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import
matplotlib
.
pyplot
as
plt
import
numpy
as
np
import
pandas
as
pd
import
scipy
as
sp
from
scipy
import
stats
###############################################################################################################
###############################################################################################################
def
reg_corr_plot
():
class
LinearRegression
:
def
__init__
(
self
,
beta1
,
beta2
,
error_scale
,
data_size
):
self
.
beta1
=
beta1
self
.
beta2
=
beta2
self
.
error_scale
=
error_scale
self
.
x
=
np
.
random
.
randint
(
1
,
data_size
,
data_size
)
self
.
y
=
(
self
.
beta1
+
self
.
beta2
*
self
.
x
+
self
.
error_scale
*
np
.
random
.
randn
(
data_size
)
)
def
x_y_cor
(
self
):
return
np
.
corrcoef
(
self
.
x
,
self
.
y
)[
0
,
1
]
fig
,
ax
=
plt
.
subplots
(
nrows
=
2
,
ncols
=
4
,
figsize
=
(
24
,
12
))
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
0.05
,
1
,
100
lrg1
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
0
,
0
].
scatter
(
lrg1
.
x
,
lrg1
.
y
)
ax
[
0
,
0
].
plot
(
lrg1
.
x
,
beta1
+
beta2
*
lrg1
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
0
,
0
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
0
,
0
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg1
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
-
0.6
,
1
,
100
lrg2
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
0
,
1
].
scatter
(
lrg2
.
x
,
lrg2
.
y
)
ax
[
0
,
1
].
plot
(
lrg2
.
x
,
2
-
0.6
*
lrg2
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
0
,
1
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
0
,
1
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg2
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
1
,
1
,
100
lrg3
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
0
,
2
].
scatter
(
lrg3
.
x
,
lrg3
.
y
)
ax
[
0
,
2
].
plot
(
lrg3
.
x
,
beta1
+
beta2
*
lrg3
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
0
,
2
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
0
,
2
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg3
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
3
,
1
,
100
lrg4
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
0
,
3
].
scatter
(
lrg4
.
x
,
lrg4
.
y
)
ax
[
0
,
3
].
plot
(
lrg4
.
x
,
beta1
+
beta2
*
lrg4
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
0
,
3
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
0
,
3
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg4
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
3
,
3
,
100
lrg5
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
1
,
0
].
scatter
(
lrg5
.
x
,
lrg5
.
y
)
ax
[
1
,
0
].
plot
(
lrg5
.
x
,
beta1
+
beta2
*
lrg5
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
1
,
0
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
1
,
0
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg5
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
3
,
10
,
100
lrg6
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
1
,
1
].
scatter
(
lrg6
.
x
,
lrg6
.
y
)
ax
[
1
,
1
].
plot
(
lrg6
.
x
,
beta1
+
beta2
*
lrg6
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
1
,
1
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
1
,
1
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg6
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
3
,
20
,
100
lrg7
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
1
,
2
].
scatter
(
lrg7
.
x
,
lrg7
.
y
)
ax
[
1
,
2
].
plot
(
lrg7
.
x
,
beta1
+
beta2
*
lrg7
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
1
,
2
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
1
,
2
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg7
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
beta1
,
beta2
,
error_scale
,
data_size
=
2
,
3
,
50
,
100
lrg8
=
LinearRegression
(
beta1
,
beta2
,
error_scale
,
data_size
)
ax
[
1
,
3
].
scatter
(
lrg8
.
x
,
lrg8
.
y
)
ax
[
1
,
3
].
plot
(
lrg3
.
x
,
beta1
+
beta2
*
lrg3
.
x
,
color
=
"#FA954D"
,
alpha
=
0.7
)
ax
[
1
,
3
].
set_title
(
r"$Y={}+{}X+{}u$"
.
format
(
beta1
,
beta2
,
error_scale
))
ax
[
1
,
3
].
annotate
(
r"$\rho={:.4}$"
.
format
(
lrg8
.
x_y_cor
()),
xy
=
(
0.1
,
0.9
),
xycoords
=
"axes fraction"
)
###############################################################################################################
###############################################################################################################
def
central_limit_theorem_plot
():
fig
,
ax
=
plt
.
subplots
(
4
,
3
,
figsize
=
(
20
,
20
))
########################################################################################
x
=
np
.
linspace
(
2
,
8
,
100
)
a
=
2
# range of uniform distribution
b
=
8
unif_pdf
=
np
.
ones
(
len
(
x
))
*
1
/
(
b
-
a
)
ax
[
0
,
0
].
plot
(
x
,
unif_pdf
,
lw
=
3
,
color
=
"r"
)
ax
[
0
,
0
].
plot
(
[
x
[
0
],
x
[
0
]], [
0
,
1
/
(
b
-
a
)],
lw
=
3
,
color
=
"r"
,
alpha
=
0.9
)
# vertical line
ax
[
0
,
0
].
plot
([
x
[
-
1
],
x
[
-
1
]], [
0
,
1
/
(
b
-
a
)],
lw
=
3
,
color
=
"r"
,
alpha
=
0.9
)
ax
[
0
,
0
].
fill_between
(
x
,
1
/
(
b
-
a
),
0
,
alpha
=
0.5
,
color
=
"r"
)
ax
[
0
,
0
].
set_xlim
([
1
,
9
])
ax
[
0
,
0
].
set_ylim
([
0
,
0.4
])
ax
[
0
,
0
].
set_title
(
"Uniform Distribution"
,
size
=
18
)
ax
[
0
,
0
].
set_ylabel
(
"Population Distribution"
,
size
=
12
)
########################################################################################
ss
=
2
# sample size
unif_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
unif_sample
=
np
.
random
.
rand
(
ss
)
unif_sample_mean
[
i
]
=
np
.
mean
(
unif_sample
)
ax
[
1
,
0
].
hist
(
unif_sample_mean
,
bins
=
20
,
color
=
"r"
,
alpha
=
0.5
)
ax
[
1
,
0
].
set_ylabel
(
"Sample Distribution， $n = 2$"
,
size
=
12
)
########################################################################################
ss
=
10
# sample size
unif_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
unif_sample
=
np
.
random
.
rand
(
ss
)
unif_sample_mean
[
i
]
=
np
.
mean
(
unif_sample
)
ax
[
2
,
0
].
hist
(
unif_sample_mean
,
bins
=
30
,
color
=
"r"
,
alpha
=
0.5
)
ax
[
2
,
0
].
set_ylabel
(
"Sample Distribution, $n = 10$"
,
size
=
12
)
########################################################################################
ss
=
1000
# sample size
unif_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
unif_sample
=
np
.
random
.
rand
(
ss
)
unif_sample_mean
[
i
]
=
np
.
mean
(
unif_sample
)
ax
[
3
,
0
].
hist
(
unif_sample_mean
,
bins
=
40
,
color
=
"r"
,
alpha
=
0.5
)
ax
[
3
,
0
].
set_ylabel
(
"Sample Distribution, $n = 1000$"
,
size
=
12
)
########################################################################################
a
=
6
b
=
2
x
=
np
.
linspace
(
0
,
1
,
100
)
beta_pdf
=
sp
.
stats
.
beta
.
pdf
(
x
,
a
,
b
)
ax
[
0
,
1
].
plot
(
x
,
beta_pdf
,
lw
=
3
,
color
=
"g"
)
ax
[
0
,
1
].
set_ylim
([
0
,
6
])
ax
[
0
,
1
].
fill_between
(
x
,
beta_pdf
,
0
,
alpha
=
0.5
,
color
=
"g"
)
ax
[
0
,
1
].
set_title
(
"Beta Distribution"
,
size
=
18
)
########################################################################################
ss
=
2
# sample size
beta_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
beta_sample
=
sp
.
stats
.
beta
.
rvs
(
a
,
b
,
size
=
ss
)
beta_sample_mean
[
i
]
=
np
.
mean
(
beta_sample
)
ax
[
1
,
1
].
hist
(
beta_sample_mean
,
color
=
"g"
,
alpha
=
0.5
)
########################################################################################
ss
=
10
# sample size
beta_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
beta_sample
=
sp
.
stats
.
beta
.
rvs
(
a
,
b
,
size
=
ss
)
beta_sample_mean
[
i
]
=
np
.
mean
(
beta_sample
)
ax
[
2
,
1
].
hist
(
beta_sample_mean
,
color
=
"g"
,
bins
=
20
,
alpha
=
0.5
)
########################################################################################
ss
=
100000
# sample size
beta_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
beta_sample
=
sp
.
stats
.
beta
.
rvs
(
a
,
b
,
size
=
ss
)
beta_sample_mean
[
i
]
=
np
.
mean
(
beta_sample
)
ax
[
3
,
1
].
hist
(
beta_sample_mean
,
color
=
"g"
,
bins
=
30
,
alpha
=
0.5
)
########################################################################################
a
=
6
x
=
np
.
linspace
(
0
,
25
,
100
)
gamma_pdf
=
sp
.
stats
.
gamma
.
pdf
(
x
,
a
)
ax
[
0
,
2
].
plot
(
x
,
gamma_pdf
,
lw
=
3
,
color
=
"b"
)
ax
[
0
,
2
].
set_ylim
([
0
,
0.34
])
ax
[
0
,
2
].
fill_between
(
x
,
gamma_pdf
,
0
,
alpha
=
0.5
,
color
=
"b"
)
ax
[
0
,
2
].
set_title
(
"Gamma Distribution"
,
size
=
18
)
########################################################################################
ss
=
2
# sample size
gamma_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
gamma_sample
=
sp
.
stats
.
gamma
.
rvs
(
a
,
size
=
ss
)
gamma_sample_mean
[
i
]
=
np
.
mean
(
gamma_sample
)
ax
[
1
,
2
].
hist
(
gamma_sample_mean
,
color
=
"b"
,
alpha
=
0.5
)
########################################################################################
ss
=
10
# sample size
gamma_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
gamma_sample
=
sp
.
stats
.
gamma
.
rvs
(
a
,
size
=
ss
)
gamma_sample_mean
[
i
]
=
np
.
mean
(
gamma_sample
)
ax
[
2
,
2
].
hist
(
gamma_sample_mean
,
bins
=
20
,
color
=
"b"
,
alpha
=
0.5
)
########################################################################################
ss
=
1000
# sample size
gamma_sample_mean
=
np
.
zeros
(
1000
)
for
i
in
range
(
1000
):
gamma_sample
=
sp
.
stats
.
gamma
.
rvs
(
a
,
size
=
ss
)
gamma_sample_mean
[
i
]
=
np
.
mean
(
gamma_sample
)
ax
[
3
,
2
].
hist
(
gamma_sample_mean
,
bins
=
30
,
color
=
"b"
,
alpha
=
0.5
)
########################################################################################
plt
.
show
()
##########################################################################################################
##########################################################################################################
def
type12_error
():
x
=
np
.
linspace
(
-
6
,
9
,
200
)
null_loc
,
alter_loc
=
0
,
3
y_null
=
sp
.
stats
.
norm
.
pdf
(
x
,
loc
=
null_loc
)
y_alter
=
sp
.
stats
.
norm
.
pdf
(
x
,
loc
=
alter_loc
)
fig
,
ax
=
plt
.
subplots
(
figsize
=
(
18
,
6
))
ax
.
plot
(
x
,
y_null
,
x
,
y_alter
)
ax
.
annotate
(
"Null"
, (
null_loc
-
0.2
,
max
(
y_null
)
/
2
),
size
=
15
)
ax
.
annotate
(
"Alternative"
, (
alter_loc
-
0.6
,
max
(
y_alter
)
/
2
),
size
=
15
)
ax
.
annotate
(
"Type I Error"
, (
2
,
max
(
y_alter
)
/
30
),
size
=
15
)
ax
.
annotate
(
"Type II Error"
, (
0
,
max
(
y_alter
)
/
30
),
size
=
15
)
ax
.
fill_between
(
x
[
-
98
:],
y_null
[
-
98
:])
ax
.
fill_between
(
x
[:
103
],
y_alter
[:
103
])
ax
.
set_ylim
([
0
,
0.5
])
plt
.
show
()
##########################################################################################################
##########################################################################################################
# def reject_region():
# data = pd.read_csv('500_Person_Gender_Height_Weight_Index.csv')
# male_mean = data[data['Gender']=='Male']['Height'].mean()
# male_std = data[data['Gender']=='Male']['Height'].std(ddof=1)
# male_std_error = male_std/np.sqrt(len(data[data['Gender']=='Male']))
# male_null = 172
# df = len(data[data['Gender']=='Male'])-1
# t_975 = sp.stats.t.ppf(.975, df=df)
# t_025 = sp.stats.t.ppf(.025, df=df)
# x = np.linspace(male_null-5, male_null+5, 200)
# df = len(data[data['Gender']=='Male'])-1
# y_t = sp.stats.t.pdf(x, df = df, loc = male_null)
# fig, ax = plt.subplots(2, 1, figsize = (18,8))
# ax[0].plot(x, y_t, color = 'tomato', lw = 3)
# rejection_lower = male_null - t_975*male_std_error
# x_rej_lower = np.linspace(rejection_lower-3, rejection_lower, 30)
# y_rej_lower = sp.stats.t.pdf(x_rej_lower, df = df, loc = male_null)
# ax[0].fill_between(x_rej_lower, y_rej_lower, color = 'tomato', alpha = .7)
# rejection_upper = male_null + t_975*male_std_error
# x_rej_upper = np.linspace(rejection_upper, rejection_upper+3, 30)
# y_rej_upper = sp.stats.t.pdf(x_rej_upper, df = df, loc = male_null)
# ax[0].fill_between(x_rej_upper, y_rej_upper, color = 'tomato', alpha = .7)
# ax[0].set_ylim([0, .45])
# x = np.linspace(-5, 5, 200)
# y_t = sp.stats.t.pdf(x, df = df, loc = 0)
# ax[1].plot(x, y_t, color = 'tomato', lw = 3)
# x_rej_lower = np.linspace(t_025-3, t_025, 30)
# y_rej_lower = sp.stats.t.pdf(x_rej_lower, df = df)
# ax[1].fill_between(x_rej_lower, y_rej_lower, color = 'tomato', alpha = .7)
# x_rej_lower = np.linspace(t_975+3, t_975, 30)
# y_rej_lower = sp.stats.t.pdf(x_rej_lower, df = df)
# ax[1].fill_between(x_rej_lower, y_rej_lower, color = 'tomato', alpha = .7)
# ax[1].set_ylim([0, .45])
# plt.show()
##########################################################################################################
# ##########################################################################################################
# def draw_something():
# x = np.linspace(0, 10, 100)
# y = np.sin(x)
# plt.plot(x, y)
##########################################################################################################
##########################################################################################################
def
anova_plot
():
def
gen_3samples
(
loc1
,
loc2
,
loc3
,
scale1
,
scale2
,
scale3
,
size1
,
size2
,
size3
):
F_statistic
,
p_value
=
[], []
for
i
in
range
(
1000
):
a
=
sp
.
stats
.
norm
.
rvs
(
loc1
,
scale1
,
size1
)
b
=
sp
.
stats
.
norm
.
rvs
(
loc2
,
scale2
,
size3
)
c
=
sp
.
stats
.
norm
.
rvs
(
loc3
,
scale3
,
size3
)
F
,
p
=
sp
.
stats
.
f_oneway
(
a
,
b
,
c
)
F_statistic
.
append
(
F
)
p_value
.
append
(
p
)
return
F_statistic
,
p_value
fig
,
ax
=
plt
.
subplots
(
nrows
=
6
,
ncols
=
3
,
figsize
=
(
17
,
34
))
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
6
,
9
,
6
,
6
,
6
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
0
,
0
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
0
,
0
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
0
,
0
].
set_title
(
"Simulation 1"
)
ax
[
0
,
0
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
1
,
0
].
hist
(
p_value
,
bins
=
50
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
3.1
,
2.9
,
6
,
6
,
6
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
0
,
1
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
0
,
1
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
0
,
1
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
1
,
1
].
hist
(
p_value
,
bins
=
50
)
ax
[
0
,
1
].
set_title
(
"Simulation 2"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
3.1
,
2.9
,
6
,
12
,
18
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
0
,
2
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
0
,
2
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
0
,
2
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
1
,
2
].
hist
(
p_value
,
bins
=
50
)
ax
[
0
,
2
].
set_title
(
"Simulation 3"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
6
,
9
,
10
,
10
,
10
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
2
,
0
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
2
,
0
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
2
,
0
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
3
,
0
].
hist
(
p_value
,
bins
=
50
)
ax
[
2
,
0
].
set_title
(
"Simulation 4"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
5
,
6
,
10
,
10
,
10
,
10
,
10
,
10
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
2
,
1
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
2
,
1
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
2
,
1
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
3
,
1
].
hist
(
p_value
,
bins
=
50
)
ax
[
2
,
1
].
set_title
(
"Simulation 5"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
5
,
6
,
10
,
10
,
10
,
5000
,
5000
,
5000
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
2
,
2
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
2
,
2
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
2
,
2
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
3
,
2
].
hist
(
p_value
,
bins
=
50
)
ax
[
2
,
2
].
set_title
(
"Simulation 6"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
3
,
3
,
100
,
100
,
100
,
10
,
10
,
10
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
4
,
0
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
4
,
0
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
4
,
0
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
5
,
0
].
hist
(
p_value
,
bins
=
50
)
ax
[
4
,
0
].
set_title
(
"Simulation 7"
)
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
3
,
3
,
1
,
1
,
2
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
4
,
1
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
4
,
1
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
4
,
1
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
5
,
1
].
hist
(
p_value
,
bins
=
50
)
ax
[
4
,
1
].
set_title
(
"Simulation 8"
)
params
=
[
3
,
3.1
,
2.9
,
0.01
,
0.01
,
0.01
,
10
,
20
,
30
]
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
=
[
3
,
3
,
3
,
1
,
1
,
2
,
10
,
20
,
30
,
]
params
=
[
mu1
,
mu2
,
mu3
,
sig1
,
sig2
,
sig3
,
size1
,
size2
,
size3
]
F_statistic
,
p_value
=
gen_3samples
(
*
params
)
n
,
bins
,
patches
=
ax
[
4
,
2
].
hist
(
F_statistic
,
bins
=
50
)
F_critical
=
sp
.
stats
.
f
.
ppf
(
0.95
,
2
,
size1
+
size2
+
size3
-
3
)
textstr
=
"
\n
"
.
join
(
(
"$\mu_1, \mu_2, \mu_3 = {}, {}, {}$"
.
format
(
mu1
,
mu2
,
mu3
),
"$\sigma_1, \sigma_2, \sigma_3 = {}, {}, {}$"
.
format
(
sig1
,
sig2
,
sig3
),
"$n_1, n_2, n_3 = {}, {}, {}$"
.
format
(
size1
,
size2
,
size3
),
r"$F_c = {:.4f}$"
.
format
(
F_critical
),
)
)
props
=
dict
(
boxstyle
=
"round"
,
facecolor
=
"wheat"
,
alpha
=
0.5
)
ax
[
4
,
2
].
text
(
max
(
bins
)
/
2
,
max
(
n
)
/
2
,
textstr
,
fontsize
=
10
,
verticalalignment
=
"top"
,
bbox
=
props
,
)
ax
[
4
,
2
].
vlines
(
F_critical
,
0
,
max
(
n
)
*
1.1
,
color
=
"r"
)
ax
[
5
,
2
].
hist
(
p_value
,
bins
=
50
)
ax
[
4
,
2
].
set_title
(
"Simulation 9"
)
#######################||Rectangle||##########################
##############################################################
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.10
,
0.633
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.3755
,
0.633
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.650
,
0.633
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.10
,
0.37
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.3755
,
0.37
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.650
,
0.37
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.650
,
0.108
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.3755
,
0.108
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
rect
=
plt
.
Rectangle
(
# (lower-left corner), width, height
(
0.1
,
0.108
),
0.2645
,
0.258
,
fill
=
False
,
color
=
"k"
,
lw
=
2
,
zorder
=
1000
,
transform
=
fig
.
transFigure
,
figure
=
fig
,
)
fig
.
patches
.
extend
([
rect
])
####################################################################
plt
.
show
()
###############################################################################################
###############################################################################################
def
two_tail_rej_region_demo
():
data
=
pd
.
read_csv
(
"500_Person_Gender_Height_Weight_Index.csv"
)
df
=
len
(
data
[
data
[
"Gender"
]
==
"Male"
])
-
1
t_975
=
sp
.
stats
.
t
.
ppf
(
0.975
,
df
=
df
)
t_025
=
sp
.
stats
.
t
.
ppf
(
0.025
,
df
=
df
)
male_mean
=
data
[
data
[
"Gender"
]
==
"Male"
][
"Height"
].
mean
()
male_std
=
data
[
data
[
"Gender"
]
==
"Male"
][
"Height"
].
std
(
ddof
=
1
)
male_std_error
=
male_std
/
np
.
sqrt
(
len
(
data
[
data
[
"Gender"
]
==
"Male"
]))
male_null
=
172
x
=
np
.
linspace
(
male_null
-
5
,
male_null
+
5
,
200
)
y_t
=
sp
.
stats
.
t
.
pdf
(
x
,
df
=
df
,
loc
=
male_null
)
fig
,
ax
=
plt
.
subplots
(
2
,
1
,
figsize
=
(
18
,
8
))
ax
[
0
].
plot
(
x
,
y_t
,
color
=
"tomato"
,
lw
=
3
)
rejection_lower
=
male_null
-
t_975
*
male_std_error
x_rej_lower
=
np
.
linspace
(
rejection_lower
-
3
,
rejection_lower
,
30
)
y_rej_lower
=
sp
.
stats
.
t
.
pdf
(
x_rej_lower
,
df
=
df
,
loc
=
male_null
)
ax
[
0
].
fill_between
(
x_rej_lower
,
y_rej_lower
,
color
=
"tomato"
,
alpha
=
0.7
)
rejection_upper
=
male_null
+
t_975
*
male_std_error
x_rej_upper
=
np
.
linspace
(
rejection_upper
,
rejection_upper
+
3
,
30
)
y_rej_upper
=
sp
.
stats
.
t
.
pdf
(
x_rej_upper
,
df
=
df
,
loc
=
male_null
)
ax
[
0
].
fill_between
(
x_rej_upper
,
y_rej_upper
,
color
=
"tomato"
,
alpha
=
0.7
)
ax
[
0
].
set_ylim
([
0
,
0.45
])
ax
[
0
].
set_title
(
"Rejection Region of Original Unit (cm)"
)
x
=
np
.
linspace
(
-
5
,
5
,
200
)
y_t
=
sp
.
stats
.
t
.
pdf
(
x
,
df
=
df
,
loc
=
0
)
ax
[
1
].
plot
(
x
,
y_t
,
color
=
"tomato"
,
lw
=
3
)
x_rej_lower
=
np
.
linspace
(
t_025
-
3
,
t_025
,
30
)
y_rej_lower
=
sp
.
stats
.
t
.
pdf
(
x_rej_lower
,
df
=
df
)
ax
[
1
].
fill_between
(
x_rej_lower
,
y_rej_lower
,
color
=
"tomato"
,
alpha
=
0.7
)
x_rej_lower
=
np
.
linspace
(
t_975
+
3
,
t_975
,
30
)
y_rej_lower
=
sp
.
stats
.
t
.
pdf
(
x_rej_lower
,
df
=
df
)
ax
[
1
].
fill_between
(
x_rej_lower
,
y_rej_lower
,
color
=
"tomato"
,
alpha
=
0.7
)
ax
[
1
].
set_ylim
([
0
,
0.45
])
ax
[
1
].
set_title
(
"Rejection Region of t-statistic"
)
plt
.
show
()
###############################################################################################
###############################################################################################
def
one_tail_rej_region_demo
():
fig
,
ax
=
plt
.
subplots
(
2
,
1
,
figsize
=
(
18
,
8
))
x
=
np
.
linspace
(
-
5
,
5
,
200
)
y_t
=
sp
.
stats
.
t
.
pdf
(
x
,
df
=
len
(
x
),
loc
=
0
)
ax
[
0
].
plot
(
x
,
y_t
,
color
=
"tomato"
)
ax
[
0
].
annotate
(
"$H_0: \mu = \mu_0$
\n
$H_1: \mu < \mu_0$"
, (
-
5
,
0.35
),
size
=
16
)
t_05
=
sp
.
stats
.
t
.
ppf
(
0.05
,
df
=
len
(
x
))
x_rej_lower
=
np
.
linspace
(
t_05
,
t_05
-
3
,
30
)
y_rej_lower
=
sp
.
stats
.
t
.
pdf
(
x_rej_lower
,
df
=
len
(
x
))
ax
[
0
].
fill_between
(
x_rej_lower
,
y_rej_lower
,
color
=
"tomato"
,
alpha
=
0.7
)
ax
[
0
].
set_ylim
([
0
,
0.45
])
x
=
np
.
linspace
(
-
5
,
5
,
200
)
y_t
=
sp
.
stats
.
t
.
pdf
(
x
,
df
=
len
(
x
),
loc
=
0
)
ax
[
1
].
plot
(
x
,
y_t
,
color
=
"tomato"
)
ax
[
1
].
annotate
(
"$H_0: \mu = \mu_0$
\n
$H_1: \mu > \mu_0$"
, (
4
,
0.35
),
size
=
16
)
t_95
=
sp
.
stats
.
t
.
ppf
(
0.95
,
df
=
len
(
x
))
x_rej_lower
=
np
.
linspace
(
t_95
,
t_95
+
3
,
30
)
y_rej_lower
=
sp
.
stats
.
t
.
pdf
(
x_rej_lower
,
df
=
len
(
x
))
ax
[
1
].
fill_between
(
x_rej_lower
,
y_rej_lower
,
color
=
"tomato"
,
alpha
=
0.7
)
ax
[
1
].
set_ylim
([
0
,
0.45
])
plt
.
show
()
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