Afinní Transformace
Dominik Heger
Masaryk University hegerdQchemi. muni. cz
STDT 04 Afinní Transformace
Dominik Heger (MU)
Afinní Transformace
□ ► « Si
STDT 04 Afinní Transformace 1/13
Transformace - přičítání
original variable
i]
5
7
s
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 2/13
Transformace - přičítání
original variable
i]
5
7
s
original variable + 4
i]
"T 2
3
~r
■5
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 2/13
Transformace - násobení
original variable
i-1-1-1
-7       -6      -5 -4
T-1-1
-3       -2 -1
n-r
3 4
n-1-r
5        6 7
2 x original variable
T-1-1-1
-7      -6      -5 -4
~i-1-1-1—1—r
-2-10        1 2
~l--1--1--1--1
3        4        5        6 7
-2 x original variable
T I I I I I I T
-7       -6      -5       -4       -3       -2       -1 0
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 3/13
Transformace - násobení
original variable
i-1-1-1
-7       -6      -5 -4
T-1-1
-3       -2 -1
n-r
3 4
n-1-r
5        6 7
2 x original variable
T-1-1-1
-7      -6      -5 -4
~i-1-1-1—1—r
-2-10        1 2
~l--1--1--1--1
3        4        5        6 7
-2 x original variable
T I I I I I I T
-7       -6      -5       -4       -3       -2       -1 0
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 3/13
Standardizované skóre - z-skóre
64
height = z-3 + 67
67
70
heights (Inches)
height - 67
í)
standard units i
Dominik Heger (MU)
Afinní Transformace
□        S1 ~ =
STDT 04 Afinní Transformace 4/13
Standardizované skóre - z-skóre
64
height = z-3 + 67
67
70
heights (Inches)
height - 67
í)
standard units i
Dominik Heger (MU)
Afinní Transformace
□        S1 ~ =
STDT 04 Afinní Transformace 4/13
Standardizované skóre - z-skóre
Standard units measures "how many SDs above average"
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 5/13
Tvar rozložení
Annual income of U.S. adults in 2010
i-1-r~
0    10 25
T"
T"
50 100
income (thousands of $)
15G
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 6/13
Příklady rozložení
Věk matek
OJ —
<  □  ►   < rS1 ►   <  -ž  ►   4        >       š      -O °s O
STDT 04 Afinní Transformace 7/13
i i i i r~
15 20 25 30 35
maternal age (years)
40
45
Dominik Heger (MU)
Afinní Transformace
Příklady rozložení
Věk matek
Váha novorozeňat
OJ
"O
i i i i r~
15 20 25 30 35
maternal age (years)
40
45
I I I I I
60 80 100 120 140
birthweight (ounces)
160
180
Dominik Heger (MU)
Afinní Transformace
< rS1 ► < -ž ► 4     >     -E    -O °s O
STDT 04 Afinní Transformace 7/13
Standardní normální krivka
density at z
y/ŤŤr
-±z2
e 2
— OG- < Z < OG
ii       i       i       i       i       i r
• total area = 1
• balance point: z = 0
• points of inflection:
z = -1, z = 1
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace 8/13
Normální křivky
Vlastnosti normálního rozdělení
table b.2; Proportions of the Norma! Curve (One-Tailed)
This table gives the proportion of the normal curve in the right-hand tail that lies at or beyond (i.e., is at least as extreme as) a given normal deviate; for example, Z— \{X\ — pt)\/(TOrZ- \{X - For example, the proportion of a normal
distribution for which 2^ 1.51 is 0.0655.
o.i n.? 0.3 o.;
0.5 0.6 0.7
as
0.9
1.0
1.1
1.2 L.3
1.4
1.5 I ú 1.7 1 S I '■>
2.■:: ■?.\
2.2
2.3
2.4
2.5
2.6
2.7
2.8 23
3.0
3.1
3.2
Í)
D.50UU 0,4602 0.4207 03821 0,3446
0,3085 0,2743 0,2420 0.2119 0.1841
0.1587 0.1357 0.1151 0.0968 0.0808
0.0668 0.054 8 0.0446 0.03 i 9 0.0287
0.0228 0-0179 0.0139 0.0107 0,0082
0.0062 0.0047 0.0035 0.0026 0.0019
0.0013 0.0010 fl.0007
0,4960 0.4562 0.4168 03783 0.3409
0.3050 0.2709 0.2389 0.2090 0.1814
0,1562 0,1335 0.1131 0.0951 0.0793
0.0655 0.0537 0.0436 0.0351 0.0281
0.0222 0.0174 0.0136 0.0104
o.ooso
0.0060 0.0045 0.0034 0.0025 0.0018
0.0013 0.0009 0.0007
0,4920 0.4522 0.4129 0.3745 0.3372
0.3015 0.2676 0.2358 0.2061 0.1788
0.1539 0.1314 0.1112 0.0934 0.0778
0.0643 0.0526 0.0427 0.0344 0.0274
0.0217 0.0170 0,0132 0.0102 0.0078
0,0059 0.0044 0.0033 0.0024 0.001 R
0.0013 0.0009 0.0006
0.4880 0.4483 0.4090 0.3707 0.3336
0.2981 0.2643 0.2327 0.2033 0.1762
0.1535 0.1292 0.1093 0.0918 0.0764
Ü.0630
IXUS 16 0.0418 0.0336 0.0268
0.0212 0.0166 0.0129 0.0099 0.0075
0,0057 0.0043 0.0032 0.0023 0.0017
0.0012 0.0009 0.0006
0.4840 0.4443 0.4052 0.3669 0.3300
11.2946 0.2611 0.2297 0.2005
0-1736
0.1492 0.1271 0.1075 0.0901 0,0749
0,0618 0.0505 0.0409
0.0262
0.0207 0.0162 0.0125 0.0096 0.0073
0.0055 0.0041 0.0031 0.0023 0.0016
0.0012 0.0008 0.0006
0.4801 0.4404 0.4013 0.3632 0.3264
0.2932 0.2578 0.2266 0.1977 0.1711
0.1469 0.1251 0.1056 0.0885 0.0735
0.0606 0.0495 0.0401 0.0322 0.0256
0.0202 0.0158 0.0122 0.0094 0.0071
0.0054 0.0040 0.0030 0-O022 0,0016
n.onn
0.0008 0.0006
0.4761 0.4364 0.3974 0.3594 0,3228
0,2877 0.2546 0.2236 0,1949 0.1685
0.1446 0.1230 0.1038 0.0869 0.0721
0.0594 0.0485 0.0392 0.0314 0.0250
0.0197 0.0154 0.0119 0.0091 0.0069
0.0052 0.0039 0.0029 0.0021 0.001 15
0,00 u 0(0)8 0.0006
0.4721 0,4325 0.3936 0.3557 0.3192
0.2843 0.2514 0.2207 0.1922 0.1660
0.1423 0.1210
0.1020 0.0853 0.O708
0.0582 O0475 0.0384 0,0307 0,0244
0,0192 0.OÍ5O 0,0116 0.0089 0.0068
0,0051. 0.0038 0,0028 0,0021 0,0015
0.0011 0.0008 0.0005
0.4681 0,4286 0,3897 0.3520 03156
0.2810 0.2483 0.2177 0.1894 0.1635
0.1401 0.1190 0.1003 0.0838 0.0694
0,0571 0.0465 0.0375 0.0301 0.0239
U.0188 0.0146 0.0113 0.0087 0.0066
0.0049 0,0037 0.0027 0.0020 0.0014
0.0010 0.0007 0.0005
0.4641 0.4247 0.3859 0.3483 0.3121
0.2776 0.2451 0.2148 0.1867 0.1611
0.1379 0.1170 0.0985 0.0823 0.0681
0.0559 0.0455 0.0367 0,0294 0.0233
.0.0183 0.0143 0.0110 O0084 0.0064
0.0048 0.0036 0.0026 0.0019 0.0014
0.0010 0.0007 0.0005
Q.Q 0.1 0.2 0.3 0 4
::■ 5 0.6 0 / 0.8 0.9
111
;.i
1.2
1.3
1.4
!.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3 24
2.5
2.6
2.7
2.8
7.9
3.0
3.1
3.2
< rS1 ► < -ž ► 4     ►    -e    >0 °s O
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace   10 / 13
Normální rozdělení a histogram
i     i     i     i     i     i i
60        80       100       120       140      160 180 birthweights (ounces)
Dominik Heger (MU) Afinní Transformace STDT 04 Afinní Transformace 11/13
Né každé rozložení je normální
Many distributions are not normal, not even approximately.
maternal age (years)
mean 27.23 years SD 5.82 years
Dominik Heger (MU) Afinní Transformace STDT 04 Afinní Transformace 12/13
Infromation sources
https://courses.edx.org/courses/BerkeleyX/Stat_2.lx Jerold Zar: Biostatistical analysis
Dominik Heger (MU)
Afinní Transformace
STDT 04 Afinní Transformace   13 / 13