movie_rating.Rmd

# 电影评分倾向 {#irt}


```{r}
library(tidyverse)
library(tidybayes)
library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())
```





```{r}
df1 <- tibble(
  movie = 1,
  y = c(3, 5)
)
df2 <- tibble(
  movie = 2,
  y = rep(c(2, 3, 4, 5), c(10, 20, 30, 40))
)

movie_data <- bind_rows(df1, df2)
movie_data
```





```{r eda-movie-rating-2, warning=FALSE, message=FALSE}
stan_program <- '
data {
  int<lower=1> N;            
  vector[N] y;               
  int movie[N];  
  int<lower=1> K;
}
parameters {
  vector[K] mu; 
  real sigma;
}

model {
  sigma ~ normal(3, 1);
  for(i in 1:N) {
    target += normal_lpdf(y[i] | mu[movie[i]], sigma);
  }
 
}

'

stan_data <- movie_data %>%
  tidybayes::compose_data(
    N          = nrow(.),
    K          = n_distinct(movie),
    movie      = movie,
    y          = y
  )

fit_normal <- stan(model_code = stan_program, data = stan_data)
```



```{r eda-movie-rating-3}
fit_normal
```


## Extending the model to J movies



```{r eda-movie-rating-4}
movie_data <-
  tibble(
    movie = 1:10,                                 # N = 10个电影
    theta = rnorm(10, 3.0, 0.5),                  # 模拟的每个电影评分均值
    N_ratings = sample(1:100, 10, replace = TRUE) # 每个电影的投票次数
  ) %>%
  mutate(
    y = map2(N_ratings, theta, rnorm, sd = 2.0)
  ) %>% 
  unnest(cols = c(y))

movie_data
```




```{r eda-movie-rating-5, warning=FALSE, message=FALSE}
stan_program <- '
data {
  int<lower=1> N;            
  vector[N] y;               
  int<lower=1> K;
  int<lower=1, upper=K> movie[N];  
}
parameters {
  vector[K] mu; 
  real sigma;
}

model {
  sigma ~ normal(3, 1);
  for(i in 1:N) {
    target += normal_lpdf(y[i] | mu[movie[i]], sigma);
  }
 
}

'

stan_data <- movie_data %>%
  tidybayes::compose_data(
    N          = nrow(.),
    K          = n_distinct(movie),
    movie      = movie,
    y          = y
  )

fit_normal2 <- stan(model_code = stan_program, data = stan_data)
```



```{r eda-movie-rating-6}
d_actual <- movie_data %>% 
  distinct(movie, theta, N_ratings)
d_actual 
```


```{r eda-movie-rating-7}
t <- fit_normal2 %>% 
  tidybayes::spread_draws(mu[i]) %>% 
  group_by(i) %>% 
  ggdist::mean_qi(.width = c(0.5, 0.95)) %>% 
  rename(movie = i) %>% 
  left_join(d_actual, by = "movie") 
t
```


```{r eda-movie-rating-8}
t %>% 
  ggplot(aes(x = mu, y = theta)) +
  geom_point() +
  ggdist::geom_pointinterval(aes(xmin = .lower, xmax = .upper)) +
  geom_abline() +
  scale_x_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) +
  scale_y_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) 
```


电影评分的人越多，我们估计它的分布精确度越高，我这里画出50%可信赖区间与样本大小的函数，样本越多，区间宽度越小，越集中



```{r eda-movie-rating-9}
t %>% 
  filter(.width == 0.5) %>% 
  mutate(intervals = .upper - .lower) %>% 
  
  ggplot(aes(x = N_ratings, intervals)) +
  geom_point() +
  ggtitle("when you have more data, you have less uncertainly.")
  
```




## item response model with parameters for rates and for movies

```{r eda-movie-rating-10}
J       <- 40  # 有多少电影
K       <- 100 # 有多少人打分，每部电影每个人都要打分

movie_quality       <- rnorm(J, 0, 1)
rater_difficulty    <- rnorm(K, 0, 1)


movie_data3 <- 
  crossing(rater = 1:100, movie = 1:40) %>% 
  mutate(
    y = map2_dbl(rater, movie, 
                 ~rnorm(1, 3 + 0.5*movie_quality[.y] - 0.5*rater_difficulty[.x]),
                 sd = 2)
  )
movie_data3
```




```{r eda-movie-rating-11}
d_rater <- tibble(
  rater = 1:100, # 有多少人打分，每部电影每个人都要打分
  rater_difficulty = rnorm(100, 0, 1)
)

d_movie <- tibble(
  movie = 1:40, # 有多少电影
  movie_quality = rnorm(40, 0, 1)
)

# set mu = 3, sigma_a = sigma_b = 0.5, sigma = 2
movie_data30 <- 
  crossing(d_rater, d_movie) %>% 
  mutate(
    y = map2_dbl(movie_quality, rater_difficulty, 
                 ~rnorm(n = 1, mean = 3 + 0.5*.x - 0.5*.y, sd = 2)
                 )
  )

movie_data30
```




```{r eda-movie-rating-12, warning=FALSE, message=FALSE}
stan_program <- '
data {
  int<lower=1> N;            
  vector[N] y;               
  int<lower=1> J;
  int<lower=1, upper=J> movie[N]; 
  int<lower=1> K;
  int<lower=1, upper=K> rater[N]; 
}
parameters {
  vector[J] alpha; 
  vector[K] beta; 
  real mu;
  real<lower=0> sigma;
  real<lower=0> sigma_a;
  real<lower=0> sigma_b;
}
transformed parameters{
  vector[N] MU;
  for(i in 1:N) {
    MU[i] = mu + sigma_a * alpha[movie[i]] - sigma_b * beta[rater[i]];
  }
}
model {
  sigma ~ normal(3, 1);
  alpha ~ normal(0, 1);
   beta ~ normal(0, 1);
  for(i in 1:N) {
    target += normal_lpdf(y[i] | MU[i], sigma);
  }
 
}

'

stan_data <- movie_data30 %>%
  tidybayes::compose_data(
    N              = nrow(.),
    movie          = movie,
    rater          = rater,
    y              = y,
    J              = n_distinct(movie),
    K              = n_distinct(rater)
  )

fit_normal3 <- stan(model_code = stan_program, data = stan_data)
```




```{r eda-movie-rating-13}
fit_normal3 %>% 
  tidybayes::spread_draws(alpha[i]) %>% 
  group_by(i) %>% 
  ggdist::mean_qi() %>% 
  
  ggplot(aes(x = alpha)) +
  geom_density()
```

```{r eda-movie-rating-14}
fit_normal3 %>% 
  tidybayes::spread_draws(beta[i]) %>% 
  group_by(i) %>% 
  ggdist::mean_qi() %>% 
  
  ggplot(aes(x = beta)) +
  geom_density()
```



```{r eda-movie-rating-15}
fit_normal3 %>% 
  tidybayes::gather_draws(mu, sigma, sigma_a, sigma_b) %>% 
  ggdist::mean_qi() 
```


可以看到，模型反演的结果，和真实结果符合的很好
mu = 3, sigma_a = sigma_b = 0.5, sigma = 2



```{r eda-movie-rating-16}
d_actual <- 
  tibble(
    i = 1:40,
    alpha_actual = movie_quality
  )


fit_normal3 %>% 
  tidybayes::spread_draws(alpha[i]) %>% 
  group_by(i) %>% 
  ggdist::median_qi(.width = c(0.5, 0.95)) %>% 
  left_join(d_actual, by = "i") %>% 

  ggplot(aes(x = alpha, y = alpha_actual)) +
  geom_point(color = "red", size = 4) +
  ggdist::geom_pointinterval(aes(xmin = .lower, xmax = .upper)) +
  geom_abline() +
  scale_x_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) +
  scale_y_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) 
```


```{r eda-movie-rating-17}
d_actual <- 
  tibble(
    i = 1:100,
    beta_actual = rater_difficulty
  )


fit_normal3 %>% 
  tidybayes::spread_draws(beta[i]) %>% 
  group_by(i) %>% 
  ggdist::median_qi(.width = c(0.5, 0.95)) %>% 
  left_join(d_actual, by = "i") %>%

  ggplot(aes(x = beta, y = beta_actual)) +
  geom_point(color = "red", size = 4) +
  ggdist::geom_pointinterval(aes(xmin = .lower, xmax = .upper)) +
  geom_abline() +
  scale_x_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) +
  scale_y_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) 
```

## more challenge

```{r eda-movie-rating-18}
d_rater <- tibble(
  rater = 1:100, # 有多少人打分，每部电影每个人都要打分
  rater_difficulty = rnorm(100, 0, 1)
)

d_movie <- tibble(
  movie = 1:40, # 有多少电影
  movie_quality = rnorm(40, 0, 1)
)

movie_data4 <- 
  crossing(d_rater, d_movie) %>% 
  mutate(
    y = map2_dbl(rater, movie, 
                 ~rnorm(1, 3 + 0.5*movie_quality - 0.5*rater_difficulty),
                 sd = 2)
  )

movie_data4
```










```{r eda-movie-rating-19, eval=FALSE}
# 我理解错误了
movie_data4 %>%
  mutate(
    movie_type = if_else(rater_difficulty > 0,
      sample(c("crime_movie", "romantic_movie"), size = 1, prob = c(0.6, 0.3)),
      sample(c("crime_movie", "romantic_movie"), size = 1, prob = c(0.3, 0.6))
    )
  )
```




正确理解如下，需要补个示意图

```{r eda-movie-rating-20}
d_rater <- tibble(
  rater = 1:100, # 有多少人打分，每部电影每个人都要打分
  rater_difficulty = rnorm(100, 0, 1)
)

d_movie <- tibble(
  movie = 1:40, # 有多少电影
  genre = rep(c("romantic", "crime"), c(20, 20)),
  movie_quality = rnorm(40, 0, 1)
)

# set mu = 3, sigma_a = sigma_b = 0.5, sigma = 2
movie_data4 <- 
  crossing(d_rater, d_movie) %>% 
  mutate(
    y = map2_dbl(movie_quality, rater_difficulty, 
                 ~rnorm(n = 1, mean = 3 + 0.5*.x - 0.5*.y, sd = 2)
    )
  )
 
movie_data4
```




相当于做了一个子集选取
- rater_difficulty > 0  genre == "romantic"  0.6 
- rater_difficulty > 0  genre == "crime"     0.3 
- rater_difficulty < 0  genre == "romantic"  0.3 
- rater_difficulty < 0  genre == "crime"     0.6 

```{r eda-movie-rating-21}
movie_data400 <- movie_data4 %>%
  mutate(
      prob_of_rated = ifelse(rater_difficulty > 0,
                        ifelse(genre == "romantic", 0.6, 0.3),
                        ifelse(genre == "romantic", 0.3, 0.6)
                        )
    ) %>% 
   mutate(
     rated = map_int(prob_of_rated , ~rbinom(1, 1, .x))
   ) %>% 
  filter(rated == 1)


movie_data400
```

文章中这个子集是一个非平衡数据，我们看看什么地方不平衡
```{r eda-movie-rating-22}
movie_data400 %>% count(rater, genre)
```




```{r eda-movie-rating-23}
movie_data4 %>% count(genre)
movie_data4 %>% count(rater, genre)
movie_data4 %>% count(rater_difficulty > 0, genre)
```

```{r eda-movie-rating-24}
movie_data400 %>% count(genre)
movie_data400 %>% count(rater, genre)
movie_data400 %>% count(rater_difficulty > 0, genre)
```
更多 `rater_difficulty > 0` 代表**挑剔严格的评分人**，

- rater_difficulty > 0  genre == "romantic"  0.6 
- rater_difficulty > 0  genre == "crime"     0.3 
- rater_difficulty < 0  genre == "romantic"  0.3 
- rater_difficulty < 0  genre == "crime"     0.6 




```{r eda-movie-rating-25, warning=FALSE, message=FALSE}
stan_program <- '
data {
  int<lower=1> N;            
  vector[N] y;               
  int<lower=1> J;
  int<lower=1, upper=J> movie[N]; 
  int<lower=1> K;
  int<lower=1, upper=K> rater[N]; 
}
parameters {
  vector[J] alpha; 
  vector[K] beta; 
  real mu;
  real<lower=0> sigma;
  real<lower=0> sigma_a;
  real<lower=0> sigma_b;
}
transformed parameters{
  vector[N] MU;
  for(i in 1:N) {
    MU[i] = mu + sigma_a * alpha[movie[i]] - sigma_b * beta[rater[i]];
  }
}
model {
  sigma ~ normal(3, 1);
  alpha ~ normal(0, 1);
   beta ~ normal(0, 1);
  for(i in 1:N) {
    target += normal_lpdf(y[i] | MU[i], sigma);
  }
 
}

'

stan_data <- movie_data400 %>%
  dplyr::select(rater, movie, y) %>% 
  tidybayes::compose_data(
    N              = nrow(.),
    movie          = movie,
    rater          = rater,
    y              = y,
    J              = n_distinct(movie),
    K              = n_distinct(rater)
  )

fit_normal4 <- stan(model_code = stan_program, data = stan_data)
```



```{r eda-movie-rating-26}
fit_normal4 %>% 
  tidybayes::gather_draws(mu, sigma, sigma_a, sigma_b) %>% 
  ggdist::mean_qi() 
```
符合的还好
我们设定的是，mu = 3, sigma_a = sigma_b = 0.5, sigma = 2




```{r eda-movie-rating-27}
d_actual <- movie_data400 %>% 
  distinct(
    movie, movie_quality
  ) %>% 
  rename(i = movie)
d_actual
```


```{r eda-movie-rating-28}
fit_normal4 %>% 
  tidybayes::spread_draws(alpha[i]) %>% 
  group_by(i) %>% 
  ggdist::median_qi(.width = c(0.5, 0.95)) %>% 
  left_join(d_actual, by = "i") %>%

  ggplot(aes(x = alpha, y = movie_quality)) +
  geom_point(color = "red", size = 4) +
  ggdist::geom_pointinterval(aes(xmin = .lower, xmax = .upper)) +
  geom_abline() +
  scale_x_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) +
  scale_y_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3)
```


```{r eda-movie-rating-29}
d_actual <- movie_data400 %>% 
  distinct(
    rater, rater_difficulty
  ) %>% 
  rename(i = rater)


fit_normal4 %>% 
  tidybayes::spread_draws(beta[i]) %>% 
  group_by(i) %>% 
  ggdist::median_qi(.width = c(0.5, 0.95)) %>% 
  left_join(d_actual, by = "i") %>%

  ggplot(aes(x = beta, y = rater_difficulty)) +
  geom_point(color = "red", size = 4) +
  ggdist::geom_pointinterval(aes(xmin = .lower, xmax = .upper)) +
  geom_abline() +
  scale_x_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) +
  scale_y_continuous(limits = c(-3,3), breaks = -3:3, labels = -3:3) 
```


- 原始设定的评分

```{r eda-movie-rating-30}
df_raw <- movie_data400 %>% 
  distinct(
    movie, movie_quality
  ) %>% 
  arrange(movie) %>% 
  mutate( mu_raw = 3 + 0.5 * movie_quality) %>% 
  select(movie, mu_raw)
df_raw
```




- 根据设定的评分作为均值，生成的随机评分（数量为评分的人数），然后做平均

```{r eda-movie-rating-31}
df_average <- movie_data400 %>% 
  group_by(movie) %>% 
  summarise(
    mean_rating =  mean(y)
  ) %>% 
  select(movie, mu_average = mean_rating)
df_average
```


- 模型给出的结果

```{r eda-movie-rating-32}
df_model <- fit_normal4 %>% 
  tidybayes::spread_draws(mu, alpha[i]) %>% 
  group_by(i) %>% 
  ggdist::median_qi() %>% 
  select(i, mu, alpha) %>% 
  mutate(mu = mu + 0.5 * alpha) %>% 
  select(movie = i, mu_model = mu)

df_model
```



```{r eda-movie-rating-33}
p1 <- df_raw %>% 
  left_join(df_average) %>% 
  left_join(df_model) %>% 
  
  ggplot(aes(x = mu_average, y = mu_raw)) +
  geom_point() +
  geom_abline() +
  scale_x_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) +
  scale_y_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) 
  

p2 <- df_raw %>% 
  left_join(df_average) %>% 
  left_join(df_model) %>% 
  
  ggplot(aes(x = mu_model, y = mu_raw)) +
  geom_point() +
  geom_abline() +
  scale_x_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) +
  scale_y_continuous(limits = c(1,5), breaks = 1:5, labels = 1:5) 
  

library(patchwork) 
p1+p2
```


<details><summary>Session Info</summary>
```{r eda-movie-rating-34, echo=FALSE}
sessioninfo::session_info()
```
</details>