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Getting it down on `paper`

Notes for Time Series Forecasting

Holt-Winters forecasting technique

F[t] = alpha * (x[t]/S[t-K]) + (1-alpha)*(F[t-1]+T[t-1])
T[t] = beta * (F[t]-F[t-1]) + (1-beta)*T[t-1]
S[t] = gamma * (x[t]/F[t]) + (1-gamma)*S[t-K]
X^[t] = (F[t-1]+T[t-1])*S[t-K]

F[t] := smoothing estimate
T[t] := trend estimate
S[t] := seasonal estimate
K := seasonal period
alpha, beta, gamma := model parameters (trial and error)
If no seasonality, gamma = 0, S[t-K] = 1

————–

Auto-regressive integrated moving average (ARIMA)
seasonal := SARIMA

X^[t] = mu + sum|i=1..Oa(A[i]*x[t-i]) + sum|j=1..Om(M[j]*e[t-j])

Oa := AR order
Om := MA order
Aj := AR coeff*
Am := MA coeff*
mu := constant*

* estimated using OLS

————–

NN with lagged inputs and direct output weights.
Avoids need for rescaling.

X^[t] = W[out,0] + sum|i=1..I(X[t-k[i]]*W[out,i]) +
sum|j=I+1..Out-1(f(sum|i=1..I(X[t-k[i]]*w[j,i] + w[j,0]))*W[out,i]

W[ij] := weight of connection from node j to i.
Out := output node
f := sigmoid function
I := number of input neurons

 

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