以下係概率論 同統計學 上嘅主要詞彙一覽。
概率論 [e 1] 數學 一個子領域，專門研究概率 （又叫機會率 ）相關嘅問題：概率係一啲描述隨機過程 嘅結果嘅數值，例如掟一個冇出千嘅銀仔 ，出公嘅概率係 50%，所以對於思考不確定性 嚟講不可或缺[1]
統計學 [e 2] 科學 領域當中搜集 、分析 同呈現 數據 ，而實證嘅科學方法 本質上就帶有不確定－理論上，淨係抽個樣本嚟睇 嘅過程就必然會有「手上個樣本有幾大機會真係代表到個總體 嘅實況」嘅問題，所以統計學嘅理論思考梗會用到概率論[2] [3]
除此之外，噉亦即係話概率論同統計學本質上就係一啲可以攞嚟「喺有不確定性嘅情況下，按過去 經驗預測未來 」嘅工具，所以呢個表入面嗰啲詞彙同概念响研究「點樣教人工智能 學習 」嘅機械學習 領域上都相當有用[3]
描述統計學 [e 126] 資訊 嘅統計數值，包括咗[69]
推論統計學 [e 155] 數據分析 ，推論數據背後反映嘅概率分佈 嘅過程。呢啲分析通常係由數據嘅個案嘅值嗰度計一啲指標出嚟，用呢啲指標評估（例如）某兩個變數之間係咪真係有關，或者個自變數 係咪真係能夠對個應變數 產生影響，甚至估計一個數學模型 出嚟描述所研究嘅現象。廿一世紀嘅統計學上有好多種推論統計分析法，每種能夠處理嘅數據類型都唔同。數據科學 等領域嘅專家一定要對呢啲唔同嘅分析法有所認識，知乜嘢時候應該用邊種分析法[78]
相關 [e 174] 統計學 上嘅定義 如下：如果話
x
{\displaystyle x}
y
{\displaystyle y}
變數 成正相關 ，即係話
x
{\displaystyle x}
y
{\displaystyle y}
x
{\displaystyle x}
y
{\displaystyle y}
x
{\displaystyle x}
y
{\displaystyle y}
負相關 ，即係話
x
{\displaystyle x}
y
{\displaystyle y}
x
{\displaystyle x}
y
{\displaystyle y}
x
{\displaystyle x}
y
{\displaystyle y}
冇明顯相關 [e 175]
x
{\displaystyle x}
y
{\displaystyle y}
[91]
皮亞遜積差相關係數 [e 176] [92]
ρ
X
,
Y
=
corr
(
X
,
Y
)
=
cov
(
X
,
Y
)
σ
X
σ
Y
=
E
[
(
X
−
μ
X
)
(
Y
−
μ
Y
)
]
σ
X
σ
Y
{\displaystyle \rho _{X,Y}=\operatorname {corr} (X,Y)={\operatorname {cov} (X,Y) \over \sigma _{X}\sigma _{Y}}={\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})] \over \sigma _{X}\sigma _{Y}}}
ρ
X
,
Y
{\displaystyle \rho _{X,Y}}
x
{\displaystyle x}
y
{\displaystyle y}
X
{\displaystyle X}
i
{\displaystyle i}
x
{\displaystyle x}
Y
{\displaystyle Y}
i
{\displaystyle i}
y
{\displaystyle y}
μ
X
{\displaystyle \mu _{X}}
x
{\displaystyle x}
平均值 ；
μ
Y
{\displaystyle \mu _{Y}}
y
{\displaystyle y}
σ
X
{\displaystyle \sigma _{X}}
x
{\displaystyle x}
標準差 ；
σ
Y
{\displaystyle \sigma _{Y}}
y
{\displaystyle y}
皮亞遜積差相關係數俾嘅資訊只係「兩個變數大致上嘅相關」，但就算兩個變數之間嘅皮亞遜積差相關係數係 0，都唔等如兩個變數之間真係冇關，好似係以下嘅一柞圖噉，每幅圖上面嗰個數表示皮亞遜積差相關係數，每一點表示一個個案，X 軸係變數
x
{\displaystyle x}
y
{\displaystyle y}
[92] 協方差 [e 177] 皮亞遜積差相關係數 條式個分子 ，即係[93]
cov
(
X
,
Y
)
=
E
[
(
X
−
E
[
X
]
)
(
Y
−
E
[
Y
]
)
]
{\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} {{\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}}}
相關唔蘊含因果 [e 178]
X
{\displaystyle X}
Y
{\displaystyle Y}
皮亞遜積差相關係數 數值大），噉可以表示三個可能性－
X
{\displaystyle X}
Y
{\displaystyle Y}
Y
{\displaystyle Y}
X
{\displaystyle X}
X
{\displaystyle X}
Y
{\displaystyle Y}
有唔少統計學嘅學生都以為兩個變數之間有相關表示咗兩者有因果關係，但呢個係一個錯誤嘅諗法，所以統計學界就有咗句噉嘅說話用嚟提醒學生要小心[94] 局部相關 [e 179]
X
{\displaystyle X}
Y
{\displaystyle Y}
n
{\displaystyle n}
混淆變數 （睇下面）
Z
=
Z
1
,
Z
2
,
.
.
Z
n
{\displaystyle Z={Z_{1},Z_{2},..Z_{n}}}
X
{\displaystyle X}
Y
{\displaystyle Y}
Z
{\displaystyle Z}
ρ
X
Y
⋅
Z
{\displaystyle \rho _{XY\cdot Z}}
e
X
{\displaystyle e_{X}}
e
Y
{\displaystyle e_{Y}}
e
X
{\displaystyle e_{X}}
線性迴歸分析 用
Z
{\displaystyle Z}
X
{\displaystyle X}
e
Y
{\displaystyle e_{Y}}
[95] 組內相關 [e 180] 數據 ，有若干個個案，而呢柞個案可以分做若干組，如果柞數據反映組內相關高，就表示同一組嘅個案嘅數值傾向彼此之間接近；要計組內相關可以有幾條唔同嘅式用[96] [97]
每個藍點係一個個案，每個個案有個
y
{\displaystyle y}
值，而 X 軸表示個個案屬邊組；左圖係 ICC 高（0.91）嘅情況，而右圖係 ICC 低（-0.07）嘅情況。
等級相關 [e 181] [98] 斯皮亞曼等級相關係數 （[e 182]
r
s
{\displaystyle r_{s}}
皮亞遜積差相關係數 ，即係[99]
r
s
=
ρ
rg
X
,
rg
Y
=
cov
(
rg
X
,
rg
Y
)
σ
rg
X
σ
rg
Y
,
{\displaystyle r_{s}=\rho _{\operatorname {rg} _{X},\operatorname {rg} _{Y}}={\frac {\operatorname {cov} (\operatorname {rg} _{X},\operatorname {rg} _{Y})}{\sigma _{\operatorname {rg} _{X}}\sigma _{\operatorname {rg} _{Y}}}},}
X
i
,
Y
i
{\displaystyle X_{i},Y_{i}}
rg
X
i
,
rg
Y
i
{\displaystyle \operatorname {rg} _{X_{i}},\operatorname {rg} _{Y_{i}}}
X
i
,
Y
i
{\displaystyle X_{i},Y_{i}}
Τ 等級相關係數 （[e 183]
τ
{\displaystyle \tau }
(
x
1
,
y
1
)
,
.
.
.
,
(
x
n
,
y
n
)
{\displaystyle (x_{1},y_{1}),...,(x_{n},y_{n})}
X
i
,
Y
i
{\displaystyle X_{i},Y_{i}}
(
x
i
,
y
i
)
{\displaystyle (x_{i},y_{i})}
(
x
j
,
y
j
)
{\displaystyle (x_{j},y_{j})}
一致 [e 184]
(
x
i
>
x
j
)
∧
(
y
i
>
y
j
)
{\displaystyle (x_{i}>x_{j})\land (y_{i}>y_{j})}
(
x
i
<
x
j
)
∧
(
y
i
<
y
j
)
{\displaystyle (x_{i}<x_{j})\land (y_{i}<y_{j})}
唔一致 [e 185]
τ
{\displaystyle \tau }
[100]
τ
=
con
−
dis
(
n
2
)
{\displaystyle \tau ={\frac {{\text{con}}-{\text{dis}}}{n \choose 2}}}
con
{\displaystyle {\text{con}}}
dis
{\displaystyle {\text{dis}}}
自相關 [e 186] 隨機過程 嘅自相關係指嗰個過程喺唔同時間點嘅數值之間嘅皮亞遜積差相關係數 ；設
{
X
t
}
{\displaystyle \left\{X_{t}\right\}}
t
{\displaystyle t}
{
X
t
}
{\displaystyle \left\{X_{t}\right\}}
行 若干次，
X
t
{\displaystyle X_{t}}
t
{\displaystyle t}
t
1
{\displaystyle t_{1}}
t
2
{\displaystyle t_{2}}
R
X
X
(
t
1
,
t
2
)
{\displaystyle \operatorname {R} _{XX}(t_{1},t_{2})}
X
t
1
{\displaystyle X_{t_{1}}}
X
t
2
{\displaystyle X_{t_{2}}}
訊號處理 上常用，可以用嚟量度一段訊號有幾接近完全隨機 [101] 交叉相關 [e 187] 時間序列
f
{\displaystyle f}
g
{\displaystyle g}
f
(
t
)
{\displaystyle f(t)}
t
{\displaystyle t}
f
{\displaystyle f}
g
(
t
)
{\displaystyle g(t)}
t
{\displaystyle t}
g
{\displaystyle g}
τ
{\displaystyle \tau }
f
(
t
)
{\displaystyle f(t)}
g
(
t
+
τ
)
{\displaystyle g(t+\tau )}
[102] 正交 [e 188] IV 「正交」，意思即係話呢兩個 IV 之間冇統計相關[103] 多重共線性 。
統計模型 [e 242] 數學模型 嘅一種。一個統計模型會帶有若干個假設，模擬一個產生數據 （觀察到嘅嘢）嘅過程；一個研究者會收數據，並且用數據估計一個統計模型嘅參數 數值，即係用數據估計世界嘅運作法則[141]
幅圖嘅兩條軸分別代表研究緊嗰兩個變數（x 同 y），每個紅點代表一個個案，每個個案都喺兩個變數上各有個值。用迴歸分析可以估計出條線（綠線同藍線都係可行嘅線）並且知道呢兩個變數大致上成正比 。 迴歸模型 [e 254] 自變數 同一個應變數 ，兩者多數會係連續變數 ，然後個演算法 就嘗試畫一條能夠表達自變數同應變數之間嘅關係嘅線[149]
例：
y
=
0.5
x
+
30
+
e
{\displaystyle y=0.5x+30+e}
y
{\displaystyle y}
x
{\displaystyle x}
e
{\displaystyle e}
殘差 [e 255] 線性迴歸模型 [e 256] 應變數 係柞自變數 嘅線性組合 [149] 多重迴歸模型 [e 259] 例：
y
=
0.2
x
1
+
1.65
x
2
+
30
+
e
{\displaystyle y=0.2x_{1}+1.65x_{2}+30+e}
x
1
{\displaystyle x_{1}}
x
2
{\displaystyle x_{2}}
e
{\displaystyle e}
多變量適應性迴歸模型 [e 260]
f
^
(
x
)
=
∑
i
=
1
k
c
i
B
i
(
x
)
{\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x)}
c
i
{\displaystyle c_{i}}
每個
B
i
(
x
)
{\displaystyle B_{i}(x)}
常數 1、
一個合頁函數 [e 261]
max
(
0
,
x
−
constant
)
{\displaystyle \max(0,x-{\text{constant}})}
max
(
0
,
constant
−
x
)
{\displaystyle \max(0,{\text{constant}}-x)}
[註 7]
兩個或者以上嘅合頁函數乘埋[150] 決定系數 （[e 262]
R
2
{\displaystyle R^{2}}
應變數 嘅變異數 有幾多可以由啲自變數 預測；最廣義上嘅定義如下：
R
2
=
1
−
S
S
r
e
s
S
S
t
o
t
{\displaystyle R^{2}=1-{SS_{\rm {res}} \over SS_{\rm {tot}}}\,}
當中
S
S
r
e
s
{\displaystyle SS_{\rm {res}}}
迴歸分析 後嘅殘差平方和 （反映「用個模型得出嘅預測值同實際值傾向差幾遠」），而
S
S
t
o
t
{\displaystyle SS_{\rm {tot}}}
變異數 同樣本大細 相乘（反映樣本整體嘅變異數）；即係話如果個模型做到完美預測，噉
S
S
r
e
s
=
0
{\displaystyle SS_{\rm {res}}=0}
R
2
=
1
{\displaystyle R^{2}=1}
[151] 多重共線性 [e 263] 多重迴歸模型 當中間中會出現嘅問題，指其中一個自變數嘅數值可以由其他自變數嘅線性噉預測，
x
1
=
f
(
x
2
,
x
3
.
.
.
x
i
)
{\displaystyle x_{1}=f(x_{2},x_{3}...x_{i})}
β
{\displaystyle \beta }
x
1
{\displaystyle x_{1}}
x
i
{\displaystyle x_{i}}
y
{\displaystyle y}
x
1
{\displaystyle x_{1}}
x
1
{\displaystyle x_{1}}
x
i
{\displaystyle x_{i}}
x
i
{\displaystyle x_{i}}
x
1
{\displaystyle x_{1}}
[152] 一般線性 [e 265] 線性迴歸模型 嘅做法，可以表達成[154]
Y
=
X
B
+
U
{\displaystyle \mathbf {Y} =\mathbf {X} \mathbf {B} +\mathbf {U} }
當中
Y
{\displaystyle \mathbf {Y} }
矩陣 ，包含啲應變數 ，
X
{\displaystyle \mathbf {X} }
自變數 嘅矩陣，
B
{\displaystyle \mathbf {B} }
參數 嘅矩陣，而最後
U
{\displaystyle \mathbf {U} }
誤差值 嘅矩陣。 邏輯 [e 266] 電子遊戲 研究當中可以用嚟靠一個玩家嘅數據嚟估計佢輸定贏[155]
p
=
P
(
Y
=
1
)
{\displaystyle p=P(Y=1)}
1
−
p
=
P
(
Y
=
0
)
{\displaystyle 1-p=P(Y=0)}
ℓ
=
log
b
p
1
−
p
=
β
0
+
β
1
x
1
+
β
2
x
2
{\displaystyle \ell =\log _{b}{\frac {p}{1-p}}=\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}}
Sigmoid 函數 ：以下呢個函數 ：
S
(
x
)
=
1
1
+
e
−
x
=
e
x
e
x
+
1
{\displaystyle S(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}}
Sigmoid 函數畫做圖嘅樣 自迴歸模型 [e 267] 時間序列 嘅一種迴歸模型；攞一個會隨住時間變化嘅變數
x
{\displaystyle x}
x
t
{\displaystyle x_{t}}
t
{\displaystyle t}
x
{\displaystyle x}
x
t
=
f
(
x
t
−
1
,
x
t
−
2
.
.
.
x
1
)
{\displaystyle x_{t}=f(x_{t-1},x_{t-2}...x_{1})}
泊淞 [e 268] 數數據 上嘅一種迴歸分析，最基本嗰個模型係噉嘅樣：
log
(
E
(
Y
∣
x
)
)
=
α
+
β
′
x
,
{\displaystyle \log(\operatorname {E} (Y\mid \mathbf {x} ))=\alpha +\mathbf {\beta } '\mathbf {x} ,}
當中
Y
{\displaystyle Y}
泊淞分佈 ），
x
{\displaystyle \mathbf {x} }
自變數 嘅向量 ，而
α
{\displaystyle \alpha }
β
′
{\displaystyle \mathbf {\beta } '}
參數 [156] 普通最小二乘法 [e 269] 殘差平方和 [e 270] 最佳化 ）。當中 RSS 係指將所有誤差值嘅平方加埋得出嘅數[149]
R
S
S
=
∑
i
=
1
n
e
i
2
{\displaystyle RSS=\sum _{i=1}^{n}e_{i}^{2}\,}
逐步 [e 271] 由一個冇自變數嘅迴歸模型開始，foreach 自變數，加個自變數入去，喺每一步都用某啲事先制定咗嘅法則講明要點決定加邊個自變數（前向）；
由一個有齊嗮啲自變數嘅迴歸模型開始，foreach 自變數，攞個自變數走，睇吓個模型嘅預測力變成點，喺每一步都用某啲事先制定咗嘅法則講明要點決定攞走邊個自變數（反向）。
喺廿一世紀嘅統計學界，逐步迴歸廣受批評，所以唔多人用[157] 線性關係 ：如果話兩個變數
x
{\displaystyle x}
y
{\displaystyle y}
直線 ，條式會係[158]
y
=
a
x
+
b
{\displaystyle y=ax+b}
a
{\displaystyle a}
參數 ，而
b
{\displaystyle b}
截距 [e 272] 固定效應 [e 273] 參數 係固定或者最少非隨機 嘅數值[159] 隨機效應 [e 274] 參數 係隨機變數 [159] 混合 [e 275] 參數 有啲係固定或者非隨機，有啲係隨機變數 [159] 嵌套 [e 276]
A
{\displaystyle A}
B
{\displaystyle B}
A
{\displaystyle A}
參數 係
B
{\displaystyle B}
子集 ；研究者可以透過比較唔同模型嘅適合度指標 ，睇吓「邊個模型能夠最有效噉描述手上攞住嘅數據」[160] 奧坎剃刀 嘅概念。等級線性模型 [e 277] 多層分析 [e 278]
i
{\displaystyle i}
例如一份管理學 上嘅研究，想分析一間公司（樣本）入面嘅員工（個體），而每個員工都有佢所屬嘅工作團隊（樣本入面嘅群組），研究者有理由相信工作團隊之間嘅差異（例如係團隊領袖嘅領導能力 ）會影響佢想研究嘅現象，所以佢就做 HLM，用類似以下噉嘅數學方程式 將唔同層面嘅變數擺入去同一條式入面[161]
Y
i
j
=
β
0
j
+
β
1
j
X
i
j
+
β
2
G
j
+
e
i
j
{\displaystyle Y_{ij}=\beta _{0j}+\beta _{1j}X_{ij}+\beta _{2}G_{j}+e_{ij}}
Y
i
j
{\displaystyle Y_{ij}}
應變數 （細階
i
{\displaystyle i}
j
{\displaystyle j}
X
i
j
{\displaystyle X_{ij}}
自變數 ；
G
j
{\displaystyle G_{j}}
β
0
j
{\displaystyle \beta _{0j}}
根 ；淨低嗰啲
β
{\displaystyle \beta }
迴歸系數 [e 279]
e
i
j
{\displaystyle e_{ij}}
誤差 。
呢條式用文字解釋係噉：
Y
i
j
{\displaystyle Y_{ij}}
X
i
j
{\displaystyle X_{ij}}
G
j
{\displaystyle G_{j}}
Y
i
j
{\displaystyle Y_{ij}}
e
i
j
{\displaystyle e_{ij}}
Y
i
j
{\displaystyle Y_{ij}}
j
{\displaystyle j}
i
{\displaystyle i}
X
i
j
{\displaystyle X_{ij}}
j
{\displaystyle j}
i
{\displaystyle i}
G
j
{\displaystyle G_{j}}
j
{\displaystyle j}
跟手個研究者就去收數據 ，做統計分析，用數據估計
β
1
j
{\displaystyle \beta _{1j}}
β
2
{\displaystyle \beta _{2}}
β
1
j
>
β
2
{\displaystyle \beta _{1j}>\beta _{2}}
管理學 等社科 領域研究，因為呢啲領域成日會遇到「樣本入面有若干個次群體」嘅情況[162] 一個潛在變數模型嘅想像圖；家陣研究者想量度
T
{\displaystyle T}
智能 ），於是就俾受試者做個測驗，有
k
{\displaystyle k}
X
1
{\displaystyle X_{1}}
X
2
{\displaystyle X_{2}}
X
3
{\displaystyle X_{3}}
X
k
{\displaystyle X_{k}}
誤差值
e
i
{\displaystyle e_{i}}
λ
i
{\displaystyle \lambda _{i}}
T
{\displaystyle T}
相關 ）。 因素分析 [e 280] 因素 [e 281]
想像家陣手上個數據集有若干個可觀察 [e 282] 隨機變數
x
1
,
x
2
,
.
.
.
,
x
p
{\displaystyle x_{1},x_{2},...,x_{p}}
μ
1
,
μ
2
,
.
.
.
,
μ
p
{\displaystyle \mu _{1},\mu _{2},...,\mu _{p}}
想像有
k
{\displaystyle k}
隱藏變數 [e 283]
F
j
{\displaystyle F_{j}}
j
∈
1
,
.
.
.
,
k
{\displaystyle j\in 1,...,k}
F
j
{\displaystyle F_{j}}
[註 8]
喺做因素分析前，
F
j
{\displaystyle F_{j}}
x
i
−
μ
i
=
l
i
1
F
1
+
⋯
+
l
i
k
F
k
+
ε
i
{\displaystyle x_{i}-\mu _{i}=l_{i1}F_{1}+\cdots +l_{ik}F_{k}+\varepsilon _{i}}
i
∈
1
,
.
.
.
,
p
{\displaystyle i\in 1,...,p}
柞
l
i
j
{\displaystyle l_{ij}}
ε
i
{\displaystyle \varepsilon _{i}}
誤差 ，平均值係 0，而變異數 係一個有限數值，唔同
i
{\displaystyle i}
ε
i
{\displaystyle \varepsilon _{i}}
假想
p
{\displaystyle p}
x
i
{\displaystyle x_{i}}
x
i
{\displaystyle x_{i}}
k
<
p
{\displaystyle k<p}
F
j
{\displaystyle F_{j}}
[163]
潛在變數模型 [e 284] 因素結構 [e 285] 因素負荷量 [e 286] 因素負荷量 [e 287] 隱藏因素 之間有嘅一個數，值喺 0 到 1 之間，係嗰個變數同個隱藏因素之間嘅統計相關 ；如果一個變數嘅因素負荷量大，就表示佢同個隱藏因素有強嘅統計相關，而如果一個變數嘅因素負荷量細，噉就表示佢同個隱藏因素之間嘅統計相關弱，通常研究者就會覺得噉表示個變數根本反映唔到個隱藏因素，會考慮將嗰個變數由個模型嗰度攞走。因素分析可以分做兩大類[164] 探索型因素分析 [e 288]
x
i
−
μ
i
{\displaystyle x_{i}-\mu _{i}}
確定型因素分析 [e 289]
主成份分析 [e 290]
x
{\displaystyle x}
y
{\displaystyle y}
變異數 －亦即係話長箭咀嗰條線所代表嗰個「成份」比較能夠用嚟分辨啲個案，所以比較「重要」。喺最簡單嗰種情況下，一個做主成份分析嘅演算法 大致上係噉[165] 攞數據；
畫條線出嚟，條線有條式，而條式包含數據當中有嘅變數 ；
計出沿呢條線嘅變異數有幾多；
改變吓條線嘅參數 ；
再計出沿條新線嘅變異數有幾多；
一路做步驟 4 同 5，做嗮所有指定咗嘅可能性，最後俾具有最大變異數嗰條線做個演算法嘅輸出。 主成分分析嘅圖解；幅圖每一點代表一個個案，兩個箭咀代表兩個成份，長啲嗰個箭咀係比較重要嗰個成份。 卡隆巴系數 （[e 291]
ρ
T
{\displaystyle \rho _{T}}
心理測量學 上成日用嚟衡量一個心理測驗 嘅信度 （睇上面）嘅數值。想像家陣有個心理測驗，有
k
{\displaystyle k}
k
{\displaystyle k}
邏輯 能力），研究者搵人做個測驗攞到數據 之後，個測驗嘅卡隆巴系數（
ρ
T
{\displaystyle \rho _{T}}
[166] [167]
ρ
T
=
k
2
σ
i
j
¯
σ
X
2
{\displaystyle \rho _{T}={k^{2}{\overline {\sigma _{ij}}} \over \sigma _{X}^{2}}}
σ
i
j
¯
{\displaystyle {\overline {\sigma _{ij}}}}
協方差 [e 292] 平均值 ；
σ
X
2
{\displaystyle \sigma _{X}^{2}}
變異數 嘅總和」加埋「題目之間嘅協方差總和」；即係話
σ
X
2
=
∑
i
=
1
k
∑
j
=
1
k
σ
i
j
=
∑
i
=
1
k
σ
i
2
+
∑
i
=
1
k
∑
j
≠
i
k
σ
i
j
{\displaystyle \sigma _{X}^{2}=\sum _{i=1}^{k}\sum _{j=1}^{k}\sigma _{ij}=\sum _{i=1}^{k}\sigma _{i}^{2}+\sum _{i=1}^{k}\sum _{j\neq {i}}^{k}\sigma _{ij}}
加總 ）；如果卡隆巴系數數值大（接近 1）嘅話，就表示呢柞題目嘅變異數主要源自佢哋之間嘅協方差，簡單講就係表示「呢柞題目之間嘅變異數主要係由佢哋之間嘅相關 引起嘅」而唔係源於佢哋各自獨立嘅變異－所以如果一柞題目嘅卡隆巴系數數值大，研究者就更有理由相信呢柞題目係量度緊同一個隱藏因素 [166]
張量分解 [e 293] 張量 形式表達嘅數據 「分解」做比較簡單嘅張量以及呢啲簡單張量之間嘅運算 ，例[168]
T
=
v
1
⊗
w
1
⊗
α
1
+
v
2
⊗
w
2
⊗
α
2
+
⋯
+
v
N
⊗
w
N
⊗
α
N
{\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}}
當中
T
{\displaystyle T}
v
{\displaystyle v}
w
{\displaystyle w}
α
{\displaystyle \alpha }
⊗
{\displaystyle \otimes }
張量積 （一種數學運算）。做張量分解目的係要等計起數上嚟冇咁撈絞[168] 生還分析嘅圖解；想像喺實驗室養咗柞老鼠 ，上面幅圖嘅 X 軸表示時間，Y 軸表示「有幾多百分比 嘅白老鼠仲係生勾勾」。 聚類分析嘅附圖；呢拃物件可以按位置座標分做三類，而聚類分析可以想像成同啲點油顏色嘅過程。 生還分析 [e 301] 動物 要幾耐先會死－事件係「死亡 」。生還分析包含咗一系列嘅技術，能夠解答以下呢啲問題：
攞一個時間數值
t
{\displaystyle t}
t
{\displaystyle t}
總體 入面有幾多百分比 嘅個體會死亡？
喺死淨嗰啲個體當中，佢哋會以幾快嘅率接近死亡？
... 等等。進階啲嘅應用仲可以將「死亡」換做第啲事件，例如電子遊戲 同人機互動 等方面嘅研究就會以「用家 放棄 隻遊戲或者產品」嚟做集中研究嘅嗰件事件，用嚟分析用家用起電子產品上嚟嘅行為 [175] 工程學 同經濟學 上都會用到[176]
生還函數 （[e 302]
S
(
t
)
{\displaystyle S(t)}
T
{\displaystyle T}
P
{\displaystyle P}
t
{\displaystyle t}
函數 ，即係話[177]
S
(
t
)
=
P
(
{
T
>
t
}
)
{\displaystyle S(t)=P(\{T>t\})}
S
(
t
)
{\displaystyle S(t)}
指數函數 就係常用嘅一種生還函數。聚類分析 [e 303] X 軸 （一個變數）同 Y 軸 度有個位置，但就噉用肉眼睇都睇得出，啲點可以分做三大類（唔同色嘅點），每個聚類 都係「個聚類入面啲點，彼此之間距離 近，同時又冚唪唥都係同聚類外嘅點距離遠嘅」；聚類分析就可以想像成「同啲點油顏色，表示每點屬邊個聚類」嘅過程[178] [179] 馬可夫鏈 [e 304] 隨機 數學模型 。喺一條馬可夫鏈當中有若干個可能狀態，而每個狀態
s
i
{\displaystyle s_{i}}
[
p
0
,
p
1
,
p
2
.
.
.
]
{\displaystyle [p_{0},p_{1},p_{2}...]}
s
i
{\displaystyle s_{i}}
機會率 ；喺統計學 上，一種簡單嘅做法係收數據 ，用數據估計
[
p
0
,
p
1
,
p
2
.
.
.
]
{\displaystyle [p_{0},p_{1},p_{2}...]}
[180] 獨立成份分析 [e 305] 訊號處理 ，會將一個受多個變數影響嘅訊號
x
=
(
x
1
,
…
,
x
m
)
T
{\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})^{T}}
x
=
(
x
1
,
…
,
x
m
)
T
{\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})^{T}}
s
=
(
s
1
,
…
,
s
n
)
T
{\displaystyle {\boldsymbol {s}}=(s_{1},\ldots ,s_{n})^{T}}
x
i
{\displaystyle x_{i}}
s
i
{\displaystyle s_{i}}
線性組合 ；簡單講就係 foreach
x
i
{\displaystyle x_{i}}
x
i
=
a
i
,
1
s
1
+
⋯
+
a
i
,
k
s
k
+
⋯
+
a
i
,
n
s
n
{\displaystyle x_{i}=a_{i,1}s_{1}+\cdots +a_{i,k}s_{k}+\cdots +a_{i,n}s_{n}}
a
i
{\displaystyle a_{i}}
s
i
{\displaystyle s_{i}}
x
i
{\displaystyle x_{i}}
當中啲
s
i
{\displaystyle s_{i}}
[181] 點過程 [e 306] 點場 [e 307] 空間 當中有隨機性噉產生一啲點 ，可以攞嚟做好多涉及空間嘅分析，例如係地質學 上分析地震 噉，一場地震嘅中心 可以想像成空間入面嘅一點，而一場地震出現可能會提升周圍嘅空間出現地震點嘅機會（可以睇餘震 ），而喺建立統計模型嚟分析地震嗰陣，分析者可以將「震央嘅出現」想像成一個會喺代表地面嘅空間嗰度產生一粒點點嘅隨機過程 [182] 線性判別分析 [e 308] 線性組合 ，嚟分別出若干個『類別』嘅嘢」。
簡單講就係冇可能同時發生嘅事件。例如家陣擲三粒骰仔，
{
1
,
3
,
5
}
{\displaystyle \{1,3,5\}}
{
2
,
4
,
6
}
{\displaystyle \{2,4,6\}}
p
{\displaystyle p}
參數 。不過除此之外仲要有幾個條件：
Y
{\displaystyle Y}
X
{\displaystyle X}
X
{\displaystyle X}
Y
{\displaystyle Y}
哈曼測試到咗 2020 年經已唔再俾人認為係一種可靠嘅做法。
喺實際應用上，考慮咁多極細嘅數值可能會出現算術下溢 嘅情況（指要處理嘅數值細過部電腦能夠表示嘅最細值），所以喺實際應用上要點樣計
Pr
(
X
|
θ
)
{\displaystyle {\text{Pr}}(X|\theta )}
如果
a
>
b
{\displaystyle a>b}
max
(
a
,
b
)
=
a
{\displaystyle \max(a,b)=a}
max
(
a
,
b
)
=
b
{\displaystyle \max(a,b)=b}
「
a
∈
b
{\displaystyle a\in b}
a
{\displaystyle a}
b
{\displaystyle b}
集 入面」。
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