the same side of the splitting line as the query point; the closest point found while exploring the first subtree may enable pruning of the second subtree.
      • k-nearest neighbors search: This method returns the k points that are closest to the query point (in any order); return all n points in the data structure if n ≤ k. It must do this in an efficient manner, i.e. using the technique from kd-tree nearest neighbor search, not from brute force.
      • BoidSimulator: Once the  k-nearest neighbors search we can simulate boids: how a flock of birds flies together and a hawk predates. Behold their flocking majesty.The following figures show the theory that are going to be used, taken from the lecture slides of the same course.

    f1f2f3f4

    f5

    Results

    The following figures and animations show how the 2-d-tree is grown with recursive space-partioning for a few sample datasets.

  • Circle 10 dataset

circle10

test1test2test3test4test6test7test8test9

  • Circle 100 dataset

circle100.gif

test1test6test16test32test64test72test99
The following figure shows the result of the range search algorithm on the same dataset after the 2d-tree is grown. The yellow points are the points found by the algorithm inside the query rectangle shown.

test-range

The next animations show the nearest neighbor search algorithm for a given query point (the fixed white point with black border: the point (0.3, 0.9)) and how the the branches are traversed and the points (nodes) are visited in the 2-d-tree until the nearest neighbor is found.

nn.gif

nearest1.gif

The next animation shows how the kd-tree is traversed for nearest-neighbor search for a different query point (0.04, 0.7).

nearest2

The next figures show the result of k-nearest-neighbor search, by extending the previous algorithm with different values of k (15, 10, 5 respectively).

test_knn2test_knn3test_knn4

 

Runtime of the algorithms with a few datasets in Python

As can be seen from the next figure, the time complexity of 2-d tree building (insertion), nearest neighbor search and k-nearest neighbor query depend  not only on the size of the datasets but also on the geometry of the datasets .

runtime.png

 

Flocking Boids simulator

Finally the flocking boids simulator is implemented with 2-d-trees and the following 2 animations (java and python respectively) shows how the flock of birds fly together, the black / white ones are the boids and the red one is the predator hawk.

birds

birds1

 

Implementing a kNN Classifier with kd tree from scratch

Training phase

Build a 2d-tree from a labeled 2D training dataset (points marked with red or blue represent 2 different class labels).

Testing phase

  • For a query point (new test point with unknown class label) run k-nearest neighbor search on the 2d-tree with the query point (for a fixed value of k, e.g., 3).
  • Take a majority vote on the class labels of the k-nearest neighbors (with known class labels) obtained by querying the 2d-tree. Label the query point with the class label that majority of its neighbors have.
  • Repeat for different values of k.

 

The following figures show how the kd tree built can be used to classify (randomly generated) 2D datasets and the decision boundaries are learnt with k=3, 5 and 10 respectively.

test_classification3test_classification5test_classification10