 EXAMPLES
FILL UP MISSING TIME SERIES DATA

# Example for the fill-up of missing time series data

Missing time series values within a time series can be filled up with the following methods:

• Linear interpolation. And optionally extrapolation at the beginning and at the end of the time series. Here a limit for the length of a data gap can be defined: Gaps are then filled with linear interpolation only up to this limit.

• Direct transfer of data of (several) other time series. Here, the optimal phase shift between the time series can be determined which yields the maximum correlation coefficient.
This method can also be combined with the linear interpolation:
First the direct transfer and then - if missing still any data - the linear interpolation.
Or first the linear interpolation and then - if missing still any data - the direct transfer.

• Linear regression with (several) reference time series. This method provides several options:

• Determination of the optimal phase shift between the time series which yields the maximum correlation coefficient.
• The fill-up with several reference time series is conducted in the order of their correlation coefficients: At first the regression with the time series with the greatest correlation. If this regression does not yield a value for fill-up then the regression with the time series with the second greatest correlation, etc., and finally the regression with the time series with the smallest correlation.
• 1-parameter regression (through the co-ordinate origin) or
2-parameter regression (with axis intercept).
• Regression differentiated according to consecutive months,
or regression differentiated according to seasonal months,
and/or regression differentiated according to wind direction sectors.

This method can also be combined with the linear interpolation:
First the linear regression and then - if missing still any data - the linear interpolation.
Or first the linear interpolation and then - if missing still any data - the linear regression.

An example for the linear regression: Time series internal

A time series with wind speeds and wind directions at 31 height levels from 50 m to 200 m above ground level, respectively, shall be filled up. This is marked with "original" in the following.
As the reference time series the time series itself is used. So, each of the height levels is filled up with regressions with the respective other 30 height levels.
For the fill-up the following options have been chosen:

• Determination of the optimal phase shift between the time series.
• 2-parameter regression (with axis intercept).
• Regression differentiated according to consecutive months
and regression differentiated according to 12 wind direction sectors.

The linear regression for wind directions is performed by WindDataSuite with a multiple multivariate linear regression.
The resulting filled up data are marked with "filled" in the following.

Fig.1 shows an excerpt of the time series for the wind speeds of the original data and of the resulting filled up data at the height of 140 m, respectively.
Fig.2 shows the same excerpt of the time series for the wind directions of the original data and of the resulting filled up data at the height of 140 m, respectively. Fig.1:
Wind speed time series at 140 m height Fig.2:
Wind direction time series at 140 m height

An example for the linear regression: With reference data

A time series with wind speeds and wind directions at the height of 120 m above ground level, respectively, shall be filled up. This is marked with "original" in the following.
As the reference time series the reanalysis data of the nearest located Merra-2 grid point is used. These are marked with "Ref" in the following. They contain wind speeds and wind directions at the heights of 10 m and 50 m above ground level.
For the fill-up the following options have been chosen:

• Determination of the optimal phase shift between the time series.
• 2-parameter regression (with axis intercept).
• Regression differentiated according to consecutive months
and regression differentiated according to 12 wind direction sectors.

The reference data at 10 m height above ground level have yielded the best correlation.
The linear regression for wind directions is performed by WindDataSuite with a multiple multivariate linear regression.
The resulting filled up data are marked with "filled" in the following.

Fig.3 shows an excerpt of the time series for the wind speeds of the original data, of the reference data, and of the resulting filled up data.
Fig.4 shows the same excerpt of the time series for the wind directions of the original data, of the reference data, and of the resulting filled up data. Fig.3:
Time series of the wind speeds Fig.4:
Time series of the wind directions