# 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