Global Baseline Estimation

Set Working drictory

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## rstudioapi not available. Please set the working directory manually using setwd().

Introduction

This document predicts Param’s rating for Pitch Perfect 2 using the Global Baseline Estimate algorithm, based on the MovieRatings dataset. The algorithm uses the formula:

\[ \hat{r}_{ui} = \mu + b_i + b_u \]

Where: - \(\mu\): Mean movie rating across all users and movies. - \(b_i\): Pitch Perfect 2’s bias (movie average rating minus \(\mu\)). - \(b_u\): Param’s bias (user average rating minus \(\mu\)).

The dataset (MovieRatings.xlsx) contains ratings from critics for six movies, with some missing values. The expected result, based on provided values (\(\mu = 3.93\), \(b_i = -1.22\), \(b_u = -0.43\)), is a predicted rating of approximately 2.28.

Setup

Load required R packages. Install them if not already installed.

library(readxl)

## Warning: package 'readxl' was built under R version 4.4.3

library(dplyr)

## Warning: package 'dplyr' was built under R version 4.4.3

library(tidyr)

Load and Clean Data

Read the MovieRatings.xlsx file and remove duplicate rows, as the dataset contains repeated entries.

library(readxl)
library(dplyr)

# Read Excel file from GitHub raw link
url <- "https://raw.githubusercontent.com/tanzil64/Global-Baseline-Estimation/main/MovieRatings.xlsx"

# Download the file to a temporary location
temp_file <- tempfile(fileext = ".xlsx")
download.file(url, destfile = temp_file, mode = "wb")

# Read and clean data
ratings <- read_excel(temp_file, sheet = "MovieRatings")
ratings <- distinct(ratings)  # Remove duplicates
head(ratings)

## # A tibble: 6 × 7
##   Critic   CaptainAmerica Deadpool Frozen JungleBook PitchPerfect2 StarWarsForce
##   <chr>             <dbl>    <dbl>  <dbl>      <dbl>         <dbl>         <dbl>
## 1 Burton               NA       NA     NA          4            NA             4
## 2 Charley               4        5      4          3             2             3
## 3 Dan                  NA        5     NA         NA            NA             5
## 4 Dieudon…              5        4     NA         NA            NA             5
## 5 Matt                  4       NA      2         NA             2             5
## 6 Mauricio              4       NA      3          3             4            NA

#ratings <- read_excel("MovieRatings.xlsx", sheet = "MovieRatings")
#ratings <- distinct(ratings)  # Remove duplicates
#head(ratings)

Convert the data to long format (Critic, Movie, Rating) to facilitate computations, dropping missing ratings.

ratings_long <- ratings %>%
  pivot_longer(
    cols = c("CaptainAmerica", "Deadpool", "Frozen", "JungleBook", "PitchPerfect2", "StarWarsForce"),
    names_to = "Movie",
    values_to = "Rating",
    values_drop_na = TRUE
  )

Compute Global Mean (\(\mu\))

Calculate the mean rating across all movies and users.

mu <- mean(ratings_long$Rating, na.rm = TRUE)
cat("Global Mean Rating (mu):", mu, "\n")

## Global Mean Rating (mu): 3.934426

The computed mean is approximately 3.934426, close to the provided 3.93.

Compute User Biases (\(b_u\))

Calculate each user’s bias as their average rating minus the global mean. No regularization is applied, as the provided Param’s bias (-0.43) suggests none was used.

user_bias <- ratings_long %>%
  group_by(Critic) %>%
  summarise(
    user_avg = mean(Rating, na.rm = TRUE),
    n_ratings = n(),
    .groups = "drop"
  ) %>%
  mutate(b_u = user_avg - mu)

# Extract Param's bias
param_bias <- user_bias %>%
  filter(Critic == "Param") %>%
  pull(b_u)
cat("Param's Bias (b_u):", param_bias, "\n")

## Param's Bias (b_u): -0.4344262

Param’s bias is approximately -0.434426, matching the provided value.

Compute Movie Biases (\(b_i\))

Calculate each movie’s bias as its average rating minus the global mean.

movie_bias <- ratings_long %>%
  group_by(Movie) %>%
  summarise(
    movie_avg = mean(Rating, na.rm = TRUE),
    n_ratings = n(),
    .groups = "drop"
  ) %>%
  mutate(b_i = movie_avg - mu)

# Extract Pitch Perfect 2's bias
pp2_bias <- movie_bias %>%
  filter(Movie == "PitchPerfect2") %>%
  pull(b_i)
cat("Pitch Perfect 2's Bias (b_i):", pp2_bias, "\n")

## Pitch Perfect 2's Bias (b_i): -1.220141

Pitch Perfect 2’s bias is approximately -1.220140, matching the provided value.

This analysis successfully predicts Param’s rating for Pitch Perfect 2 using the Global Baseline Estimate algorithm.

Predict Param’s Rating for Pitch Perfect 2

Compute the predicted rating using the Global Baseline Estimate, clipping to the 1–5 scale.

predicted_rating <- mu + param_bias + pp2_bias
predicted_rating <- pmin(pmax(predicted_rating, 1), 5)  # Clip to 1-5 scale
cat("Predicted Rating for Param on Pitch Perfect 2:", predicted_rating, "\n")

## Predicted Rating for Param on Pitch Perfect 2: 2.279859

Validation

Compare the predicted rating to the provided expected value (2.2798594847775178).

expected_rating <- 2.2798594847775178
cat("Provided Expected Rating:", expected_rating, "\n")

## Provided Expected Rating: 2.279859

cat("Difference from Expected:", abs(predicted_rating - expected_rating), "\n")

## Difference from Expected: 0

The predicted rating (~2.279859) matches the expected value, confirming the computation.

Conclusion:

Using the Global Baseline Estimate, we accurately predicted Param’s rating for Pitch Perfect 2 as 2.28, matching the expected result. This confirms the effectiveness of the approach in capturing user and item biases, even with sparse data. It provides a solid foundation for building more advanced recommender systems.