ARMAEstimateMissing Class

Estimates missing values in a time series collected with equal spacing. Missing values can be replaced by these estimates prior to fitting a time series using the ARMA class.

For a list of all members of this type, see ARMAEstimateMissing Members.

System.Object
Imsl.Stat.ARMAEstimateMissing

public class ARMAEstimateMissing

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

Traditional time series analysis as described by Box, Jenkins and Reinsel (1994) requires the observations be made at equidistant time points $t_0,t_1,\ldots,t_n$ where . When observations are missing, ARMA requires that they be replaced with suitable estimates. Class ARMAEstimateMissing offers 4 methods for estimating missing values: Median, CubicSpline, AR_1, and AR_p

The centering method Median estimates the missing observations in a gap by the median of the last four time series values before and the first four values after the gap. If not enough values are available before or after the gap then the number is reduced accordingly. This method is very fast and simple, but its use is limited to stationary ergodic series without outliers and level shifts.

Centering method CubicSpline uses a cubic spline interpolation method to estimate missing values. Here the interpolation is again done over the last four time series values before and the first four values after the gap. The missing values are estimated by the resulting interpolant. This method gives smooth transitions across missing values.

Method AR_1 assumes that the time series before the gap can be approximated using an AR(1) process. If the last observation prior to the gap is made at time point then this method uses values at $t_0,t_1,\ldots,t_m$ to compute the one-step-ahead forecast at origin . This value is used to estimate the missing value at time point . If the value at is also missing then the values at time points $t_0,t_1,\ldots,t_m + 1$ are used to recompute the AR(1) model, and then estimate the value at and so on. The coefficient $\phi_1$ in the AR(1) model is computed internally by the method of least squares from class ARMA.

Finally, method AR_p uses an AR(p) model to estimate missing values using a one-step-ahead forecast similar to method AR_1. First, class ARAutoUnivariate, is applied to the time series values just prior to the missing values to determine the optimum p from the set $\{0,1,\ldots,\rm{maxlag}\}$ of possible values and to compute the parameters $\phi_1,\ldots,\phi_p$ of the resulting AR(p) model. The parameters are estimated by the least squares method based on Householder transformations as described in Kitagawa and Akaike (1978). Denoting the mean of the series $y_{t_0}, y_{t_1},\ldots,y_{t_m}$ by $\mu$ the one-step-ahead forecast at origin $t_m,\,\, \hat{y_{t_m}}(1)$ , can be computed by the formula

$\hat{y_{t_m}}(1)=\mu(1 - \sum\nolimits_{j=1}^p\phi_j)+\sum\nolimits_{j=1}^p\phi_j y_{t_m+1-j}\rm{.}$

This value is used as an estimate for the missing value at $t_{m+1}$ . The procedure starting with ARAutoUnivariate is then repeated for every further missing value in the gap. All four estimation methods treat gaps of missing values in increasing time order.

Requirements

Namespace: Imsl.Stat

Assembly: ImslCS (in ImslCS.dll)

ARMAEstimateMissing Class

Thread Safety

Remarks

Requirements

See Also