Web開發會經常遇到給實體打分的需求,比如論壇用戶的聲望分、電商系統的類目和產品熱度分,新聞的熱度分等。有些分數只需要排序使用,有些分數需要顯示給用戶,讓用戶看到分數之後能直觀的感受到這個分數所處的位置。往往熱度分的計算並不只是參考單一維度,會有很多維度的參考。比如,如果我們想計算一個論壇的用戶的綜合貢獻值,需要參考回帖數量、發帖數量、被點贊數量等指標。下面以論壇用戶貢獻值爲例子來演示一個熱度分的計算過程。
首先,初始化一個用戶表,200條記錄,字段爲id
,post_count
, reply_count
, faver_count
,值爲隨機整數。
SELECT row_number() over() as id,
(generate_series * random())::integer as post_count,
(generate_series * random())::integer as reply_count,
(generate_series * random())::integer as faver_count
INTO TABLE users
FROM (SELECT * FROM generate_series(1, 200)) AS r;
users
表數據如下:
select * from users limit 10;
id | post_count | reply_count | faver_count
----+------------+-------------+-------------
1 | 0 | 0 | 0
2 | 2 | 1 | 1
3 | 2 | 2 | 1
4 | 3 | 0 | 1
5 | 3 | 3 | 4
6 | 4 | 3 | 3
7 | 4 | 0 | 3
8 | 5 | 4 | 3
9 | 7 | 2 | 5
10 | 6 | 3 | 5
Row Number
最簡單的想法是單指標的排名相加,比如,按post_count
從大到小排序,算出post_num
,reply
和faver
同理:
SELECT *,
row_number() over(order by post_count desc) as post_num,
row_number() over(order by reply_count desc) as reply_num,
row_number() over(order by faver_count desc) as faver_num
FROM users;
計算結果如下,score = a * post_num + b * reply_num + c * faver_num
,其中abc
爲加權係數。這樣的計算方法存在一個問題,score
的範圍不確定,一個用戶打了99分的話,我們無法從99這個數值看出他處於什麼位置。
id | post_count | reply_count | faver_count | post_num | reply_num | faver_num
-----+------------+-------------+-------------+----------+-----------+-----------
187 | 62 | 25 | 173 | 70 | 122 | 1
169 | 6 | 57 | 167 | 176 | 69 | 2
171 | 2 | 46 | 162 | 193 | 84 | 3
172 | 41 | 152 | 162 | 101 | 6 | 4
200 | 76 | 193 | 149 | 51 | 1 | 5
156 | 62 | 116 | 144 | 69 | 28 | 6
166 | 114 | 31 | 144 | 20 | 109 | 7
153 | 127 | 97 | 138 | 10 | 39 | 8
135 | 25 | 131 | 135 | 135 | 16 | 9
186 | 112 | 147 | 134 | 23 | 7 | 10
163 | 106 | 134 | 133 | 28 | 13 | 11
155 | 124 | 3 | 132 | 14 | 188 | 12
173 | 74 | 92 | 132 | 53 | 42 | 13
133 | 119 | 78 | 132 | 17 | 53 | 14
NTile
一個改進的方案是,排名之後按區間分段,比如1-10打1分,11-20打2分,以此類推。這樣可以把每個指標的範圍確定,再加權之後範圍也是可以計算的。
SELECT *,
ntile(10) over(order by post_count desc) as post_num,
ntile(10) over(order by reply_count desc) as reply_num,
ntile(10) over(order by faver_count desc) as faver_num
FROM users;
按區間分段存在的問題是,結果不夠平滑,也不能反映不同用戶之間的差別,比如第一名和第二名分別爲10000和100,分段之後他們得到相同的分數,體現不出差異。
id | post_count | reply_count | faver_count | post_num | reply_num | faver_num
-----+------------+-------------+-------------+----------+-----------+-----------
187 | 62 | 25 | 173 | 4 | 7 | 1
169 | 6 | 57 | 167 | 9 | 4 | 1
171 | 2 | 46 | 162 | 10 | 5 | 1
172 | 41 | 152 | 162 | 6 | 1 | 1
200 | 76 | 193 | 149 | 3 | 1 | 1
156 | 62 | 116 | 144 | 4 | 2 | 1
166 | 114 | 31 | 144 | 1 | 6 | 1
153 | 127 | 97 | 138 | 1 | 2 | 1
135 | 25 | 131 | 135 | 7 | 1 | 1
186 | 112 | 147 | 134 | 2 | 1 | 1
163 | 106 | 134 | 133 | 2 | 1 | 1
155 | 124 | 3 | 132 | 1 | 10 | 1
173 | 74 | 92 | 132 | 3 | 3 | 1
133 | 119 | 78 | 132 | 1 | 3 | 1
174 | 42 | 14 | 131 | 5 | 8 | 1
181 | 120 | 10 | 130 | 1 | 8 | 1
148 | 38 | 120 | 129 | 6 | 2 | 1
176 | 84 | 105 | 128 | 3 | 2 | 1
128 | 113 | 103 | 128 | 2 | 2 | 1
179 | 114 | 118 | 127 | 1 | 2 | 1
157 | 66 | 120 | 127 | 4 | 2 | 2
198 | 122 | 35 | 126 | 1 | 6 | 2
195 | 166 | 112 | 118 | 1 | 2 | 2
192 | 175 | 124 | 117 | 1 | 1 | 2
Z Score
標準分數(Standard Score,又稱z-score,中文稱爲Z-分數或標準化值)在統計學中是一種無因次值,就是一種純數字標記,是藉由從單一(原始)分數中減去母體的平均值,再依照母體(母集合)的標準差分割成不同的差距,按照z值公式,各個樣本在經過轉換後,通常在正、負五到六之間不等。
WITH post AS (SELECT avg(post_count) as mean, stddev(post_count) as sd from users),
reply AS (SELECT avg(reply_count) as mean, stddev(reply_count) as sd from users),
faver AS (SELECT avg(faver_count) as mean, stddev(faver_count) as sd from users)
SELECT users.*,
((post_count - post.mean) / post.sd)::numeric(6,3) AS z_score_post,
((reply_count - reply.mean) / reply.sd)::numeric(6,3) AS z_score_reply,
((faver_count - faver.mean) / faver.sd)::numeric(6,3) AS z_score_faver
FROM users,
post,
faver,
reply
ORDER BY 4 DESC
結果如下:
id | post_count | reply_count | faver_count | z_score_post | z_score_reply | z_score_faver
-----+------------+-------------+-------------+--------------+---------------+---------------
187 | 62 | 25 | 173 | 0.251 | -0.543 | 2.835
169 | 6 | 57 | 167 | -1.076 | 0.150 | 2.697
172 | 41 | 152 | 162 | -0.247 | 2.208 | 2.582
171 | 2 | 46 | 162 | -1.171 | -0.088 | 2.582
200 | 76 | 193 | 149 | 0.582 | 3.096 | 2.283
156 | 62 | 116 | 144 | 0.251 | 1.428 | 2.168
166 | 114 | 31 | 144 | 1.483 | -0.413 | 2.168
153 | 127 | 97 | 138 | 1.791 | 1.016 | 2.031
135 | 25 | 131 | 135 | -0.626 | 1.753 | 1.962
186 | 112 | 147 | 134 | 1.435 | 2.099 | 1.939
163 | 106 | 134 | 133 | 1.293 | 1.818 | 1.916
133 | 119 | 78 | 132 | 1.601 | 0.605 | 1.893
173 | 74 | 92 | 132 | 0.535 | 0.908 | 1.893
155 | 124 | 3 | 132 | 1.720 | -1.019 | 1.893
174 | 42 | 14 | 131 | -0.223 | -0.781 | 1.870
181 | 120 | 10 | 130 | 1.625 | -0.868 | 1.847
148 | 38 | 120 | 129 | -0.318 | 1.515 | 1.824
128 | 113 | 103 | 128 | 1.459 | 1.146 | 1.801
176 | 84 | 105 | 128 | 0.772 | 1.190 | 1.801
179 | 114 | 118 | 127 | 1.483 | 1.471 | 1.778
Z-Score
的意義是樣本值到均值之間有多少個標準差,它的取值理論上也是沒有範圍的,但如果樣本數值服從正態分佈,會有99%以上的值落在[-3, 3]這個區間。如圖:
其實從上面我們計算的結果也可以觀察出這個結論。對於落在[-3, 3]
區間外的數據,我們可以調整爲3或-3,這個影響完全可以忽略不計。使用Z-Score
之後,我們可以保證了分數值既平滑又有範圍區間。