Saya tidak tahu tentang Anda, tetapi statistik tidak mudah bagi saya. Selain itu, itu "diberikan" - masih diucapkan dengan keras. Ya, ternyata Anda dapat mempelajari manual pelatihan untuk waktu yang agak lama, entah bagaimana mempelajari arti dari formula empat lantai, dan terkadang bahkan tidak memahami hasilnya, tetapi tetap melanjutkan. Pergi dan tidak mendapatkan kesenangan apa pun - semuanya tampak jelas, tetapi perasaan bahwa Anda "tidak cukup dalam subjek" masih tidak pergi. Untuk sementara saya mencoba membaca buku tentang R dan bukannya itu tidak akan berhasil sama sekali, tetapi tidak juga "api". Saya menemukan buku paling keren "Statistics for All" oleh Sarah Boslaf, membacanya ... masih ada beberapa nuansa, artinya belum sepenuhnya dipahami.
Secara umum, seperti yang Anda tebak, artikel ini berasal dari seri "Saya mencoba menjelaskan dengan jari saya untuk mengetahuinya sendiri." Jadi jika Anda tidak peduli dengan statistik, maka silakan, di bawah cat.
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import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
#import seaborn as sns
from pylab import rcParams
#sns.set()
rcParams['figure.figsize'] = 10, 6
%config InlineBackend.figure_format = 'svg'
np.random.seed(42)
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norm_rv = stats.norm(loc=30, scale=5)
samples = np.trunc(norm_rv.rvs(365))
samples[:30]
array([32., 29., 33., 37., 28., 28., 37., 33., 27., 32., 27., 27., 31.,
20., 21., 27., 24., 31., 25., 22., 37., 28., 30., 22., 27., 30.,
24., 31., 26., 28.])
:
samples.mean(), samples.std()
(29.52054794520548, 4.77410133275075)
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sns.histplot(x=samples, discrete=True);
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norm_rv = stats.norm(loc=30, scale=5)
beta_rv = stats.beta(a=5, b=5, loc=14, scale=32)
gamma_rv = stats.gamma(a = 20, loc = 7, scale=1.2)
tri_rv = stats.triang(c=0.5, loc=17, scale=26)
x = np.linspace(10, 50, 300)
sns.lineplot(x = x, y = norm_rv.pdf(x), color='r', label='norm')
sns.lineplot(x = x, y = beta_rv.pdf(x), color='g', label='beta')
sns.lineplot(x = x, y = gamma_rv.pdf(x), color='k', label='gamma')
sns.lineplot(x = x, y = tri_rv.pdf(x), color='b', label='triang')
sns.histplot(x=samples, discrete=True, stat='probability',
alpha=0.2);
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unif_rv = stats.uniform(loc=0, scale=4)
unif_rv.rvs()
0.8943833540778106
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unif_rv.rvs(size=3)
array([3.85289016, 0.0486179 , 3.87951531])
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Y_samples = [unif_rv.rvs(size=15).sum() for i in range(10000)]
sns.histplot(x=Y_samples);
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N = 5000
t_data = norm_rv.rvs(N)
t_data[(25 < t_data) & (t_data < 35)].size/N
0.7008
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t_data[t_data > 40].size/N
0.0248
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8.000000000000001e-06
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fig, ax = plt.subplots()
x = np.linspace(norm_rv.ppf(0.001), norm_rv.ppf(0.999), 200)
ax.vlines(40.5, 0, 0.1, color='k', lw=2)
sns.lineplot(x=x, y=norm_rv.pdf(x), color='r', lw=3)
sns.histplot(x=samples, stat='probability', discrete=True);
samples, , . . 40 . , . - , .. , , 5000 , , :
np.random.seed(42)
N = 10000
values = np.trunc(norm_rv.rvs(N))
fig, ax = plt.subplots()
v_le_41 = np.histogram(values, np.arange(9.5, 41.5))
v_ge_40 = np.histogram(values, np.arange(40.5, 51.5))
ax.bar(np.arange(10, 41), v_le_41[0]/N, color='0.8')
ax.bar(np.arange(41, 51), v_ge_40[0]/N)
p = np.sum(v_ge_40[0]/N)
ax.set_title('P(Y>40min) = {:.3}'.format(p))
ax.vlines(40.5, 0, 0.08, color='k', lw=1);
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fig, ax = plt.subplots()
x = np.linspace(norm_rv.ppf(0.0001), norm_rv.ppf(0.9999), 300)
ax.plot(x, norm_rv.pdf(x))
ax.fill_between(x[x>41], norm_rv.pdf(x[x>41]), np.zeros(len(x[x>41])))
p = 1 - norm_rv.cdf(41)
ax.set_title('P(Y>40min) = {:.3}'.format(p))
ax.hlines(0, 10, 50, lw=1, color='k')
ax.vlines(41, 0, 0.08, color='k', lw=1);
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fig, ax = plt.subplots(nrows=1, ncols=2, figsize = (12, 5))
nrv_hobbit = stats.norm(91, 8)
nrv_gnome = stats.norm(134, 6)
for i, (func, h) in enumerate(zip((nrv_hobbit, nrv_gnome), (99, 143))):
x = np.linspace(func.ppf(0.0001), func.ppf(0.9999), 300)
ax[i].plot(x, func.pdf(x))
ax[i].fill_between(x[x>h], func.pdf(x[x>h]), np.zeros(len(x[x>h])))
p = 1 - func.cdf(h)
ax[i].set_title('P(H > {} ) = {:.3}'.format(h, p))
ax[i].hlines(0, func.ppf(0.0001), func.ppf(0.9999), lw=1, color='k')
ax[i].vlines(h, 0, func.pdf(h), color='k', lw=2)
ax[i].set_ylim(0, 0.07)
fig.suptitle(' , ', fontsize = 20);
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fig, ax = plt.subplots()
N_rv = stats.norm()
x = np.linspace(N_rv.ppf(0.0001), N_rv.ppf(0.9999), 300)
ax.plot(x, N_rv.pdf(x))
ax.hlines(0, -4, 4, lw=1, color='k')
ax.vlines(1, 0, 0.4, color='r', lw=2, label='')
ax.vlines(1.5, 0, 0.4, color='g', lw=2, label='')
ax.legend();
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fig, ax = plt.subplots()
N_rv = stats.norm()
x = np.linspace(N_rv.ppf(0.0001), N_rv.ppf(0.9999), 300)
ax.plot(x, N_rv.pdf(x))
ax.hlines(0, -4, 4, lw=1, color='k')
ax.vlines(2, 0, 0.4, color='g', lw=2);
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sns.histplot(np.trunc(norm_rv.rvs(size=(N, 3))).mean(axis=1),
bins=np.arange(19, 42));
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1 - N_rv.cdf(2)
0.02275013194817921
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(1 - N_rv.cdf(2))**3
1.1774751750476762e-05
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N = 10000
means = np.trunc(norm_rv.rvs(size=(N, 3))).mean(axis=1)
means[(means>=25)&(means<=35)].size/N
0.9241
N = 10000
fig, ax = plt.subplots()
means = np.trunc(norm_rv.rvs(size=(N, 3))).mean(axis=1)
h = np.histogram(means, np.arange(19, 41))
ax.bar(np.arange(20, 25), h[0][0:5]/N, color='0.5')
ax.bar(np.arange(25, 36), h[0][5:16]/N)
ax.bar(np.arange(36, 41), h[0][16:22]/N, color='0.5')
p = np.sum(h[0][6:16]/N)
ax.set_title('P(25min < Y < 35min) = {:.3}'.format(p))
ax.vlines([24.5 ,35.5], 0, 0.08, color='k', lw=1);
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x, n, mu, sigma = 35, 3, 30, 5
z = abs((x - mu)/(sigma/n**0.5))
N_rv = stats.norm()
fig, ax = plt.subplots()
x = np.linspace(N_rv.ppf(0.0001), N_rv.ppf(0.9999), 300)
ax.plot(x, N_rv.pdf(x))
ax.hlines(0, x.min(), x.max(), lw=1, color='k')
ax.vlines([-z, z], 0, 0.4, color='g', lw=2)
x_z = x[(x>-z) & (x<z)] # & (x<z)
ax.fill_between(x_z, N_rv.pdf(x_z), np.zeros(len(x_z)), alpha=0.3)
p = N_rv.cdf(z) - N_rv.cdf(-z)
ax.set_title('P({:.3} < z < {:.3}) = {:.3}'.format(-z, z, p));
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. 5, 30 100 , , [29; 31]? :
fig, ax = plt.subplots(nrows=1, ncols=3, figsize = (12, 4))
for i, n in enumerate([5, 30, 100]):
x, mu, sigma = 31, 30, 5
z = abs((x - mu)/(sigma/n**0.5))
N_rv = stats.norm()
x = np.linspace(N_rv.ppf(0.0001), N_rv.ppf(0.9999), 300)
ax[i].plot(x, N_rv.pdf(x))
ax[i].hlines(0, x.min(), x.max(), lw=1, color='k')
ax[i].vlines([-z, z], 0, 0.4, color='g', lw=2)
x_z = x[(x>-z) & (x<z)] # & (x<z)
ax[i].fill_between(x_z, N_rv.pdf(x_z), np.zeros(len(x_z)), alpha=0.3)
p = N_rv.cdf(z) - N_rv.cdf(-z)
ax[i].set_title('n = {}, z = {:.3}, p = {:.3}'.format(n, z, p));
fig.suptitle(r'Z- $\bar{x} = 31$', fontsize = 20, y=1.02);
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x, n, mu, sigma = 41, 3, 30, 5
z = abs((x - mu)/(sigma/3**0.5))
N_rv = stats.norm()
fig, ax = plt.subplots()
x = np.linspace(N_rv.ppf(1e-5), N_rv.ppf(1-1e-5), 300)
ax.plot(x, N_rv.pdf(x))
ax.hlines(0, x.min(), x.max(), lw=1, color='k')
ax.vlines([-z, z], 0, 0.4, color='g', lw=2)
x_z = x[(x>-z) & (x<z)] # & (x<z)
ax.fill_between(x_z, N_rv.pdf(x_z), np.zeros(len(x_z)), alpha=0.3)
p = N_rv.cdf(z) - N_rv.cdf(-z)
ax.set_title('P({:.3} < z < {:.3}) = {:.3}'.format(-z, z, p));
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x, n, mu, sigma = 31, 5, 30, 5
z = abs((x - mu)/(sigma/n**0.5))
N_rv = stats.norm()
fig, ax = plt.subplots()
x = np.linspace(N_rv.ppf(0.0001), N_rv.ppf(0.9999), 300)
ax.plot(x, N_rv.pdf(x))
ax.hlines(0, x.min(), x.max(), lw=1, color='k')
ax.vlines([-z, z], 0, 0.4, color='g', lw=2)
x_le_z, x_ge_z = x[x<-z], x[x>z]
ax.fill_between(x_le_z, N_rv.pdf(x_le_z), np.zeros(len(x_le_z)), alpha=0.3, color='b')
ax.fill_between(x_ge_z, N_rv.pdf(x_ge_z), np.zeros(len(x_ge_z)), alpha=0.3, color='b')
p = N_rv.cdf(z) - N_rv.cdf(-z)
ax.set_title('P({:.3} < z < {:.3}) = {:.3}'.format(-z, z, p));
p-value , . p-value , .. . - , p-value , . , , , 
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1 - (N_rv.cdf(5) - N_rv.cdf(-5))
5.733031438470704e-07
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Sepertinya tidak terlalu masuk akal, tapi mari kita lihat. Kami akan menghasilkan 1000 nilai dari distribusi seragam, eksponensial dan Laplace, dan kemudian, secara berurutan, untuk setiap distribusi, kami akan membuat grafik kde dari distribusi nilai rata-rata sampel dengan ukuran berbeda:
Kode GIF
import matplotlib.animation as animation
fig, axes = plt.subplots(nrows=2, ncols=3, figsize = (12, 7))
unif_rv = stats.uniform(loc=10, scale=10)
exp_rv = stats.expon(loc=10, scale=1.5)
lapl_rv = stats.laplace(loc=15)
np.random.seed(42)
unif_data = unif_rv.rvs(size=1000)
exp_data = exp_rv.rvs(size = 1000)
lapl_data = lapl_rv.rvs(size=1000)
title = ['Uniform', 'Exponential', 'Laplace']
data = [unif_data, exp_data, lapl_data]
y_max = [60, 310, 310]
n = [3, 10, 30, 50]
for i, ax in enumerate(axes[0]):
sns.histplot(data[i], bins=20, alpha=0.4, ax=ax)
ax.vlines(data[i].mean(), 0, y_max[i], color='r')
ax.set_xlim(10, 20)
ax.set_title(title[i])
def animate(i):
for ax in axes[1]:
ax.clear()
for j in range(3):
rand_idx = np.random.randint(0, 1000, size=(1000, n[i]))
means = data[j][rand_idx].mean(axis=1)
sns.kdeplot(means, ax=axes[1][j])
axes[1][j].vlines(means.mean(), 0, 2, color='r')
axes[1][j].set_xlim(10, 20)
axes[1][j].set_ylim(0, 2.1)
axes[1][j].set_title('n = ' + str(n[i]))
fig.set_figwidth(15)
fig.set_figheight(8)
return axes[1][0], axes[1][1],axes[1][2]
dist_animation = animation.FuncAnimation(fig,
animate,
frames=np.arange(4),
interval = 200,
repeat = False)
# gif :
dist_animation.save('dist_means.gif',
writer='imagemagick',
fps=1)
Tentu saja, gambar tersebut bukanlah bukti sama sekali, tetapi jika Anda mau, Anda dapat menjalankan sendiri uji normalitas. Namun pada artikel selanjutnya kita hanya akan melakukan pengujian dan pembuktian hipotesis saja.
Secara umum, terima kasih atas perhatiannya. Saya menekan F5 dan menantikan komentar Anda dan Anda.