def var_ratio_ci_est(data1, data2, alpha):
n1 = len(data1)
n2 = len(data2)
f_lower_value = f.ppf(alpha/2, n1-1, n2-1)
f_upper_value = f.ppf(1-alpha/2, n1-1, n2-1)
var_ratio = variance(data1)/variance(data2)
return var_ratio/f_upper_value, var_ratio/f_lower_value
def t_test(data1, data2=None, tail='both', mu=0, equal=True):
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 var_ratio_ci_est(data1, data2, alpha):
"""两个总体方差未知，求方差比的置信区间"""
n1 = len(data1)
n2 = len(data2)
sample_ratio = variance(data1) / variance(data2)
f_low_value = f.ppf(alpha/2, n1-1, n2-1)
f_high_value = f.ppf(1-alpha/2, n1-1, n2-1)
return round(sample_ratio / f_high_value, 3), round(sample_ratio / f_low_value, 3)
def var_ci_est(data, alpha):
"""均值未知，方差的区间估计"""
n = len(data)
s2 = variance(data)
chi2_low_value = chi2.ppf(alpha/2, n-1)
chi2_high_value = chi2.ppf(1-alpha/2, n-1)
return round((n-1)*s2/chi2_high_value,2), round((n-1)*s2/chi2_low_value,2)
def f_test(data1, data2, tail="both", ratio=1):
assert tail in ['both', 'left',
'right'], 'tail should be one of "both","left","right"'
n1 = len(data1)
n2 = len(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
f_val = sample1_var/sample2_var/ratio
df1 = n1-1
df2 = n2-1
if tail == "both":
p = 2*min(1-f.cdf(f_val, df1, df2), f.cdf(f_val, df1, df2))
elif tail == "left":
p = f.cdf(f_val, df1, df2)
else:
p = 1-f.cdf(f_val, df1, df2)
return f_val, df1, df2, p
def simple_linear_reg(x, y):
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 t_test(data1, data2=None, tail='both', mu=0.0, equal=True):
assert tail in ['both', 'left',
'right'], 'tail should be one of "both", "left", "right"'
if data2 is None:
# 单个总体的情况
mean_val = mean(data1)
se = std(data1) / np.sqrt(len(data1))
t_val = (mean_val - mu) / se
df = len(data1) - 1
else:
# 两个总体的情况
n1 = len(data1)
n2 = len(data2)
mean_diff = mean(data1) - mean(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
if equal:
# 方差相等的情况
sw = np.sqrt((((n1 - 1) * sample1_var + (n2 - 1) * sample2_var)) /
(n1 + n2 - 2))
t_val = (mean_diff - mu) / (sw * np.sqrt(1 / n2 + 1 / n2))
df = n1 + n2 - 2
else:
# 方差不等的情况
se = np.sqrt(sample1_var / n1 + sample2_var / n2)
t_val = (mean_diff - mu) / se
df = (sample1_var / n1 + sample2_var / n2)**2 / (
(sample1_var / n1)**2 / (n1 - 1) + (sample2_var / n2)**2 /
(n2 - 1))
if tail == 'both':
# 双尾检验
p = 2 * (1 - t.cdf(abs(t_val), df))
elif tail == 'left':
# 左尾检验
p = t.cdf(t_val, df)
else:
# 右尾检验
p = 1 - t.cdf(t_val, df)
return round(t_val, 2), round(df, 2), p
def mean_diff_ci_t_est(data1, data2, alpha, equal=True):
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_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
else:
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
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_t_est(data1, data2, alpha, equal=True):
"""总体方差未知， 求均值差的置信区间"""
n1 = len(data1)
n2 = len(data2)
mean_diff = mean(data1) - mean(data2)
sample1_var = variance(data1)
sample2_var = variance(data2)
if equal:
"""两总体方差未知且相等"""
sw = np.sqrt(((n1-1)*sample1_var + (n2-1)*sample2_var) / (n1+n2-2))
t_value = abs(t.ppf(alpha/2, n1+n2-2))
return round(mean_diff - sw*np.sqrt(1/n1+1/n2)*t_value, 2), \
round(mean_diff + sw*np.sqrt(1/n1+1/n2)*t_value, 2)
else:
"""两总体方差未知且不等"""
df = (sample1_var/n1 + sample2_var/n2)**2 / ((sample1_var/n1)**2 / (n1-1) + (sample2_var/n2)**2 / (n2-1))
t_value = abs(t.ppf(alpha/2, df))
return round(mean_diff - np.sqrt(sample1_var/n1 + sample2_var/n2) * t_value, 2), \
round(mean_diff + np.sqrt(sample1_var/n1 + sample2_var/n2) * t_value, 2)
def var_ci_est(data, alpha):
"""
总体均值未知，求方差的置信空间
data为传入的样本; alpha为需要传入的置信水平的值
"""
n = len(data) #求样本容量
s2 = variance(data) #求样本方差
chi2_lower_value = chi2.ppf(
alpha / 2, n - 1) #求坐标左侧Z面积，没错你没看错，因为数学证明的过程中是以右侧为基准的，但是scipy是以左侧为基准的
chi2_upper_value = chi2.ppf(1 - alpha / 2, n - 1) #求坐标右侧Z面积
return (n - 1) * s2 / chi2_upper_value, (n - 1) * s2 / chi2_lower_value
def f_test(data1, data2, tail='both', ratio=1):
"""两个总体"""
assert tail in ['both', 'left',
'right'], 'tail should be one of "both", "left", "right"'
n1 = len(data1)
n2 = len(data2)
sample_var1 = variance(data1)
sample_var2 = variance(data2)
f_val = sample_var1 / sample_var2 / ratio
if tail == 'both':
# 双尾检验
p = 2 * min(1 - f.cdf(f_val, n1 - 1, n2 - 1),
f.cdf(f_val, n1 - 1, n2 - 1))
elif tail == 'left':
# 左尾检验
p = f.cdf(f_val, n1 - 1, n2 - 1)
else:
# 右尾检验
p = 1 - f.cdf(f_val, n1 - 1, n2 - 1)
return round(f_val, 4), n1 - 1, n2 - 1, round(p, 5)
def chi2_test(data, tail="both", sigma2=1):
assert tail in ['both', 'left',
'right'], 'tail should be one of "both","left","right"'
n = len(data)
sample_var = variance(data)
chi2_val = (n-1)*sample_var/sigma2
if tail == "both":
p = 2*min(1-chi2.cdf(chi2_val, n-1), chi2.cdf(chi2_val, n-1))
elif tail == "left":
p = chi2.cdf(chi2_val, n-1)
else:
p = 1-chi2.cdf(chi2_val, n-1)
return chi2_val, n-1, p
def chi2_test(data, tail='both', sigma2=1):
"""单个总体"""
assert tail in ['both', 'left',
'right'], 'tail should be one of "both", "left", "right"'
n = len(data)
sample_var = variance(data)
chi2_val = (n - 1) * sample_var / sigma2
if tail == 'both':
# 双尾检验
p = 2 * min(1 - chi2.cdf(chi2_val, n - 1), chi2.cdf(chi2_val, n - 1))
elif tail == 'left':
# 左尾检验
p = chi2.cdf(chi2_val, n - 1)
else:
# 右尾检验
p = 1 - chi2.cdf(chi2_val, n - 1)
return round(chi2_val, 2), n - 1, p
def var_ci_est(data, alpha):
n = len(data)
s2 = variance(data)
chi2_lower_value = chi2.ppf(alpha/2, n-1)
chi2_upper_value = chi2.ppf(1-alpha/2, n-1)
return (n-1)*s2/chi2_upper_value, (n-1)*s2/chi2_lower_value
def var_one_sided_upper_ci_est(data,alpha):
n = len(data)
s2 = variance(data)
chi2_lower_value = chi2.ppf(alpha, n - 1) # 接受的是左边的面积 求上限，分母是下分位点
return -inf, (n - 1) * s2 / chi2_lower_value
def var_one_sided_lower_ci_est(data,alpha):
n = len(data)
s2 = variance(data)
chi2_upper_value = chi2.ppf(1 - alpha, n - 1) # 接受的是左边的面积 求下限，分母是上分位点
return (n - 1) * s2 / chi2_upper_value, inf
# 测试频率
print(frequency(data))
# 测试众数
print(mode(data))
# 测试中位数
print(median(data))
# 测试均值
print(mean(data))
# 测试极差
print(rng(data))
# 测试四分位数
print(quartile(data))
# 测试方差
print(variance(data))
# 测试标准差
print(std(data))
import random
from playStats.descriptive_stats import mean, variance
import matplotlib.pyplot as plt
if __name__ == "__main__":
sample_means = []
sample_vars = []
indices = []
for sz in range(20, 10001, 50):
indices.append(sz)
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)
plt.show()
import random
from playStats.descriptive_stats import mean
from playStats.descriptive_stats import variance
chi2 = []
for i in range(50000):
#x = random.random() #返回一个介于左闭右开[0.0, 1.0)区间的浮点数
x1 = random.normalvariate(0, 1) #返回一个均值是0，方差是1的正态分布随机数
x2 = random.normalvariate(0, 1)
x3 = random.normalvariate(0, 1)
x4 = random.normalvariate(0, 1)
x5 = random.normalvariate(0, 1)
x6 = random.normalvariate(0, 1)
x7 = random.normalvariate(0, 1)
x8 = random.normalvariate(0, 1)
chi2.append(x1**2) # 演示一个自由度chi2分布
#chi2.append(x1**2+x2**2+x3**2+x4**2+x5**2+x6**2+x7**2+x8**2) # 演示多个自由度chi2分布
#chi2.append(random.normalvariate(0,1)) # 演示正态分布
#chi2.append(x) #演示uniform分布
print(variance(chi2)) #打印相关卡方分布方差
#plt.figure(num= "不同自由度卡方分布图")
plt.hist(chi2, bins=30)
plt.show()
'''
x = np.linspace(-5,5,1000)
f = 1/np.sqrt(2*np.pi)*np.exp(-x**2/2)
plt.plot(x,f)
plt.show()
'''
总体均值未知，求方差的置信空间
data为传入的样本; alpha为需要传入的置信水平的值
"""
n = len(data) #求样本容量
s2 = variance(data) #求样本方差
chi2_lower_value = chi2.ppf(
alpha / 2, n - 1) #求坐标左侧Z面积，没错你没看错，因为数学证明的过程中是以右侧为基准的，但是scipy是以左侧为基准的
chi2_upper_value = chi2.ppf(1 - alpha / 2, n - 1) #求坐标右侧Z面积
return (n - 1) * s2 / chi2_upper_value, (n - 1) * s2 / chi2_lower_value
if __name__ == '__main__':
salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113,
1601] #18岁月收入数据
salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345,
2182] #35岁月收入数据
print(mean(salary_18)) #平均月收入的点估计
print(mean_ci_est(salary_18, 0.05)) #平均月收入的区间估计
print(mean(salary_35)) #平均月收入的点估计
print(mean_ci_est(salary_35, 0.05)) #平均月收入的区间估计
print()
print(std(salary_18)) #整体方差的点估计开根
print(variance(salary_18)) #整体方差的点估计(样本方差)
print(var_ci_est(salary_18, 0.05)) #区间估计
print(std(salary_35)) #整体方差的点估计开根
print(variance(salary_35)) #整体方差的点估计(样本方差)
print(var_ci_est(salary_35, 0.05)) #区间估计
print()
import random
from playStats.descriptive_stats import mean
from playStats.descriptive_stats import variance
import matplotlib.pyplot as plt
if __name__ == '__main__':
indices = []
data_mean = []
data_varvance = []
for sample_sz in range(20, 10001, 50):
indices.append(sample_sz)
sample = [random.gauss(0.0, 1.0) for _ in range(sample_sz)]
data_mean.append(mean(sample))
data_varvance.append(variance(sample))
plt.plot(indices, data_mean)
plt.axhline(0, c='r')
plt.plot(indices, data_varvance)
plt.axhline(1, c='b')
plt.show()
