🤝 第2课：协同过滤（UserCF / ItemCF）

🎯 学习目标

🗂️ 课程定位

📖 核心理论

协同过滤（Collaborative Filtering）是推荐系统最经典的方法，核心假设：过去偏好相似的用户，未来也会偏好相似的物品。它的优势在于无需领域知识，仅依赖用户行为数据即可完成推荐。

协同过滤起源于1992年Xerox的Tapestry系统，1994年GroupLens将其应用于新闻推荐。Amazon在2001年提出ItemCF方法，成为电商推荐的标配。Netflix Prize竞赛（2006-2009）进一步推动了协同过滤方法的发展。

📐 数学公式

💻 代码实现

import numpy as np
from collections import defaultdict
np.random.seed(42)
users=[f"U{i}"for i in range(1,16)];items=[f"I{j}"for j in range(1,21)]
ratings={}
for u in users:
    n=np.random.poisson(5)+2
    for item in np.random.choice(items,size=min(n,len(items)),replace=False):
        ratings[(u,item)]=np.random.choice([1,2,3,4,5],p=[.05,.1,.2,.35,.3])
print("="*60+"\n协同过滤推荐系统\n"+"="*60)
print(f"用户:{len(users)} 物品:{len(items)} 交互:{len(ratings)} 密度:{len(ratings)/(len(users)*len(items)):.4f}")
def cosine_sim(a,b):
    ck=set(a.keys())&set(b.keys())
    if not ck:return 0.0
    va=np.array([a[k]for k in ck]);vb=np.array([b[k]for k in ck])
    d=np.linalg.norm(va)*np.linalg.norm(vb)
    return np.dot(va,vb)/d if d>0 else 0.0
def pearson_sim(a,b):
    ck=set(a.keys())&set(b.keys())
    if len(ck)<2:return 0.0
    va=np.array([a[k]for k in ck]);vb=np.array([b[k]for k in ck])
    va_c,vb_c=va-va.mean(),vb-vb.mean()
    d=np.linalg.norm(va_c)*np.linalg.norm(vb_c)
    return np.dot(va_c,vb_c)/d if d>0 else 0.0
uv=defaultdict(dict);iv=defaultdict(dict)
for(u,i),r in ratings.items():uv[u][i]=r;iv[i][u]=r
# UserCF
target="U1";K=5
sims={u:pearson_sim(uv[target],uv[u])for u in users if u!=target}
top_k=sorted(sims.items(),key=lambda x:-x[1])[:K]
print(f"\n与{target}最相似的{K}个用户(Pearson):")
for u,s in top_k:print(f"  {u}:{s:.4f}")
ti=set(items)-set(uv[target].keys());preds={}
for item in ti:
    num=den=0
    for u,sim in top_k:
        if item in uv[u]:num+=sim*uv[u][item];den+=abs(sim)
    if den>0:preds[item]=num/den
urec=sorted(preds.items(),key=lambda x:-x[1])[:5]
print(f"UserCF推荐Top-5:");[print(f"  {i}:{s:.3f}")for i,s in urec]
# ItemCF
ti2="I1";isims={i:cosine_sim(iv[ti2],iv[i])for i in items if i!=ti2}
top_ki=sorted(isims.items(),key=lambda x:-x[1])[:K]
print(f"\n与{ti2}最相似的{K}个物品:")
for i,s in top_ki:print(f"  {i}:{s:.4f}")
ipreds={}
for item in ti:
    num=den=0
    for ri,rv in uv[target].items():
        s=cosine_sim(iv.get(item,{}),iv.get(ri,{}));num+=s*rv;den+=abs(s)
    if den>0:ipreds[item]=num/den
irec=sorted(ipreds.items(),key=lambda x:-x[1])[:5]
print(f"ItemCF推荐Top-5:");[print(f"  {i}:{s:.3f}")for i,s in irec]
# 评估
test_data={};train_data=dict(ratings)
for u in users:
    ui=[(i,r)for(uu,i),r in ratings.items()if uu==u]
    if ui:h=ui[np.random.randint(len(ui))];test_data[(u,h[0])]=h[1];del train_data[(u,h[0])]
def eval_cf(data,method,Kn=5,Kr=5):
    u2=defaultdict(dict);i2=defaultdict(dict)
    for(u,i),r in data.items():u2[u][i]=r;i2[i][u]=r
    hits=0;ndcgs=[]
    for(u,ti3),tr in test_data.items():
        if u not in u2:continue
        ts=set(items)-set(u2[u].keys())
        if ti3 not in ts:ts.add(ti3)
        ps={}
        for item in ts:
            num=den=0
            if method=='user':
                sl=[(o,pearson_sim(u2[u],u2[o]))for o in users if o!=u and o in u2]
                for o,s in sorted(sl,key=lambda x:-x[1])[:Kn]:
                    if item in u2[o]:num+=s*u2[o][item];den+=abs(s)
            else:
                for ri,rv in u2[u].items():
                    s=cosine_sim(i2.get(item,{}),i2.get(ri,{}));num+=s*rv;den+=abs(s)
            if den>0:ps[item]=num/den
        rec=[i for i,_ in sorted(ps.items(),key=lambda x:-x[1])[:Kr]]
        if ti3 in rec:hits+=1;ndcgs.append(1.0/np.log2(rec.index(ti3)+2))
        else:ndcgs.append(0)
    n=len(test_data);return hits/n if n else 0,np.mean(ndcgs)
u_hr,u_ndcg=eval_cf(train_data,'user');i_hr,i_ndcg=eval_cf(train_data,'item')
print(f"\n{'方法':<10}{'HitRate@5':<12}{'NDCG@5':<10}")
print("-"*32);print(f"{'UserCF':<10}{u_hr:<12.4f}{u_ndcg:<10.4f}");print(f"{'ItemCF':<10}{i_hr:<12.4f}{i_ndcg:<10.4f}")
# Jaccard对比
u1,u2_id="U1","U2"
if u1 in uv and u2_id in uv:
    cs=cosine_sim(uv[u1],uv[u2_id]);ps=pearson_sim(uv[u1],uv[u2_id])
    sa=set(uv[u1].keys());sb=set(uv[u2_id].keys())
    js=len(sa&sb)/len(sa|sb)if sa|sb else 0
    print(f"\n{u1}与{u2_id}相似度:Cosine={cs:.4f} Pearson={ps:.4f} Jaccard={js:.4f}")

🔴 运行结果

✅ 验证通过

🔍 代码解读

💡 关键洞察

📝 本章小结

🧪 练习

📚 延伸阅读

💼 面试考点

🏭 工业案例