def z_test(data1, data2=None, tail="both", mu=0, sigma1=1, sigma2=None):
"""
z检验
:param data1: 样本 1
:param data2: 样本 2
:param tail: 是否双尾检验， 默认是
:param mu: μ值
:param sigma1:
:param sigma2:
:return:
"""
assert tail in ["both", "left", "right"], \
'tail should be one of "both", "left", "right"'
if data2 is None:
# 样本均值
mean_value = mean(data1)
# 标准误
se = sigma1 / sqrt(len(data1))
z_value = (mean_value - mu) / se
else:
assert sigma2 is not None
mean_diff = mean(data1) - mean(data2)
se = sqrt(sigma1**2 / len(data1) + sigma2**2 / len(data2))
z_value = (mean_diff - mu) / se
if tail == "both":
# 计算面积
p = 2 * (1 - norm.cdf(abs(z_value)))
elif tail == "left":
p = norm.cdf(z_value)
else:
p = 1 - norm.cdf(z_value)
return z_value, p
def mean_diff_ci_t_est(data1, data2, alpha, equal=True):
# 样本容量 1
n1 = len(data1)
# 样本容量 2
n2 = len(data2)
# 均值差
mean_diff = mean(data1) - mean(data2)
# 样本方差
sample1_var = variance(data1)
sample2_var = variance(data2)
# 两个总体方差未知且相等，求均值差的置信区间
if equal:
# 联合标准差
sw = sqrt(
((n1 - 1) * sample1_var + (n2 - 1) * sample2_var) / (n1 + n2 - 2))
t_value = abs(t.ppf(alpha / 2, n1 + n2 - 2))
return mean_diff - sw * sqrt(1 / n1 + 1 / n2) * t_value, \
mean_diff + sw * sqrt(1 / n1 + 1 / n2) * t_value
# 两个总体方差未知且不等，求均值差的置信区间
else:
# 自由度
# 分子
df_numerator = (sample1_var / n1 + sample2_var / n2)**2
# 分母
df_denominator = (sample1_var / n1)**2 / (n1 - 1) + (sample2_var /
n2)**2 / (n2 - 1)
df = df_numerator / df_denominator
t_value = abs(t.ppf(alpha / 2, df))
return mean_diff - sqrt(sample1_var / n1 + sample2_var / n2) * t_value, \
mean_diff + sqrt(sample1_var / n1 + sample2_var / n2) * t_value
def mean_diff_ci_z_est(data1, data2, alpha, sigma1, sigma2):
# 样本容量 1
n1 = len(data1)
# 样本容量 2
n2 = len(data2)
# 均值差
mean_diff = mean(data1) - mean(data2)
z_value = abs(norm.ppf(alpha / 2))
return mean_diff - sqrt(sigma1 ** 2 / n1 + sigma2 ** 2 / n2) * z_value, \
mean_diff + sqrt(sigma1 ** 2 / n1 + sigma2 ** 2 / n2) * z_value
def anova_oneway(data):
"""单因素方差分析"""
k = len(data)
assert k > 1
# 组均值
group_means = [mean(group) for group in data]
# 组样本容量
group_szs = [len(group) for group in data]
n = sum(group_szs)
assert n > k
# 总平均
grand_mean = sum(
group_mean * group_sz
for group_mean, group_sz in zip(group_means, group_szs)) / n
# 平方和
sst = sum(sum((y - grand_mean)**2 for y in group) for group in data)
ssg = sum((group_mean - grand_mean)**2 * group_sz
for group_mean, group_sz in zip(group_means, group_szs))
sse = sst - ssg
dfg = k - 1
dfe = n - k
# 均方和
msg = ssg / dfg
mse = sse / dfe
f_value = msg / mse
p = 1 - f.cdf(f_value, dfg, dfe)
return f_value, dfg, dfe, p
def sample(num_of_samples, sample_sz):
data = []
# 遍历样本
for _ in range(num_of_samples):
# 从 0-1的均匀分布中抽取 sample_sz 的个体组成的样本,mean 计算样本均值
data.append(mean([random.uniform(0.0, 1.0) for _ in range(sample_sz)]))
return data
def variance_bias(data):
"""无偏性方差"""
n = len(data)
if n <= 1:
return None
mean_value = mean(data)
return sum((e - mean_value)**2 for e in data) / n
def t_test(data1, data2=None, tail="both", mu=0, equal=True):
"""
t检验
:param data1: 样本 1
:param data2: 样本 2
:param tail: 是否双尾检验， 默认是
:param mu: μ值
:param equal:
:return:
"""
assert tail in ["both", "left", "right"], \
'tail should be one of "both", "left", "right"'
if data2 is None:
mean_val = mean(data1)
se = std(data1) / sqrt(len(data1))
t_val = (mean_val - mu) / se
df = len(data1) - 1
else:
n1 = len(data1)
n2 = len(data2)
mean_diff = mean(data1) - mean(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
if equal:
sw = sqrt(((n1 - 1) * sample1_var + (n2 - 1) * sample2_var) /
(n1 + n2 - 2))
t_val = (mean_diff - mu) / (sw * sqrt(1 / n1 + 1 / n2))
df = n1 + n2 - 2
else:
se = sqrt(sample1_var / n1 + sample2_var / n2)
t_val = (mean_diff - mu) / se
df_numerator = (sample1_var / n1 + sample2_var / n2)**2
df_denominator = (sample1_var / n1)**2 / (n1 - 1) + (
sample2_var / n2)**2 / (n2 - 1)
df = df_numerator / df_denominator
if tail == "both":
p = 2 * (1 - t.cdf(abs(t_val), df))
elif tail == "left":
p = t.cdf(t_val, df)
else:
p = 1 - t.cdf(t_val, df)
return t_val, df, p
def simple_linear_reg(y, x):
"""一元线性回归"""
assert len(x) == len(y)
n = len(x)
assert n > 1
mean_x = mean(x)
mean_y = mean(y)
beta1 = covariance(x, y) / variance(x)
beta0 = mean_y - beta1 * mean_x
y_hat = [beta0 + beta1 * e for e in x]
ss_residual = sum((e1 - e2)**2 for e1, e2 in zip(y, y_hat))
se_model = sqrt(ss_residual / (n - 2))
t_value = beta1 / (se_model / sqrt((n - 1) * variance(x)))
p = 2 * (1 - t.cdf(abs(t_value), n - 2))
return beta0, beta1, t_value, n - 2, p
def mean_ci_est(data, alpha, sigma=None):
"""ci-置信区间, est-均值置信区间"""
n = len(data)
sample_mean = mean(data)
if sigma is None:
# 方差未知
s = std(data)
se = s / sqrt(n)
t_value = abs(t.ppf(alpha / 2, n - 1))
return sample_mean - se * t_value, sample_mean + se * t_value
else:
# 方差已知
se = sigma / sqrt(n) # 标准误
# 计算 Z 值
z_value = abs(norm.ppf(alpha / 2))
return sample_mean - se * z_value, sample_mean + se * z_value
return sum((e - mean_value)**2 for e in data) / n
def sample(num_of_samples, sample_sz, var):
data = []
# 遍历样本
for _ in range(num_of_samples):
# 从 0-1的均匀分布中抽取 sample_sz 的个体组成的样本,mean 计算样本均值
data.append(var([random.uniform(0.0, 1.0) for _ in range(sample_sz)]))
return data
if __name__ == '__main__':
"""有偏"""
data1 = sample(1000, 40, variance_bias)
plt.hist(data1, bins="auto", rwidth=0.8)
# 样本方差均值 实验值
plt.axvline(x=mean(data1), c="000")
# 总体方差均值 (b-a)^2/12 0.0, 1.0 理论值
plt.axvline(x=1 / 12, c="red")
print("bias: ", mean(data1), 1 / 12)
plt.show()
"""无偏"""
data2 = sample(1000, 40, variance)
plt.hist(data1, bins="auto", rwidth=0.8)
# 样本方差均值 实验值
plt.axvline(x=mean(data2), c="000")
# 总体方差均值 (b-a)^2/12 0.0, 1.0 理论值
plt.axvline(x=1 / 12, c="red")
print("un_bias: ", mean(data2), 1 / 12)
plt.show()
# 测试众数
zs, zs_count = mode(data)
print("zs", zs, zs_count)
# 测试中位数
data_zws1 = [1, 2, 3, 4]
data_zws2 = [1, 2, 3, 4, 5]
data_zws3 = [1, 2, 3, 4, 99]
zws1 = median(data_zws1)
zws2 = median(data_zws2)
zws3 = median(data_zws3)
print("zws", zws1, zws2, zws3) # 再次认证中位数和极端值没有关联，是集中趋势
# 测试均值
data_jz = [1, 2, 3, 4, 5]
jz = mean(data_jz)
print("jz", jz)
# 测试极差
data_jc = [1, 2, 3, 999]
jc = rng(data_jc)
print("jc", jc)
# 测试四分位数
data_sfws = [1, 4, 2, 3, 5]
sfws = quartile(data_sfws)
print(sfws)
# 测试方差
data_fc = [1, 4, 2, 3, 5]
fc = variance(data_fc)
import random
import matplotlib.pyplot as plt
from stats.descriptive_stats import mean, variance
if __name__ == '__main__':
"""相合性"""
# 样本均值
sample_means = []
# 样本方差
sample_vars = []
# 样本容量
indices = []
for sz in range(20, 10001, 50):
indices.append(sz)
# 调用高斯分布
sample = [random.gauss(0.0, 1.0) for _ in range(sz)]
sample_means.append(mean(sample))
sample_vars.append(variance(sample))
plt.plot(indices, sample_means)
plt.plot(indices, sample_vars)
"""结论，当样本越大时，样本均值逐渐趋向于样本方差,这就是相合性。"""
plt.show()
import random, matplotlib.pyplot as plt
from stats.descriptive_stats import mean
def sample(num_of_samples, sample_sz):
data = []
# 遍历样本
for _ in range(num_of_samples):
# 从 0-1的均匀分布中抽取 sample_sz 的个体组成的样本,mean 计算样本均值
data.append(mean([random.uniform(0.0, 1.0) for _ in range(sample_sz)]))
return data
"""中心极限定理"""
if __name__ == '__main__':
data = sample(1000, 40)
plt.hist(data, bins="auto", rwidth=0.8)
# 绘制均值线
plt.axvline(x=mean(data), c="red")
plt.show()
