rangkuman_202603011304

Week 1: Pendahuluan dan Metode Sederhana

Definition: Cross-section Data

Tip

“Table” of subjects and variables at specific single “slice” of time.

Definition: Time Series Data

Tip

The “raw” data itself, related to time. Usually for stochastic process

Definition: Panel/Pooled Data

Tip

Definition Cross-section Data but in an interval “slice” of time rather than a single slice

Definition: Stochastic Process

Tip

Random variable indexed with time

Cheatsheet: Time Series Components (Trend, Seasonal, Cyclical, Irregular)

Tip

If time series is a kite,

• Trend: String that “pulls” or “leads” the kite to its “expected” path
• Seasonal: The wind is always stronger at noon because the sun is out. (Predictable by time).
• Cyclical: A “Storm System” is passing through. The wind howls for 3 days, then it’s calm for 10 days, then another storm lasts for 5 days. It is a long-term wave of energy that lingers, but there is no calendar for when the next storm starts.
• Irregular: unpredictable deviations

Definition: Autocovariance and Autocorrelation

Tip

Respectively, cov and corr of 2 random variables from same sequence at two time points

Definition: Weakly Stationary

Let $\gamma$ : Function of the lag $k$ and is independent of $t$.

A stochastic process $\{Y_t\}$ is weakly stationary

If it satisfies three conditions:

1. $E[Y_t]=\mu$ (constant mean)
2. $\mathrm{Var}(Y_t)<\infty ,\forall t$ (finite variance)
3. $\mathrm{Cov}(Y_t,Y_{t-k})={\gamma }_k$ (cov independent of $t$)

Definition: Strictly Stationary

A stochastic process $\{Y_t\}$ is strictly stationary

If for any $n\in {\mathbb{Z}}^{+}$, any set of time points $\{t_1,t_2,\dots ,t_n\}$, lag $k$ the joint cumulative distribution function of $(Y_{t_1},Y_{t_2},\dots ,Y_{t_n})$ is identical to that of $(Y_{t_1+k},Y_{t_2+k},\dots ,Y_{t_n+k})$

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If $\{Y_t\}$ is strictly stationary and $\mathrm{Var}(Y_t)$ of is finite for all $t$, then $\{Y_t\}$ is also weakly stationary

Some simple time series forecasting methods

Naive method

Future prediction is based on the last observation. $F_{t+1}=X_t$

Averaging method

Forecasting is based on the average of all past observations. $F_{t+1}=\frac{1}{t}{\sum }_{i=1}^t{X}_i$

Smoothing methods

Methods to identify data patterns by smoothing local variation, generally using the average principle.

Formula: Single Moving Average (SMA)

$$\begin{array}{rl}{S}_t& =\frac{1}{m}\sum _{i=t-m+1}^t{X}_i\\ F_{t+1}& =S_t\end{array}$$

Tip

Average of the last $m$ data points. Suitable for stationary data patterns.

Formula: Double Moving Average (DMA)

$$\begin{array}{rl}{S}_{1,t}& =\text{SMA of }X\\ S_{2,t}& =\text{SMA of }{S}_1\\ F_{t+h}& =A_t+h{B}_t\\ A_t& =2S_{1,t}-S_{2,t}\\ B_t& =\frac{2}{m-1}(S_{1,t}-S_{2,t})\end{array}$$

Tip

SMA process performed twice. Suitable for data with trends.

Formula: Single Exponential Smoothing (SES)

$$\begin{array}{rl}{S}_t& =\alpha X_t+(1-\alpha )S_{t-1}\\ F_{t+1}& =S_t\end{array}$$

Assigns exponentially decreasing weights to older data.

Formula: Double Exponential Smoothing (Holt’s Linear Trend)

$$\begin{array}{rl}{S}_t& =\alpha X_t+(1-\alpha )(S_{t-1}+T_{t-1})\\ T_t& =\gamma (S_t-S_{t-1})+(1-\gamma )T_{t-1}\\ F_{t+h}& =S_t+h{T}_t\end{array}$$

Involves two smoothing parameters ($\alpha$ and $\gamma$) for level and trend.

Formula: Holt-Winter Seasonal Method

Smoothing method for data with seasonal patterns with period length $p$.

Additive

$$\begin{array}{rl}{S}_t& =\alpha (X_t-M_{t-p})+(1-\alpha )(S_{t-1}+T_{t-1})\\ T_t& =\gamma (S_t-S_{t-1})+(1-\gamma )T_{t-1}\\ M_t& =\delta (X_t-S_t)+(1-\delta )M_{t-p}\\ F_{t+h}& =S_t+h{T}_t+M_{t-p+h}\end{array}$$

Used when the seasonal component is constant relative to the level.

Multiplicative

$$\begin{array}{rl}{S}_t& =\alpha (X_t/M_{t-p})+(1-\alpha )(S_{t-1}+T_{t-1})\\ T_t& =\gamma (S_t-S_{t-1})+(1-\gamma )T_{t-1}\\ M_t& =\delta (X_t/S_t)+(1-\delta )M_{t-p}\\ F_{t+h}& =(S_t+h{T}_t)M_{t-p+h}\end{array}$$

Used when seasonal variation changes along with the data level.

Procedure: Model Selection (Training vs Testing Data Splitting)

Cheatsheet: Forecasting Model Accuracy Measures (MAD, MSD, MAPE)

Definition: Univariate vs Multivariate Models

Procedure: Box-Jenkins Strategy

Week 2: Konsep Dasar dan Sifat Proses Stokastik

Definition: Lag Definition

Definition: White Noise

Definition: Moving Average Process (Stochastic)

Definition: Random Walk

Cheatsheet: Forecast Accuracy Measures (MAD, MSE, MAPE, MPE)

Definition: Stochastic vs Deterministic Trend

Definition: Linear and Quadratic Deterministic Trends

Procedure: Estimating Constant Mean

Week 3: Metode Regresi dan Model AR(1)

Procedure: Least Squares for Linear Trend Estimation

Procedure: Least Squares for Quadratic Trend Estimation

Definition: Seasonal Average Model

Definition: Sample Autocorrelation (ACF)

Cheatsheet: Correlogram

Definition: General Linear Process

Property: Stationarity Condition for General Linear Process

Definition: Autoregressive (AR) Process Definition

Definition: AR(1) Process Model

Property: Stationarity Condition for AR(1)

Property: ACF of AR(1)

Definition: Backshift Operator (B)

Definition: AR Characteristic Equation

Week 4: Model AR(p), MA(q), dan ARMA(p,q)

Definition: AR(p) Process Model

Property: Stationarity Condition for AR(p)

Property: Yule-Walker Equations for AR(p)

Property: Variance of AR(p)

Definition: Moving Average Process (MA(q))

Definition: MA(1) Process Model

Property: Bounds of MA(1) Autocorrelation

Property: Non-uniqueness of MA(1) Model

Definition: MA(2) Process Model

Property: ACF of MA(q)

Definition: ARMA(p,q) Process Model

Definition: ARMA(1,1) Process Model

Property: Variance and ACF of ARMA(1,1)

Property: Invertibility Condition for MA(1)

Week 5: Model ARIMA dan Non-Stasioneritas

About Rationale for Non-Stationary Models

Example: Explosive AR(1) Process

Procedure: Differencing to Achieve Stationarity

Definition: ARIMA(p,d,q) Model Definition

Definition: ARIMA(p,1,q) Formulation

Property: Characteristic Polynomial of ARIMA(p,1,q)

Week 6: Model IMA, ARI, dan Transformasi Data

Definition: IMA(d,q) Model

Definition: ARI(p,d) Model

Definition: IMA(1,1) Model

Definition: IMA(2,2) Model

Definition: ARI(1,1) Model

Procedure: Determining Weights for ARI(1,1)

Definition: Constant Term in ARIMA Models

Procedure: Log Transformation for Variance Stabilization

Procedure: Percentage Changes Transformation