統計模型

統計模型（粵拼：tung2 gai3 mou4 jing4 ）係一大類嘅數學模型，建基於統計學技術。統計模型通常包含一組假設，用嚟解釋數據點之間嘅規律，並透過參數嚟刻劃呢啲規律。例如迴歸模型可以用嚟分析一組變數點對點之間嘅關聯。統計模型喺好多領域裏便都有用，譬如機械學習、經濟學、心理學等領域，幫助人理解數據同作出預測。

基礎概念

統計模型係一啲用嚟描述數據點樣產生出嚟嘅數學模型。呢啲模型通常會包含若干個參數，而呢啲參數可以根據實際情況作調整。舉個例子，想像依家研究擲銀仔，擲銀仔有兩個可能情況，公或者字，而呢個過程可以用伯努利分佈嚟表示，其概率質量函數 $f$ 係^[1]：

f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}

呢個模型模擬緊手上嘅現象，將現象用數學方式表達出嚟。呢個模型中只有一個參數 $p$ ，代表公（或者字）出現嘅概率。喺現實世界，絕大部分嘅模型，都只能夠做到近似產生數據嘅過程，亦未必可以包含晒所有會影響結果嘅因素。

無論係咩統計模型，都可以大致想像成以下呢啲結構：

應變數 = 自變數預測嘅大概趨勢 + 隨機上落

應變數 = 平均 + 喺平均周圍嘅變化

應變數 = 系統性質嘅變化 + 隨機性質嘅變化

呢種結構最簡單個款，就係線性嘅迴歸分析。

可辨識度

喺統計學同計量經濟學等學科上，可辨識度係指一個模型係咪具備足夠嘅資訊，去唯一噉確定佢啲參數嘅值。如果某個模型係不可辨識嘅，意思即係話縱使分析者手上有無限咁多嘅數據，佢都冇辦法分辨出邊一組參數至係真相，因為唔同嘅參數組合，都可以產生一模一樣嘅實際分佈，所以齋睇數據顯示嘅實際分佈，無法分辨邊個參數組合至係真正嗰個。^[11]^[12]

統計學界有好多方法應付可辨識度不足嘅問題，例如係

固定參數嘅值：睇返過往嘅數據或者相關嘅學術文獻，判斷某啲模型參數嘅值似會係咩數值，然後就按此嚟決定嗰啲參數係幾多，跟住先至郁手做估計—要估計嘅參數少咗，就有可能補救可辨識度不足嘅問題。
區局可辨識度 (暫譯) ^{[註 1]}：將參數嘅可能值，局限喺某啲特定嘅區間，縮窄搜索範圍，就有較大機會得以將可能性減少到得一個^[11]。

模型類型

迴歸模型
因素分析
結構方程模型
生還分析
聚類分析
混合物模型：簡單講就係指將個模型設做「群體入面有若干個子群體，每個子群體之間都有顯著嘅差異」，而且唔使拃數據事先指定每個個案屬邊個子群體^[13]。睇埋聚類分析嘅概念。

睇埋

獨立同分佈：好多統計建模做法都會做嘅假設。
統計分析
混合模型
電腦模擬

註釋

↑ 英文：local idenfiability

引咗

↑ Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific.
1 2 Similarities of Univariate & Multivariate Statistical Analysis.
↑ Cox, D. R. (2006), Principles of Statistical Inference, Cambridge University Press, p. 197.
↑ Huber-Carol, C.; Balakrishnan, N.; Nikulin, M. S.; Mesbah, M., eds. (2002), Goodness-of-Fit Tests and Model Validity, Springer
↑ Singh, R. (2009). Does my structural model represent the real phenomenon?: a review of the appropriate use of Structural Equation Modelling (SEM) model fit indices. The Marketing Review, 9(3), 199-212.
↑ Sarstedt, M. , Henseler, J. and Ringle, C. (2011), "Multi-group analysis in partial least squares (PLS) path modeling: alternative methods and empirical results", Advances in International Marketing, Vol. 22 No. 1, pp. 195-218.
↑ Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. p. 61.
↑ Sarstedt, M. , Henseler, J. and Ringle, C. (2011), "Multi-group analysis in partial least squares (PLS) path modeling: alternative methods and empirical results", Advances in International Marketing, Vol. 22 No. 1, pp. 195-218.
↑ 嵌套模型（nested model）
↑ Inness, M., Turner, N., Barling, J., & Stride, C. B. (2010). Transformational leadership and employee safety performance: a within-person, between-jobs design. Journal of occupational health psychology, 15(3), 279，呢份管理學研究用咗嵌套模型，剖析（簡化講）轉工同管理者嘅領導能力點影響打工仔嘅某啲行為。
1 2 Model Identifiability - By Guan-Hua Huang，佢哋一開始就開宗明義噉講："In some statistical models, different parameter values can give rise to identical probability distributions. When this happens, there will be a number of different parameter values associated with the maximum likelihood of any set of observed data."
↑ Olukan, D., 2023. Heterogeneity in Agent-based models (Doctoral dissertation, University of Leeds), also see Jurnal of Artificial Societies and Social Simulation (JASSS).
↑ Everitt, B.S.; Hand, D.J. (1981). Finite mixture distributions. Chapman & Hall.

[13] 英文：local idenfiability

[1] Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific.

[unimultivariate-2] 1 2 Similarities of Univariate & Multivariate Statistical Analysis.

[3] Cox, D. R. (2006), Principles of Statistical Inference, Cambridge University Press, p. 197.

[4] Huber-Carol, C.; Balakrishnan, N.; Nikulin, M. S.; Mesbah, M., eds. (2002), Goodness-of-Fit Tests and Model Validity, Springer

[5] Singh, R. (2009). Does my structural model represent the real phenomenon?: a review of the appropriate use of Structural Equation Modelling (SEM) model fit indices. The Marketing Review, 9(3), 199-212.

[6] Sarstedt, M. , Henseler, J. and Ringle, C. (2011), "Multi-group analysis in partial least squares (PLS) path modeling: alternative methods and empirical results", Advances in International Marketing, Vol. 22 No. 1, pp. 195-218.

[7] Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. p. 61.

[8] Sarstedt, M. , Henseler, J. and Ringle, C. (2011), "Multi-group analysis in partial least squares (PLS) path modeling: alternative methods and empirical results", Advances in International Marketing, Vol. 22 No. 1, pp. 195-218.

[9] 嵌套模型（nested model）

[10] Inness, M., Turner, N., Barling, J., & Stride, C. B. (2010). Transformational leadership and employee safety performance: a within-person, between-jobs design. Journal of occupational health psychology, 15(3), 279，呢份管理學研究用咗嵌套模型，剖析（簡化講）轉工同管理者嘅領導能力點影響打工仔嘅某啲行為。

[GHH-11] 1 2 Model Identifiability - By Guan-Hua Huang，佢哋一開始就開宗明義噉講："In some statistical models, different parameter values can give rise to identical probability distributions. When this happens, there will be a number of different parameter values associated with the maximum likelihood of any set of observed data."

[12] Olukan, D., 2023. Heterogeneity in Agent-based models (Doctoral dissertation, University of Leeds), also see Jurnal of Artificial Societies and Social Simulation (JASSS).

[14] Everitt, B.S.; Hand, D.J. (1981). Finite mixture distributions. Chapman & Hall.

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[註 1]

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基礎概念

相關概念

可辨識度

模型類型

睇埋

註釋

引咗