Time Series
A time series $$Y_t$$ $$(t=1,2, \ldots, n)$$ is a set of $$n$$ observations recorded through time $$t$$, (eg. days, weeks, months). The arithmetic mean of blocks of $$k$$ successive values \[\frac{Y_1+Y_{2}\ldots +Y_k}{k}, ~\frac{Y_2+Y_3\ldots +Y_{k+1}}{k},\ldots\] is a simple moving average of order $$k$$, itself a time series which is smoother than $$Y_t$$ and can be used to track, or estimate, the underlying level, $$\mu_t$$, of $$Y_t$$.
If $$0<\alpha<1$$ an exponentially weighted moving average (EWMA) at time $$t$$ uses a discounted weighted average of current and past data to estimate $$\mu_t$$ with \[\hat{\mu}_t =\alpha Y_t+ \alpha(1-\alpha)Y_{t-1}+ \alpha(1-\alpha)^2Y_{t-2}+\ldots .\] This is equivalent to the recurrence relation \[\hat{\mu}_t =\alpha Y_t+ (1-\alpha)\hat{\mu}_{t-1}.\] Moving averages are often plotted on the same graph as $$Y_t$$.
If $$Y_t$$ additionally contains trend, $$R_t$$, the rate of change of data per unit time, and $$\mu_t= \mu_{t-1}+R_{t-1}$$, then the recurrence relation is \[\hat{\mu}_t =\alpha Y_t+ (1-\alpha)\left(\hat{\mu}_{t-1}+\hat{R}_{t-1}\right).\] If $$0< \beta <1$$ the trend smoothing equation is \[\hat{R}_{t}=\beta\left(\hat{\mu}_t -\hat{\mu}_{t-1}\right)+(1-\beta)\hat{R}_{t-1}\] known as Holt's Linear Exponential Smoothing.
If $$Y_t$$ also contain seasonality, $$S_t$$, a smoothing constant $$\gamma$$, $$(0< \gamma <1)$$ is used in a seasonal smoothing equation, $$\hat{S}_{t}= \gamma Y_t/\hat{\mu}_{t} + (1-\gamma)\hat{S}_{t-k}$$, assuming the periodicity is $$k$$, with multiplicative seasonality. For monthly data $$k=12$$.
Forecasting from time $$n$$ (now) to time $$n+h$$ $$(h=1,2,\ldots)$$
Level only, $$\hat{Y}_{n+h}=\hat{\mu}_{n}$$ , the latest EWMA.
Level and constant trend, $$\hat{Y}_{n+h}= a + b(n+h)$$, the simple linear regression trend line of $$Y_t$$ against $$t$$.
Level and changing trend, $$\hat{Y}_{n+h}=\hat{\mu}_{n}+h\hat{R}_{n}.$$
Level, changing trend and seasonality
$$\hat{Y}_{n+h}= \left(\hat{\mu}_{n}+h\hat{R}_{n}\right)S_{n+h-k}$$, $$h=1, 2, \ldots, k$$;
$$\hat{Y}_{n+h}= \left(\hat{\mu}_{n}+h\hat{R}_{n}\right)S_{n+h-2k}$$, $$h=k+1, k+2, \ldots, 2k$$;
and so on, where $$\hat{\mu}_{n}= \alpha Y_n/\hat{S}_{t-k} +(1-\alpha)(\hat{\mu}_{n-1} + \hat{R}_{n-1})$$.