ARIMA the power spectrum remain constant over the entire

ARIMA (p,d,q) forecasting equation:

Auto-Regressive
Integrated Moving Average models are efficient class of models used for forecasting time
series data which can be transformed to be “stationary” by differencing. A
random variable is said to be stationary if the statistical properties of that
variable remains constant over the whole time series.  A series that is
stationary in nature has a negligible trend and the fluctuations around its
mean have similar constant amplitude and it squirm in a regular patterns
i.e. its short term random patterns of time always look similar. The stationary
series Autocorrelation factor also remain constant over that particular period
of time and the power spectrum remain constant over the entire series. A random
variable composed of Autocorrelation factor and power spectrum series and
contains both signal and noise, the signal pattern could be fast or slow
reversion or a sinusoid oscillation or it could be a rapid alteration in sign
containing a seasonal component. ARIMA model is considered as “filter” that
segregates the signal from noise to gain valuable information for the future
movements.

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ARIMA forecasting
mathematical equation for stationary time series is linear like regression
equation in that the predictors/variables consists of lags of dependent
variables and the lags of forecast errors. The mathematical formulation of
model is,

Forecasted value of Y = constant/weighted sum of single or more
values of Y and/ weighted sum of single or more values of errors.

Non-Seasonal ARIMA model
is defined as ARIMA (p,q,d) where,

p is the number of
autoregressive terms,
q is the number of lagged
forecast errors in the prediction equation
d is the number of non-seasonal
differences needed for stationary, and

The forecasting equation
is constructed as follows.

First, let y denote
the dth difference of Y, which means:

If d=0:  yt =  Yt

If d=1:  yt =  Yt – Yt-1

If d=2:  yt = (Yt – Yt-1) – (Yt-1 – Yt-2)  =  Yt –
2Yt-1 + Yt-2

Note that the second
difference of Y (the d=2 case) is not the difference from two periods
ago.  Rather, it is the first-difference-of-the-first
difference, which is the discrete analog of a second derivative, i.e., the
local acceleration of the series rather than its local trend.

In order to identify the
appropriate ARIMA model for Y, the first step involves the determining of the
differencing order (d) needed to stationary the series and also to remove the
gross features of seasonality in conjunction with a variance-stabilizing
transformation such as logging or deflating the stationary series also may
contains auto correlated errors representing some number of AR terms (p>1)
and some number MA terms (q>1) would also be need to formulate the
forecasting equation.