text:
Q: Python lists and and loops I need to print out the number of a's, b's, c's etc. in the following statement and have it cascade down and have the number of times it appeared in the statement next to it.
"you can have data without information, but you cannot have information without data."
I have got so far and could do with some help?
This is day 3 of me learning python so take it easy.
from collections import defaultdict

text = 'You can have data without information, but you cannot have information without data.'
lowerCaseText = text.lower()

print(lowerCaseText)
chars = defaultdict(int)

for char in text:
chars[char] += 1

print(chars)

It needs to look like this
a : 10
b : 1
c : 2
d : 2
e : 2
f : 2
h : 4
i : 6
m : 2
n : 7
o : 9
r : 2
t : 10
u : 5
v : 2
w : 2
y : 2

A: If you're doing a homework assignment, then this isn't going to help - but if you're looking for a quick way to achieve your goal, then the Counter function from the collections module should be of interest:
text = 'You can have data without information, but you cannot have information without data.'
from collections import Counter
Counter(text.lower())

>> Counter({ "y" : 2,
             "o" : 9,
             "u" : 5,
             ...etc

A: I think you asked for occurences in a string ,first example is occurences of the
words in a list,and the other example is occurences of letters in a string using dictionary but you can also use Counter there!
First Example:
   from collections import Counter

   text = 'You can have data without information, but you cannot have information without data.'
   text = text.split()
   text = Counter(text)
   print(text)

Second Example:
    text = 'You can have data without information, but you cannot have information
without data.'
    text = text.lower()
    text_dictionary = {}
    for element in text:
        if element not in text_dictionary and element.isalpha():
            text_dictionary[element] = 1
        else:
            if element.isalpha():
                text_dictionary[element] += 1
    print(text_dictionary)

A: list(set(lowerCaseText) will return all of the unique characters in your string.
You can then go through those unique characters and get a count of their occurrences in your initial string by using lowerCaseText.count(char)
Put it all together and you get something like this:
text = 'You can have data without information, but you cannot have information without data.'
lowerCaseText = text.lower()
uniq = sort(list(set(lowerCaseText)))
for char in uniq:
    print(char, ':', lowerCaseText.count(char))

A: Code:
import re      #This package to remove space and special char from text
import string  #This package to alphabets

text = 'You can have data without information, but you cannot have information without data.'
Alpha = dict.fromkeys(string.ascii_lowercase, 0)   #Output {a:0,b:0 ..}

for t in re.sub('[^A-Za-z0-9]+', '', text):        #using re avoiding the spaces and special characters
    Alpha[t.lower()] = Alpha[t.lower()]+1          #each time add 1 on specific alphabet

Alpha

Output:
{'a': 10,
 'b': 1,
 'c': 2,
 'd': 2,
 'e': 2,
 'f': 2,
 'g': 0,
 'h': 4,
 'i': 6,
 'j': 0,
 'k': 0,
 'l': 0,
 'm': 2,
 'n': 7,
 'o': 9,
 'p': 0,
 'q': 0,
 'r': 2,
 's': 0,
 't': 10,
 'u': 5,
 'v': 2,
 'w': 2,
 'x': 0,
 'y': 2,
 'z': 0}

To get your desire output {k:v for k,v in Alpha.items() if v>0}
output:
{'a': 10, 'b': 1, 'c': 2, 'd': 2, 'e': 2, 'f': 2, 'h': 4, 'i': 6, 'm': 2, 'n': 7, 'o': 9, 'r': 2, 't': 10, 'u': 5, 'v': 2, 'w': 2, 'y': 2}

A: Here's a different and rough idea for a different way to do things:
So what you could do is have an array of 26 values, 1 for each letter of the alphabet. And then a 2nd array with 26 values, all starting with 0.
So then you can loop through the 1st array, figure out how many times that value is found in the text, and then set that value in the 2nd array.
a = ['a', 'b', to 'z']
b = [0,0,0,…0]
for each letter in the alphabet:
B[character value] = text.count(a[character value])
Then you could do a for loop to get things printed the way you want.

A: You can use collections.Counter. I also used collections.OrderedDict to sort values.
from collections import Counter, OrderedDict

text = 'You can have data without information, but you cannot have information without data.'
lower_case_text = text.lower()
lower_case_text = ''.join(filter(str.isalnum, lower_case_text)) # this line is required for removing non alpha numeric characters.
counter = Counter(lower_case_text)
counter_ordered=OrderedDict(counter.most_common())
for k, v in counter_ordered.items():
    print(k + " : " + str(v))

A: first you need to filter out everything that is not a letter from your string. You also need to preferably sort your string before you convert it into a dict so that the values get added in the right order. It's also possible to sort the dict afterwards but that's more difficult in this case.
We do this using this line of code:
filteredText = sorted(''.join(filter(str.isalnum, text)).lower())
(str.isalnum, text) removes all non letters but it comes out as a list so we need to join it later again.
then when we need to go over the text and check if it is already in our dictionary. If not we need to add it with value 1, otherwise we increment with 1.
Full code:
text = 'You can have data without information, but you cannot have information without data.'
filteredText = sorted(''.join(filter(str.isalnum, text)).lower())

chars = {}

for char in filteredText:
  if char in chars:
      chars[char] += 1
  else :
      chars[char] = 1

print(chars)

A: Nice problem here is solution for your defaultdict
Please let me know where are you learning python problems!
from collections import defaultdict
text = 'You can have data without information, but you cannot have information without data.'

lowerCaseText = text.lower()
chars = defaultdict(int)

for char in lowerCaseText:
   if char.isalpha():      # Only taking alphabet removing . , space
      chars[char] += 1

for i, j in sorted(chars.items()):
   print(i, ":", j)

A: We can take advantage that a character is also a number: subtracting 97 from the ascii code gives us an number between 0 and 25 that we can use as the index of a list:
text = 'You can have data without information, but you cannot have information without data'.lower()
lst  = [0] * 26

#counts iterations
for char in text:
    if not char.isalpha(): continue
    lst[ord(char) - 97] += 1

#prints the result:
for i in range(26):
    if lst[i] > 0:
        string = chr(i + 97) + " : " + str(lst[i])
        print(string)

#=>
a : 10
b : 1
c : 2
d : 2
e : 2
f : 2
h : 4
i : 6
m : 2
n : 7
o : 9
r : 2
t : 10
u : 5
v : 2
w : 2
y : 2

A: This 3 workarounds uses ascii_lowercase from string module.
First, print while iterating:
from string import ascii_lowercase
text = "You can have data without information, but you cannot have information without data."

for char in ascii_lowercase:
    count = 0
    for letter in text.lower():
        if letter == char:
            count += 1
    if count:
        print(f'{char} : {count}')

Second, using dictionary to update the counted value using dict.update():
from string import ascii_lowercase
text = "You can have data without information, but you cannot have information without data."

count = dict()
for char in ascii_lowercase:
    if char in text:
        count.update({char: text.lower().count(char)})
for char in count:
    print(f'{char} : {count[char]}')

Third, using dict comprehension:
from string import ascii_lowercase
text = "You can have data without information, but you cannot have information without data."
count = {char: text.lower().count(char) for char in ascii_lowercase if char in text}
for char in count: print(f'{char} : {count[char]}')

Results:
a : 10
b : 1
c : 2
d : 2
e : 2
f : 2
h : 4
i : 6
m : 2
n : 7
o : 9
r : 2
t : 10
u : 5
v : 2
w : 2
y : 2


I urge you to make appointments at a local school to spend several hours observing three classrooms. That way you'll learn first hand about daily activities in these classrooms. I'm sure the teachers you observe will be happy to answer your questions and share their time-management tips.
I can not this week. Schools are out for spring break and so are the teachers. That is why I am asking for your help.
OK. I'll write up my daily routine as a middle school social studies teacher. I'll post it within a half hour.
Repeat above for each of the other 5 periods.
Sometimes we spent all period doing hands-on projects. Especially valuable were the seventh-grade draw-a-map projects. Students were presented with blank papers with latitude and longitude lines. They then copied a prepared map on their grid sheets and finished by labeling and coloring.
The major eighth-grade projects were illustrated timelines. This was a year-long project on paper about 8" by 110".
* Homework assignments were usually several questions from the book, answered in complete sentences. To check their answers, I called on students at random to read their answers for which they got credit if their answers were correct.
* At the beginning of each unit (4 to 6 weeks), I gave each student a day-by-day syllabus which outlined each day's lesson and the homework assignment.
* The English teacher and I worked together in assigning research papers that counted for both classes.
In Michigan, a person can teach 7th and 8th grade with either an elementary or secondary certificate.
I don't know of any elementary teachers who frequent this board.
a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg/day for males. Express this quantity in ?g/s. Express this quantity in b) For adults, the RDA of the amino acid lysine is 12 mg per kg of body weight.
If a building contractor hires 5 day laborers and 2 concrete finishers, his payroll for the day is $1200. If he hire 1 day laborer and 4 concrete finishers, his daily cost is $924. Find the daily wage for each type of worker.


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text:
Perspective //
Like all grand narratives, the one of language is one of falling empires, warring monarchs and conquest.
by William Tandany
Nested in the trend of self-productivity, language apps have achieved significant popularity. Amongst the most popular are Duolingo and Babbel, whose gamified interface has become a trademark of, not only online language apps, but skill learning apps in general.
If you've started using a language learning app, pause for a moment to question how that language came to be on your device. As an introductory tool, breaking down complex linguistic systems into bite-size grammatical rules and vocabulary is invaluable to beginners. But normalising this low level commitment and endorsing a schematic pedagogy of language education risks overlooking the socio-cultural dimensions of language and its ever-evolving processes that make it more than just a mode of communication.
Every day new words are created, innovations in grammar finds its way into common usage and old words are given new meanings. This happens idiosyncratically all around the world, across all languages. How then, might one explain the harmonisation of hundreds upon thousands of distinctive tongues and dialects into a nationalised and standardised language coherent enough to be experienced through a freemium app?
Competing perspectives on this question centres upon history. Like all grand narratives, the one of language is one of falling empires, warring monarchs and conquest.
Consider for example, that today the Institute of Cervantes estimates around 477 million native Spanish speakers worldwide, and unsurprisingly, it is the most learnt language on Duolingo. Yet the naked statistic obscures the diversity of Spanish speakers and 'variants' of Spanish spoken globally. For instance, many non-Castilian Spaniards, even those mainly residing outside the nation's capital Madrid, consider Spanish a second language to the regional dialect which includes Valencian, Andalusian or Aragonese. None of these are available on Duolingo.
These dialects are distinct from Spanish, often with Latin-Romance roots as the only common ground. Their coexistence and autonomy in today's era defined by the constitution of the centralised nation state, is a situation some 500 years in the making, dating back to the standardisation of Castillian-Spanish in 1492 under Queen Isabella. After centuries of continental war during the crusades, exchanging the peninsula between caliphate and monarch and the fall of the rival Nasrid kingdom of Granada, Queen Isabella well understood the role of language in establishing hegemony. Her gambit in standardising national Spanish proved fortuitous, arming Columbus shortly after with an indispensable tool in the administration of colonialism and provision of genocide in Latin America.
Regional languages in Spain survived, in part from political compromises and in part clandestinely as private resistance to the violence of state-craft. The Catalunyan language today serves as a fulcrum for the Catalonian independence movement, combining diverse groups from Barcelonian urbanites to veteran anti-Franco Anarchists against the overweening conservative Spanish government, Partido Popular. Indeed Catalunya and much of peripheral Spain are still haunted by the memory of Franco's dictatorship, whose fascistic nationalism and sycophantic Catholicism repressed Spanish cultural and linguistic diversity.
The twentieth century left a legacy of murderous despots who invigorated nationalistic vitriol with the institution of language. More than just a tool for the mediation of communication or culture, language has proven to be a mode of power in itself.
Today its political implications are yet again palpable. Regional and traditional identities, along with their languages, are wielded by people marginalised by globalisation to make sense of an incessantly changing word.

redpajama_set_name: RedPajamaCommonCrawl


| ismayil — 03.02.2015 at 22:14:53 Provider may prescribe testosterone to men for 10 seconds, then that we know violate the rules. Peace of thoughts.
| FULL_GIRL — 03.02.2015 at 10:58:23 Many related points, especially banned acquired - research not financially supported by Novo Nordisk and is unbiased of any.


text:
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After weeks of mystery and intrigue, The Sun has tracked down Baltimore's "dancing girl."
A mysterious dancing woman has captivated Baltimore for weeks.
Prancing through the streets, jamming out to various radio stations on her iPod, Betsy Mills captured the attention of the city recently when her moves brought a smile to the faces of passersby in Mount Vernon, Canton and the Inner Harbor.
Little did Mills know a YouTube video of her dancing in Canton (unintentionally in sync with the video's music) had garnered thousands of views, and residents were posting videos, tweets and impassioned pleas to local neighborhood Facebook groups to find the "#baltimoredancinggirl." Mills had no idea anyone had taken notice until an acquaintance passed along the Canton video, posted on Aug. 14.
"It was very surprising," Mills, 30, said. "I know sometimes people take videos on their phone, but I just assume that they look at them themselves."
Mills lived in Baltimore for eight years and used to dance when she went on runs. A few years ago, however, she injured herself and could no longer run.
"Recently, I realized that I could just walk and dance and it was just as fun," she said.
Reporter Justin Fenton records video of Betsy Mills, formerly of Mount Vernon, as she dances by The Baltimore Sun building on July 31.
Although the most popular documented instance of Mills' moves took place in Canton, spending time in the neighborhood was a rarity -- Mills lived in Mount Vernon and primarily pranced down Calvert Street and around the Inner Harbor, she said.
Perhaps the reason Baltimoreans' interest ramped up in recent weeks is because Mills was out and about more often than usual; she moved to Michigan for a new job over the weekend, and dancing was her farewell to Charm City. Mills said she hopes to keep dancing in her new home, though she'll have to gauge the community reaction.
"It just makes me really happy and it makes other people happy," she said. "It seemed like a really good way to say goodbye to the city."
Though combing through Facebook comments reveals the "dancing girl's" fans have had a hard time tracking her down, Mills said she did have frequent participants in her impromptu dance parties.
"Lots of times people would join in," she said. "They often asked what I was listening to. No one ever really asked why."


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text:
Feudale Email Exposes Beemer, Others
AG First Deputy, Bruce Beemer's claim that Kane's conduct in releasing private emails was unethical seemingly pales in comparison to other potential misconduct in the case of the PSU 3.
In a memo to top attorneys in the Pennsylvania Attorney General's Office (OAG), Bruce Beemer alleged that AG Kathleen Kane had conducted herself unethically for producing the so-called "private emails" of former (bumbling) grand jury judge, Barry Feudale.
In what should come as no surprise to anyone who has closely followed this case, Beemer has his own ethical issues regarding his handling of evidence in the cases of the PSU 3 (i.e., Timothy Curley, Gary Schultz, and Graham Spanier).
And what also should come as no surprise, the evidence involves emails.
Spanier Discovery Request
On September 9, 2014, counsel for Spanier joined a motion by the attorneys for Curley and Schultz requesting discovery of emails between Feudale and Fina regarding his removal as grand jury judge AND any correspondence about the Conspiracy of Silence case. See below.
On September 9, 2014, the Commonwealth responded and clarified that First Deputy Attorney Bruce Beemer was the "lead prosecutor" since the filing of charges on November 1st, 2012. Prior to the filing of charges, Senior Deputy Attorney General Jonelle Eshbach was the lead prosecutor on the case and was supervised by Chief Deputy Attorney General Frank G. Fina.
The Spanier filing also revealed the Beemer was instrumental in directing the searches for emails and in responding to requests of the status of the discovery.
3/545 = Playing "hide the ball"
Bruce Beemer: Kept
playing"hide the ball?"
The Commonwealth's response stated it found 545 emails related to the search terms provided by the defendants and that the majority of the emails dealt with case logistics. Only three dealt with matters that were material to the case.
The "material" emails were from December 16 and 17, 2011 and referenced expanded Information Technology (IT) evidence collection to be conducted by the PSU IT staff for hard drives, portable devices, and copies of emails from persons of interest.
When the Spanier defense team stated it "strained credulity" that only three emails were found, the OAG indignantly responded that the case prosecutors had been "forthright, cooperative, and direct" and that it "strained credulity" for them to be accused of playing "hide the ball."
As reported in early 2014, substantial evidence in the case showed that the prosecutors had been playing "hide the ball" all along.
The Feudale emails simply added to the existing mountain of evidence that has been suppressed in the case of the PSU 3.
Feudale Email
The July 18, 2013 email from Judge Barry Feudale to the Inky's Craig R. McCoy revealed that his email to Frank Fina regarding his removal as judge contained grand jury information. Given that the email included various rants about AG Kathleen Kane over his removal -- and that Kane had stirred the hornet's nest with the Mouton investigation - there is little doubt it referenced the Sandusky and/or Conspiracy of Silence grand jury. See below.
AG Kane told reporters that Feudale mistakenly sent emails to Fina at his old email address in the OAG's office and those emails resided on OAG servers. Feudale's emails were discovered after his removal as a grand jury judge in July of 2013 -- uncovered during the investigation of the Sandusky investigation, conducted by Special Deputy AG, Geoffrey Moulton. The Moulton Report revealed that emails from the OAG accounts were recovered beginning in November 2013 and continuing into March 2014.
In summary, the OAG's attorneys, including Beemer, Laura Ditka, and James Barker deceived the legal teams of Curley, Schultz, and Spanier regarding the existence of email evidence.
It is extremely rich that Bruce Beemer wrote a memo accusing AG Kathleen Kane of unethical conduct regarding the release of Feudale's so-called "private emails," given his possible misconduct in the PSU 3 case.
Duke Redeaux
Former Durham County, North Carolina, District Attorney Mike Nifong was disbarred for similar misrepresentations to opposing counsel and the court in the Duke Lacrosse case.
94. On December 13, 2006, the Duke Defendants filed a Motion to Compel Discovery: Expert DNA Analysis, detailing their discovery of the existence of DNA from multiple unidentified males on the rape kit items and explaining that this evidence had not been included DSI's written report. The motion did not allege any attempt or agreement to conceal the potentially exculpatory DNA evidence or test results. The Motion to Compel Discovery: Expert DNA Analysis was addressed by the Honorable Osmond W. Smith III, Superior Court Judge presiding, at a hearing on December 15, 2006.
95. At the December 15 hearing, both in chambers and again in open court, Nifong stated or implied to Judge Smith that he was unaware of the existence of DNA from multiple unidentified males on the rape kit items until he received the December 13 motion and/or was unaware that the results of any DNA testing performed by DSI had been excluded from DSI's written report. Nifong stated to Judge Smith in open court: "The first I heard of this particular situation was when I was served with these reports -- this motion on Wednesday of this week."
96. Nifong's representations that he was unaware of the existence of DNA from multiple unidentified males on the rape kit items and/or that he was unaware of the exclusion of such evidence from DSI's written report, were intentional misrepresentations and intentional false statements of material fact to the Court and to opposing counsel.
Violations of the Code of Ethics
(c) By not providing to the Duke Defendants prior to November 16, 2006, a complete report setting forth the results of all tests and examinations conducted by DSI, including the existence of DNA from multiple unidentified males on the rape kit items and including written or recorded memorializations of Dr. Meehan's oral statements, Nifong:
ii. failed to make a reasonably diligent effort to comply with a legally proper discovery request, in violation of former Rule 3.4(d) of the Revised Rules of Professional Conduct;
(d) By never providing the Duke Defendants on or after November 16, 2006, and prior to his recusal on January 12, 2007, a report setting forth the results of all tests or examinations conducted by DSI, including the existence of DNA from multiple unidentified males on the rape kit items and including written or recorded memorializations of Dr. Meehan's oral statements, Nifong:
ii. failed to disclose evidence or information that he knew, or reasonably should have known, was subject to disclosure under applicable law, rules of procedure or evidence, or court opinions, in violation of current Rule 3.4(d)(3) of the Revised Rules of Professional Conduct.
(e) By falsely representing to the Court and to counsel for the Duke Defendants that he had provided all discoverable material in his possession and that the substance of all Dr. Meehan's oral statements to him concerning the results of all examinations and tests conducted by DSI were included in DSI's written report, Nifong made false statements of material fact or law to a tribunal in violation of Rule 3.3(a)(1), made false statements of material fact to a third person in the course of representing a client in violation of Rule 4.1, and engaged in conduct involving dishonesty, fraud, deceit or misrepresentation in violation of Rule 8.4(c) of the Revised Rules of Professional Conduct.
(f) By representing or implying to the Court that he was not aware of the existence on rape kit items of DNA from multiple unidentified males who were not members of the lacrosse team and/or that he was not aware of the exclusion of that evidence from DSI's written report at the beginning of the December 15, 2006, hearing, Nifong made false statements of material fact or law to a tribunal in violation of Rule 3.3(a)(1) and engaged in conduct involving dishonesty, fraud, deceit or misrepresentation in violation of Rule 8.4(c) of the Revised Rules of Professional Conduct.
Media Defends Corruption
"Duke Lacrosse on Steroids" is an apt description of the Sandusky and Conspiracy of Silence cases because the degree of possible misconduct and the number of players who were likely involved.
Judges, prosecutors, and politicians all have taken part in abusing the so-called system of justice in the Commonwealth.
AG Kathleen Kane, who made good on her promise to investigate the Sandusky investigation, likely had no idea what Fina, Feudale, Beemer, and others feared her investigation would uncover about their misconduct -- and it wasn't pornography.
The media, who should be a watch dog to root out corruption, continues to be a lap dog of the corrupt. Instead of recognizing how officers of the court have compromised the system of justice, it has aligned itself with them.
To make an analogy, instead of being against the "crack dealers" who sell drugs to children, it has turned against the "undercover cops" who identified them. As for the children, the media is content to leave them to fend for themselves.
Kane and the corrupt are now engaged in a scorched earth policy in which few will be left unscathed.
Unfortunately, those among the scorched will be every citizen of the Commonwealth who will have the unfortunate circumstance to be abused by this corrupt system of justice.
Posted by Ray Blehar at 10:38 PM 5 comments:
Labels: Beemer, corruption, discovery, Duke Lax, emails, exposes, Feudale, Fina, Kane, misconduct, OAG, scorched earth


text:
name             'iscsiStorage01'
maintainer       'YOUR_COMPANY_NAME'
maintainer_email 'YOUR_EMAIL'
license          'All rights reserved'
description      'Installs/Configures iscsiStorage01'
long_description IO.read(File.join(File.dirname(__FILE__), 'README.md'))
version          '0.1.0'


text:
Bruce Fowler (*10. července 1947) je americký pozounista a hudební skladatel. Byl členem skupin The Mothers of Invention, The Magic Band a Fowler Brothers Band. Ve skupině Mothers of Invention hrál i jeho mladší bratr Tom Fowler.

Diskografie

Frank Zappa
 Over-Nite Sensation - 1973
 Apostrophe (') - 1974
 Roxy & Elsewhere - 1974
 Bongo Fury - 1975
 Studio Tan - 1978
 Sleep Dirt - 1979
 You Can't Do That on Stage Anymore, Vol. 1 - 1988
 Broadway the Hard Way - 1989
 The Best Band You Never Heard in Your Life - 1991
 You Can't Do That on Stage Anymore, Vol. 4 - 1991
 Make a Jazz Noise Here - 1991
 Beat the Boots - 1992
 You Can't Do That on Stage Anymore, Vol. 6 - 1992
 Imaginary Diseases - 2006
 Trance-Fusion - 2006
 Wazoo - 2007

Captain Beefheart
 Shiny Beast (Bat Chain Puller) - 1978
 Doc at the Radar Station - 1980
 I'm Going to Do What I Wanna Do: Live at My Father's Place 1978 - 2000

 The Toshiko Akiyoshi - Lew Tabackin Big Band
 Farewell - 1980
 From Toshiko With Love - 1981
 European Memoirs - 1982

 Eric Clapton
 The Road to Escondido - 2006

 Stu Nevitt
 The Marion Kind - 2007

 Jon Larsen
 Strange News from Mars - 2007

 Stan Ridgway
 The Big Heat Sólo
 Entropy - 1995
 Synthetic Division - 1995
 Ants Can Count - 1995
 Inception'' - 2010

Reference

Členové The Magic Band
Členové The Mothers of Invention
Narození v roce 1947
Žijící lidé
Američtí pozounisté
Narození 10. července
Muži

redpajama_set_name: RedPajamaWikipedia


Charming Andalusian farmhouse on 4000 sq plot with stunning views of the sea and mountains ready to turn it into your usual or holiday home.
It has a lot of potential to turn it into a rural hotel, because it can increase 4 more rooms without problem in the semi basement with views to one of the gardens.
It is 1.5 km from the beach, shopping centers, gas station, restaurants, supermarkets and the A-7 road that will take you to Puerto Banús in 15 minutes, Marbella 20 minutes, Malaga Airport 45 minutes and town center 10 minutes.
It's like getting into a story about farmers, every room, porches, mega equipped kitchen, well-kept gardens and decorated in rustic style with high ceilings on solid wooden beams.
The first impression you get when entering through the front door is WAO !! What a lovely place, what orderly, how well taken care of, how many details, what views, what a relaxing feeling, can breathe peace !!!
Entering through the main door you will find the parking on the right, on the left there is a huge garden with lawn and flower bushes, in front you will see the impressive house that you have to go through an arch to get to a large patio, on the right find 3 independent rooms with bathroom, 2 of them with living room inclusive.
On the left is the main entrance of the house, once inside you will see the dining room on the right and a toilet, on the left is the living room, with a super high ceiling with stained glass windows that give a special light and views of the garden by the windows.
You will find a room on the ground floor with views of the garden, and a wooden staircase that will take you to the 1st floor where there are 2 rooms with bathrooms and a small open library.
From the dining room you have direct access to the porch with 2 sets of sofas and a dining set protected by a pergola decorated with bougainvillea. To the right is a very large kitchen very well equipped. From the kitchen you have access to another porch with 3 sets of outdoor dining tables with a pergola that covers half for those who prefer to eat under the sun; with views to the sea, pool and mountains.
From the 2 porches has access to the beautiful gardens and a pool, at the bottom has a semi basement overlooking the garden, is where you have the option of increasing rooms, cellar, homecinema, sauna, gym or whatever you can think of.
Under the pool there is a room for storage or employees.
Ready to move in or start a good business !!


text:
\section{Introduction}
One of the challenging issues for statisticians and computer scientists is dealing with large datasets containing hundreds of variables which some of them may have interesting but unexplored relationships with each other. This is due to
examples of massive datasets in different areas such as: social networks, astronomy, genomics, medical records, and political science. Hence, it is an interesting topic to try to come up with methods which help us to discover these relationships.

Measuring the amount of dependence among two variables has been extensively studied in the literature and several methods have been proposed for it. We review some of them in the following. In \cite{Renyi:2005}, the author has suggested seven properties to be satisfied by any measure $\phi(x,y)$ used for determining the amount of dependence between $x$ and $y$. These properties known as R\'{e}nyi's axioms are:
\begin{itemize}
 \item In defining $\phi(x,y)$ between any two random variables, neither $x$ nor $y$ should be constant with probability 1.
\item $0\leq\phi(x,y)\leq1$.
\item $\phi(x,y)=0$ if and only if $x$ and $y$ are independent from each other.
\item $\phi(x,y)=1$ if there is an arbitrary functional dependency between $x$ and $y$. In other words, if $y=f(x)$ or $x=g(y)$ where $f(.)$ and $g(.)$ are Borel-measurable functions.
\item $\phi(x,y)=\phi(y,x)$
\item if $f(.)$ and $g(.)$ are strictly monotonic functions, then $\phi(x,y)=\phi(f(x),g(y))$.
\item if $x$ and $y$ are jointly Gaussian random variables, then $\phi(x,y)=|{\rm PCC}(x,y)|$ where ${\rm PCC}$ is the Pearson correlation coefficient.
\end{itemize}

The Pearson correlation coefficient (PCC) is the most well known dependency measure. However, it is unable to detect non-linear associations. In other words, the PCC is only able to capture linear associations between two variables.

As another measure of dependency, correlation ratio of random variable $y$ (if $\sigma^2(y)$ exists and $\sigma(y)>0$) on random variable $x$, introduced in \cite{cramer:1999} and \cite{kolmogorov:1933}, is defined as
\begin{eqnarray}
 \Theta(y)=\frac{\sigma(\mathbb{E}(y|x))}{\sigma(y)}.
\end{eqnarray}

It is easy to show that $0\leq \Theta(y) \leq 1$ where $\Theta(y)=1$ if and only if $y=f(x)$ in which $f(x)$ is a Borel-measurable function and $\Theta(y)=0$ if $x$ and $y$ are independent. The alternative formula of the correlation ratio mentioned in \cite{Renyi:2005} is
\begin{eqnarray}
 \Theta(y) = \sup_{(g)}{|{\rm PCC}(y,g(x))|}.
\end{eqnarray}
This alternative formula leads to another measure of dependency called \textit{maximal correlation}\cite{Gabelein:1941}:
\begin{eqnarray}
  \mathbb{S}(x,y) = \sup_{f,g}{{\rm PCC}(f(x),g(y))},
 \end{eqnarray}
where $f(.)$ and $g(.)$ are Borel-measurable functions. The author in \cite{Renyi:2005} has shown that $\mathbb{S}(x,y)=0$ if and only if $x$ and $y$ are independent. Furthermore, if there is an arbitrary functional relationship between $x$ and $y$, then $\mathbb{S}(x,y)=1$. The authors in \cite{breiman1985estimating} have introduced the \textit{alternating conditional expectation (ACE)} algorithm to find the optimal transformations.

The Spearman correlation coefficient \cite{spearman2004} is defined similar to the PCC; however, it is defined between the two ranked variables. By ranked variables we mean replacing each data point by its rank (or the average rank for equal sample points) in the ascending order. Therefore, if $\tilde{x}_i$ and $\tilde{y}_i$ denote the ranked versions of $x_i$ and $y_i$, the Spearman correlation coefficient would be
\begin{eqnarray}
 \rho=\frac{\sum_{i}{(\tilde{x}_i-\bar{x})(\tilde{y}_i-\bar{y})}}{\sqrt{\sum_{i}{(\tilde{x}_i-\bar{x})^2} \sum_{i}{(\tilde{y}_i-\bar{y})^2}}}.
\end{eqnarray}

The authors in \cite{delicado2009measuring} have expressed covariance and linear correlation in terms of principal components and generalized them for variables
distributed along a curve. They have estimated their measures using principal curves.

Mutual Information \cite{cover1991elements} is another measure that can be used for quantification of dependency between two variables since it satisfies some common properties of other dependency measures. As an example $I(x,y)=0$ if and only if $x$ and $y$ are independent.  The authors in \cite{moon1995estimation} have used kernel density estimation of probability density functions in order to estimate the mutual information between two variables. In \cite{kraskov2004estimating}, a method of mutual information estimation based on binning and estimating entropy from k-nearest neighbors is proposed.

The MIC \cite{MIC:2011} is recently proposed for quantifying dependency between two random variables. It is based on binning the dataset using dynamic programming technique to compute mutual information between different variables. It has two main properties which makes it superior in comparison with the aforementioned measures. First, it has \textit{generality} meaning that if the sample size is large enough, it is able to detect different kinds of associations rather than specific types. Second, it is an \textit{equitable} measure meaning that it gives similar scores to equally noisy associations no matter what type the association is.

One of the problems with the MIC is the fact that its computational cost grows rapidly as a function of the dataset size. Since this computational cost may become infeasible, the authors in \cite{MIC:2011} have applied a heuristic so as not to compute the mutual information for all possible grids. This heuristic application may result in finding a local maximum.

In this paper, we develop a computationally efficient approximation to the MIC. This approximation is based on replacing the dynamic programming application used in computation of the MIC with a very efficient technique that is uniformly binning the data. We show that our proposed method is able to detect both functional and non-functional associations between different variables, similar to the MIC while more efficiently. In addition, it has a better performance in recognizing the independence between different variables.

The rest of this paper is organized as the following. In section (\ref{sec:MIC}), we review the MIC and the algorithm used to compute it from \cite{MIC:2011}. In section (\ref{sec:sub-optimal}), we introduce our new measure of dependency that is a modification to the MIC. We present simulation results in section (\ref{sec:simul}). Finally, section (\ref{sec:con}) includes the conclusion of the paper.

\section{The Maximal Information Coefficient (MIC)}\label{sec:MIC}

\subsection{MIC Definition and Properties}\label{subsec:MICDEF}
For any finite dataset $D$ which contains ordered pairs of two random variables, one can partition the first element, i.e., $x$-value of these pairs into $\ell_x$ bins and similarly partition the second element or $y$-value of these pairs into $\ell_y$ bins. As a result of this partitioning, we will have an $\ell_x$-by-$\ell_y$ grid $G$. Every cell of this grid may or may not contain some sample points from the set $D$. This grid induces a probability distribution on the cells of $G$ where the corresponding probability of each cell is equal to the portion of sample points located in that cell. That is to say
\begin{eqnarray}
p_{ij}=\frac{|D_{ij}|}{|D|},
\end{eqnarray}
where $p_{ij}$ denotes the probability corresponding to the cell located at the $i^{th}$ row and the $j^{th}$ column and $|D_{ij}|$ denotes the number of sample points falling into the $i$-th row and the $j$-th column (See Figure \ref{fig:Dij} for a graphical view of the grid G).
It is obvious that for each $(\ell_x,\ell_y)$, we will have a grid that induces a new probability distribution and hence results in a different mutual information between the two variables.

\begin{figure}[t]
 \centering
 \includegraphics[width=60mm]{Dij.eps}
 \caption{Partitioning of dataset $D$ into $\ell_x$ columns and $\ell_y$ rows. $D_{ij}$ denotes the set of sample points located in the $i$-th row and the $j$-th column.}
\label{fig:Dij}
\end{figure}

Let $I_{D|G}^{*}(P;Q) = \max_G I_{D|G}(P;Q)$ be the largest possible mutual information achievable by an $\ell_x$-by-$\ell_y$ grid $G$ on a set $D$ of sample points. $P$ and $Q$ are the partitions of X-axis and Y-axis of grid $G$, respectively.  In order to have a fair comparison among different grids, the computed values of mutual information should be normalized. Since $I(P;Q)=H(Q)-H(Q|P)=H(P)-H(P|Q)$, we divide $I_{D|G}^{*}(P;Q)$ by $\log(\min(\ell_x,\ell_y))$. Therefore, we have
\begin{eqnarray}
 0\leq \frac{I_{D|G}^{*}(P;Q)}{\log(\min(\ell_x,\ell_y))} \leq 1.
\end{eqnarray}
This inequality motivates the definition of the MIC as a measure of dependency between two variables. For a dataset $D$ containing $n$ samples of two variables, we have
\begin{eqnarray}
 \text{MIC}(D) = \max_{\ell_x\ell_y<B(n)}\frac{I_{D|G}^{*}(P;Q)}{\log(min(\ell_x,\ell_y))}.
\end{eqnarray}
where $B(n)=n^{0.6}$ \cite{MIC:2011} or more generally $\omega(1)\leq B(n)\leq O(n^{1-\epsilon})$.
According to this definition, the MIC has the following properties:
\begin{itemize}
 \item $0\leq \text{MIC}(D)\leq 1$.
\item $\text{MIC}(x,y)=\text{MIC}(y,x)$.
\item It is invariant under order-preserving transformation applied to the dataset $D$.
\item It is not invariant under the rotation of coordinate axes, e.g., if $y=x$, then $\text{MIC}(D)=1$. However, after a $45^{\circ}$ clockwise rotation of coordinate axes, instead of $y=x$ we have $y=0$ and hence $\text{MIC}(D)=0$.
\end{itemize}

\subsection{MIC Algorithm}\label{subsec:MICALG}
Although the algorithm for computing the MIC is fully described in \cite{MIC:2011}, here we only review the \textit{OptimizeXAxis} algorithm which is used in computation of the highest mutual information achievable by an $\ell_x$-by-$\ell_y$ grid. Any $\ell_x$-by-$\ell_y$ grid imposes two sets of partitions on $x$-values (columns of grid) and $y$-values (rows of grid). We indicate columns of the grid by $\langle c_1,c_2,\ldots,c_{\ell_x}\rangle$ where $c_i$ denotes the endpoint (largest $x$-value) of the $i$-th column.

Since $I(P,Q)$ is upper-bounded by $H(P)$ and $H(Q)$, in order to maximize it, one can equipartition either the Y or X axis, i.e., impose a discrete uniform distribution on either $Q$ or $P$. Without loss of generality, we consider the version of the algorithm that equipartitions the Y-axis. However, it is obvious that we should check both of the cases (equipartitioning either the X or Y axis) separately for each $\ell_x$-by-$\ell_y$ grid and choose the maximum resulting mutual information.

Let $H(P)$ denote the entropy of distribution imposed by $m$ sample points ($m\leq|D|=n$) on the partition of X-axis. Similarly, let $H(Q)$ denote the entropy of distribution imposed by $m$ sample points ($m\leq|D|=n$) on the partition of Y-axis. Since we have assumed that the Y-axis is equipartitioned, $H(Q)$ is constant and equal to $\log(|Q|)$. Finally, let $H(P,Q)$ denote the entropy of distribution imposed by $m$ sample points ($m<|D|=n$) on the cells of grid $G$ which has X-axis partition $P$ and Y-axis partition $Q$. Since $I(P;Q)=H(Q)-H(Q|P)$ and we have already maximized $H(Q)$ by equipartitioning the Y-axis, to achieve the highest mutual information, we have to minimize the $H(Q|P)$. This is done by the \textit{OptimizeXAxis} algorithm \cite{MIC:2011}.

An alternative formula for the mutual information is $I(P;Q)=H(Q)+H(P)-H(P,Q)$. Since $H(Q)$ is constant, the \textit{OptimizeXAxis} only needs to maximize $H(P)-H(P,Q)$. The following theorem \cite{MIC:2011} is the key to solve this problem.

\begin{figure}[t]
 \centering
 \includegraphics[width=45mm]{clumps.eps}
 \caption{\textit{OptimizeXAxis} \cite{MIC:2011} considers only consecutive points falling into the same row and draw partitions between them. The set of consecutive points falling into the same row is called \textit{clump}.}
\label{fig:clump}
\end{figure}

\begin{theorem}
For a dataset $D$ of size $n$ and a fixed row partition $Q$, and for every $m,l\in \mathbb{N}$, if we define $F(m,l)=\max_{D(1:m),|P|=l}\{H(P)-H(P,Q)\}$ then for $l>1$ and $1 < m \leq n$ we would have the following recursive equation
\begin{eqnarray}
 F(m,l)=\max_{1\leq i<m}\{\frac{i}{m}F(i,l-1)-\frac{m-i}{m}H(\langle i,m \rangle,Q)\}.
\end{eqnarray}
\label{th:recursive}
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{th:recursive}]
 See proposition 3.2. in \cite{MIC:2011}.
\end{proof}
The \textit{OptimizeXAxis} uses dynamic programming technique motivated by Theorem \ref{th:recursive}. It ensures $F(n,l)$ that is the desired partition of dataset $D$ (which has $n$ sample points) having $l$ columns imposing partition $P$ over X-axis.
In order to minimize the $H(Q|P)$, \textit{OptimizeXAxis} considers only consecutive points falling into the same row and draw partitions between them. The set of consecutive points falling into the same row is called \textit{clump} (See Figure \ref{fig:clump} for a graphical view of clump). In Algorithm \ref{alg:OptimizeXAxis}, the GetClumpsPartition subroutine is responsible for finding and partitioning the clumps. Moreover, $P_{t,l}$ is an optimal partition of size $l$ for the first $t$ clumps.

\begin{algorithm}[!t]
\caption{OptimizeXAxis($D,Q,\ell_x$) \cite{MIC:2011} \label{alg:OptimizeXAxis}}
\textbf{Require:} $D$ is a set of ordered pairs sorted in increasing order by x-values

\textbf{Require:} $Q$ is a Y-axis partition of $D$

\textbf{Require:} $\ell_x$ is an integer greater than 1

\textbf{Ensure:} Returns a list of scores $(I_2,\ldots,I_{\ell_x})$ such that each $I_l$ is the maximum value of $I(P;Q)$ over all partitions $P$ of size $l$
\begin{algorithmic}[1]
\STATE $\langle c_0,\ldots,c_k \rangle \gets \text{GetClumpsPartition($D$,$Q$)}$
\STATE
\STATE {Find the optimal partition of size 2}
\FOR {$t=2$ to $k$}
\STATE Find $s\in \{1,\ldots,t\}$ maximizing $H(\langle c_s,c_t\rangle)-H(\langle c_s,c_t\rangle,Q)$.
\STATE $P_{t,2} \gets \langle c_s,c_t \rangle$
\STATE $I_{t,2} \gets H(Q)+H(P_{t,2})-H(P_{t,2},Q)$
\ENDFOR
\STATE
\STATE {Inductively build the rest of the table of optimal partitions}
\FOR {$l=3$ to $\ell_x$}
\FOR {$t=2$ to $k$}
\STATE Find $s \in \{1,\ldots,t\}$ maximizing $F(s,t,l):=\frac{c_s}{c_t}(I_{s,l-1}-H(Q))+\sum_{i=1}^{|Q|}\frac{\#_{i,l}}{c_t}\log{\frac{\#_{i,l}}{\#_{*,l}}}$

where $\#_{*,j}$ is the number of points in the $j$-th column of $P_{s,l-1}\cup c_t$ and $\#_{i,j}$ is the number of points in the $j$-th column of $P_{s,l-1}\cup c_t$ that fall in the $i$-th row of $Q$
\STATE $P_{t,l} \gets P_{s,l-1} \cup c_t$
\STATE $I_{t,l} \gets H(Q)+H(P_{t,l})-H(P_{t,l},Q)$
\ENDFOR
\ENDFOR
\RETURN  $(I_{k,2},\ldots,I_{k,\ell_x})$
\end{algorithmic}
\end{algorithm}
\section{The Uniform-MIC (U-MIC)}\label{sec:sub-optimal}
\subsection{Noiseless Setting}
The major drawback of the Algorithm \ref{alg:OptimizeXAxis} is its computational complexity. If there exists $k$ clumps in the given partition of an $\ell_x$-by-$\ell_y$ grid, the runtime of this algorithm would be $O(k^2\ell_x \ell_y)$. If there is a functional association between the two variables, the number of clumps in the corresponding grid is pretty small. However, for noisy or random datasets it is easy to imagine that the number of clumps is very large and hence the computational complexity of the Algorithm {alg:OptimizeXAxis} would be large.

Furthermore and due to this problem, this algorithm cannot be generalized in order to detect associations between more than two variables. As an example, if we want to detect whether or not three variables are related to each other, we may write the formula for the \textit{generalized} mutual information as:
\begin{align}
  I(P;Q;R)=&H(P)+H(Q)+H(R)-H(P,Q)\\
&-H(P,R)-H(Q,R)+H(P,Q,R). \nonumber
 \end{align}
Hence, intuitively and like the case for two random variables, in order to maximize the generalized mutual information, we have to equipartition one axis to maximize the entropy. Nevertheless, we should partition the two other axes with respect to the places of the clumps in them. if we equipartition the first axis and there exists $k_1$ clumps in the second axis and $k_2$ clumps in the third axis, then the runtime of this algorithm would be $O(k_1^2k_2^2\ell_x\ell_yl_z^2)$ where $\ell_x,\ell_y,l_z$ are the sizes of partitioning. This runtime is not acceptable for large datasets. Therefore, we have to modify the algorithm in order to decrease its runtime and as a result make it generalizable to higher dimensions.

The algorithm we propose in here for replacing the Algorithm \ref{alg:OptimizeXAxis} is \textit{uniform partitioning} (Algorithm \ref{alg:UMIC}). Let $y_{\min}=\min_i y_i$, $y_{\max}=\max_j y_j$, and similarly $x_{\min}=\min_i x_i$ and $x_{\max}=\max_j x_j$. we then partition both $X$ and $Y$ axes such that all the columns have length $\frac{x_{\max}-x_{\min}}{\ell_x}$ and similarly all the rows have length $\frac{y_{\max}-y_{\min}}{\ell_y}$. We call this new measure, that is derived by replacing the Algorithm \ref{alg:OptimizeXAxis} with Algorithm \ref{alg:UMIC}, by the U-MIC (Uniform Maximal Information Coefficient).  In the following we prove that the U-MIC will approach 1 as the sample size grows for when there exists a functional association between two variables (with finite derivative). Without loss of generality, we do all the proofs in the case that $(x,y) \in [0,1]\times[0,1]$. These proofs could be generalized to other cases easily.

\begin{algorithm}
\caption{UniformPartition($\ell_x,\ell_y$)\label{alg:UMIC}}
\textbf{Require:} Dataset $D$

\textbf{Require:} $\ell_x$ and $\ell_y$ are integers greater than 1

\textbf{Ensure:} Returns a score $I^*$ which is the value of $I(P;Q)$ where $P$ are $Q$ are distributions from uniform partitioning of both axes.

\begin{algorithmic}[1]
\STATE $P \gets$ Uniform partition of $X$-axis by $\ell_x$ columns each has length $\frac{x_{\max}-x_{\min}}{\ell_x}$
\STATE $Q \gets$ Uniform partition of $Y$-axis by $\ell_y$ rows each has length $\frac{y_{\max}-y_{\min}}{\ell_y}$
\STATE $I^*=\frac{H(P)+H(Q)-H(P,Q)}{\log(\min(\ell_x,\ell_y))}$
\RETURN $I^*$
\end{algorithmic}
\end{algorithm}

\begin{proposition}
If $D=\{(x_i,y_i)\}_{i=1}^n$ where $y_i=h(x_i)$ and $|h'(x)|<\infty$, then $\lim_{n\to \infty}\text{U-MIC}(D)=1$. \label{pro:UMIC}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{pro:UMIC}]
We denote by $g_h(\alpha)$ the sub-level function of function $h(.)$, i.e.,
\begin{eqnarray}
 g_h(\alpha)=\lambda(\{x:h(x)\leq \alpha\}),
\end{eqnarray}
where $\lambda(\mathbb{T})$ denotes the fraction of sample points in the set $\mathbb{T}$. Consequently
\begin{eqnarray}
 g_h(\alpha)=F_y(\alpha)=\mathbb{P}(y\leq \alpha)=\mathbb{P}(h(x)\leq \alpha),
\end{eqnarray}
where $F_y(.)$ denotes the cumulative distribution function (CDF) and $\mathbb{P}(.)$ denotes the probability function. Using this notation and assuming that we uniformly partition $Y$-axis by $\ell_y$ rows, we can write the entropy of $Q$ which is the uniform partition of Y-axis as
\begin{align}\label{eq:uniformH(y)}
 &H(Q) \\&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}(Q=i)\log(P(Q=i))\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}\right)\log \left(\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}\right)\right)\nonumber\\
&=-\sum_{i=0}^{\ell_y-1}g'_h(\alpha_i)\frac{1}{\ell_y}\log \left(g'_h(\alpha_i)\frac{1}{\ell_y}\right)\nonumber\\
&=-\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y}g'_h(\alpha_i)\log(g'_h(\alpha_i))-\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y}g'_h(\alpha_i)\log\left(\frac{1}{\ell_y}\right)\nonumber,
\end{align}
where $\frac{i}{\ell_y} \leq \alpha_i < \frac{i+1}{\ell_y}$ for each $i$ ($0\leq i \leq \ell_y-1$) is derived according to the mean value Theorem. If without loss of generality we assume that $\min(\ell_x,\ell_y)=\ell_y$ then we can write
\begin{align}\label{eq:H(y) over ly}
 \frac{H(Q)}{\log(\ell_y)}=-\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y\log(\ell_y)}g'_h(\alpha_i)\log(g'_h(\alpha_i))+\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y}g'_h(\alpha_i).
\end{align}
As a result, in the asymptotic setting we can write
\begin{align}
 \lim_{\ell_y\to \infty}\frac{H(Q)}{\log(\ell_y)}=\sum_i\frac{1}{\ell_y}g'_h(\alpha_i)=1,
\end{align}
where the last equality holds since $ \lim_{\ell_y \to \infty}\sum_i\frac{1}{\ell_y}g'_h(\alpha_i)$ is the Riemann integral of the function $g'_h(\alpha_i)$.
If we assume that $|h'(.)|<c$, then according to the mean value Theorem we have
\begin{eqnarray}\label{eq:upperbound-derivative}
 \left|h\left(\frac{i+1}{\ell_y}\right)-h\left(\frac{i}{\ell_y}\right)\right|\leq\frac{c}{\ell_y}.
\end{eqnarray}
Equation (\ref{eq:upperbound-derivative}) states that for a particular column of the X-axis partition, the curve of the function passes through at most $c+1$ cells of that column. We use this fact in upper-bounding the $H(Q|P)$. Similar to (\ref{eq:uniformH(y)}) and (\ref{eq:H(y) over ly}) we have

\begin{align}\label{eq:uniformH(y|x)}
 \displaybreak[0]
 H(Q|P=k)&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}(Q=i|P=k) \\
&~~~~~~~~~\times\log(\mathbb{P}(Q=i|P=k))\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}|P=k\right) \nonumber \\
&~~~~~~~~~\times\log\left(\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}|P=k\right)\right)\nonumber \\
\displaybreak[0]
&=-\sum_{i=0}^{\ell_y-1}f_{y|x}(\alpha_i|P=k)\frac{1}{\ell_y}  \nonumber \\
&~~~~~~~~~\times \log\left(f_{y|x}(\alpha_i|P=k)\frac{1}{\ell_y}\right)\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k) \nonumber \\
&~~~~~~~~~ \times \log(f_{y|x}(\alpha_i|P=k))\nonumber \\
&~~~~~~-\sum_{i=0}^{\ell_y-1}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k)\log\left(\frac{1}{\ell_y}\right),\nonumber
\end{align}
where $f_{y|x}$ denotes the conditional probability density function. Because of equation (\ref{eq:upperbound-derivative}), we can simplify (\ref{eq:uniformH(y|x)}) as
\begin{align}
 H(Q|P=k)&=-\sum_{i=j_1}^{j_{c+1}}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k) \\
&~~~~~~~~~\times \log(f_{y|x}(\alpha_i|P=k))\nonumber \\
&~~~~~~-\sum_{i=j_1}^{j_{c+1}}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k)\log\left(\frac{1}{\ell_y}\right).\nonumber
\end{align}
If we define $k^{*} =\arg\max_{k}H(Q|P=k)$, then since $H(Q|P)=\sum_k\mathbb{P}(P=k)H(Q|P=k)$, we can write
\begin{align}
H(Q|P)&\leq-\sum_{i=j_1}^{j_{c+1}}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k^*) \\
&~~~~~~~~~\times \log(f_{y|x}(\alpha_i|P=k^*)\nonumber \\
&~~~~~~-\sum_{i=j_1}^{j_{c+1}}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k^*)\log\left(\frac{1}{\ell_y}\right),\nonumber
\end{align}
and hence
\begin{align}
 \lim_{\ell_y \to \infty}\frac{H(Q|P)}{\log(\ell_y)}\leq \sum_{i=j_1}^{j_{c+1}}\frac{1}{\ell_y}f_{y|x}(\alpha_i|P=k^*)=0.
\end{align}
The last equality holds since $\frac{1}{\ell_y} \to 0$ but $c<\infty$.
As a result
\begin{align}
\lim_{\ell_y\rightarrow\infty} \text{U-MIC}(D)&=\frac{I(P;Q)}{\log(\min\{\ell_x,\ell_y\})}\\
&=\frac{H(Q)-H(Q|P)}{\log(\min\{\ell_x,\ell_y\})}=1.\nonumber
\end{align}
\end{proof}

If $x$ and $y$ are independent, then according to the following Proposition we have $\text{U-MIC}(D)=0$.

\begin{proposition}
If $D=\{(x_i,y_i)\}_{i=1}^n$ where $x_i \independent y_i$ for $1\leq i \leq n$, then $\text{U-MIC}(D)=0$.
\label{pro:UMIC_ind}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{pro:UMIC_ind}]
The line of reasoning is straight forward and similar to the proof of Proposition \ref{pro:UMIC}. Since $x$ and $y$ are independent from each other, we can write
\begin{align}
&H(Q) \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}(Q=i)\log(P(Q=i))\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}\right)\log \left(\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}\right)\right)\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}|P=k\right) \nonumber \\
&~~~~~~~~~~~\times\log\left(\mathbb{P}\left(\frac{i}{\ell_y}\leq Y < \frac{i+1}{\ell_y}|P=k\right)\right)\nonumber \\
&=-\sum_{i=0}^{\ell_y-1}\mathbb{P}(Q=i|P=k) \log(\mathbb{P}(Q=i|P=k))\nonumber \\
&=H(Q|P=k).\nonumber
\end{align}
Therefore, $H(Q)=H(Q|P=k)$ for every $k$ where $0\leq k\leq \ell_x-1$. Now since $H(Q|P)=\sum_k\mathbb{P}(P=k)H(Q|P=k)$, we have $H(Q)=H(Q|P)$ and as a result U-MIC($D$)=0.
\end{proof}

\subsection{Noisy Setting}
In this section we study performance of the U-MIC in noisy setting. We first give a lower-bound on it when the two variables $x$ and $y$ have a noisy functional association in which the noise is bounded. After that, we study the case of unbounded noise.

For the bounded noise case, without loss of generality we assume that $x\sim U[0,1]$ and the noise has a uniform distribution. Specifically, we assume that sample points $(x_i,y_i)$ have the form $(x_i,h(x_i)+z_\epsilon)$ where $z_\epsilon \sim U[-\epsilon,\epsilon]$. We define $y_{\rm mid} = \frac{y_{\max}+y_{\min}}{2}$. In Algorithm \ref{alg:UMIC}, we divide the Y-axis into two rows by drawing a horizontal line at $y_{\rm mid}$. In addition, we divide the X-axis into $\ell_x$ columns each having the length $\frac{1}{\ell_x}$ (since $x\sim U[0,1]$). Let $D_1 = \{ (x_i,y_i)|y_i<y_{\rm mid} \}$ and $D_2 = \{(x_i,y_i)|y_i>y_{\rm mid} \}$. We use $\mathcal{P}$ and $\mathcal{Q}$ to denote the partition of X-axis and Y-axis of the grid in this setting. Having this setting and notations in mind, the following Corollary gives a simple lower-bound for U-MIC($D$) in this case.

\begin{corollary}
 Let $m$ be the number of columns in $\mathcal{P}$ in which there exists a sample point $(\hat{x},\hat{y})$ such that $|\hat{y}-y_{\rm mid}|\leq \epsilon$. Then, U-MIC($D$) is lower-bounded by
\begin{eqnarray}
\frac{|D|\log(|D|)-|D_1|\log(|D_1|)-|D_2|\log(|D_2|)}{|D|}- \frac{m}{\ell_x}. \nonumber
\label{equ:boundUMIC}
\end{eqnarray}
\label{pro:UMICNoisy}
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{pro:UMICNoisy}]
Since $I(\mathcal{P},\mathcal{Q})=H(\mathcal{Q})-H(\mathcal{Q}|\mathcal{P})$, we need to have an upper-bound on $H(\mathcal{Q}|\mathcal{P})$ in order to determine a lower-bound on $I(\mathcal{P},\mathcal{Q})$. According to the entropy definition we can write
\begin{align}\label{equ:entropy}
H(\mathcal{Q}) &= -\frac{|D_1|}{|D|}\log\left(\frac{|D_1|}{|D|}\right)-\frac{|D_2|}{|D|}\log\left(\frac{|D_2|}{|D|}\right) \\
&=\frac{|D|\log(|D|)-|D_1|\log(|D_1|)-|D_2|\log(|D_2|)}{|D|}. \nonumber
\end{align}
Let $\mathcal{M}=\{l_{p_1},\ldots,l_{p_m}\}$ denote the columns in which there exists a data point $(\hat{x},\hat{y})$ such that $|\hat{y}-y_{\rm mid}|\leq \epsilon$. Since $\mathcal{Q}$ has only two rows, we can upper-bound the $H(\mathcal{Q}|\mathcal{P})$ as the following
\begin{align}\label{equ:upp}
H(\mathcal{Q}|\mathcal{P})&=\sum_{k=0}^{\ell_x-1}\mathbb{P}(\mathcal{P}=k)H(\mathcal{Q}|\mathcal{P}=k) \\
&\overset{(a)}{=}\frac{1}{\ell_x}\left( \sum_{k \in \mathcal{M}} H(\mathcal{Q}|\mathcal{P}=k) +  \sum_{k \notin \mathcal{M}} H(\mathcal{Q}|\mathcal{P}=k)\right) \nonumber \\
&\overset{(b)}{=}\frac{1}{\ell_x} \sum_{k \in \mathcal{M}} H(\mathcal{Q}|\mathcal{P}=k) \nonumber \\
&\leq \frac{|\mathcal{M}|}{\ell_x}=\frac{m}{\ell_x}, \nonumber
\end{align}
where (a) holds since $x\sim U[0,1]$ and (b) holds because $z_\epsilon \sim U[-\epsilon,\epsilon]$. The lower-bound is then derived by combining \eqref{equ:entropy} and \eqref{equ:upp}.
\end{proof}

The main issue with generalizing this lower-bounding idea to other noise distributions is that noise values could be unbounded. Hence, we use the idea of \textit{$k$-nearest neighbors} to bound the noise so as to come up with a consistent version of the association detector. We study this idea for the case that noise is drawn from a Gaussian distribution with 0 mean and variance of $\sigma^2$.

For each sample point, we consider its $\delta_n$-neighborhood (we use subscript $n$ to show the dependency on the size of the dataset $n$). We replace each data point with the average of sample points located in its $\delta_n$-neighborhood. The following lemma characterizes the number of sample points in this neighborhood.

\begin{figure}[t]
 \centering
 \includegraphics[width=50mm]{noisy.eps}
 \caption{Using \textit{k-nearest neighbors} method to bound the noise in noisy relationships. We replace each point with the average of its neighbors in its $\delta_n$-neighborhood.}
\label{fig:kNearestNeighbors}
\end{figure}

\begin{lemma}
Let $x$ be uniformly distributed, i.e., $x \sim U[0,1]$ and $(x_i,y_i)$ denote the $i$-th data point in $D$ where $y_i=h(x_i)+z_i$. If $\mathcal{N}=\{(x_j,y_j)|(x_i-x_j)^2\leq \delta_n^2\}$, then $\lim_{n \to \infty}|\mathcal{N}|=2n\delta_n$.
\label{lem:neighbor}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:neighbor}]
Let $\mathbb{I}(.)$ denote the indicator function. Then we can write
\begin{eqnarray}
2\epsilon_n=|\mathcal{N}|=\sum_{j=1}^{n}\mathbb{I}(x_i-\delta_n\leq x_j\leq x_i+\delta_n).
\end{eqnarray}

As a result $\mathbb{E}[2\epsilon_n]=2n\delta_n$. Using the Hoeffding inequality we have
\begin{eqnarray}
 \mathbb{P}(|2\epsilon_n-\mathbb{E}[2\epsilon_n]|\geq t) \leq 2e^{-2ct^2n^2},
\end{eqnarray}
for some constant $c$. If we let $t=\frac{1}{\log n}$, then $\lim_{n \to \infty}(\epsilon_n)=n\delta_n$ or equivalently,  $\lim_{n \to \infty}|\mathcal{N}|=2n\delta_n$.
\end{proof}
Assume that $h(.)$ is a Lipschitz continuous function of order $\beta$, i.e.,$~|h(v)-h(w)|\leq k|v-w|^{\beta}$ where $k$ is a constant depends on the function $h(.)$. If we estimate (or replace) the y-value of each noisy sample point with the average of sample points in its $\delta_n$-neighborhood, in the case of Gaussian noise (0 mean and variance of $\sigma^2$) we can write the estimation mean squared error as
\begin{align}
 \Delta_n&=\frac{1}{n}\sum_{i=1}^n\mathbb{E}(\bar{h}(x_i)-h(x_i))^2\\
&=\frac{1}{n}\sum_{i=1}^n\mathbb{E}\left[\frac{\sum_{j=-\epsilon_n}^{\epsilon_n}(h(x_{i-j})+z_{i-j})}{2\epsilon_n+1}-h(x_i)\right]^2 \nonumber \\ &\leq \frac{k^2\epsilon_n^{2\beta}}{n^{2\beta}}+\frac{\sigma^2}{2\epsilon_n+1}. \nonumber
\end{align}
In order to minimize the estimation error we can take derivative with respect to $\epsilon_n$ and set it to 0. Therefore, $\epsilon_n$ which minimizes the mean squared error is
\begin{eqnarray}
 \epsilon_n^*=\sqrt[2\beta+1]{\frac{\sigma^2}{4k^2\beta}}n^{1-\frac{1}{2\beta+1}}.
\label{equ:optimalNeighbors}
\end{eqnarray}
We use this $ \epsilon_n^*$ later to to bound the noise. The following lemma gives a probabilistic bound on the noise values.

\begin{lemma}
 If $z_1,z_2,\ldots,z_n$ are i.i.d. drawn from $N(0,\sigma^2)$, then $\mathbb{P}\{\max_{1\leq i \leq n} |z_i|>t\}\leq 2ne^{\frac{-t^2}{2\sigma^2}}$.
\label{lem:boundGaussian}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:boundGaussian}]
First of all, for a zero mean Gaussian random variable $z_i$ we prove that $\mathbb{P}\{ |z_i|>t\}\leq 2ne^{\frac{-t^2}{2\sigma^2}}$. Let $u=z_i-t$ and hence $u\thicksim N(-t,\sigma^2)$. We have
\begin{eqnarray}
\int_0^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-u^2-2ut}{2\sigma^2}}du\leq\int_0^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-u^2}{2\sigma^2}}du\leq1.
\end{eqnarray}
As a result we can write
\begin{eqnarray}
 \int_t^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-z_i^2}{2\sigma^2}}dz_i=\int_0^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(u-t)^2}{2\sigma^2}}du\leq e^{\frac{-t^2}{2\sigma^2}}.
\label{equ:boundGaussian}
\end{eqnarray}
Similarly, (\ref{equ:boundGaussian}) holds for  $[-\infty,-t]$ and hence $\mathbb{P}\{ |z_i|>t\}\leq 2e^{\frac{-t^2}{2\sigma^2}}$. The result of lemma then follows from using union-bound on the $z_i$s.
\end{proof}
By using the $k$-nearest neighbors method, each $z_i$ is replaced by $\bar{z}_i$ which is the average of $2\epsilon_n+1$ i.i.d. noise values and hence its variance is decreased by $2\epsilon_n+1$. This idea motivates the following corollary which lets us to bound the noise.
\begin{corollary}
By using the $k$-nearest neighbors method, $\bar{z}_i=\frac{\sum_{j=-\epsilon_n}^{\epsilon_n}z_{i-j}}{2\epsilon_n+1}$, and as a result $\lim_{n \to \infty}\max_{1\leq i \leq n}|\bar{z}_i| = 0$.
\label{Cor:limNoise}
\end{corollary}
\begin{proof}[Proof of Corollary \ref{Cor:limNoise}]
According to the Lemma \ref{lem:boundGaussian}, we can write $\mathbb{P}\{\max_{1\leq i \leq n} |\bar{z}_i|>t\}\leq 2ne^{\frac{-t^2(2\epsilon_n+1)}{2\sigma^2}}$. The result then follows from letting $t=\frac{1}{\log n}$ and $\epsilon_n=\epsilon_n^*$ which was derived in (\ref{equ:optimalNeighbors}).
\end{proof}
In the next section we show how the U-MIC works in practice comparing to the MIC.
\section{Simulation Results}\label{sec:simul}

In this section, we study the performance of our proposed measure. We first show how it works for functional associations. Second, we study its performance for non-functional associations. Finally, we do some experiments for the case of noisy relationships. As mentioned previously, the authors in \cite{MIC:2011} apply a heuristic to compute the MIC which may not result in the true MIC. On the other hand, we do not apply any heuristic in the simulation results in order to have a precise comparison with our proposed method.

\begin{figure}[t]
\centering
\includegraphics[width=90mm]{funcs.eps}
\caption{Test functional relationships for Tables \ref{tab:funcs1},\ref{tab:funcs2}, and \ref{tab:timefuncs}.}
\label{fig:funcs}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=60mm]{funcs2.eps}
\caption{Test non-functional relationships for Tables \ref{tab:non-funcs} and \ref{tab:timenon-funcs}. }
\label{fig:non-funcs}
\end{figure}

Figure \ref{fig:funcs} shows the functional associations that we have tested the performance of the MIC and U-MIC algorithms on. Table \ref{tab:funcs1} summarizes the results for the case that there are 200 sample points. One interesting point in Table \ref{tab:funcs1} is the value of the U-MIC for sinusoidal function with different frequencies. As we can see, MIC($D$)=1 while U-MIC($D$)=0.75 for this function. One interpretation of this difference is that in the proof of Proposition \ref{pro:UMIC}, we have assumed that the absolute value of derivative of function $h(.)$ is upper-bounded by constant $c$. However, this is not the case for sinusoidal function with different frequencies since there is a discontinuity in this function. If we increase the sample size, as reported in Table \ref{tab:funcs2}, this issue is alleviated as we can see.

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  c  c  c | }
 \hline
 &  Linear &  Parabolic & Periodic &  Cubic  &  Sin (Diff. Freq.) & Sin (Single Freq.) \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 1 & 1 &1 & 1 & 1 & 1 \\
 \hline
 \textbf{U-MIC}& 1 &1 & 0.93 & 0.95 & 0.75  & 0.91 \\
 \hline
 \end{tabular}}
\caption{ MIC($D$) and U-MIC($D$) for different functional relationships in Figure \ref{fig:funcs}. For this set of experiments, $|D|=200$.}
\label{tab:funcs1}
\end{center}
\end{table}

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  c  c  c | }
 \hline
 &  Linear &  Parabolic & Periodic &  Cubic  &  Sin (Diff. Freq.) & Sin (Single Freq.) \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 1 & 1 &1 & 1 & 1 & 1 \\
 \hline
 \textbf{U-MIC}& 1 &1 & 0.99 & 0.99 & 0.93  & 0.95 \\
 \hline
 \end{tabular}}
\caption{ MIC($D$) and U-MIC($D$) for different functional relationships in Figure \ref{fig:funcs}. For this set of experiments, $|D|=5000$.}
\label{tab:funcs2}
\end{center}
\end{table}

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  c  c  c | }
 \hline
 &  Linear &  Parabolic & Periodic &  Cubic  &  Sin (Diff. Freq.) & Sin (Single Freq.) \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 0.1 & 0.5 &0.1 & 0.2 & 2 & 0.4 \\
 \hline
 \textbf{U-MIC}& 0.01 &0.01 & 0.01 & 0.01 & 0.01  & 0.01 \\
 \hline
 \end{tabular}}
\caption{ Run time (in sec.) for calculation of MIC($D$) and U-MIC($D$) for different functional relationships in Figure \ref{fig:funcs}. For this set of experiments, $|D|=200$.}
\label{tab:timefuncs}
\end{center}
\end{table}

Although the same issue holds for periodic function in Figure \ref{fig:funcs}, we do not see that much effect. Qualitatively, the derivative of continuous pieces of the periodic function in Figure \ref{fig:funcs} $(y=x)$ is smaller than the maximum of the derivative of sinusoidal function with different frequencies $(y=\sin(10x),~y=\sin(20x))$. Hence, if we uniformly partition the X-axis in the case of periodic function, there would be fewer sample points in rows of a certain column and more probably higher entropy (resulting in higher U-MIC), as the case in Table \ref{tab:funcs1}.

Table \ref{tab:timefuncs} summarizes the runtime for calculation of the MIC and U-MIC for different functional associations in Figure \ref{fig:funcs}. As we can see, the U-MIC is at least 10 times faster in these cases. This is expected since the MIC uses dynamic programming to find a close to optimal grid for the data while the U-MIC just uniformly partitions the axes.

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  c | }
 \hline
 &  Circle &  Sinusoidal Mixture & Two Lines &  Random  \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 0.68 &0.72 &0.71 & 0.16  \\
 \hline
 \textbf{U-MIC}& 0.64 &0.69 & 0.68 & 0.06 \\
 \hline
\end{tabular}}
\caption{ MIC($D$) and U-MIC($D$) for different non-functional relationships in Figure \ref{fig:non-funcs}. For this set of experiments, $|D|=200$.}
\label{tab:non-funcs}
\end{center}
\end{table}

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  c | }
 \hline
 &  Circle &  Sinusoidal Mixture & Two Lines &  Random  \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 26.38 &13.41 & 16.60 & 61.00  \\
 \hline
 \textbf{U-MIC}& 0.01 & 0.02 &  0.01 & 0.02 \\
 \hline
\end{tabular}}
\caption{ Run time (in sec.) for calculation of MIC($D$) and U-MIC($D$) for different non-functional relationships in Figure \ref{fig:non-funcs}. For this set of experiments, $|D|=200$.}
\label{tab:timenon-funcs}
\end{center}
\end{table}

Table \ref{tab:non-funcs} summarizes the results for non-functional associations presented in Figure \ref{fig:non-funcs}. One important point about Table \ref{tab:non-funcs} is that the U-MIC has a better performance in the case of random sample points (i.e., $x \independent y$). In this case, the ideal MIC and U-MIC is 0; however, as we can see MIC($D$)=0.16 and U-MIC(D)=0.06. This issue is related to one of the criticisms made about MIC in the literature \cite{simon2014comment}. One of the drawbacks of the MIC is the fact that as a statistical test it has a lower power than other measures of dependency such as distance correlation \cite{simon2014comment}. In other words, it gives more false positives in detecting associations. However, according to our simulation results and Proposition \ref{pro:UMIC_ind} this issue is alleviated in the U-MIC.

Table \ref{tab:timenon-funcs} shows the runtime for calculation of the MIC and U-MIC. In the case of non-functional relationships we have more clumps in the initial grid of sample points for calculation of the MIC. Hence, Algorithm \ref{alg:OptimizeXAxis} which is basically running dynamic programming over initial grid to find the optimal grid, would have larger runtime as we can see in Table \ref{tab:timenon-funcs}. On the other hand, since the U-MIC is dealing with uniform partitioning of the grid of sample points, it does not matter what type of relationship the two random variables have. The runtime is almost constant and similar to the cases that there is a functional association between two variables.

Tables \ref{tab:noisy-non-funcs} and \ref{tab:noisy-timenon-funcs} summarize the results for noisy non-functional associations presented in Figure  \ref{fig:noisy-non-funcs}. Figure \ref{fig:noisy-non-funcs} is similar to Figure \ref{fig:non-funcs} except for the fact that we have added noise drawn from uniform distribution, i.e., $U[-0.5,0.5]$ to the sample points. Comparing Table \ref{tab:noisy-non-funcs}  with Table \ref{tab:non-funcs}, we can see that the range of decrease for different associations is almost the same for the both MIC and U-MIC. We expected this for MIC since it has an important property called equitability \cite{MIC:2011}. On the other hand, we can observe that at least according to the simulation results reported here, U-MIC has approximately the same equitability property.
\section{Conclusion}\label{sec:con}
In this paper we introduced a novel measure of dependency between two variables. This measure is called the uniform maximal information coefficient (U-MIC) because it is a modification of the original MIC \cite{MIC:2011}. It is derived from uniform partitioning of the both $X$ and $Y$ axes. Therefore, it is not dealing with dynamic programming similar to what the MIC does and hence, is much faster. We proved that asymptotically, U-MIC equals to 1 if there is a functional relationship between two variables. If two variables are truly independent from each other, then we showed that the U-MIC would be equal to 0. Specifically, according to the simulation results, we showed that the U-MIC does a better job in recognizing independence between variables comparing to the MIC.

\begin{figure}[t]
\centering
\includegraphics[width=80mm]{noisy_rel.eps}
\caption{Test noisy non-functional relationships for Tables \ref{tab:noisy-non-funcs} and \ref{tab:noisy-timenon-funcs}. }
\label{fig:noisy-non-funcs}
\end{figure}

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  | }
 \hline
 &  Circle &  Sinusoidal Mixture & Two Lines   \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 0.54 &0.60 &0.57  \\
 \hline
 \textbf{U-MIC}& 0.52 &0.48 & 0.54 \\
 \hline
\end{tabular}}
\caption{ MIC($D$) and U-MIC($D$) for different non-functional relationships in Figure \ref{fig:non-funcs}. For this set of experiments, $|D|=200$ and noise is uniformly distributed in [-0.05,0.05].}
\label{tab:noisy-non-funcs}
\end{center}
\end{table}

\begin{table}[ht!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c || c  c  c  | }
 \hline
 &  Circle &  Sinusoidal Mixture & Two Lines \\ [0.5ex]
 \hline\hline
 \textbf{MIC}& 35 &16 & 27 \\
 \hline
 \textbf{U-MIC}& 0.01 & 0.02 &  0.01  \\
 \hline
\end{tabular}}
\caption{ Run time (in sec.) for calculation of MIC($D$) and U-MIC($D$) for different noisy non-functional relationships in Figure \ref{fig:non-funcs}. For this set of experiments, $|D|=200$ and noise is uniformly distributed in [-0.05,0.05].}
\label{tab:noisy-timenon-funcs}
\end{center}
\end{table}

\bibliographystyle{IEEETran}

redpajama_set_name: RedPajamaArXiv


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Top with grapefruit sections and sliced strawberries.
Chill the fruit first for cold juice or add some ice cubes while blending.
In order to better help you identify recipes that feature a high concentration of nutrients for the calories they contain, we created a Recipe Rating System. This system allows us to highlight the recipes that are especially rich in particular nutrients. The following chart shows the nutrients for which Grapefruit Sunrise is either an excellent, very good, or good source (below the chart you will find a table that explains these qualifications). If a nutrient is not listed in the chart, it does not necessarily mean that the recipe doesn't contain it. It simply means that the nutrient is not provided in a sufficient amount or concentration to meet our rating criteria. (To view this recipe's in-depth nutritional profile that includes values for dozens of nutrients - not just the ones rated as excellent, very good, or good - please use the link below the chart.) To read this chart accurately, you'll need to glance back up to see the ingredients used in the recipe and the number of serving sizes provided by the recipe. Our nutrient ratings are based on a single serving. For example, if a recipe makes 4 servings, you would be receiving the nutrient amounts listed in the chart by eating 1/4th of the combined ingredients found in the recipe. Now, returning to the chart itself, you can look next to the nutrient name in order to find the nutrient amount it offers, the percent Daily Value (DV%) that this amount represents, the nutrient density that we calculated for this recipe and nutrient, and the rating we established in our rating system. For most of our nutrient ratings, we adopted the government standards for food labeling that are found in the U.S. Food and Drug Administration's "Reference Values for Nutrition Labeling." Read more background information and details of our rating system.


The Altherm JCR Yamaha team was riding high at the annual Bay of Plenty Motocross's Whakatane Summercross.
There were two riders on track but the usual team make-up differed slightly, as after winning the Huka Honda MX Fest's MX1 class in October, Kayne Lamont tore the ACL in his knee and sat out today's racing. Team Manager Josh Coppins stepped in for this one race meeting and showed the young bucks he still has what it takes, finishing third overall in MX1.
Altherm JCR Yamaha's MX2 rider, Mangakino's Maximus Purivs totally dominated the day – winning five from five races. Riding his Yamaha YZ250F, Purvis raced the back-to-back format of two motos in a row – twice - followed by a single race at the end of the day.
The up-and-coming 18-year-old is in fine form for his first full season in MX2 after taking home the winner's trophy from the recent MX Fest.
Attracting hundreds of riders from New Zealand, Australia and England, plus thousands of spectators, the Awakaponga circuit, near Whakatane, is one of the best motocross viewing venues in the country. The fans were in for a special treat, watching former international motocross GP star Coppins provide the Altherm JCR Yamaha team with a MX1 presence and his outings in the five races were a credible 4-3-3-4-4.
"It was a very good day for Yamaha and at 33 degrees, Summercross really lived up to its name. I felt like I filled in alright. Third was about as good as I could expect with my lack of fitness these days. I enjoyed it and it was fun but I'm more than happy to hand it over to the professionals for the next race," Coppins laughs.
The team will expand for the Woodville Grand Prix at the end of January.
"We have Australian Kurt Gibbs, who is the defending NZ champion starting racing with us in Woodville. We will also have Kayne back from injury, so we have a really strong MX1 outfit for Altherm JCR Yamaha moving into 2019," Coppins says.
"Yamaha also has Rhys Carter, who finished second in MX1 today, racing under the Yamalube label, so the brand is looking solid for the four-round nationals. Rhys will be heading up our coaching programme in April too," Coppins adds.
For him, the Summercross meeting was about more than just the individual results and looking at the bigger picture, what was really positive for Coppins, was Yamaha's overall presence.
"We were dominant in Women's' racing with 1, 2, 3 on the podium and were really strong winning some of the Vet's racing. Then despite not taking the 125cc class win, we made up the other top placings through to fifth. There were a lot of good customers getting great results, which is something we really want to build on and improve on in 2019," Coppins says.


Listen to another ferocious new song from the Canadian punks' forthcoming Deep Fantasy.
White Lung'sDeep Fantasy is rapidly becoming one of our most anticipated records of 2014, if only because, based on every song we've heard so far, it'll be the riff-heaviest album in recent memory. Plus, no one spits bile like the almighty Mish Way. If you needed more proof, check out the band's new song "Face Down" and its accompanying video, which stars the band headbanging around L.A. in a Volvo, burning scarecrows and playing air guitar. It's a great track too, with guitarist Kenneth William laying down smart, concise blasts of noise as Way sneers her way through lyrics like "all the world's pretend" and Anne Marie Vassilou's drums careen towards trainwreck. Check it out - Deep Fantasy is out June 17th.
REC'D: White Lung, "Snake Jaw"


text:
Q: Session timeout in Python Here is the code I am working with.
import requests

headers = { 'Accept':'*/*',
            'Accept-Language':'en-US,en;q=0.8',
            'Cookie':'Cookie:PHPSESSID=vev1ekv3grqhh37e8leu1coob1',
            'Cache-Control':'max-age=0',
            'Connection':'keep-alive',
            'Proxy-Authorization':'Basic ZWRjZ3Vlc3Q6ZWRjZ3Vlc3Q=',
            'If-Modified-Since':'Fri, 13 Nov 2015 17:47:23 GMT',
            'User-Agent': 'Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/48.0.2564.116 Safari/537.36'
            }

with requests.Session() as c:
    url = 'http://172.31.13.135/tpo/spp/'
    c.get(url, headers=headers)
    payload = {'regno': 'myregno', 'password': 'mypassword'}
    c.post(url, data = payload, headers=headers)
    r = c.get('http://172.31.13.135/tpo/spp/home.php', headers=headers)
    print r.content

I get the following message when I run this script.
<script>
alert("Session timeout !");
window.location = "logout.php";
</script><script>
alert("Unauthorised Access!");
window.location = "index.php";
</script>

<!DOCTYPE html>
<html lang="en">

How do I deal with this "session timeout" issue ?
Many thanks in advance.

A: It really makes tough to answer when I can't visit the website to scrape.
So here's my guess,
1) Try removing cookies from your headers you don't need that.
Because requests.Session() will generate cookies of its own when you visit url = 'http://172.31.13.135/tpo/spp/' for the first time.
So your headers will be,

headers = { 'Accept':'*/*',
            'Accept-Language':'en-US,en;q=0.8',
            'Cache-Control':'max-age=0',
            'Connection':'keep-alive',
            'Proxy-Authorization':'Basic ZWRjZ3Vlc3Q6ZWRjZ3Vlc3Q=',
            'If-Modified-Since':'Fri, 13 Nov 2015 17:47:23 GMT',
            'User-Agent': 'Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/48.0.2564.116 Safari/537.36'
            }

2) Make sure that 'If-Modified-Since' field in header is static to what you have mentioned and it doesn't change. If it does change then please code it accordingly to set the date and time on realtime basis.
3) I am not sure why you have 'Proxy-Authorization':'Basic ZWRjZ3Vlc3Q6ZWRjZ3Vlc3Q=' in headers. Try headers without it.
But, if you have to have it then please make sure that this auth code is static too and it doesn't change everytime.
Let me know if that helps

A: You could also just pass the timeout variable in your get and post method:
timeout = 10  # ten sec.
with requests.Session() as c:
    url = 'http://172.31.13.135/tpo/spp/'
    c.get(url, headers=headers, timeout=timeout)
    payload = {'regno': 'myregno', 'password': 'mypassword'}
    c.post(url, data = payload, headers=headers, timeout=timeout)
    r = c.get('http://172.31.13.135/tpo/spp/home.php', headers=headers, timeout=timeout)
    print r.content

You could tweak the max time you wan to wait for a response by doing this
For more about request see the docs

redpajama_set_name: RedPajamaStackExchange


Yesterday was a quiet day. David blogged about beer, Jason blogged about Hype Cycles, Gartner, and Spend Management, The iPod: Contract Manufactured With Love (well, almost), and Measuring the Impact of War in Israel and Lebanon, and there was silence over at Vendor Management. The only activity was Eric Strovink's comment to Dave's post where he noted that we need to step back from the RFP "blunt instrument" a bit since simple but smart actions sometimes generate surprising results. More importantly, he asked Do we clearly understand the demand side of spend?. This is a very good question, and one I'd bet that we rarely really know the answer to. After all, if we did – we wouldn't need "forecasting" tools, "demand measurement" tools would do just fine.
On a different note, I can tell you that not only are more leading bloggers crafting their predictions even as I type this but that you can look forward to at least one surprise guest contribution on the future of sourcing here at Sourcing Innovation in the next week. Note that my four-part (yes, 4) series on Optimization begins today on eSourcing Forum. As I mentioned earlier, it will explain why I agree with Tim that decision-optimization guided sourcing is about to enter into the sourcing mainstream in a big way.
Getting back to the future of sourcing, I meant what I said when I said that I believe that sourcing five years from now will not be the same as sourcing today. For starters, it will be a lot more strategic. After all, once the 800 pound gorillas incorporate basic procurement and sourcing functionality into their application suites and the open source start-ups provide you with free applications for the basic information gathering, negotiation, and contract tracking portions of your sourcing cycle, the only way a(n) (e-)Sourcing provider will survive is by taking sourcing to the next level and developing new applications that allow you to define, analyze, execute, and access a databank of multiple sophisticated sourcing strategies with ease. Whereas today RFXs are templated, in the sourcing world of tomorrow entire sourcing cycles will be templated with integrated optimization-based decision support analytical tools that will help you determine make-vs-buy, short-cycle vs long-cycle, internal vs shared innovation, etc. Total Value Management will be the norm, Lead Time Optimization will be automatically accounted for, and demand forecasts will be based on dynamically updating collaborative multi-channel analysis. Tactics will be automated … strategy will be everything. And I'm eagerly anticipating Jason's response to this prediction!
This entry was posted in Sourcing Innovation, Technology on August 25, 2006 by thedoctor.


text:
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Akhil Akkineni, the latest sensation of Telugu Cinema is all set to make his grand debut on screen in three days.
Akhil's first film titled Akhil is releasing on the Diwali day and the buzz around this film is growing by the day. Akkineni fans are kicked about Akhil's debut and even the movie lovers are keen to watch the youngster in action.
Although Akhil is very young he is showing a lot of maturity in his interviews. He has clarity over what he is doing and what he wanted to do. Akhil is targeting to become a big star and he is not underplaying it.
He is frankly saying that his aim is to establish himself as a star. He respects the taste of audience and is not taking them for granted.
Akhil openly stated that Chiranjeevi has inspired him as a dancer. Also Prabhu Deva dances inspired him a lot to get thorough training in this department. Though his father and brother are not known to be good dancers Akhil has surprised everyone with his dancing skills.
Even Nagarjuna was shocked to see AKhil dance like a dream. Akhil is yet to take a decision about his second film. He is currently focused on promoting Akhil and is eager to see the response of audience.


text:
YLS Today
Professor Kate Stith Receives Educator Award
Analysis: The Lesson From Pence's (And Biden's) Closets: The Government Classifies Way Too Many Things Los Angeles Times
Too Many Top Secrets The New York Times
Sorry, That's Classified On the Media
It's Not JPMorgan's Fault If Frank Lied — A Commentary by Stephen L. Carter '79 The Washington Post
The Connecticut Bar Association (CBA) presented Kate Stith with the 2018 Tapping Reeve Legal Educator Award on April 12, 2018. The award is presented to educators who have made significant contributions to the cause of legal education over a period of years and have distinguished themselves as a legal educator of the highest quality.
The award was presented at the CBA's annual "Celebrate with the Stars" awards ceremony in April at the Aqua Turf Club in Plantsville.
Kate Stith is the Lafayette S. Foster Professor of Yale Law School, where she teaches and writes in the areas of criminal law, criminal procedure, constitutional law, sentencing, and prosecutorial ethics. Professor Stith joined the Yale Law School faculty in 1985 and has served as deputy dean and acting dean of the law school. Prior to joining the faculty at Yale, Professor Stith was an Assistant United States Attorney for the Southern District of New York, where she prosecuted white-collar and organized-crime cases. A graduate of Dartmouth College, the Kennedy School of Government, and Harvard Law School, she clerked for Judge Carl McGowan of the U.S. Court of Appeals for the District of Columbia and for Supreme Court Justice Byron R. White.
"At Yale, Kate's devotion to students is renowned," said Yale Law School Dean Heather K. Gerken. "Since 1985, she has guided generations through law school and beyond. Her students have described her as inspiring, knowledgeable, and brilliant."
"Kate Stith is a respected teacher and scholar. Her focus on criminal procedure and sentencing guidelines are particularly relevant at a time when there is an active national conversation on criminal reforms," said Karen DeMeola, president of the CBA. "As lawyers, members of the CBA have a particular interest in maintaining the rule of law and protecting the liberty of all Americans. Professor Stith's work embodies those values and should be recognized."
About the Connecticut Bar Association
The Connecticut Bar Association, founded in 1875, is the preeminent organization for lawyers and the legal profession in Connecticut. With its 70+ sections and committees, the CBA produces more than 300 programs each year, including the Connecticut Legal Conference. The CBA is a nonprofit member service organization dedicated to advancing the legal profession and the principles of law and justice.
Harvard and Yale to Hold Law School Conference on Best Practices for Law School Data
Professor Guido Calabresi at a celebration of his 75th birthday in 2007.


Once cultivated by Persian kings and believed to have healing powers, saffron is now fuelling the growth of a small German business that imports tons of the spice from Iran to make fine food products for sale in Europe and the Gulf.
"We try to capture the soul of saffron and the magic it contains," says Michael Sabet, an Iranian-German business executive who quit his banking job six years ago to found Miasa GmbH, which is now doubling its revenues every year. Ream More via Reuters.