Time Series Decomposition¶

		Advantages	Disadvantages
Classical		Easy to understand & interpret	1. Estimate of trend is unavailable in the first few and last few observations 2. Assumes that seasonal component repeats 3. Not robust to outliers due to usage of means
X-11		1. Relatively robust to outliers 2. Completely automated choices for trend and seasonal changes 3. Tried & tested method	1. No prediction/confidence intervals 2. Ad hoc method with no underlying model 3. Only for quarterly & monthly data
X-12-ARIMA/ X-13-ARIMA		1. Allow adjustments for trading days and explanatory variables 2. Known outliers can be omitted 3. Level shifts & ramp effects can be modelled 4. Missing values estimated and replaced 5. Holiday factors can be estimated
X-13-ARIMA-SEATS		1. Model-based 2. Smooth trend estimate 3. Allows estimates at end-points 4. Incorporates changing seasonality
STL Seasonal & Trend Decomposition using Loess	- Iterative alogirthm - Starts with $\hat T = 0$ - Uses mixture of loess and moving averages to successively refine trend & seasonal estimates - Trend window controls loess bandwidth applied to de-seasonalized values - Season window controls loess bandwidth applied to detrended subseries - Seasonal component allowed to change over time; Rate of change controlled by analyst - Smoothness of trend controlled by analyst	- Versatile - Robust - Handle any type of seasonality	- Only additive (Use log/Box-Cox transformations for other) - No training day/calendar adjustments

Classical Decomposition¶

	$y_t$	Appropriate when Magnitude of seasonal fluctuations proportional to level of series
Addititive	$S_t + T_t + R_t$	❌
Multiplicative	$S_t \times T_t \times R_t$	✅

Alternatively, use Box-Cox transformation, and then use additive decomposition. Logs turn multiplicative relationship into additive

\[ \begin{aligned} y_t &= S_t \times T_t \times R_t \\ \implies \ln y_t &= \ln S_t + \ln T_t + \ln R_t \end{aligned} \]

Trend Estimation¶

Centered moving averages, to combat odd order

\[ \begin{aligned} \hat T_t &= \dfrac{1}{2m} \left( \sum_{i = -(k+1)}^k y_{t+i} + \sum_{i = -k}^{k+1} y_{t+i} \right) \\ \text{where } k &= \dfrac{m-1}{2} \end{aligned} \]

Order ($m$)	Curve	Data Retention
Larger	Smoother, flatter	Less (end points are lost)
Smaller	Noisy	More

Moving average of the same length of a season/cycle removes its pattern

Seasonal Adjusted Data¶

Component excluding the seasonal component

Detrended Series¶

\[ \begin{aligned} y_t - \hat T_t &= \hat S_t + \hat R_t \\ \frac{y_t}{\hat T_t} &= \hat S_t \times \hat R_t \end{aligned} \]

Seasonal component¶

Average of de-trended series for that season. For eg, average of all values in Januaries

You can constraint the seasonal components such that

\[ \hat S_1 + \hat S_2 + \dots + \hat S_{n} = 0 \\ \hat S_1 \times \hat S_2 \times \dots \times \hat S_{n} = m \]

Remainder Component¶

\[ \begin{aligned} \hat R_t &= y_t - (\hat T_t + \hat S_t) \\ \hat R_t &= \dfrac{y_t}{\hat T_t \hat S_t} \end{aligned} \]

Fourier Transforms¶

FT’s limitation: FT is completely blind to time, in accordance with Heisenberg’s Uncertainty principle. There’s a tradeoff between correctly estimating the value of function in the frequency & time domain.

It is 1D representation

Types of Fourier Transforms¶

Type
Continuous Time & Frequency	Functional form of time series is known analytically	$\hat x(f) = \int \limits_{-\infty}^\infty x(t) e^{-2\pi i f t} dt$
Continuous Time, Discrete Frequency (Fourier Series)		$\hat x(f_n) = \dfrac{1}{T} \int \limits_{0}^T x(t) e^{-2\pi i f_n t} dt; f_n = \dfrac{n}{T}$
Discrete Time & Frequency (Fourier Frequencies)		$\hat x(f_n) = \sum \limits_{k=0}^{N-1} x_t e^{- 2 \pi i f_n (k \Delta t)} \Delta t; f_n = \dfrac{n}{N \Delta t}; \hat x_n = \hat x^*_{-n}$
FFT (Fast Fourier Transform)

Denoising using FFT¶

Apply FFT
Filter it to only the frequencies with the highest amplitude
Take inverse FFT

Wavelet Transform¶

Overcomes FT’s limitation: FT is completely blind to time, by obtaining an optimal balance between accuracy in frequency & time domain

Wavelet¶

Short-lived oscillation, localized in time

Zero mean: $E[\phi(t)]=0; \int \phi(t) \cdot dt = 0$ (Admissibility condition)
Finite energy: $\int [\phi(t)]^2 \cdot dt = k, k < \infty$

Type	$\phi(t)$
Daubechies
Coiflet
Symlet
Haar
Morlet	$k_0 \cdot e^{i w_0 t} \cdot e^{-t^2/2}$
Gaussian
Shannon
Meyer
Mexican Hat

IDK¶

2D representation: $y(t) \to T(t, f)$ represents the contribution of frequency $f$ at time $t$

Scaled Wavelet $\phi(t, a, b) = \phi \left(\dfrac{t-b}{a} \right)$

The value of $T(a, b) =$ contribution of $\phi(t, a, b)$ to comprising the signal $$ T(a, b) = \int y(t) \cdot \phi(t, a, b) \cdot dt $$ Demonstrates the goodness of fit: local similarity

Signals¶

	Time Resolution	Frequency Resolution
Raw Time Series	High	$\approx 0$
Fourier Transform	$\approx 0$	High
Wavelet Transform	Low for small frequencies High for high frequencies This is intuitive, as high freq signals are usually short-lived, and small freq signals are usually long-lived

Last Updated: 2024-05-12 ; Contributors: AhmedThahir

	\(y_t\)	Appropriate when Magnitude of seasonal fluctuations proportional to level of series
Addititive	\(S_t + T_t + R_t\)	❌
Multiplicative	\(S_t \times T_t \times R_t\)	✅

Type
Continuous Time & Frequency	Functional form of time series is known analytically	\(\hat x(f) = \int \limits_{-\infty}^\infty x(t) e^{-2\pi i f t} dt\)
Continuous Time, Discrete Frequency (Fourier Series)		\(\hat x(f_n) = \dfrac{1}{T} \int \limits_{0}^T x(t) e^{-2\pi i f_n t} dt; f_n = \dfrac{n}{T}\)
Discrete Time & Frequency (Fourier Frequencies)		\(\hat x(f_n) = \sum \limits_{k=0}^{N-1} x_t e^{- 2 \pi i f_n (k \Delta t)} \Delta t; f_n = \dfrac{n}{N \Delta t}; \hat x_n = \hat x^*_{-n}\)
FFT (Fast Fourier Transform)

	Time Resolution	Frequency Resolution
Raw Time Series	High	\(\approx 0\)
Fourier Transform	\(\approx 0\)	High
Wavelet Transform	Low for small frequencies High for high frequencies This is intuitive, as high freq signals are usually short-lived, and small freq signals are usually long-lived