Time Series Analysis

References http://www.math.psu.edu/wysocki/M412/Notes412_8.pdf https://zhuanlan.zhihu.com/p/19763358 https://www.math.psu.edu/tseng/class/Math251/Notes-PDE%20pt2.pdf http://www.oulu.fi/sites/default/files/content/files/series.pdf http://www.engr.uconn.edu/~lanbo/G377FFTYC.pdf Fourier Series Theorem (references: Second Order Linear Partial Differential Equations, 复数形式傅立叶变换的物理意义，相位究竟指的是什么？) \(\text{Suppose } f(x) \text{ is a periodic function with period } T \text{ and is an integrable function on } [0, T]. \\ \text{Then, the Fourier Series of } f(x) \text{ can be written as }\) \[ \begin{align} f(x) & = \frac{c_0}{2} + \sum_{n=1}^{\infty} c_ncos(n \cdot \frac{2\pi}{T} \cdot x + \varphi_n) \\ &= \frac{c_0}{2} + \sum_{n=1}^{\infty} c_ncos(\varphi _n)cos(n \cdot \frac{2\pi}{T} \cdot x)+ (-c_n)sin(\varphi _n)sin(n \cdot \frac{2\pi}{T} \cdot x) \\ &\text{( let } a_0 = c_0, \;a_n = c_ncos(\varphi _n) \text{ and } b_n = (-c_n)sin(\varphi _n) \;) \\ &= \frac{a_0}{2} + \sum_{n=1}^{\infty} a_ncos(n \cdot \frac{2\pi}{T} \cdot x)+ b_nsin(n \cdot \frac{2\pi}{T} \cdot x) \\ \\ \text{where } c_n &= \sqrt{a_n^2 + b_n^2} = \sqrt{c_n^2(cos^2(\varphi _n) + sin^2(\varphi _n))} = \sqrt{c_n^2} \;\; (Amplitude)\\ \varphi_n &= tan^{-1}(-\frac{b_n}{a_n}) \;\; (Phase)\\ a_0 &= \frac{1}{T}\int_{0}^{T}f(x)dx \\ a_n &= \frac{1}{T}\int_{0}^{T}f(x) \cdot cos(n \cdot \frac{2\pi}{T} \cdot x)dx \\ b_n &= \frac{1}{T}\int_{0}^{T}f(x) \cdot sin(n \cdot \frac{2\pi}{T} \cdot x)dx \end{align} \]

BATS model is Exponential Smoothing Method + Box-Cox Transformation + ARMA model for residuals. The Box-Cox Transformation here is for dealing with non-linear data and ARMA model for residuals can de-correlated the time series data. Alysha M.(2010) has proved that BATS model can improve the prediction performance compared to the simple Sate Space Model. However, BATS model does not do well when the the seaonality is complex and high frequency.

The main weakness of original Exponential Smoothing Method is that it can only provide point estimation. Hyndman (2002) proposed using state space framework to rewrite the original exponential smoothing algorithm and then give distribution assumption on the error terms to calculate the prediction interval. There are two types of error terms in the state space model: Additive and Multiplicative. The point estimator for these two models are the same but the prediction intervals are different.

This post is about Exponential Smoothing method, a prediction method for time series data. There are many forms of Exponential Smoothing method and the most basic ones are Single, Double and Triple (Holt-Winters) Exponential Smoothing. Some of the Exponential Smoothing forms can be written as ARIMA model; some of them can not and vice versa. Compared to ARIMA model, Exponential Smoothing method do not have strong model assumptions and it also can not add explanatory variables in the algorithm.