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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Advanced Recommender Systems with Python
Welcome to the code notebook for creating Advanced Recommender Systems with Python. This is an optional lecture notebook for you to check out. Currently there is no video for this lecture because of the level of mathematics used and the heavy use of SciPy here.
Recommendation Systems usually rely on larger data sets and specifically need to be organized in a particular fashion. Because of this, we won't have a project to go along with this topic, instead we will have a more intensive walkthrough process on creating a recommendation system with Python with the same Movie Lens Data Set.
Note
Step1:  We can then read in the u.data file, which contains the full dataset. You can read a brief description of the dataset here.
Note how we specify the separator argument for a Tab separated file.
Step2:  Let's take a quick look at the data.
Step3:  Note how we only have the item_id, not the movie name. We can use the Movie_ID_Titles csv file to grab the movie names and merge it with this dataframe
Step4:  Then merge the dataframes
Step5:  Now let's take a quick look at the number of unique users and movies.
Step6:  Train Test Split
Recommendation Systems by their very nature are very difficult to evaluate, but we will still show you how to evaluate them in this tutorial. In order to do this, we'll split our data into two sets. However, we won't do our classic X_train,X_test,y_train,y_test split. Instead we can actually just segement the data into two sets of data
Step7:  Memory-Based Collaborative Filtering
Memory-Based Collaborative Filtering approaches can be divided into two main sections
Step8:  You can use the pairwise_distances function from sklearn to calculate the cosine similarity. Note, the output will range from 0 to 1 since the ratings are all positive.
Step9:  Next step is to make predictions. You have already created similarity matrices
Step10:  Evaluation
There are many evaluation metrics but one of the most popular metric used to evaluate accuracy of predicted ratings is Root Mean Squared Error (RMSE). 
<img src="https
Step11:  Memory-based algorithms are easy to implement and produce reasonable prediction quality. 
The drawback of memory-based CF is that it doesn't scale to real-world scenarios and doesn't address the well-known cold-start problem, that is when new user or new item enters the system. Model-based CF methods are scalable and can deal with higher sparsity level than memory-based models, but also suffer when new users or items that don't have any ratings enter the system. I would like to thank Ethan Rosenthal for his post about Memory-Based Collaborative Filtering. 
Model-based Collaborative Filtering
Model-based Collaborative Filtering is based on matrix factorization (MF) which has received greater exposure, mainly as an unsupervised learning method for latent variable decomposition and dimensionality reduction. Matrix factorization is widely used for recommender systems where it can deal better with scalability and sparsity than Memory-based CF. The goal of MF is to learn the latent preferences of users and the latent attributes of items from known ratings (learn features that describe the characteristics of ratings) to then predict the unknown ratings through the dot product of the latent features of users and items. 
When you have a very sparse matrix, with a lot of dimensions, by doing matrix factorization you can restructure the  user-item matrix into low-rank structure, and you can represent the matrix by the multiplication of two low-rank matrices, where the rows contain the latent vector. You fit this matrix to approximate your original matrix, as closely as possible, by multiplying the low-rank matrices together, which fills in the entries missing in the original matrix.
Let's calculate the sparsity level of MovieLens dataset
Step12:  To give an example of the learned latent preferences of the users and items
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Python Code:
import numpy as np
import pandas as pd
Explanation: Advanced Recommender Systems with Python
Welcome to the code notebook for creating Advanced Recommender Systems with Python. This is an optional lecture notebook for you to check out. Currently there is no video for this lecture because of the level of mathematics used and the heavy use of SciPy here.
Recommendation Systems usually rely on larger data sets and specifically need to be organized in a particular fashion. Because of this, we won't have a project to go along with this topic, instead we will have a more intensive walkthrough process on creating a recommendation system with Python with the same Movie Lens Data Set.
Note: The actual mathematics behind recommender systems is pretty heavy in Linear Algebra.
Methods Used
Two most common types of recommender systems are Content-Based and Collaborative Filtering (CF). 
Collaborative filtering produces recommendations based on the knowledge of users’ attitude to items, that is it uses the "wisdom of the crowd" to recommend items. 
Content-based recommender systems focus on the attributes of the items and give you recommendations based on the similarity between them.
Collaborative Filtering
In general, Collaborative filtering (CF) is more commonly used than content-based systems because it usually gives better results and is relatively easy to understand (from an overall implementation perspective). The algorithm has the ability to do feature learning on its own, which means that it can start to learn for itself what features to use. 
CF can be divided into Memory-Based Collaborative Filtering and Model-Based Collaborative filtering. 
In this tutorial, we will implement Model-Based CF by using singular value decomposition (SVD) and Memory-Based CF by computing cosine similarity. 
The Data
We will use famous MovieLens dataset, which is one of the most common datasets used when implementing and testing recommender engines. It contains 100k movie ratings from 943 users and a selection of 1682 movies.
You can download the dataset here or just use the u.data file that is already included in this folder.
Getting Started
Let's import some libraries we will need:
End of explanation
column_names = ['user_id', 'item_id', 'rating', 'timestamp']
df = pd.read_csv('u.data', sep='\t', names=column_names)
Explanation: We can then read in the u.data file, which contains the full dataset. You can read a brief description of the dataset here.
Note how we specify the separator argument for a Tab separated file.
End of explanation
df.head()
Explanation: Let's take a quick look at the data.
End of explanation
movie_titles = pd.read_csv("Movie_Id_Titles")
movie_titles.head()
Explanation: Note how we only have the item_id, not the movie name. We can use the Movie_ID_Titles csv file to grab the movie names and merge it with this dataframe:
End of explanation
df = pd.merge(df,movie_titles,on='item_id')
df.head()
Explanation: Then merge the dataframes:
End of explanation
n_users = df.user_id.nunique()
n_items = df.item_id.nunique()
print('Num. of Users: '+ str(n_users))
print('Num of Movies: '+str(n_items))
Explanation: Now let's take a quick look at the number of unique users and movies.
End of explanation
from sklearn.cross_validation import train_test_split
train_data, test_data = train_test_split(df, test_size=0.25)
Explanation: Train Test Split
Recommendation Systems by their very nature are very difficult to evaluate, but we will still show you how to evaluate them in this tutorial. In order to do this, we'll split our data into two sets. However, we won't do our classic X_train,X_test,y_train,y_test split. Instead we can actually just segement the data into two sets of data:
End of explanation
#Create two user-item matrices, one for training and another for testing
train_data_matrix = np.zeros((n_users, n_items))
for line in train_data.itertuples():
    train_data_matrix[line[1]-1, line[2]-1] = line[3]  
test_data_matrix = np.zeros((n_users, n_items))
for line in test_data.itertuples():
    test_data_matrix[line[1]-1, line[2]-1] = line[3]
Explanation: Memory-Based Collaborative Filtering
Memory-Based Collaborative Filtering approaches can be divided into two main sections: user-item filtering and item-item filtering. 
A user-item filtering will take a particular user, find users that are similar to that user based on similarity of ratings, and recommend items that those similar users liked. 
In contrast, item-item filtering will take an item, find users who liked that item, and find other items that those users or similar users also liked. It takes items and outputs other items as recommendations. 
Item-Item Collaborative Filtering: “Users who liked this item also liked …”
User-Item Collaborative Filtering: “Users who are similar to you also liked …”
In both cases, you create a user-item matrix which built from the entire dataset.
Since we have split the data into testing and training we will need to create two [943 x 1682] matrices (all users by all movies). 
The training matrix contains 75% of the ratings and the testing matrix contains 25% of the ratings.  
Example of user-item matrix:
<img class="aligncenter size-thumbnail img-responsive" src="http://s33.postimg.org/ay0ty90fj/BLOG_CCA_8.png" alt="blog8"/>
After you have built the user-item matrix you calculate the similarity and create a similarity matrix. 
The similarity values between items in Item-Item Collaborative Filtering are measured by observing all the users who have rated both items.  
<img class="aligncenter size-thumbnail img-responsive" style="max-width:100%; width: 50%; max-width: none" src="http://s33.postimg.org/i522ma83z/BLOG_CCA_10.png"/>
For User-Item Collaborative Filtering the similarity values between users are measured by observing all the items that are rated by both users.
<img class="aligncenter size-thumbnail img-responsive" style="max-width:100%; width: 50%; max-width: none" src="http://s33.postimg.org/mlh3z3z4f/BLOG_CCA_11.png"/>
A distance metric commonly used in recommender systems is cosine similarity, where the ratings are seen as vectors in n-dimensional space and the similarity is calculated based on the angle between these vectors. 
Cosine similiarity for users a and m can be calculated using the formula below, where you take dot product of  the user vector $u_k$ and the user vector $u_a$ and divide it by multiplication of the Euclidean lengths of the vectors.
<img class="aligncenter size-thumbnail img-responsive" src="https://latex.codecogs.com/gif.latex?s_u^{cos}(u_k,u_a)=\frac{u_k&space;\cdot&space;u_a&space;}{&space;\left&space;\|&space;u_k&space;\right&space;\|&space;\left&space;\|&space;u_a&space;\right&space;\|&space;}&space;=\frac{\sum&space;x_{k,m}x_{a,m}}{\sqrt{\sum&space;x_{k,m}^2\sum&space;x_{a,m}^2}}"/>
To calculate similarity between items m and b you use the formula:
<img class="aligncenter size-thumbnail img-responsive" src="https://latex.codecogs.com/gif.latex?s_u^{cos}(i_m,i_b)=\frac{i_m&space;\cdot&space;i_b&space;}{&space;\left&space;\|&space;i_m&space;\right&space;\|&space;\left&space;\|&space;i_b&space;\right&space;\|&space;}&space;=\frac{\sum&space;x_{a,m}x_{a,b}}{\sqrt{\sum&space;x_{a,m}^2\sum&space;x_{a,b}^2}}
"/>
Your first step will be to create the user-item matrix. Since you have both testing and training data you need to create two matrices.
End of explanation
from sklearn.metrics.pairwise import pairwise_distances
user_similarity = pairwise_distances(train_data_matrix, metric='cosine')
item_similarity = pairwise_distances(train_data_matrix.T, metric='cosine')
Explanation: You can use the pairwise_distances function from sklearn to calculate the cosine similarity. Note, the output will range from 0 to 1 since the ratings are all positive.
End of explanation
def predict(ratings, similarity, type='user'):
    if type == 'user':
        mean_user_rating = ratings.mean(axis=1)
        #You use np.newaxis so that mean_user_rating has same format as ratings
        ratings_diff = (ratings - mean_user_rating[:, np.newaxis]) 
        pred = mean_user_rating[:, np.newaxis] + similarity.dot(ratings_diff) / np.array([np.abs(similarity).sum(axis=1)]).T
    elif type == 'item':
        pred = ratings.dot(similarity) / np.array([np.abs(similarity).sum(axis=1)])     
    return pred
item_prediction = predict(train_data_matrix, item_similarity, type='item')
user_prediction = predict(train_data_matrix, user_similarity, type='user')
Explanation: Next step is to make predictions. You have already created similarity matrices: user_similarity and item_similarity and therefore you can make a prediction by applying following formula for user-based CF:
<img class="aligncenter size-thumbnail img-responsive" src="https://latex.codecogs.com/gif.latex?\hat{x}_{k,m}&space;=&space;\bar{x}_{k}&space;&plus;&space;\frac{\sum\limits_{u_a}&space;sim_u(u_k,&space;u_a)&space;(x_{a,m}&space;-&space;\bar{x_{u_a}})}{\sum\limits_{u_a}|sim_u(u_k,&space;u_a)|}"/>
You can look at the similarity between users k and a as weights that are multiplied by the ratings of a similar user a (corrected for the average rating of that user). You will need to normalize it so that the ratings stay between 1 and 5 and, as a final step, sum the average ratings for the user that you are trying to predict. 
The idea here is that some users may tend always to give high or low ratings to all movies. The relative difference in the ratings that these users give is more important than the absolute values. To give an example: suppose, user k gives 4 stars to his favourite movies and 3 stars to all other good movies. Suppose now that another user t rates movies that he/she likes with 5 stars, and the movies he/she fell asleep over with 3 stars. These two users could have a very similar taste but treat the rating system differently. 
When making a prediction for item-based CF you don't need to correct for users average rating since query user itself is used to do predictions.
<img class="aligncenter size-thumbnail img-responsive" src="https://latex.codecogs.com/gif.latex?\hat{x}_{k,m}&space;=&space;\frac{\sum\limits_{i_b}&space;sim_i(i_m,&space;i_b)&space;(x_{k,b})&space;}{\sum\limits_{i_b}|sim_i(i_m,&space;i_b)|}"/>
End of explanation
from sklearn.metrics import mean_squared_error
from math import sqrt
def rmse(prediction, ground_truth):
    prediction = prediction[ground_truth.nonzero()].flatten() 
    ground_truth = ground_truth[ground_truth.nonzero()].flatten()
    return sqrt(mean_squared_error(prediction, ground_truth))
print('User-based CF RMSE: ' + str(rmse(user_prediction, test_data_matrix)))
print('Item-based CF RMSE: ' + str(rmse(item_prediction, test_data_matrix)))
Explanation: Evaluation
There are many evaluation metrics but one of the most popular metric used to evaluate accuracy of predicted ratings is Root Mean Squared Error (RMSE). 
<img src="https://latex.codecogs.com/gif.latex?RMSE&space;=\sqrt{\frac{1}{N}&space;\sum&space;(x_i&space;-\hat{x_i})^2}" title="RMSE =\sqrt{\frac{1}{N} \sum (x_i -\hat{x_i})^2}" />
You can use the mean_square_error (MSE) function from sklearn, where the RMSE is just the square root of MSE. To read more about different evaluation metrics you can take a look at this article. 
Since you only want to consider predicted ratings that are in the test dataset, you filter out all other elements in the prediction matrix with prediction[ground_truth.nonzero()].
End of explanation
sparsity=round(1.0-len(df)/float(n_users*n_items),3)
print('The sparsity level of MovieLens100K is ' +  str(sparsity*100) + '%')
Explanation: Memory-based algorithms are easy to implement and produce reasonable prediction quality. 
The drawback of memory-based CF is that it doesn't scale to real-world scenarios and doesn't address the well-known cold-start problem, that is when new user or new item enters the system. Model-based CF methods are scalable and can deal with higher sparsity level than memory-based models, but also suffer when new users or items that don't have any ratings enter the system. I would like to thank Ethan Rosenthal for his post about Memory-Based Collaborative Filtering. 
Model-based Collaborative Filtering
Model-based Collaborative Filtering is based on matrix factorization (MF) which has received greater exposure, mainly as an unsupervised learning method for latent variable decomposition and dimensionality reduction. Matrix factorization is widely used for recommender systems where it can deal better with scalability and sparsity than Memory-based CF. The goal of MF is to learn the latent preferences of users and the latent attributes of items from known ratings (learn features that describe the characteristics of ratings) to then predict the unknown ratings through the dot product of the latent features of users and items. 
When you have a very sparse matrix, with a lot of dimensions, by doing matrix factorization you can restructure the  user-item matrix into low-rank structure, and you can represent the matrix by the multiplication of two low-rank matrices, where the rows contain the latent vector. You fit this matrix to approximate your original matrix, as closely as possible, by multiplying the low-rank matrices together, which fills in the entries missing in the original matrix.
Let's calculate the sparsity level of MovieLens dataset:
End of explanation
import scipy.sparse as sp
from scipy.sparse.linalg import svds
#get SVD components from train matrix. Choose k.
u, s, vt = svds(train_data_matrix, k = 20)
s_diag_matrix=np.diag(s)
X_pred = np.dot(np.dot(u, s_diag_matrix), vt)
print('User-based CF MSE: ' + str(rmse(X_pred, test_data_matrix)))
Explanation: To give an example of the learned latent preferences of the users and items: let's say for the MovieLens dataset you have the following information: (user id, age, location, gender, movie id, director, actor, language, year, rating). By applying matrix factorization the model learns that important user features are age group (under 10, 10-18, 18-30, 30-90), location and gender, and for movie features it learns that decade, director and actor are most important. Now if you look into the information you have stored, there is no such feature as the decade, but the model can learn on its own. The important aspect is that the CF model only uses data (user_id, movie_id, rating) to learn the latent features. If there is little data available model-based CF model will predict poorly, since it will be more difficult to learn the latent features. 
Models that use both ratings and content features are called Hybrid Recommender Systems where both Collaborative Filtering and Content-based Models are combined. Hybrid recommender systems usually show higher accuracy than Collaborative Filtering or Content-based Models on their own: they are capable to address the cold-start problem better since if you don't have any ratings for a user or an item you could use the metadata from the user or item to make a prediction. 
SVD
A well-known matrix factorization method is Singular value decomposition (SVD). Collaborative Filtering can be formulated by approximating a matrix X by using singular value decomposition. The winning team at the Netflix Prize competition used SVD matrix factorization models to produce product recommendations, for more information I recommend to read articles: Netflix Recommendations: Beyond the 5 stars and Netflix Prize and SVD.
The general equation can be expressed as follows:
<img src="https://latex.codecogs.com/gif.latex?X=USV^T" title="X=USV^T" />
Given m x n matrix X:
* U is an (m x r) orthogonal matrix
* S is an (r x r) diagonal matrix with non-negative real numbers on the diagonal
* V^T is an (r x n) orthogonal matrix
Elements on the diagnoal in S are known as singular values of X. 
Matrix X can be factorized to U, S and V. The U matrix represents the feature vectors corresponding to the users in the hidden feature space and the V matrix represents the feature vectors corresponding to the items in the hidden feature space.
<img class="aligncenter size-thumbnail img-responsive" style="max-width:100%; width: 50%; max-width: none" src="http://s33.postimg.org/kwgsb5g1b/BLOG_CCA_5.png"/>
Now you can make a prediction by taking dot product of U, S and V^T.
<img class="aligncenter size-thumbnail img-responsive" style="max-width:100%; width: 50%; max-width: none" src="http://s33.postimg.org/ch9lcm6pb/BLOG_CCA_4.png"/>
End of explanation
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