Transcendental regression

For statistical samples of small volume, an increase in the order of the polynomial regression of variables can produce a serious increase in the residual variance. We can reduce the number of the coefficients from the model but then we must introduce a transcendental regression relationship for the variables of the process. From the general theory of statistical process modelling (relations (5.1)-(5.9)) we can claim that the use of these types of relationships between dependent and independent process variables is possible. However, when using these relationships between the variables of the process, it is important to obtain an excellent ensemble of statistical data (i.e. with small residual and relative variances). 

It is well known that using an exponential or power function can also describe the portion of a polynomial curve. Indeed, these types of functions, which can represent the relationships between the process variables, accept to be developed into a Taylor expansion. This procedure can also be applied to the example of the statistical process modelling given by the general relation (5.3) [5.20]. 

---

Coefficients Pq, P/ can easily be obtained by using the method of least squares. Nevertheless, the interest is to have the original coefficients of the transcendental regression. To do so, we apply an inverse operator transformation to Po and Pj. Here, we can note that Pq and P/ are the bypassed estimations for their correspondents Po and Pi. 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series from which the trend is to be identified is assumed to be generated by a nonstationary process where the nonstationarity results from a deterministic trend. A classical model is the regression or error model (Anderson, 1971) where the observed series is treated as the sum of a systematic part or trend and a random part or irregular. This model can be written as... 

If the function f(x) is not linear in x, then nonlinear regression is followed The transcendental equation is recast in a linear form by suitable (if often approximate) transformations (e.g., computing its logarithm), and one proceeds as in the linear case, with a suitable back-transformation at the end. 

---