協同過濾用戶相似度度量

閔氏距離(Minkowski Distance)

當r=1時，曼哈頓距離(Manhatten)

當r=2時，歐氏距離(Euclidean)

r=無窮大，上確界距離(Supermum Distance)

皮爾遜相關系數(Pearson CORRELATION Coeffcient)，取值[-1,1]，1表示完全相關，-1表示完全不相關

近似計算公式

余弦相似度計算，取值[-1,1]，1表示完全相似，-1表示完全不相似

 

users = {"Angelica": {"Blues Traveler": 3.5, "Broken Bells": 2.0, "Norah Jones": 4.5, "Phoenix": 5.0, "Slightly Stoopid": 1.5, "The Strokes": 2.5, "Vampire Weekend": 2.0},
         "Bill":{"Blues Traveler": 2.0, "Broken Bells": 3.5, "Deadmau5": 4.0, "Phoenix": 2.0, "Slightly Stoopid": 3.5, "Vampire Weekend": 3.0},
         "Chan": {"Blues Traveler": 5.0, "Broken Bells": 1.0, "Deadmau5": 1.0, "Norah Jones": 3.0, "Phoenix": 5, "Slightly Stoopid": 1.0},
         "Dan": {"Blues Traveler": 3.0, "Broken Bells": 4.0, "Deadmau5": 4.5, "Phoenix": 3.0, "Slightly Stoopid": 4.5, "The Strokes": 4.0, "Vampire Weekend": 2.0},
         "Hailey": {"Broken Bells": 4.0, "Deadmau5": 1.0, "Norah Jones": 4.0, "The Strokes": 4.0, "Vampire Weekend": 1.0},
         "Jordyn":  {"Broken Bells": 4.5, "Deadmau5": 4.0, "Norah Jones": 5.0, "Phoenix": 5.0, "Slightly Stoopid": 4.5, "The Strokes": 4.0, "Vampire Weekend": 4.0},
         "Sam": {"Blues Traveler": 5.0, "Broken Bells": 2.0, "Norah Jones": 3.0, "Phoenix": 5.0, "Slightly Stoopid": 4.0, "The Strokes": 5.0},
         "Veronica": {"Blues Traveler": 3.0, "Norah Jones": 5.0, "Phoenix": 4.0, "Slightly Stoopid": 2.5, "The Strokes": 3.0}
        }#{用戶：{作品：評分}}
def manhattan(rating1, rating2):#計算曼哈頓距離
    """Computes the Manhattan distance. Both rating1 and rating2 are dictionaries
       of the form {'The Strokes': 3.0, 'Slightly Stoopid': 2.5}"""
    distance = 0
    commonRatings = False 
    for key in rating1:
        if key in rating2:
            distance += abs(rating1[key] - rating2[key])
            commonRatings = True
    if commonRatings:
        return distance
    else:
        return -1
def pearson(rating1, rating2):#計算Pearson相關系數
    sum_xy = 0
    sum_x = 0
    sum_y = 0
    sum_x2 = 0
    sum_y2 = 0
    n = 0
    for key in rating1:
        if key in rating2:
            n += 1
            x = rating1[key]
            y = rating2[key]
            sum_xy += x * y
            sum_x += x
            sum_y += y
            sum_x2 += pow(x, 2)
            sum_y2 += pow(y, 2)
    # now compute denominator
    denominator = sqrt(sum_x2 - pow(sum_x, 2) / n) * sqrt(sum_y2 - pow(sum_y, 2) / n)
    if denominator == 0:
        return 0
    else:
        return (sum_xy - (sum_x * sum_y) / n)/denominator

 

相似度的選擇：

當不同用戶對不同商品評價標准的范圍不一樣時，使用皮爾遜相關系數；

當數據稠密，且屬性值大小十分重要，使用歐氏或者曼哈頓距離；

當數據稀疏，存在很多零值，考慮余弦相似度。

來自《A Programmer's Guide To Data Mining》