What Are Non-Profit Models?

A non-parametric model is a mathematical model of a system that does not explicitly include estimable parameters. For example, the frequency response, impulse response, and step response of the system are all non-parametric models.

Compare with the parameters of the known model structure, and then get through theoretical analysis

The general expression of a non-parametric model is

Where Y is the response variable, T is a covariate and is independent of the random error , m (T) = E (Y / T) is an unknown smooth function, and the error satisfies E () = 0, var () = 1 . Standard deviation function (.) Is always positive.

For non-parametric models, there are many estimation methods to choose from, such as kernel estimation method, spline method, Fourier series expansion method and local polynomial method. ^[1]

Non-parametric models of the system are estimated by directly recording or analyzing the system's input and output signals. Nonparametric models are usually expressed as response curves or discrete values. Non-parametric model identification can be performed by directly recording the response of the system output to the input; it can also be analyzed by analyzing the autocorrelation and cross-correlation functions of the input and output (see

The advantage of a parametric model is its flexibility, without making any specific assumptions about the structure of the model. However, non-parametric models have significant drawbacks.

First, the curse of dimensionality is an essential problem that non-parametric estimation cannot escape.

Second, it is difficult to include discrete predictors in nonparametric models.

Third, when the dimension of the predictor is high, it is difficult to draw an image of the estimation function and give a reasonable explanation of the estimation.

As a kind of model between non-parametric and parametric models, semi-parametric models inherit both the flexibility of non-parametric models and the interpretability of parametric models, which can further improve the shortcomings of non-parametric models. ^[3]

With the advent of fast Fourier transform instruments, pseudo-random signal generators, and correlators, non-parametric models of identification systems have become easier. However, the application of non-parametric models to real-time control and adaptive control is not as convenient as parameterized models. Non-parametric models can be transformed into parametric models in some cases. For example, if the transfer function of a system can be expressed as a rational fraction H ( s ) = K / ( a + s ), the model of the system can be expressed by the ordinary differential equation y '+ ay = ku , a and k are to be estimated This is a parameterized model. For another example, for a discrete function weight sequence (discrete impulse response sequence) { hi, i = 0,1,}, if i is sufficiently large (such as i > N 0) and hi is sufficiently small, the model It can be expressed as and an estimate of the sequence of finite weight functions { hi, i = 0,1, ... N 0} can be given by the method of least squares. Generally speaking, it is easy to obtain non-parametric impulse or frequency response from a parametric model, but it is much more difficult to convert a non-parametric model into a parametric model. ^[3]