lkh

.gitignore

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+.Rproj.user
+.Rhistory
+.RData
+.Ruserdata

README.txt

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+intentionally left blank

movielens-recommendation-system.R

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+# Calculate 10th percentile of movieId by rating
+movieId_cutoff <-
+  edx %>%
+  group_by(movieId) %>%
+  summarize(n = n()) %>%
+  .$n %>%
+  quantile(.10)
+
+paste("Movies below the 10th percentile with fewer than",
+      as.character(movieId_cutoff),
+      "ratings will be ignored.",
+      sep=" ")
+
+#Calculate 10th percentile of userId by rating
+userId_cutoff <-
+  edx %>% 
+  group_by(userId) %>%
+  summarize(n = n()) %>%
+  .$n %>%
+  quantile(.10)
+
+paste("Users below the 10th percentile with fewer than",
+      as.character(userId_cutoff),
+      "ratings will be ignored.",
+      sep=" ")
+
+# Remove any movie below the 10th percentile in terms of ratins
+edx2 <-
+  edx %>%
+  group_by(movieId) %>%
+  filter(n()>=movieId_cutoff) %>%
+  ungroup()
+
+# Remove any user below the 10th percentile in terms of ratings
+edx2 <-
+  edx2 %>%
+  group_by(userId) %>%
+  filter(n()>=userId_cutoff) %>%
+  ungroup()
+
+# Create partition, reserving 20% of edx set for testing purposes
+test_index <-
+  createDataPartition(y = edx2$rating,
+                      times = 1, p = 0.2, list = FALSE)
+train_set <- edx2[-test_index,] 
+test_set <- edx2[test_index,] 
+
+# Define loss function, and throw away any NA values that result from 
+RMSE <-
+  function(true_ratings, predicted_ratings){
+    sqrt(mean((true_ratings - predicted_ratings)^2, na.rm=TRUE))} 
+
+# Calcualte the average rating for the training set.
+mu_hat <- mean(train_set$rating)
+
+paste("Penalized User and Movie Effect Least Squares approach with a sample mean rating of",
+      as.character(mu_hat),
+      ", optimizing lambda to the nearest integer.",
+      sep=" ")
+
+# Using Movie and User Effects with Penalization on both, optimize lambda to the nearest integer.
+
+lambdas <- seq(1, 10, 1)
+
+RMSEs <- sapply(lambdas, function(l){
+  b_i <-
+    train_set %>%
+    group_by(movieId) %>%
+    summarize(b_i = sum(rating - mu_hat)/(n()+l))
+  
+  b_u <-
+    train_set %>%
+    left_join(b_i, by="movieId") %>%
+    group_by(userId) %>%
+    summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+  
+  predicted_ratings <-
+    test_set %>%
+    left_join(b_i, by = "movieId") %>%
+    left_join(b_u, by = "userId") %>%
+    mutate(pred = mu_hat + b_i + b_u) %>%
+    pull(pred)
+  
+  return(RMSE(predicted_ratings, test_set$rating))
+  
+})
+
+lambda_best <- lambdas[which.min(RMSEs)]
+
+paste("Penalized LSE Model with lambda =",
+      as.character(lambda_best),
+      "gives an RMSE of",
+      as.character(min(RMSEs)),
+      "on the test set.",
+      sep=" ")
+
+# Use this optimized lambda value to test against the validation set.
+paste("Now testing this optimized lambda against the vaildation set.")
+
+l <- lambda_best
+b_i <- train_set %>%
+  group_by(movieId) %>%
+  summarize(b_i = sum(rating - mu_hat)/(n()+l))
+b_u <- train_set %>%
+  left_join(b_i, by="movieId") %>%
+  group_by(userId) %>%
+  summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+final_predicted_ratings <-
+  validation %>%
+  left_join(b_i, by = "movieId") %>%
+  left_join(b_u, by = "userId") %>%
+  mutate(pred = mu_hat + b_i + b_u) %>%
+  pull(pred)
+final_rmse <-
+  RMSE(final_predicted_ratings, validation$rating)
+
+paste("Penalized LSE Model with optimized lambda =",
+      as.character(lambda_best),
+      "gives an RMSE of",
+      as.character(final_rmse),
+      "on the validation set.",
+      sep=" ")

movielens-recommendation-system.Rmd

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+---
+title: "MovieLens Recommendation System"
+author: "Kaylee Robert Tejeda"
+date: "9/13/2021"
+output: pdf_document
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+## Introduction and Overview
+
+  One common application of machine learning is the creation of recommendation systems.  These are mathematical models that take as input the parameters associated with a particular observation to give as output the value of an unknown parameter associated with the observation as a prediction.  The more accurate this prediction is, the more useful and valuable it is to specific industries.  These industries use these predictions to make recommendations to users based on past behavior and the behavior of other similar users under similar circumstances.  For example, video distribution systems might use such a recommendation algorithm to create lists of suggested movies based on what is currently being watched by a particular user.
+
+### Goal Summary 
+
+  Our goal is to create such a recommendation algorithm.  Our data set containing movie ratings is provided by GroupLens known as the ["10M version of the MovieLens dataset"](https://grouplens.org/datasets/movielens/10m/), which contains 5 base variables per observation.  We will measure the accuracy of our model by comparing the ratings predicted for a subset of our data to the actual ratings for those observations, and calculate the root mean square error (RMSE) for the final set of predictions.  An RMSE less than 0.86490 will indicate a successful model for the purposes of this report.
+
+### Data
+
+The MovieLens data set contains 10000054 rows, 10677 movies, 797 genres and 69878 users. Each observation in the data set has the following variables associated with it:
+
+* `userId`: Unique user identification number.
+* `movieId`: Unique movielens movie identification number.
+* `rating`: User-provided ratings on a 5-star scale with half-star increments starting at 0.5
+* `timestamp`: Time of user-submitted review in epoch time, 
+* `title`: Movie titles including year of release as identified in [IMDB](http://www.imdb.com)
+* `genres`: A pipe-separated list of film genres
+
+
+### Preparation
+
+The main data set is split into a working set called "edx" and a second set representing 10% of the original set, called "validation", which will be only used for checking the model once it has been created, and will not be used at all during the model formation process.
+
+```{r data preparation, message=FALSE, echo=FALSE, warning=FALSE}
+
+##########################################################
+# Create edx set, validation set (final hold-out test set)
+##########################################################
+
+# Note: this process could take a couple of minutes
+
+if(!require(tidyverse)) install.packages("tidyverse", repos = "http://cran.us.r-project.org")
+if(!require(caret)) install.packages("caret", repos = "http://cran.us.r-project.org")
+if(!require(data.table)) install.packages("data.table", repos = "http://cran.us.r-project.org")
+
+library(tidyverse)
+library(caret)
+library(data.table)
+
+# MovieLens 10M dataset:
+# https://grouplens.org/datasets/movielens/10m/
+# http://files.grouplens.org/datasets/movielens/ml-10m.zip
+
+
+dl <- tempfile()
+download.file("http://files.grouplens.org/datasets/movielens/ml-10m.zip", dl)
+
+ratings <- fread(text = gsub("::", "\t", readLines(unzip(dl, "ml-10M100K/ratings.dat"))),
+                 col.names = c("userId", "movieId", "rating", "timestamp"))
+
+movies <- str_split_fixed(readLines(unzip(dl, "ml-10M100K/movies.dat")), "\\::", 3)
+colnames(movies) <- c("movieId", "title", "genres")
+
+movies <- as.data.frame(movies) %>% mutate(movieId = as.numeric(movieId),
+                                            title = as.character(title),
+                                            genres = as.character(genres))
+
+
+movielens <- left_join(ratings, movies, by = "movieId")
+
+# Validation set will be 10% of MovieLens data
+set.seed(1, sample.kind="Rounding") # if using R 3.5 or earlier, use `set.seed(1)`
+test_index <- createDataPartition(y = movielens$rating, times = 1, p = 0.1, list = FALSE)
+edx <- movielens[-test_index,]
+temp <- movielens[test_index,]
+
+# Make sure userId and movieId in validation set are also in edx set
+validation <- temp %>% 
+      semi_join(edx, by = "movieId") %>%
+      semi_join(edx, by = "userId")
+
+# Add rows removed from validation set back into edx set
+removed <- anti_join(temp, validation)
+edx <- rbind(edx, removed)
+
+rm(dl, ratings, movies, test_index, temp, movielens, removed)
+
+```
+
+The resulting working set "edx" is further split into a training set and a test set which will be used to evaluate various methods.  The test set comprises of 20% of the edx working set.  No attempt was made to ensure that all the users and movies in the test set are also in the training set.  This leads to a fortunate discovery later.
+
+## Analysis
+
+### Outline of Steps Taken
+
+1)  Baseline Naive-RMSE
+
+2)  Movie Effect Model
+
+3)  Notice NA's in predictions, need to adjust somehow (leads to insight that is worth mentioning)
+
+4)  Movie and User Effects Model with manual outlier pruning.
+
+5)  Penalized User and Movie effect with Lambda to the hundredth's place (overkill, but interesting).
+
+6)  Simplified Penalized Effect method with integer Lambda and statistical outlier pruning.
+
+7)  RMSE is now in range, test against validation set.
+
+
+### Exploration 
+
+#### Baseline Naive-RMSE
+
+  Just to get a baseline, a first attempt is made by calculating the arithmetical mean of all movie ratings in the test set, and that is used as a prediction for all movie ratings.  This is not a very good approach, but it will give us a worst-case error, of sorts.
+  
+```{r exploratory-data-analysis, message=FALSE, warning=FALSE, echo=FALSE}
+
+# Split the edx data set into a training set and a test set
+
+test_index <- createDataPartition(y = edx$rating, times = 1, p = 0.2, list = FALSE)
+train_set <- edx[-test_index,]
+test_set <- edx[test_index,]
+
+# Loss function that computes the RMSE for vectors of ratings and their corresponding predictors:
+
+RMSE <- function(true_ratings, predicted_ratings){
+sqrt(mean((true_ratings - predicted_ratings)^2))
+} 
+
+# Find the average of all ratings in the training set
+
+mu_hat <- mean(train_set$rating)
+
+#  First attempt:  Naive RMSE
+
+naive_rmse <- RMSE(test_set$rating, mu_hat)
+
+#  Keep track of RMSE results in a tibble
+
+
+method_string <- paste("Naive method using average rating of", as.character(mu_hat), sep=" ")
+rmse_results <- tibble(method = method_string, RMSE = naive_rmse)
+
+
+#rmse_results
+```
+
+####  Movie Effect Model
+The next approach was to account for a movie bias effect, since some movies naturally get higher ratings than others.  This is when "NA" values started showing up in the predicted ratings. To get past this error for now, the option "na.rm=TRUE" was added to the mean part of the loss function.  This effectively throws out any observation that does not have a matching user ID or movie title in the training set.  This was only a temporary fix and merited a closer look.
+
+```{r Notice NAs, echo=FALSE}
+#  Factor in movie bias the quick way
+
+movie_avgs <- train_set %>%
+group_by(movieId) %>%
+summarize(b_i = mean(rating - mu_hat))
+
+#  Calculate RMSE after accounting for movie bias
+
+predicted_ratings <- mu_hat + test_set %>%
+left_join(movie_avgs, by='movieId') %>%
+pull(b_i)
+
+bi_rmse <- RMSE(predicted_ratings, test_set$rating)
+
+rmse_results <- bind_rows(rmse_results,
+                          tibble(method="Movie Effect Model",  
+                                     RMSE = bi_rmse))
+
+# How many are we throwing out? 
+nas <- sum(is.na(predicted_ratings))
+paste("There are", as.character(nas), "NA values in our predictions, which need to be removed.", sep= " ")
+
+edx %>% group_by(userId) %>% summarize(reviews = n()) %>% ggplot(aes(reviews)) + geom_histogram(color = "black", bins = 50) + scale_x_log10() + ggtitle("Number of reviews per user") + xlab("Number of reviews (log scale)") +
+  ylab("Count of users") 
+
+edx %>% group_by(movieId) %>% summarize(reviews = n()) %>% ggplot(aes(reviews)) + geom_histogram(color = "black", bins = 50) + scale_x_log10() + ggtitle("Number of reviews per movie") + xlab("Number of reviews (log scale)") +
+  ylab("Count of movies") 
+
+```
+
+Not very many NA values are being generated.  The assumption is that movies with only a few ratings affect the mean and therefore the error without contribution much to the overall effect.  The same could be said for users who have only rated a few movies.  Removing these NA values allowed the RMSEs to be calculated, however a more rigorous approach needs to be formulated as this is only a temporary workaround.
+
+````{r remove NAs, echo=FALSE, message=FALSE}
+RMSE <- function(true_ratings, predicted_ratings){
+sqrt(mean((true_ratings - predicted_ratings)^2, na.rm=TRUE))
+} 
+
+# Calculate movie effect method after throwing out NAs from predicted ratings
+
+predicted_ratings <- mu_hat + test_set %>%
+left_join(movie_avgs, by='movieId') %>%
+pull(b_i)
+
+bi_rmse <- RMSE(predicted_ratings, test_set$rating)
+
+rmse_results <- bind_rows(rmse_results,
+                          tibble(method="Movie Effect Model, na.rm=TRUE",  
+                                     RMSE = bi_rmse))
+
+#rmse_results
+```
+
+####  Move and User Effect Model
+
+Similar to the Movie Effect model, user bias can also be accounted for.
+
+````{r more effects, echo=FALSE, message=FALSE}
+#  Now account for user bias as well, quick method:
+
+user_avgs <- train_set %>%
+left_join(movie_avgs, by='movieId') %>%
+group_by(userId) %>%
+summarize(b_u = mean(rating - mu_hat - b_i))
+
+#  Check the new RMSE
+
+predicted_ratings <- test_set %>%
+left_join(movie_avgs, by='movieId') %>%
+left_join(user_avgs, by='userId') %>%
+mutate(pred = mu_hat + b_i + b_u) %>%
+pull(pred)
+
+bi_bu_rmse <- RMSE(predicted_ratings, test_set$rating)
+
+rmse_results <- bind_rows(rmse_results,
+                          tibble(method="Movie and User Effects Model",  
+                                     RMSE = bi_bu_rmse))
+
+
+#rmse_results
+```
+
+####  Penalized Movie Effect model.
+
+Now that we have developed a model that performs better than the Naive RMSE approach that we began with, we can start to regularize the effects by penalizing with a parameter we will call lambda.  We start by penalizing both the Movie and User effects with the penalization factor lambda arbitrarily fixed at 3.
+
+````{r penalized approach, echo=FALSE, message=FALSE}
+
+#  Now try a Penalized Movie and User Effects estimate with lambda = 3
+
+lambda <- 3
+#movie_reg_avgs <- train_set %>%
+#group_by(movieId) %>%
+#summarize(b_i = sum(rating - mu_hat)/(n()+lambda), n_i = n())
+
+b_i <-  train_set %>% group_by(movieId) %>% summarize(b_i = sum(rating - mu_hat)/(n()+lambda))
+b_u <- train_set %>% left_join(b_i, by="movieId") %>% group_by(userId) %>% summarize(b_u = sum(rating - b_i - mu_hat)/(n()+lambda))
+
+#  Check RMSE
+
+#predicted_ratings <- test_set %>%
+#left_join(movie_reg_avgs, by = "movieId") %>%
+#mutate(pred = mu_hat + b_i) %>%
+#pull(pred)
+predicted_ratings <- test_set %>% left_join(b_i, by = "movieId") %>% left_join(b_u, by = "userId") %>% mutate(pred = mu_hat + b_i + b_u) %>% pull(pred)
+
+
+
+lambda_rmse <- RMSE(predicted_ratings, test_set$rating)
+
+
+rmse_results <- bind_rows(rmse_results,
+                          tibble(method="Penalized Movie Effect Model, lambda=3",  
+                                     RMSE = lambda_rmse))
+#rmse_results
+```
+
+#### Step-Wise Cross-Validation with Lambda optimized to two decimal places.
+
+Iterated cross-validation can be used to quickly optimize the penalization factor lambda to two decimal places of precision.  First, lambda is optimized to the nearest integer, then it is optiized again to the nearest tenth, and then again to the nearest hundredth.
+
+````{r optimized penalized approach, echo=FALSE, message=FALSE}
+#  Now optimize lambda using step-wise cross-validation
+
+lambdas <- seq(1, 10, 1)
+RMSEs <- sapply(lambdas, function(l){
+b_i <-  train_set %>% group_by(movieId) %>% summarize(b_i = sum(rating - mu_hat)/(n()+l))
+b_u <- train_set %>% left_join(b_i, by="movieId") %>% group_by(userId) %>% summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+predicted_ratings <- test_set %>% left_join(b_i, by = "movieId") %>% left_join(b_u, by = "userId") %>% mutate(pred = mu_hat + b_i + b_u) %>% pull(pred)
+return(RMSE(predicted_ratings, test_set$rating))
+})
+
+lambda_best <- lambdas[which.min(RMSEs)]
+
+lambdas <- seq(lambda_best - 1 , lambda_best + 1 , 0.1)
+RMSEs <- sapply(lambdas, function(l){
+b_i <-  train_set %>% group_by(movieId) %>% summarize(b_i = sum(rating - mu_hat)/(n()+l))
+b_u <- train_set %>% left_join(b_i, by="movieId") %>% group_by(userId) %>% summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+predicted_ratings <- test_set %>% left_join(b_i, by = "movieId") %>% left_join(b_u, by = "userId") %>% mutate(pred = mu_hat + b_i + b_u) %>% pull(pred)
+return(RMSE(predicted_ratings, test_set$rating))
+})
+
+lambda_best <- lambdas[which.min(RMSEs)]
+
+
+lambdas <- seq(lambda_best - 0.1 , lambda_best + 0.1 , 0.01)
+RMSEs <- sapply(lambdas, function(l){
+b_i <-  train_set %>% group_by(movieId) %>% summarize(b_i = sum(rating - mu_hat)/(n()+l))
+b_u <- train_set %>% left_join(b_i, by="movieId") %>% group_by(userId) %>% summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+predicted_ratings <- test_set %>% left_join(b_i, by = "movieId") %>% left_join(b_u, by = "userId") %>% mutate(pred = mu_hat + b_i + b_u) %>% pull(pred)
+return(RMSE(predicted_ratings, test_set$rating))
+})
+
+
+lambdas_rmse <- min(RMSEs)					
+lambda_best <- lambdas[which.min(RMSEs)]
+
+final_method <- paste("Optimized Penalized Movie & User Effect Model, lambda =", as.character(lambda_best), sep=" ")
+rmse_results <- bind_rows(rmse_results, tibble(method=final_method, RMSE = lambdas_rmse))
+
+
+#  Optimize lambda to the integer level only
+
+lambdas <- seq(1, 10, 1)
+
+RMSEs <- sapply(lambdas, function(l){
+b_i <-
+train_set %>%
+group_by(movieId) %>%
+summarize(b_i = sum(rating - mu_hat)/(n()+l))
+
+b_u <-
+train_set %>%
+left_join(b_i, by="movieId") %>%
+group_by(userId) %>%
+summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+
+predicted_ratings <-
+test_set %>%
+left_join(b_i, by = "movieId") %>%
+left_join(b_u, by = "userId") %>%
+mutate(pred = mu_hat + b_i + b_u) %>%
+pull(pred)
+
+return(RMSE(predicted_ratings, test_set$rating))
+
+})
+
+lambdas_rmse <- min(RMSEs)					
+lambda_best <- lambdas[which.min(RMSEs)]
+
+final_method <- paste("OPM&UE Model, integer lambda =", as.character(lambda_best), sep=" ")
+rmse_results <- bind_rows(rmse_results, tibble(method=final_method, RMSE = lambdas_rmse))
+
+
+```
+
+The final RMSE does not seem to depend greatly on lambda to a high degree of precision.  Optimizing lambda to the integer level seems sufficient.
+
+```{r rmse results, echo=FALSE}
+#print.data.frame(rmse_results)
+
+```
+
+
+### Insights
+
+* `NAs in predicted_ratings`: I was getting NAs in the predicted_ratings output.  This lead to an investigation of outliers that helped reduce the overall RMSE.
+
+* `Lambda can be refined step-wise`: Lambda can be optimized to two decimal places using 30 tests instead of 100.  This ultimately proved to be worthless, but interesting.
+
+
+### Final Modeling Approach
+
+Ultimately, what helped more than fine-tuning lambda was to throw away the outliers that were adding more noise than was worth in the overall weighted model.  I experimented with a few hard-coded values, starting by removing any movie with three or fewer ratings submitted, and removing any user that has reviewed nineteen or fewer movies.
+
+```{r hard-coded outlier filter, message=FALSE, echo=FALSE, warning=FALSE}
+edx2 <- edx %>% group_by(movieId) %>% filter(n()>4) %>% ungroup()
+edx2 <- edx2 %>% group_by(userId) %>% filter(n()>20) %>% ungroup()
+test_index <- createDataPartition(y = edx2$rating, times = 1, p = 0.2, list = FALSE)
+train_set <- edx2[-test_index,] 
+test_set <- edx2[test_index,] 
+
+
+lambdas <- seq(1, 10, 1)
+
+RMSEs <- sapply(lambdas, function(l){
+b_i <-
+train_set %>%
+group_by(movieId) %>%
+summarize(b_i = sum(rating - mu_hat)/(n()+l))
+
+b_u <-
+train_set %>%
+left_join(b_i, by="movieId") %>%
+group_by(userId) %>%
+summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+
+predicted_ratings <-
+test_set %>%
+left_join(b_i, by = "movieId") %>%
+left_join(b_u, by = "userId") %>%
+mutate(pred = mu_hat + b_i + b_u) %>%
+pull(pred)
+
+return(RMSE(predicted_ratings, test_set$rating))
+
+})
+
+lambdas_rmse <- min(RMSEs)					
+lambda_best <- lambdas[which.min(RMSEs)]
+
+final_method <- paste("Hard-Coded Pruned OPM&UE Model, lambda =", as.character(lambda_best), sep=" ")
+rmse_results <- bind_rows(rmse_results, tibble(method=final_method, RMSE = lambdas_rmse))
+
+
+
+```
+
+
+
+```{r rmse results_2, echo=FALSE}
+print.data.frame(rmse_results)
+
+```
+
+The slight reduction in RMSE was a surprise.  The reason for pruning out the lowest number of ratings per movie and per user was to no longer need to remove NA values from the predictions, since that arose from either movies or users with a low count winding up in the training set but not the test set.  It was assumed that if every movie has been rated more than three times, and if every user had rated at least nineteen movies, then the possibility of either not being represented in the test set was much lower.  It was realized after the fact that this can be controlled during the partition creation by using semi_join, as was done with the original data set.  Nonetheless, this attempt at removing the NA values resulted in a reduction of the RMSE value that was unexpected, and merited further investigation.
+
+Rather than hard-code arbitrary values, a statistical approach based on ignoring the lowest 10th percentile of both movies and users (in terms of numbers of ratings associated with each unique value) was developed.
+
+10% seemed like a good number to start with, although this could be refined more in the future, and is itself an arbitrary cutoff point not based on any deeper investigation.
+
+## Results
+
+  The fine tuning of lambda beyond the integer level does not seem to provide enough reduction in error to justify the increased cost in execution time.  Instead, the statistical approach to pruning lower outliers seems to provide the needed optimization.  A simple cutoff of the 10th percentile and integer lambda yields the following:
+
+```{r final model, message=FALSE, echo=FALSE, warning=FALSE}
+
+# Calculate 10th percentile of movieId by rating
+movieId_cutoff <-
+edx %>%
+group_by(movieId) %>%
+summarize(n = n()) %>%
+.$n %>%
+quantile(.10)
+
+paste("Movies below the 10th percentile with fewer than",
+as.character(movieId_cutoff),
+"ratings will be ignored.",
+sep=" ")
+
+#Calculate 10th percentile of userId by rating
+userId_cutoff <-
+edx %>% 
+group_by(userId) %>%
+summarize(n = n()) %>%
+.$n %>%
+quantile(.10)
+
+paste("Users below the 10th percentile with fewer than",
+as.character(userId_cutoff),
+"ratings will be ignored.",
+sep=" ")
+
+# Remove any movie below the 10th percentile in terms of ratings
+edx2 <-
+edx %>%
+group_by(movieId) %>%
+filter(n()>=movieId_cutoff) %>%
+ungroup()
+
+# Remove any user below the 10th percentile in terms of ratings
+edx2 <-
+edx2 %>%
+group_by(userId) %>%
+filter(n()>=userId_cutoff) %>%
+ungroup()
+
+# Create partition, reserving 20% of edx set for testing purposes
+test_index <-
+createDataPartition(y = edx2$rating,
+times = 1, p = 0.2, list = FALSE)
+train_set <- edx2[-test_index,] 
+test_set <- edx2[test_index,] 
+
+# Define loss function, and throw away any NA values that result from 
+RMSE <-
+function(true_ratings, predicted_ratings){
+	sqrt(mean((true_ratings - predicted_ratings)^2, na.rm=TRUE))} 
+
+# Calcualte the average rating for the training set.
+mu_hat <- mean(train_set$rating)
+
+paste("Penalized User & Movie Effect method using a mean rating of",
+as.character(mu_hat),
+".",
+sep=" ")
+
+# Using Movie and User Effects with Penalization on both, optimize lambda to the nearest integer.
+
+lambdas <- seq(1, 10, 1)
+
+RMSEs <- sapply(lambdas, function(l){
+b_i <-
+train_set %>%
+group_by(movieId) %>%
+summarize(b_i = sum(rating - mu_hat)/(n()+l))
+
+b_u <-
+train_set %>%
+left_join(b_i, by="movieId") %>%
+group_by(userId) %>%
+summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+
+predicted_ratings <-
+test_set %>%
+left_join(b_i, by = "movieId") %>%
+left_join(b_u, by = "userId") %>%
+mutate(pred = mu_hat + b_i + b_u) %>%
+pull(pred)
+
+return(RMSE(predicted_ratings, test_set$rating))
+
+})
+
+lambda_best <- lambdas[which.min(RMSEs)]
+
+paste("Optimized lambda =",
+as.character(lambda_best),
+"gives an RMSE of",
+as.character(min(RMSEs)),
+"on the test set.",
+sep=" ")
+
+
+```
+
+
+### Performance
+
+We can measure the performance of this final algorithm against the validation set (for the first time):
+
+```{r validation set, echo=FALSE}
+# Use this optimized lambda value to test against the validation set.
+paste("Now testing this optimized lambda against the vaildation set.")
+
+l <- lambda_best
+b_i <- train_set %>%
+group_by(movieId) %>%
+summarize(b_i = sum(rating - mu_hat)/(n()+l))
+b_u <- train_set %>%
+left_join(b_i, by="movieId") %>%
+group_by(userId) %>%
+summarize(b_u = sum(rating - b_i - mu_hat)/(n()+l))
+final_predicted_ratings <-
+validation %>%
+left_join(b_i, by = "movieId") %>%
+left_join(b_u, by = "userId") %>%
+mutate(pred = mu_hat + b_i + b_u) %>%
+pull(pred)
+final_rmse <-
+RMSE(final_predicted_ratings, validation$rating)
+
+paste("Optimized lambda =",
+as.character(lambda_best),
+"gives an RMSE of",
+as.character(final_rmse),
+"on the validation set.",
+sep=" ")
+
+```
+
+
+## Conclusion
+
+   Our penalized movie and user effect model gives an error in the desired range when lower outlier pruning is applied at the 10th percentile of ratings for both users and movie titles.  
+
+### Summary
+
+We began a linear estimate accounting for both movie and user effect as well as a penalized term for both effects.  This got us close to our goal, and further exploration was warranted.  The original mistake of not using semi_join when generating the training and test sets lead to an investigation of low-value outliers.  
+
+These add noise without contributing to predictability.  Our model is based on the mean rating for each movie across userId, which is NOT resistant to outliers.  Hence, the users and movies with low frequency add noise to the mean without adding much weight to the predictions.  an attempt to remove these (to eliminate NAs in the final predictions) inadvertently lead to a reduction in RMSE, which is what makes this method powerful.
+
+### Limitations
+
+This seems to depend on large data sets.  Eliminating below a certain percentile might work against you with a smaller data set.  This needs to be explored.
+
+### Future Work
+
+Optimizing the percentile removed from each predictor (movieId and userId) would be interesting, as it is unlikely that the 10th percentile is always the ideal cutoff.  It would be nice to keep as much of the original data set as possible while still training a good solid model.
+
+Many predictors were not considered, such as timestamps or genres.  No factor analysis was done of any kind.  These could be included in the model to decrease the overall error if desired.
+
+

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