大家好，我是正在实战各种AI项目的程序员晚枫。
今天继续NumPy进阶内容，学习统计函数和线性代数运算。
这些函数是数据分析的核心工具，掌握它们，你就能处理90%的数值计算需求。
统计函数
基础统计量
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| import numpy as np
data = np.array([85, 92, 78, 90, 88, 95, 82])
print(np.sum(data))
print(np.mean(data))
print(np.median(data))
print(np.std(data))
print(np.var(data))
print(np.min(data), np.max(data))
print(np.argmin(data), np.argmax(data))
print(np.ptp(data))
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百分位数
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| scores = np.array([65, 70, 75, 80, 85, 90, 95, 98, 100])
print(np.percentile(scores, 25))
print(np.percentile(scores, 50))
print(np.percentile(scores, 75))
print(np.percentile(scores, 90))
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多维数组统计
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scores = np.array([
[85, 92, 78, 90],
[88, 85, 92, 87],
[90, 88, 85, 95]
])
print(np.mean(scores, axis=0))
print(np.sum(scores, axis=1))
print(np.max(scores, axis=0))
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常用数学函数
四舍五入与取整
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| arr = np.array([3.14, 2.718, 1.414, 4.669])
print(np.round(arr, 2))
print(np.floor(arr))
print(np.ceil(arr))
print(np.trunc(arr))
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条件统计
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| sales = np.array([120, 150, 80, 200, 95, 180, 110])
print(np.sum(sales > 100))
print(np.sum(sales[sales > 100]))
filtered = sales[sales >= 50]
print(np.mean(filtered))
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累积运算
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| revenue = np.array([100, 120, 90, 150, 130])
print(np.cumsum(revenue))
print(np.cumprod(revenue))
print(np.diff(revenue))
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线性代数运算
矩阵创建
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A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
print(A)
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矩阵运算
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print(A @ B)
print(A.T)
print(np.linalg.inv(A))
print(np.linalg.det(A))
eigenvalues, eigenvectors = np.linalg.eig(A)
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解线性方程组
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A = np.array([[2, 1], [1, -1]])
b = np.array([5, 1])
solution = np.linalg.solve(A, b)
print(solution)
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随机数生成
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np.random.seed(42)
np.random.rand(3, 3)
np.random.uniform(1, 10, 5)
np.random.randn(1000)
np.random.normal(0, 1, 100)
np.random.randint(1, 7, 10)
np.random.choice(['A', 'B', 'C'], size=5, p=[0.5, 0.3, 0.2])
arr = np.arange(10)
np.random.shuffle(arr)
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实战：销售数据分析
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| import numpy as np
np.random.seed(42)
daily_sales = np.random.normal(1000, 200, 30).astype(int)
print("=== 销售数据报告 ===")
print(f"总销售额: ¥{np.sum(daily_sales):,}")
print(f"日均销售: ¥{np.mean(daily_sales):.0f}")
print(f"最高单日: ¥{np.max(daily_sales)}")
print(f"最低单日: ¥{np.min(daily_sales)}")
print(f"标准差: ¥{np.std(daily_sales):.0f} (波动程度)")
top_10_threshold = np.percentile(daily_sales, 90)
top_days = daily_sales[daily_sales >= top_10_threshold]
print(f"\n优秀天数(前10%): {len(top_days)}天")
print(f"优秀标准: ¥{top_10_threshold:.0f}以上")
weekly_sales = daily_sales.reshape(5, 6).sum(axis=1)
print(f"\n每周销售: {weekly_sales}")
print(f"最佳周: 第{np.argmax(weekly_sales)+1}周")
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性能优化技巧
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result = []
for i in range(len(arr)):
result.append(arr[i] * 2)
result = arr * 2
standardized = (arr - np.mean(arr)) / np.std(arr)
result = np.zeros_like(arr)
for i in range(len(arr)):
result[i] = complex_operation(arr[i])
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性能对比：NumPy vs 纯Python实现
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| import numpy as np
import time
import math
data = np.random.randn(1000000)
py_list = data.tolist()
start = time.time()
mean_val = sum(py_list) / len(py_list)
var_val = sum((x - mean_val)**2 for x in py_list) / len(py_list)
std_val = math.sqrt(var_val)
py_time = time.time() - start
start = time.time()
np_std = np.std(data)
np_time = time.time() - start
print(f"纯Python: {py_time:.4f}秒, 结果={std_val:.6f}")
print(f"NumPy: {np_time:.4f}秒, 结果={np_std:.6f}")
print(f"NumPy快了{py_time/np_time:.0f}倍")
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进阶用法
矩阵分解与应用
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A = np.random.randn(5, 3)
U, S, Vt = np.linalg.svd(A, full_matrices=False)
print("奇异值:", S)
A_reconstructed = U @ np.diag(S) @ Vt
print("还原误差:", np.linalg.norm(A - A_reconstructed))
k = 2
A_low_rank = U[:, :k] @ np.diag(S[:k]) @ Vt[:k, :]
print("压缩后误差:", np.linalg.norm(A - A_low_rank))
print(f"压缩率: {A.size / (U[:,:k].size + S[:k].size + Vt[:k,:].size):.1f}倍")
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多维数组的轴操作
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data = np.random.randn(30, 24, 5)
daily_mean = data.mean(axis=1)
hourly_mean = data.mean(axis=0)
max_per_metric = data.max(axis=(0, 1))
best_hour_per_day = data.mean(axis=2).argmax(axis=1)
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向量化实现移动平均
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def moving_avg_cumsum(arr, window):
cumsum = np.cumsum(arr)
cumsum[window:] = cumsum[window:] - cumsum[:-window]
return cumsum[window-1:] / window
def moving_avg_convolve(arr, window):
return np.convolve(arr, np.ones(window)/window, mode='valid')
def moving_avg_loop(arr, window):
result = []
for i in range(len(arr) - window + 1):
result.append(arr[i:i+window].mean())
return np.array(result)
data = np.random.randn(100000)
window = 20
%timeit moving_avg_cumsum(data, window)
%timeit moving_avg_convolve(data, window)
%timeit moving_avg_loop(data, window)
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避坑指南
❌ 坑1：除以零的警告
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| arr = np.array([1, 2, 0, 4])
result = 1 / arr
print(result)
with np.errstate(divide='ignore', invalid='ignore'):
result = np.divide(1, arr)
result[arr == 0] = 0
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❌ 坑2：浮点数精度问题
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a = np.array([0.1, 0.2])
print(a.sum() == 0.3)
print(np.isclose(a.sum(), 0.3))
b = np.array([0.3])
print(np.allclose(a.sum(), b))
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❌ 坑3：随机数的可重复性
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print(np.random.rand(3))
np.random.seed(42)
print(np.random.rand(3))
rng = np.random.RandomState(42)
print(rng.rand(3))
rng = np.random.default_rng(42)
print(rng.random(3))
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实战案例：分析电商平台用户消费数据
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| import numpy as np
np.random.seed(42)
n_users = 10000
user_id = np.arange(1, n_users + 1)
register_days = np.random.exponential(365, n_users).astype(int)
total_spend = np.random.exponential(2000, n_users)
order_count = np.random.poisson(8, n_users)
avg_order_value = total_spend / np.maximum(order_count, 1)
last_purchase_days = np.random.exponential(30, n_users).astype(int)
print("=" * 50)
print("电商平台用户消费分析报告")
print("=" * 50)
print(f"总用户数: {n_users:,}")
print(f"平均消费: ¥{total_spend.mean():,.0f}")
print(f"消费中位数: ¥{np.median(total_spend):,.0f}")
print(f"消费标准差: ¥{total_spend.std():,.0f}")
is_active = last_purchase_days < 30
is_high_value = total_spend > np.percentile(total_spend, 80)
vip = is_active & is_high_value
potential = is_active & ~is_high_value
at_risk = ~is_active & is_high_value
lost = ~is_active & ~is_high_value
print(f"\nVIP用户: {vip.sum()} ({vip.mean()*100:.1f}%)")
print(f"潜力用户: {potential.sum()} ({potential.mean()*100:.1f}%)")
print(f"风险用户: {at_risk.sum()} ({at_risk.mean()*100:.1f}%)")
print(f"流失用户: {lost.sum()} ({lost.mean()*100:.1f}%)")
for name, mask in [("VIP", vip), ("潜力", potential), ("风险", at_risk), ("流失", lost)]:
print(f"{name}用户平均客单价: ¥{avg_order_value[mask].mean():.0f}")
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下节预告
下一课我们将进入Pandas，这是数据分析最核心的库。
你将学会：
- Series和DataFrame数据结构
- 如何创建和操作表格数据
👉 继续阅读：Pandas入门-Series和DataFrame
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课程导航
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