Three Forcasting Models

Arithmetric mean

$$ \bar{x}t = \frac{1}{t} \bigg( \sum{\tau=1}^{t-1} x_\tau + x_t \bigg) $$

EWMA

$$ \bar{y}t=\alpha y_t+(1-\alpha) \bar{y}{t-1} \text{ , where } 0 \le \alpha \le 1 $$

Simple Exponential Smoothing (SES)

• Non-trending, non-seasonal

• Equations

  • Forecast Equation: $\hat{y}_{t+h|t} = l_t \text{ , } h=1,2,3...$
  • Smoothing Equation: $l_t = \alpha y_t+(1-\alpha)l_{t-1}$

• Codes

  from statsmodels.tsa.holtwinters. import SimpleExpSmoothing
  df.index.freq = 'MS'
  ses = SimpleExpSmoothing(df['Passengers'], initialization_method = 'legacy-heuristic')
  alpha = 0.2
  res = ses.fit(smoothing_level = alpha, optimized = False)
  df['SES'] = res.predict(start = df.index[0], end = df.index[-1])
  np.allclose(df['SES'], res.fittedvalues)  # res.fittedvalues and res.predict(start = df.index[0], end = df.index[-1]) are same
  df.plot()
  
  

  

  N_test = 12
  train = df.iloc[:-N_test]
  test = df.iloc[-N_test:]
  ses = SimpleExpSmoothing(train['Passengers'], initialization_method='legancy-heuristic')
  res = ses.fit()
  
  train_idx = df.index <= train.index[-1]
  test_idx = df.index > train.index[-1]
  df.loc[train_idx, 'SESfitted'] = res.fittedvalues
  df.loc[test_idx, 'SESfitted] = res.forecast(N_test)
  df[['Passengers', 'SESfitted']].plot()
  

  

  res.params
  

  

Holt Model

• Trending, but non-seasonal

• Equations

  • Forecast Equation: $\hat{y}_{t+h|t} = l_t+hb_t$
  • Level Equation: $l_t=\alpha y_t+(1-\alpha)(l_{t-1}+b_{t-1})$
  • Trend Equation: $b_t=\beta(l_t-l_{t-1})+(1-\beta)b_{t-1}$

• Decide $\alpha,\beta$

  $$ Error=\sum_{t=1}^T(y_t-\hat{y}_{t|t-1})^2 \\ \alpha,\beta=\text{arg } \underset{\alpha,\beta}{min} Error $$

• Codes

  from statsmodels.tsa.holtwinters.import Holt
  
  holt = Holt(train['Passengers'], initialization_method='legacy-heuristic')
  res_h = holt.fit()    # no parameters, optimized itself
  df.loc[train_idx, 'Holt'] = res_h.fittedvalues
  df.loc[test_idx, 'Holt'] = res_h.forecast(N_test)
  df[['Passengers', 'Holt']].plot()
  

  

Holt-Winters Model

• Trending & seasonal
• Additive Method v.s. Multiplicative Method
  • Additive: $y=level+trend+seasonal$
  • Multiplicative: $y=(level+trend)\times seasonal$
• Equations (Additive Model)

  • Forecast:

    $$ \hat{y}{t+h|t}=l_t+hb_t+s{t+h-mk} $$

  • Level:

    $$ l_t=\alpha(y_t-s_{t-m})+(1-\alpha)(l_{t-1}+b_{t-1}) $$

  • Trend:

    $$ b_t=\beta(l_t-l_{t-1})+(1-\beta)b_{t-1} $$

  • Seasonality:

    $$ s_t=\gamma(y_t-l_{t-1}-b_{t-1})+(1-\gamma)s_{t-m} $$