recommend | Microsoft SAR Analysis of nearest neighbor collaborative filtering algorithm （ One ） The previous article introduces the whole SAR Algorithm , The algorithm itself is easier to understand . This article is mainly about the interesting small functions .
This happened in $C$ matrix co-occurence matrix
become $S$ matrix item similarity matrix
, It's about the correlation between each other .
Once we have a co-occurrence matrix , The similarity matrix can be obtained by rescaling the co-occurrence according to the given measure ：Jaccard
, lift
, and counts
( It's counting , In fact, it has not changed , No compression / The zoom ).
If $c_{ii}$ and $c_{jj}$ yes matrix $C$ Of the $i$ individual and The first $j$ A diagonal element , Then the rescale option is :
Jaccard
: $s_{ij}=(c_{ii}+c_{jj}−c_{ij})c_{ij} $lift
: $s_{ij}=(c_{ii}×c_{jj})c_{ij} $counts
: $s_{ij}=c_{ij}$The formula is as follows ：
import numpy as np
def jaccard(cooccurrence):
"""Helper method to calculate the Jaccard similarity of a matrix of co-occurrences.
Args:
cooccurrence (np.array): the symmetric matrix of co-occurrences of items.
Returns:
np.array: The matrix of Jaccard similarities between any two items.
"""
diag = cooccurrence.diagonal()
diag_rows = np.expand_dims(diag, axis=0)
diag_cols = np.expand_dims(diag, axis=1)
with np.errstate(invalid="ignore", divide="ignore"):
result = cooccurrence / (diag_rows + diag_cols - cooccurrence)
return np.array(result)
def lift(cooccurrence):
"""Helper method to calculate the Lift of a matrix of co-occurrences.
Args:
cooccurrence (np.array): the symmetric matrix of co-occurrences of items.
Returns:
np.array: The matrix of Lifts between any two items.
"""
diag = cooccurrence.diagonal()
diag_rows = np.expand_dims(diag, axis=0)
diag_cols = np.expand_dims(diag, axis=1)
with np.errstate(invalid="ignore", divide="ignore"):
result = cooccurrence / (diag_rows * diag_cols)
return np.array(result)
Here must be the calculation of two diagonal matrices ,
import numpy as np
from scipy import sparse
data = [[1,3,5],[3,1,6],[5,6,1]]
x = np.array(data)
x
# For a diagonal matrix jaccard 、 lift
jaccard(x) # It has to be square
lift(x)
among jaccard Output ：
jaccard(x) # It has to be square
Out[63]:
array([[ 1. , -3. , -1.66666667],
[-3. , 1. , -1.5 ],
[-1.66666667, -1.5 , 1. ]])
def get_top_k_scored_items(scores, top_k, sort_top_k=False):
"""Extract top K items from a matrix of scores for each user-item pair,
{optionally sort results per user.
Args:
scores (np.array): score matrix (users x items).
top_k (int): number of top items to recommend.
sort_top_k (bool): flag to sort top k results.
Returns:
np.array, np.array: indices into score matrix for each users top items, scores corresponding to top items.
"""
# ensure we're working with a dense ndarray
if isinstance(scores, sparse.spmatrix):
scores = scores.todense()
if scores.shape[1] < top_k:
logger.warning(
"Number of items is less than top_k, limiting top_k to number of items"
)
k = min(top_k, scores.shape[1])
test_user_idx = np.arange(scores.shape[0])[:, None]
# get top K items and scores
# this determines the un-ordered top-k item indices for each user
top_items = np.argpartition(scores, -k, axis=1)[:, -k:]
top_scores = scores[test_user_idx, top_items]
if sort_top_k:
sort_ind = np.argsort(-top_scores)
top_items = top_items[test_user_idx, sort_ind]
top_scores = top_scores[test_user_idx, sort_ind]
return np.array(top_items), np.array(top_scores)
In a user-item In the matrix of , Take out the most important top-k Of item.
Example :
import numpy as np
from scipy import sparse
data = [[1,3,5],[3,1,6],[5,6,1]]
x = np.array(data)
get_top_k_scored_items(x, 1, sort_top_k=False) # Extract the maximum value by row
give the result as follows ：
(array([[2],
[2],
[1]], dtype=int64),
array([[5],
[6],
[6]]))
first line （ first ） The maximum number of users is 2（ Matrix one ）, The value is 5（ Matrix two ）.
Before that, I also studied sparse matrix ,scipy.sparse、pandas.sparse、sklearn The use of sparse matrices , Just by the way SAR How to use ：
utilize coo_matrix Form a matrix -> Turn into csr Calculate
Intercept sar_singlenode.py The code in ：
# generate pseudo user affinity using seed items
pseudo_affinity = sparse.coo_matrix(
(ratings, (user_ids, item_ids)), shape=(n_users, self.n_items)
).tocsr()
# calculate raw scores with a matrix multiplication
test_scores = pseudo_affinity.dot(self.item_similarity)
# remove items in the seed set so recommended items are novel
test_scores[user_ids, item_ids] = -np.inf
top_items, top_scores = get_top_k_scored_items(
scores=test_scores, top_k=top_k, sort_top_k=sort_top_k
)
get_top_k_scored_items(pseudo_affinity, 1, sort_top_k=False)
Here is a simple example of generation ：
# sparse matrix
ratings = [5,4,3]
user_ids = [0,1,2]
item_ids = [1,2,1]
n_users = len(user_ids)# Scribble , It's better than max(user_ids) Big
n_items = 10 # Scribble , It's better than max(item_ids) Big
pseudo_affinity = sparse.coo_matrix(
(ratings, (user_ids, item_ids)), shape=(n_users, n_items)
).tocsr()
pseudo_affinity.todense()
pseudo_affinity.toarray()
The result is ：
array([[0, 5, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 4, 0, 0, 0, 0, 0, 0, 0],
[0, 3, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=int32)
among ( Reference resources ：python scipy Sparse matrix details )：
csr_matrix It can be used in various arithmetic operations ： It supports addition , Subtraction , Multiplication , Division and matrix idempotent operations . It has five instantiation methods , The first four initialization methods are similar coo_matrix
, That is, by building dense matrices 、 Through other types of sparse matrix transformation 、 Build a certain shape The empty matrix of 、 adopt (row, col, data)
Build the matrix . Its fifth initialization mode is directly reflected in csr_matrix Storage characteristics of ：csr_matrix((data, indices, indptr), [shape=(M, N)])
, intend , In matrix i The column number of a non-zero row element is indices[indptr[i]:indptr[i+1]]
, The corresponding value is data[indptr[i]:indptr[i+1]]
>>> import numpy as np
>>> from scipy.sparse import csr_matrix
>>> indptr = np.array([0, 2, 3, 6])
>>> indices = np.array([0, 2, 2, 0, 1, 2])
>>> data = np.array([1, 2, 3, 4, 5, 6])
>>> csr = csr_matrix((data, indices, indptr), shape=(3, 3)).toarray()
array([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
csr_matrix There are also many ways , among tobytes()
,tolist()
, tofile()
,tostring()
It is worth noting that , For other details, please refer to the official documents ,csr_matrix The first five attributes of an object are the same as coo_matrix
, There are also properties as follows ：
indices
And properties data One-to-one correspondence , The element value represents the column number in a row
indptr
csr_matrix The starting value of each row ,length(csr_object.indptr) == csr_object.shape[0] + 1
has_sorted_indices
Judge the line of indices Whether it's orderly , return bool value
csr_matrix The advantages of ：
csr_matrix The shortcomings of ：
stay top-K In recommendation ,HR Is a commonly used index to measure the recall rate , The formula is ：
The denominator is all the test sets , The numerator represents each user top-K The sum of the number of test sets in the list .
A simple example , The number of products in the test set for the three users are respectively 10,12,8, The model gets top-10 In the recommended list , There were 6 individual ,5 individual ,4 It's in the test set , So at this time HR The value of is
(6+5+4)/(10+12+8) = 0.5.
Related definitions ：
def hit(gt_items, pred_items):
count = 0
for item in pred_items:
if item in gt_items:
count += 1
return count
Average accuracy AP, If we use google Search for a keyword , Back to 10 results . The best case, of course, is this 10 The results are all the relevant information we want . But if only part of it is relevant , such as 5 individual , So this 5 A result is also a relatively good result if it is shown in the front . But if this 5 Related information from 6 Results returned , Then this situation is relatively poor . This is AP The indicators reflected , And recall There's something similar about the concept of , But is “ Sequence sensitive recall.
For the user uu, Recommend something to him , that uu The average accuracy of is ：
def AP(ranked_list, ground_truth):
"""Compute the average precision (AP) of a list of ranked items
"""
hits = 0
sum_precs = 0
for n in range(len(ranked_list)):
if ranked_list[n] in ground_truth:
hits += 1
sum_precs += hits / (n + 1.0)
if hits > 0:
return sum_precs / len(ground_truth)
else:
return 0
MAP Represents all users uu Of AP Take the mean again , The calculation formula is as follows ：
Cumulative gain CG, In the recommendation system CG It means that the score of the relevance of each recommendation result is accumulated as the score of the whole recommendation list ：
among ,rel Indicate location i The relevance of the recommended results of ,k Indicates the size of the recommended list .
CG The effect of different positions of each recommendation result on the whole recommendation result is not considered , for example , We always want results that are highly relevant to come first , Low relevancy at the top affects the user experience .
Suppose search “ Basketball ” result , The ideal result is ：B1, B2, B3; And the result is B3, B1,
B2 Words ,CG There is no change in the value of , So we need the following DCG.
DCG stay CG On the basis of the introduction of location factors , The calculation formula is as follows ：
From the above formula we can get ：1） The more relevant the recommendation results are ,DCG The bigger it is .2） Those with good relevance are at the top of the recommendation list , Better recommendation ,DCG The bigger it is .
DCG It's difficult to evaluate horizontally between different recommendation lists , However, it is impossible for us to evaluate a recommendation system only by using a user's recommendation list and corresponding results , Instead, it evaluates the users and their recommended list results throughout the test set . that , The evaluation scores of different users' recommendation lists need to be normalized , That is to say NDCG.
IDCG Represents a list of the best recommendation results returned by a user of the recommendation system , That is, suppose the returned results are sorted according to the relevance , The most relevant results come first , Of this sequence DCG by IDCG. therefore DCG The value is between (0,IDCG] , so NDCG The value is between (0,1], So users u Of NDCG@K Defined as ：
Average NDCG The value of is ：
For example ：
Suppose the search comes back 5 results , The correlation scores are 3、2、3、0、1、2
that CG = 3+2+3+0+1+2
It can be seen that only a related score is given for the relevant score , There is no recall location effect on ranking result score . And we see DCG：
therefore DCG = 3+1.26+1.5+0+0.38+0.71 = 6.86
Next we normalize , Normalization needs to be settled first IDCG, If we actually recall 8 Items , Except for the top 6 individual , There are two other results , Hypothesis number 1 7 The correlation is 3, The first 8 The correlation is 0. So in the ideal case, the correlation score should be ：3、3、3、2、2、1、0、0. Calculation IDCG@6:
therefore IDCG = 3+1.89+1.5+0.86+0.77+0.35 = 8.37
so Final NDCG@6 = 6.86/8.37 = 81.96%
You can go through %time
To time , Here's a ：
from timeit import default_timer
from datetime import timedelta
import time
class Timer(object):
"""Timer class.
`Original code <https://github.com/miguelgfierro/pybase/blob/2298172a13fb4a243754acbc6029a4a2dcf72c20/log_base/timer.py>`_.
Examples:
>>> import time
>>> t = Timer()
>>> t.start()
>>> time.sleep(1)
>>> t.stop()
>>> t.interval < 1
True
>>> with Timer() as t:
... time.sleep(1)
>>> t.interval < 1
True
>>> "Time elapsed {}".format(t) #doctest: +ELLIPSIS
'Time elapsed 1...'
"""
def __init__(self):
self._timer = default_timer
self._interval = 0
self.running = False
def __enter__(self):
self.start()
return self
def __exit__(self, *args):
self.stop()
def __str__(self):
return "{:0.4f}".format(self.interval)
def start(self):
"""Start the timer."""
self.init = self._timer()
self.running = True
def stop(self):
"""Stop the timer. Calculate the interval in seconds."""
self.end = self._timer()
try:
self._interval = self.end - self.init
self.running = False
except AttributeError:
raise ValueError(
"Timer has not been initialized: use start() or the contextual form with Timer() as t:"
)
@property
def interval(self):
"""Get time interval in seconds.
Returns:
float: Seconds.
"""
if self.running:
raise ValueError("Timer has not been stopped, please use stop().")
else:
return self._interval
After this execution, we can see that ：
# perform
with Timer() as t:
time.sleep(1)
# The elapsed time
t.interval
In terms of structure, there are many aspects , You can use it later
Recommendation algorithm commonly used evaluation index ：NDCG、MAP、MRR、HR、ILS、ROC、AUC、F1 etc.
Search for metrics ——NDCG