Study of Cubic Splines and Fourier Series as Interpolation Techniques for Filling in Short Periods of Missing Building Energy Use and Weather Data

Abstract

To define a single technique that can be reliably used for evaluation of missing data in time series of weather and building energy use data is a complex task. This thesis evaluates the application of spline and Fourier series mathematical techniques for filling in 1-6 hour periods of missing building energy use and weather data. The procedure followed created artificially missing points in measured data sets and was based on the local behavior of the data set around those pseudo-gaps. Seventeen approaches, five related to the spline technique and 12 belonging to the Fourier series, were tested over 20 yearlong samples of energy use and weather data. Two primary factors have been used to determine the preferred interpolation technique and approach to be used for filing data gaps. The principal factor is the reliability of the values yielded. The primary measure of reliability used was the mean bias error (MBE), although the Coefficient of Variation of the Root Mean Square (CV-RMSE) was also evaluated. The other factor is the simplicity of the interpolation technique. Simplicity is of great value when a technique must be applied to a large database where the data are managed and stored. The CV-RMSE is the more stable statistical parameter, produces values very similar for each of the techniques, and is independent of the number of points that are being evaluated. In contrast, the MBE is highly dependent on the number of gaps evaluated and shows a little more stability as the pseudo-gap increases. From the results of the normalized MBE, it is recommended that linear interpolation be used for tilling in missing data in time series of dry bulb and dew point temperature data. For building energy use data sets, the Fourier series approach with 24 data points before and after each gap and six constants was found to be the most suitable. In cases where there are not enough data points for the application of this approach, simple linear interpolation is recommended. Also linear interpolation gives the smallest relative errors of CV-RMSE when used to fill pseudo-gaps for all variables analyzed in this thesis.