## Thursday, 25 May 2017

### Residual Plot in Regression, ACF, PACF in ARIMA

What is Constant Mean for a time series- If we draw mean value line on time series, half of the points are above live and half of the points are below line, so we can say that mean is constant over the period of time for this series.

What is the important of residual plot in modelling linear relationshipThe first plot shows a random pattern, indicating a good fit for a linear model. The other plot patterns are non-random (U-shaped and inverted U), suggesting a better fit for a non-linear model.

What is constant variance of a time-series-
1.       The mean E(xt) is the same for all t.
2.       The variance of xt is the same for all t.

In other words, Mean of series x_t and x_t-h is same.
Standard deviation of series x_t is same as standard deviation of series x_t-h.
An interesting property of a stationary series is that theoretically it has the same structure forwards as it does backwards.

what is ACF for a time series-  The correlation between xt to all xt-h is ACF. If there is .3 correlation between xt anf xt-1 then by simple multiplicative rule, there will be .3*.3 correlation between xt and xt-2 and .3*.3*.3 between xt and xt-3.

in the above image correlation with lag 1 is .6, with lag 2 is .36 and so on. Based on above graph we can say that it is AR(1) process where y=.6*yt-1 +constant +error.

What is Moving Average and how is it related to PACF -  MA is time series of past errors(multiplied by some constant). PACF is more difficult to understand. It is a conditional correlation between variables( series).It is correlation between 2 variables conditioning that the correlation is coming from some other variables.

If we have regression y= x1 and x2 so PACF between y and x2 will be-
covariance(y,x2/x1)/sd(y/x1)*sd(x2/x1)

For time series, PACF between yt and yt-2 is given by-
covariance (yt, yt-2/yt-1)/sd(yt,yt-1)* sd(yt-2.yt-1)

ACF is used to identify order of MA and PACF is used to identify order of AR terms in stationary time series.

know more about the relation between time-series and regression-
Regression and time series