Description
1 HW2: The Distribution of Movie Ratings
Movie ratings are numeric scores summarizing the quality of a movie. In this assignment, we study two
sources of ratings:
• up-to-date, recently tweeted movie ratings (from MovieTweetings)
• historical movie rating averages (from IMDb).
Ratings from both of these sources are numeric values ranging from 0 to 10.
Perhaps remarkably, the distribution for the two kinds of ratings look the same. The goal of this assignment is to characterize the movie ratings distribution.
You are supposed to produce an output file answering four sets of questions (11 questions in all):
1. characterizing the distribution of live MovieTweetings movie ratings
2. studying the differences between average and median MovieTweetings ratings.
3. characterizing the distribution of archival IMDb movie ratings
4. analyzing skewness of the Gamma distribution.
The details of these questions are laid out in the notebook. And as it explains, your program should produce
a CSV file HW2_output.csv answering the questions. A correct output file could look like this:
lognormal,5.55555,1.11111
skewness,2.22222,
kurtosis,3.33333,
Batman: The Dark Knight,3.33333,8.88888
Batman v Superman: Dawn of Justice,9.11111,9.55555
lognormal,5.22222,1.44444
skewness,2.55555,
kurtosis,3.44444,
1,,
False,,
This is just an example of the format of an output file; your output file will be different.
This file characterizes the distribution of ratings as a lognormal distribution. This is not correct: the
ratings distribution clearly cannot be lognormal, since it is negatively skewed (it leans to the right) whereas
the lognormal distribution is positively skewed (it leans to the left).
Another distribution is needed. The notebook suggests some candidate distributions as possibilities, but
you job is to identify one, and obtain the best fit (i.e., maximal likelihood parameters) for the data.
The notebook does not give as much guidance as the earlier assignment notebooks. However, this is also
a short assignment. To complete this assignment, please upload two files to CCLE:
1
1. your output CSV file HW2_output.csv
2. your notebook file HW2_Movie_Ratings.ipynb (to show your work).
The notebook should have the commands you used to produce the output file. All assignment grading in
this course will be automated, so please assume that when uploading files.
We will use Paul Eggert’s Late Policy: The number of days late is N = 0 for the first 24 hrs, N = 1
for the next 24 hrs, etc., and if you submit an assignment H hours late, 2bH/24c points are deducted.
2 Part 1: Live Movie Ratings – extracted from Tweets
2.1 Live Movie Ratings are available at GitHub
These ratings are updated automatically online by a process scanning current Tweets; see the Movie Tweetings page of Simon Dooms. The information in the tweets has been digested into three tables – about movies,
users, and ratings. (Up-to-date snapshots and archives are also available.)
In [142]: URL = “https://raw.githubusercontent.com/sidooms/MovieTweetings/master/latest/ratings.dat”
Ratings = read.table( URL, sep = “:”, header=FALSE )[,c(1,3,5,7)]
colnames(Ratings) = c(“UserID”, “MovieID”, “Rating”, “TwitterID”)
head(Ratings)
# if your connection to github fails when retrieving this dataset, persist in trying
Out[142]:
UserID MovieID Rating TwitterID
1 1 68646 10 1381620027
2 1 113277 10 1379466669
3 2 422720 8 1412178746
4 2 454876 8 1394818630
5 2 790636 7 1389963947
6 2 816711 8 1379963769
In [143]: dim(Ratings) # not a tiny dataset
Out[143]:
1. 489378
2. 4
2.2 Exploration of the Live Ratings
In [144]: # Summary statistics
summary( Ratings$Rating )
Out[144]: Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000 6.000 7.000 7.304 9.000 10.000
In [145]: # Count rating values with table()
CountOfRatings = as.data.frame(table( Ratings$Rating, dnn=”rating” ), responseName=”count”)
CountOfRatings
2
Out[145]:
rating count
1 0 95
2 1 5844
3 2 5105
4 3 8625
5 4 16114
6 5 34932
7 6 64206
8 7 109957
9 8 118612
10 9 69420
11 10 56468
In [146]: options( repr.plot.width=4, repr.plot.height=4 ) # control plot dimensions
In [147]: # Histogram of Rating values (integer values from 0 to 10)
h = hist( Ratings$Rating, breaks = 0:11, col=”cyan” ) # save and plot the histogram
mtext( sprintf(“histogram of %d tweeted ratings”, length(Ratings$Rating)) )
In [148]: barplot( h$counts, names.arg=0:10, col=”aquamarine”,
main=”latest MovieTweetings rating values” )
3
In [149]: hist( Ratings$Rating, probability=TRUE, col=”palegreen”, breaks=0:11 )
rating_avg = mean(Ratings$Rating)
rating_sd = sd(Ratings$Rating)
curve( dnorm(x, mean=rating_avg, sd=rating_sd),
col=”red”, add=TRUE ) # add a curve with the normal MLE
mtext(sprintf(“Ratings with superimposed normal fit (mean %4.1f, sd %4.1f)”,
rating_avg, rating_sd), cex=0.65)
4
3 Problem 1: Find a distribution that fits the histogram of MovieTweetings Rating values
Question 1: Give the name of a specific distribution (pdf), with maximum likelihood parameter values,
that resembles the MovieTweetings Rating values (as closely as you can). To permit distributions like the
Beta distribution to be considered, you can scale the rating values. For example, dividing the values by 10
puts them in the interval [0,1], as the Beta distribution requires.
Question 2: determine the skewness of the MovieTweetings ratings.
Question 3: determine the (excess) kurtosis of the MovieTweetings ratings.
The skewness and excess kurtosis values ought to be near zero if the data is normally distributed.
Inspecting them is a simple check of whether the data follows a normal distribution.
3.1 You can use the fitdistr() function in this assignment
In [150]: not.installed = function(package_name) !is.element(package_name, installed.packages()[,1])
if (not.installed(“MASS”)) install.packages(“MASS”)
library(MASS)
# example(fitdistr) # run examples showing use of the fitdistr() function
In [151]: # A start at analysis, which needs work:
hist( Ratings$Rating, probability=TRUE, col=”palegreen”, breaks=0:11 )
5
theta = fitdistr( Ratings$Rating+1, “lognormal” )
curve( dlnorm(x+0.5, meanlog=theta$estimate[1], sdlog=theta$estimate[2]),
col=”red”, add=TRUE ) # add a curve with the MLE fit for the lognormal density
mtext( “the lognormal distribution doesn’t fit well”, col=”red” )
mtext( sprintf(“MLE parameters: %5.2f, %5.2f”,
theta$estimate[1], theta$estimate[2]), side=1, col=”blue”)
3.2 “Trending” Movies: movies with more than 50 current ratings
In [152]: NumberOfRatings = data.frame(aggregate( Rating ~ MovieID, length, data=Ratings ))
colnames(NumberOfRatings) = c(“MovieID”, “NumberOfRatings”)
TrendingMovies = subset( NumberOfRatings, NumberOfRatings > 50 )
head(TrendingMovies)
Out[152]:
MovieID NumberOfRatings
225 21749 51
235 22100 51
451 27977 60
555 31381 87
590 32138 61
655 33467 83
In [153]: nrow(TrendingMovies)
6
Out[153]:
1615
In [154]: max(TrendingMovies$NumberOfRatings)
Out[154]:
2951
3.3 Averaged Ratings of Trending Movies
In [155]: RatingsOfTrendingMovies = merge( Ratings, TrendingMovies, by=”MovieID” ) # join of tables
head(RatingsOfTrendingMovies)
Out[155]:
MovieID UserID Rating TwitterID NumberOfRatings
1 21749 31658 7 1411560342 51
2 21749 14766 10 1408217153 51
3 21749 3724 9 1419690175 51
4 21749 13477 10 1455471595 51
5 21749 2564 10 1460041235 51
6 21749 4577 8 1422001419 51
In [156]: AveragedRatingsOfTrendingMovies = aggregate( Rating ~ MovieID, mean,
data=RatingsOfTrendingMovies )
head(AveragedRatingsOfTrendingMovies)
hist(AveragedRatingsOfTrendingMovies$Rating, breaks=25,
col=”orange”, main=”avg trending MovieTweeting ratings”)
Out[156]:
MovieID Rating
1 21749 8.705882
2 22100 8.529412
3 27977 8.633333
4 31381 8.804598
5 32138 8.721311
6 33467 8.662651
7
In [157]: maxAveragedRating = max(AveragedRatingsOfTrendingMovies$Rating)
maxAveragedRating
Out[157]:
9.43649635036496
In [158]: HotTrendingMovies = subset(AveragedRatingsOfTrendingMovies, Rating == maxAveragedRating)
HotTrendingMovies
Out[158]:
MovieID Rating
188 111161 9.436496
4 Download the corresponding MovieTweetings movie Name &
Genre information
In [159]: URL = “https://raw.githubusercontent.com/sidooms/MovieTweetings/master/latest/movies.dat”
MovieText = readLines( URL )
Movies = matrix( sapply( MovieText,
function(x) unlist(strsplit(sub(” [(]([0-9]+)[)]”, “::\\1″,x),”::”))[1:4] ),
nrow=length(MovieInformation), ncol=4, byrow=TRUE )
colnames(Movies) = c(“MovieID”, “MovieTitle”, “Year”, “Genres”)
head(Movies)
8
Out[159]:
MovieID MovieTitle Year Genres
0000008 Edison Kinetoscopic Record of a Sneeze 1894 Documentary—Short
0000010 La sortie des usines Lumi¡c3¿¡a8¿re 1895 Documentary—Short
0000012 The Arrival of a Train 1896 Documentary—Short
0000091 Le manoir du diable 1896 Short—Horror
0000417 Le voyage dans la lune 1902 Short—Adventure—Fantasy
0000439 The Great Train Robbery 1903 Short—Action—Crime
4.1 joining the Ratings and Trending Movie information
In [160]: Ratings_and_Movies = merge( RatingsOfTrendingMovies, Movies, by=”MovieID” )
head(Ratings_and_Movies)
Out[160]:
MovieID UserID Rating TwitterID NumberOfRatings MovieTitle Year Genres
1 1001526 32702 6 1453606362 59 Megamind 2010 Animation—Action—Comedy
2 1001526 17195 8 1375450650 59 Megamind 2010 Animation—Action—Comedy
3 1001526 22716 7 1387735390 59 Megamind 2010 Animation—Action—Comedy
4 1001526 2086 7 1367107409 59 Megamind 2010 Animation—Action—Comedy
5 1001526 38072 7 1452036480 59 Megamind 2010 Animation—Action—Comedy
6 1001526 22420 4 1383345146 59 Megamind 2010 Animation—Action—Comedy
5 Problem 2: Compare Average vs. Median Rating values for
Trending Movies
After computing Average and Median Rating values for each Trending Movie in the MovieTweetings data:
Question 4: find the name of the movie with the highest Average Rating (and also list its Median
Rating and Average Rating). If there is more than one such movie, select any one.
Question 5: find the name of the movie with the largest difference |Median Rating – Average Rating|
(and also list its Median Rating and Average Rating). If there is more than one such movie, select any one.
For describing a skewed distribution, the median can be more informative than the mean.
In [161]: # Hint:
# ? aggregate
# ? by
6 Part 2. Historical Movie Ratings – from IMDb
In this part we analyze an historical dataset of movies with ratings from IMDb.
In [162]: # source is at: https://github.com/hadley/ggplot2movies
if (not.installed(“ggplot2movies”)) install.packages(“ggplot2movies”)
library(ggplot2movies)
data(movies)
dim(movies) # also not a tiny dataset
9
Out[162]:
1. 58788
2. 24
In [163]: summary(movies)
?movies
Out[63]: title year length budget
Length:58788 Min. :1893 Min. : 1.00 Min. : 0
Class :character 1st Qu.:1958 1st Qu.: 74.00 1st Qu.: 250000
Mode :character Median :1983 Median : 90.00 Median : 3000000
Mean :1976 Mean : 82.34 Mean : 13412513
3rd Qu.:1997 3rd Qu.: 100.00 3rd Qu.: 15000000
Max. :2005 Max. :5220.00 Max. :200000000
NA’s :53573
rating votes r1 r2
Min. : 1.000 Min. : 5.0 Min. : 0.000 Min. : 0.000
1st Qu.: 5.000 1st Qu.: 11.0 1st Qu.: 0.000 1st Qu.: 0.000
Median : 6.100 Median : 30.0 Median : 4.500 Median : 4.500
Mean : 5.933 Mean : 632.1 Mean : 7.014 Mean : 4.022
3rd Qu.: 7.000 3rd Qu.: 112.0 3rd Qu.: 4.500 3rd Qu.: 4.500
Max. :10.000 Max. :157608.0 Max. :100.000 Max. :84.500
r3 r4 r5 r6
Min. : 0.000 Min. : 0.000 Min. : 0.000 Min. : 0.00
1st Qu.: 0.000 1st Qu.: 0.000 1st Qu.: 4.500 1st Qu.: 4.50
Median : 4.500 Median : 4.500 Median : 4.500 Median :14.50
Mean : 4.721 Mean : 6.375 Mean : 9.797 Mean :13.04
3rd Qu.: 4.500 3rd Qu.: 4.500 3rd Qu.: 14.500 3rd Qu.:14.50
Max. :84.500 Max. :100.000 Max. :100.000 Max. :84.50
r7 r8 r9 r10
Min. : 0.00 Min. : 0.00 Min. : 0.000 Min. : 0.00
1st Qu.: 4.50 1st Qu.: 4.50 1st Qu.: 4.500 1st Qu.: 4.50
Median : 14.50 Median : 14.50 Median : 4.500 Median : 14.50
Mean : 15.55 Mean : 13.88 Mean : 8.954 Mean : 16.85
3rd Qu.: 24.50 3rd Qu.: 24.50 3rd Qu.: 14.500 3rd Qu.: 24.50
Max. :100.00 Max. :100.00 Max. :100.000 Max. :100.00
mpaa Action Animation Comedy
Length:58788 Min. :0.00000 Min. :0.00000 Min. :0.0000
Class :character 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.0000
Mode :character Median :0.00000 Median :0.00000 Median :0.0000
Mean :0.07974 Mean :0.06277 Mean :0.2938
3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:1.0000
Max. :1.00000 Max. :1.00000 Max. :1.0000
Drama Documentary Romance Short
Min. :0.000 Min. :0.00000 Min. :0.0000 Min. :0.0000
1st Qu.:0.000 1st Qu.:0.00000 1st Qu.:0.0000 1st Qu.:0.0000
Median :0.000 Median :0.00000 Median :0.0000 Median :0.0000
Mean :0.371 Mean :0.05906 Mean :0.0807 Mean :0.1609
3rd Qu.:1.000 3rd Qu.:0.00000 3rd Qu.:0.0000 3rd Qu.:0.0000
Max. :1.000 Max. :1.00000 Max. :1.0000 Max. :1.0000
Out[63]:
Movie information and user ratings from IMDB.com.movies
The internet movie database, http://imdb.com/, is a website devoted to collecting movie data
supplied by studios and fans. It claims to be the biggest movie database on the web and is run
by amazon. More about information imdb.com can be found online, http://imdb.com/help/
show_leaf?about, including information about the data collection process, http://imdb.com/
help/show_leaf?infosource.
10
movies
Format: a data frame with 28819 rows and 24 variables
• title. Title of the movie.
• year. Year of release.
• budget. Total budget (if known) in US dollars
• length. Length in minutes.
• rating. Average IMDB user rating.
• votes. Number of IMDB users who rated this movie.
• r1-10. Multiplying by ten gives percentile (to nearest 10%) of users who rated this movie a
1.
• mpaa. MPAA rating.
• action, animation, comedy, drama, documentary, romance, short. Binary variables representing if movie was classified as belonging to that genre.
Movies were selected for inclusion if they had a known length and had been rated by at least one
imdb user. http://had.co.nz/data/movies/
Out[65]:
title year length budget rating votes r1 r2 r3 r4 ellip.h r9 r10 mpaa Action Animation Comedy Drama Documentary Romance Short
1 $ 1971 121 NA 6.4 348 4.5 4.5 4.5 4.5 … 4.5 4.5 0 0 1 1 0 0 0
2 $1000 a Touchdown 1939 71 NA 6 20 0 14.5 4.5 24.5 … 4.5 14.5 0 0 1 0 0 0 0
3 $21 a Day Once a Month 1941 7 NA 8.2 5 0 0 0 0 … 24.5 24.5 0 1 0 0 0 0 1
4 $40,000 1996 70 NA 8.2 6 14.5 0 0 0 … 34.5 45.5 0 0 1 0 0 0 0
5 $50,000 Climax Show, The 1975 71 NA 3.4 17 24.5 4.5 0 14.5 … 0 24.5 0 0 0 0 0 0 0
6 $pent 2000 91 NA 4.3 45 4.5 4.5 4.5 14.5 … 14.5 14.5 0 0 0 1 0 0 0
In [66]: hist( movies$rating, col=”lawngreen”, main=”IMDb ratings”, breaks=50 )
mtext( sprintf(“histogram of %d movie average rating values”, nrow(movies)) )
11
7 Problem 2: Find a distribution that fits the histogram of IMDb
Ratings
Question 6: Give the name of a specific distribution (pdf), with maximum likelihood parameter values,
that resembles the IMDb Rating values (as closely as you can). Again, to permit distributions like the Beta
distribution to be considered, you can scale the rating values. For example, dividing the values by 10 puts
them in the interval [0,1], as the Beta distribution requires.
Question 7: determine the skewness of the IMDb ratings.
Question 8: determine the (excess) kurtosis of the IMDb ratings.
7.1 Possibilities: the Extreme Value Distribution, or related distributions
The (Smallest) Extreme Value Distribution models the minimum of a set of values drawn from a single
distribution. It is sometimes used for modeling sets of identical independent processes that can fail – and
the time of the first failure is the failure time of the entire set. In other words, the distribution models the
failure time of the weakest link. The Extreme Value Distribution is related to the Weibull Distribution.
7.2 Possibility: the Beta Distribution
The Beta distribution is a general model for random values of percentages and proportions. It is used
very heavily in Bayesian methods. The distribution Beta(α,β) leans to the right when α > β. Note: As
mentioned above, movie rating values are in the interval [0,10] . . . The Beta distribution requires all values
to be nonnegative, and in the interval [0,1]. Thus the ratings would need to be rescaled here. Warning: the
fitdistr() function appears to be fragile when fitting a beta distribution. If you consider the Beta distribution,
obtaining MLE parameter values may require a different method.
7.3 Not a Possibility: the Negative Gamma Distribution
The Negative Gamma distribution is a “mirror image” of the Gamma Distribution, defined for x < 0 instead of x > 0. In other words, the value of the Negative Gamma distribution at −x is defined to be the value of
the Gamma distribution at +x. Do not consider it in this assignment.
8 Problem 3: Answer two Multiple-Answer questions about the
Gamma distribution
Question 9: Give the number of the following expressions that is a formula for the skewness of the Gamma
distribution with parameters α > 0, β > 0: (1) α/β (2) α − β (3) p
α/β (4) √
αβ/2 (5) 2/
√
α (6) 6/α (7)
log(α)/β (8) None of the above
Question 10: True or False: the Gamma distribution is not negatively skewed, for any parameter values
α > 0, β > 0.
(Again, for this assignment please consider only the usual Gamma Distribution, defined for x > 0.) Hint:
the skewness measure is the result of an integral for E[((x − µ)/σ)
3
], so both of these questions could be
answered with a symbolic algebra system, like SymPy or Wolfram Alpha. Some online resources also might
give a formula for skewness.
12
9 Finally: Produce a CSV file “HW2 output.csv” including your
answers
9.1 Your output CSV file “HW2 output.csv” should look like this:
lognormal,5.55555,1.11111
skewness,2.22222,
kurtosis,3.33333,
Batman: The Dark Knight,3.33333,8.88888
Batman v Superman: Dawn of Justice,9.11111,9.55555
lognormal,5.22222,1.44444
skewness,2.55555,
kurtosis,3.44444,
1,,
False,,
If your program had been given the Table above as input, it should print the following CSV file, a table
with 10 rows, and three columns:
There should be NO header line in this file. There should be 10 rows, one for each question. Each row
should have three fields. You can enter any text description of a distribution, such as lognormal or beta
or Beta or whatever. However: the lognormal distribution is just provided as an example; it is not a good
description of Rating distributions. Also: do not use the gamma distribution.
10 Submit your output CSV file and your notebook on CCLE.
Upload your .csv file for this assignment, and also upload your .ipynb file (to show your work). Both files
are required.
13
