📏 第19课：离线评估指标

🎯 学习目标

🗂️ 课程定位

📖 核心理论

离线评估是推荐系统开发的基础环节，用历史数据评估模型效果。但离线指标和在线效果之间常有差距，理解各种指标的适用场景和局限至关重要。

📐 数学公式

💻 代码实现

import numpy as np
np.random.seed(42)
print("="*60+"\n离线评估指标\n"+"="*60)
n_users=100;n_items=50
# 生成预测分数和真实标签
y_true=np.random.random((n_users,n_items))>0.85
y_scores=np.random.random((n_users,n_items))
# 让分数和标签有一定相关性
y_scores=y_scores*0.3+y_true*0.7+np.random.random((n_users,n_items))*0.3
print(f"用户:{n_users} 物品:{n_items} 正例率:{y_true.mean():.4f}")
K=5
# Precision@K
def precision_at_k(y_true,y_scores,k=5):
    precs=[]
    for u in range(len(y_true)):
        top_k=np.argsort(-y_scores[u])[:k]
        precs.append(y_true[u,top_k].mean())
    return np.mean(precs)
# Recall@K
def recall_at_k(y_true,y_scores,k=5):
    recs=[]
    for u in range(len(y_true)):
        pos=np.where(y_true[u])[0]
        if len(pos)==0:continue
        top_k=set(np.argsort(-y_scores[u])[:k])
        recs.append(len(top_k&set(pos))/len(pos))
    return np.mean(recs) if recs else 0
# NDCG@K
def ndcg_at_k(y_true,y_scores,k=5):
    ndcgs=[]
    for u in range(len(y_true)):
        top_k=np.argsort(-y_scores[u])[:k]
        rel=y_true[u,top_k].astype(float)
        dcg=np.sum((2**rel-1)/np.log2(np.arange(len(rel))+2))
        ideal_rel=sorted(y_true[u].astype(float),reverse=True)[:k]
        idcg=np.sum((2**np.array(ideal_rel)-1)/np.log2(np.arange(len(ideal_rel))+2))
        ndcgs.append(dcg/idcg if idcg>0 else 0)
    return np.mean(ndcgs)
# HitRate@K
def hit_rate_at_k(y_true,y_scores,k=5):
    hits=0;n=0
    for u in range(len(y_true)):
        pos=np.where(y_true[u])[0]
        if len(pos)==0:continue
        top_k=set(np.argsort(-y_scores[u])[:k])
        if top_k&set(pos):hits+=1
        n+=1
    return hits/n if n else 0
# AUC
def auc_score(y_true,y_scores):
    aucs=[]
    for u in range(len(y_true)):
        pos=np.where(y_true[u])[0];neg=np.where(~y_true[u])[0]
        if len(pos)==0 or len(neg)==0:continue
        sp=y_scores[u,pos];sn=y_scores[u,neg]
        aucs.append(np.mean(sp[:,None]>sn[None,:]))
    return np.mean(aucs) if aucs else 0.5
# MRR
def mrr_score(y_true,y_scores):
    mrrs=[]
    for u in range(len(y_true)):
        pos=np.where(y_true[u])[0]
        if len(pos)==0:continue
        order=np.argsort(-y_scores[u])
        for rank,idx in enumerate(order):
            if y_true[u,idx]:mrrs.append(1/(rank+1));break
    return np.mean(mrrs) if mrrs else 0
# 覆盖率
def coverage(y_scores,k=5):
    recommended=set()
    for u in range(len(y_scores)):
        top_k=np.argsort(-y_scores[u])[:k]
        recommended.update(top_k)
    return len(recommended)/y_scores.shape[1]
# 计算所有指标
print(f"\n--- 离线评估指标(K={K}) ---")
for k in [3,5,10,20]:
    p=precision_at_k(y_true,y_scores,k)
    r=recall_at_k(y_true,y_scores,k)
    n=ndcg_at_k(y_true,y_scores,k)
    h=hit_rate_at_k(y_true,y_scores,k)
    c=coverage(y_scores,k)
    print(f"  K={k}:P={p:.4f} R={r:.4f} NDCG={n:.4f} HR={h:.4f} Cov={c:.2%}")
print(f"  AUC={auc_score(y_true,y_scores):.4f}")
print(f"  MRR={mrr_score(y_true,y_scores):.4f}")
# 随机基线对比
y_random=np.random.random((n_users,n_items))
print(f"\n--- 与随机基线对比 ---")
for k in [5,10]:
    p1=precision_at_k(y_true,y_scores,k)
    p0=precision_at_k(y_true,y_random,k)
    print(f"  K={k}:模型P={p1:.4f} 随机P={p0:.4f} 提升={p1/max(p0,1e-6):.1f}x")

🔴 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