Mathematical Expressions

Abstract

Statistical methods are useful tools to deal with data on socioeconomic-technological systems. In this chapter, we will address fundamental expressions used in statistics and methods of data analysis: time series analysis, network analysis and spatial analysis.

Appendices

Appendix A: Proof of $0\ln 0$

Let us consider

$$\begin{aligned} h = \lim _{x\rightarrow +0} x \ln x. \end{aligned}$$

(3.254)

Putting $x = e^{-z}$ one has

$$\begin{aligned} h = -\lim _{x\rightarrow +0} z e^{-z}. \end{aligned}$$

(3.255)

By using the Taylor expansion of $e^{z} = \sum _{k=0}^{\infty }\frac{1}{k!}z^k$, one obtains

$$\begin{aligned} \nonumber h&= -\lim _{z\rightarrow \infty }z e^{-z} \\ \nonumber&= -\lim _{z\rightarrow \infty } z / e^{z} \\ \nonumber&= -\lim _{z\rightarrow \infty } \frac{z}{\sum _{k=0}\frac{1}{k!}z^k} \\ \nonumber&= -\lim _{z\rightarrow \infty } \frac{1}{\sum _{k=0}\frac{1}{k!}z^{k+1}} \\&= 0. \end{aligned}$$

(3.256)

Therefore, we gets

$$\begin{aligned} h = \lim _{x\rightarrow 0} x \ln x = 0 \ln 0 = 0. \end{aligned}$$

(3.257)

Appendix B: Derivation of the Mean Square Error of RMA Regression

The mean square error $MSE$ of the RMA regression is defined as

$$\begin{aligned} MSE = \frac{1}{T-2}\sum _{i=1}^T (y_i - \hat{a}x_i - \hat{b})^2. \end{aligned}$$

(3.258)

Inserting Eq. (3.70) into Eq. (3.258), we get

$$\begin{aligned} \nonumber MSE&= \frac{1}{T-2}\sum _{i=1}^T (y_i - \hat{a}x_i - \hat{b})^2 \\ \nonumber&= \frac{1}{T-2}\sum _{i=1}^T \Bigl \{y_i - \hat{a}x_i - \Bigl (\frac{\sum _{i=1}^T y_i}{T} - \hat{a} \frac{\sum _{i=1}^T x_i}{T}\Bigr )\Bigr \}^2 \\ \nonumber&= \frac{1}{T-2}\sum _{i=1}^T \Bigl \{\Bigl (y_i - \frac{\sum _{i=1}^T y_i}{T}\Bigr ) - \hat{a}\Bigl (x_i - \frac{\sum _{i=1}^T x_i}{T}\Bigr )\Bigr \}^2 \\ \nonumber&= \frac{1}{T-2}\sum _{i=1}^T \Bigl \{\Bigl (y_i - \frac{\sum _{i=1}^T y_i}{T}\Bigr )^2 + \hat{a}^2\Bigl (x_i - \frac{\sum _{i=1}^T x_i}{T}\Bigr )^2\\&\qquad - 2 \hat{a}\Bigl (x_i - \frac{\sum _{i=1}^T x_i}{T}\Bigr )\Bigl (y_i - \frac{\sum _{i=1}^T y_i}{T}\Bigr )\Bigr \}. \end{aligned}$$

(3.259)

This is also written as

$$\begin{aligned} MSE = \frac{T}{T-2}\Bigl \{\mathrm {Var}[Y] + \hat{a}^2\mathrm {Var}[X] - 2\hat{a}\mathrm {Cov}[X,Y]\Bigr \}. \end{aligned}$$

(3.260)

Inserting Eq. (3.74) into Eq. (3.260), consequently we obtain

$$\begin{aligned} MSE = \Bigl (\mathrm {Var}[Y] - \hat{a}\mathrm {Cov}[X,Y]\Bigr )\frac{2T}{T-2}. \end{aligned}$$

(3.261)