first of all it can not be guaranteed that the parameter estimates will converge to the true values for non-linear models. Furthermore, most often the non-linear estimation equations have no closed form solutions, therefore the estimates have to be found by numerical optimization.
Non-linear models have therefore only been considered briefly, an exam-ple is demonstrated in PaperE. As the conditionally parametric models outperformed the non-linear model, this approach was not pursued fur-ther. Instead of using non-linear models conditionally parametric models estimated via local regression have been considered. These models and a recursive estimation technique is described in the Papers A and B.
In this model class the non-linearity is not explicitly parameterized, the shape of a specific relation is determined via the estimation method.
3.4 Bibliographics notes
A general introduction to time series analysis and estimation of time series models can be found in (Box & Jenkins 1976). An introduction to non-linear time series models, and a description of the Kalman filter approach applied for the non-linear model in Paper E, can be found in (Madsen & Holst 1999).
The conditional parametric model defined in Section 3.1 is similar to the varying-coefficient model defined in (Hastie & Tibshirani 1993). The varying-coefficient model defined in (Hastie & Tibshirani 1993) considers coefficients that are functions of time only, while in (3.3) the coefficients may be functions of several variables. In (Anderson, Fang & Olkin 1994) (3.3) is denoted a conditional parametric model, because when ui is constant the model reduces to an ordinary linear model as in (3.2).
The models and estimation methods described in the Papers A and B are based on a combination of the recursive methods described in (Ljung
& S¨oderstr¨om 1983) and local regression. Early work on local regression includes (Stone 1977, Cleveland 1979, Cleveland 1981), although, as de-scribed by (Cleveland & Loader 1996), it dates back to the 19’th century.
A comprehensive overview of local regression can be found in (Cleveland
& Devlin 1988, Cleveland, Devlin & Grosse 1988).
In the estimations methods used in this thesis, selection of smoothing pa-rameters has not been a real issue. The recursive nature of the methods implies that the data used to calculate the prediction errors is never used in the estimation. In traditional local regression cross-validation and re-lated methods have traditionally been used for selection of smoothing pa-rameters (Hastie & Tibshirani 1990). Also leave-one-out cross-validation is used, although (Breiman & Spector 1992, Shao 1993) argue that this is not optimal. Specifically for time series (Hart 1996) considers the subject of selection of smoothing parameters.
Chapter
4
Models and methods
The objective of this thesis has been to develop models for short-term prediction of wind power, with special emphasis on the integration of statistical and physical models and methods. Before considering such combined models, it should first be considered what statistics and physics is all about.
4.1 Initial considerations
Basically, a physical model and a statistical model are similar, in the way that both models describe a relation between some input and some output to and from a system.
The physical approach is to consider the nature of the system, and based on which laws can be determined to govern the system, a relation is derived. If the derivation was sound, the relation can be verified on measurements of the input, the system states and the output from the system.
The statistical approach works the other way around. It starts with the measurements, and based on what relations can be seen in the measure-ments, a model is derived.
In reality, the distinction is not as separate as stated above. Both physi-cists and statisticians would argue, that their approaches consists of an iterative use of both the physical and statistical approach as stated above.
If the statistician has knowledge about which laws govern the system, then he knows what to look for in the data. Furthermore, in practice there is rarely an sufficient amount of data available as to describe the system response to all combinations of input data.
In the physical approach there has to be observations at some point, oth-erwise the physical model would just be guesswork. Also, the verification process of a physical model typically leads to identification of week points in the model derivation, and as a consequence of this, a modified model is derived.
Another important issue with regard to the physical, or deductive, ap-proach is pointed out by (Hasselmann 1981). As described in Chapter2, the meteorological system represents a very complex structure of coupled subprocesses of which each subprocess represents a detailed discipline in its own right. These subprocesses interact across a wide spectra of space and time scales in a complicated manner, which ultimately determines the dynamics of the complete system. If detailed models of the dynam-ics of each subprocess existed, and the necessary computing power was available, one could try to obtain a total model by coupling the individ-ual subprocesses together in a very comprehensive numerical model. It is, however, questionable whether such a deductive approach would be successful (Hasselmann 1981)
It is argued (Hasselmann 1981) that a larger probability for finding a useful model is obtained by the opposite approach, i.e. the inductive approach. This approach contains an attempt to identify the governing interactions in the system by statistical methods. Once such a model is obtained for the most important variations, the model can be iteratively improved by a combination of more detailed comparison with data and use of well-established physical facts.
4.2 What are the options? 39