System (1), containing a single linear differential equation and two linear algebraic equations, allows the selection of one ordinary differential equation (ODE) with respect to the variable pressure p (t):

In the model of Frank's blood is the source of input f₀(t) given as a periodic function of time. Rectangular pulses described by the function:

in formula (1) the brackets denote the function "integer part", 2 nd formula determines the value of U in the stroke volume of blood. Some authors represent systolic flow pulses parabolic shape, then instead of (3) have a flow:

Let the initial time p(0) ≡ p₀ known end-diastolic pressure (ESP), and p(T₁) ≡ p₁ unknown end systolic pressure (ESP). Then, solving the ODE (2) for pulse flows (3) and (4), we get a couple of functions that describe the change in pressure in the adjacent phases of systole and diastole,

where Exp(t) 1 - exp(t), and the function F (t) - contribution of the systolic pulse, which for a rectangular pulse and a parabolic form is defined, respectively, as follows:

Piecewise Exponential description of the two phases of relaxation oscillations of pressure provided by a pair of functions (5a) and (5b), belong to the same cycle. In order to obtain high cycle description of these fluctuations, suitable for algorithmic (computer) charting, you should use the above-introduced "sawtooth" function of time θ(t), augmented by two logical implication:

From the linear relation of the output stream with the pressure, see the 3rd equation (1), should be similar to a functional description of the relaxation oscillations of the flow:

Heref₀≡ (p₀- P)/ r final diastolic flow (FDF), f₁≡ (p₀- P)/ r final systolic flow (FSF).

   3.1.2 Fluctuations of pressure and flow

Stroke volume of the parabolic pulse, according to (4), U = 80 ml - this value is typical for hemodynamic norms of a healthy person. For rectangular pulses provide the same stroke volume, their amplitude, see (3) and (4), should be reduced in half. Using the solutions (5) and (6), as well as the parameters (7), we can construct a recursive way graphics-cycle oscillations of pressure and flow for the two forms of the systolic pulse (Fig. 1). The above graph shows that with the additional condition of equality of stroke volume, output fluctuations in the flow for the two forms of input pulses practically coincide. In view of the linear relation, are almost identical and graphs of pressure oscillations on the right graph. The theoretical evaluation of relations CDS / CRT≡ p_es / p_ed = 128/82 gives a value p_ed = 82 mmHg, which almost coincides with the initial pressure p₀ = 80 mmHg.

3.2 Methods for Time Series Prediction

   3.2.1 Average and sliding average

The simplest model based on a simple averaging is:

In the above formula assumes that the series is averaged over a sufficiently long interval of time. However, as a rule, the values of the time series from the recent past better describe the prognosis than the older values of the same series. Then you can use to predict the moving average

When predicting often used method of exponential average, which is constantly adapting to the data from the new values. The formula describing this model is written as:

This method is an internal parameter a, which determines the dependence of the forecast from the older data, and the impact of data on the forecast decreases exponentially with the "age" of data.

If the forecasts using the model of exponential smoothing, usually on some test set predictions are constructed with a = [0.01, 0.02, ..., 0.98, 0.99] and tracked under which a prediction accuracy of the above. This value a is then used to predict the future.

Although the model described above (methods based on the average, moving average and exponential smoothing) is used to predict in the not very difficult situations is not recommended to use these methods in forecasting problems in mind the apparent primitiveness and the inadequacy of the models.

Along with this I would like to note that the algorithms described quite successfully be used as collateral and support for pre-processing of data in forecasting problems. For example, to predict in most cases it is necessary to decompose the time series (ie, allocate a separate trend, seasonal and irregular components). One of the methods of allocating trend components is the use of exponential smoothing.

   3.2.2 Methods Holt and Brown

In the middle of last century, Holt proposed an improved method of exponential smoothing, later named after him. In the proposed algorithm, the values of the level and trend are smoothed using exponential smoothing. And smoothing parameters may differ.

Standing in the smoothing method Holt ideologically play the same role as the constant in the simple exponential smoothing. They are selected, for example, by sorting on these parameters with a certain step. You can use less sophisticated in terms of computational algorithms. The main thing that you can always pick up a pair of parameters, which gives greater accuracy of the model on the test set and then use this pair of parameters in real forecasting.
A special case of the method of Holt is the method of Brown, when a = β.

   3.2.3 Method Winters

Although the above-described method of Holt (method of two-parameter exponential smoothing) and is not very simple (with respect to models based on averaging), it does not allow to take into account seasonal variations in forecasting. Speaking more strictly, this method can not make them "see" in the background. There is increased to a three-parametric method is Holt exponential smoothing. This algorithm is called the method of Winters. It attempts to take into account the seasonal components in the data. The system of equations describing the method of Winters as follows:

Fraction in the first equation serves to eliminate the seasonality of Y (t). After excluding seasonal algorithm works with a "clean" data, in which there is no seasonal variations. They appear already in the final prediction, when "pure" forecast, counted almost by the method of Holt is multiplied by a seasonal factor.

   3.2.4 Regression methods for forecasting

Along with the above described methods based on exponential smoothing, is long enough to predict the regression algorithms are used. Briefly the essence of this class of algorithms can be described as follows. There is a predicted variable Y (dependent variable) and pre-selected set of variables on which it depends - X1, X2, ..., XN (independent variables).

Model of multiple regression in the general case described by

In a simpler version of the linear regression model, the dependence of the dependent variable from independent has the form:

Here β₀, β₁, β₂, β_N matched by the regression coefficients, εcomponent errors. It is assumed that all errors are independent and normally distributed.

To construct the regression models should have a database of observations that looks like this:

Using the table values of past observations, we can choose (for example, the method of least squares) regression coefficients, thereby setting the model.

When working with the regression must comply with some caution and be sure to check on the adequacy of the model found. There are different ways to the test. The essential one is the statistical analysis of residuals, Durbin-Watson test. It is useful, as is the case with neural networks, have an independent set of examples, where you can check the quality of the model.

   3.2.5 Methods for Box-Jenkins (ARIMA)

In the mid 90-ies of the last century has been developed fundamentally new and fairly powerful class of algorithms for predicting time series. Most of the work on the study methodology and validation of models was carried out by two statisticians, G.E.P. Boxing (G. E. P. Box) and GM Jenkins (G. M. Jenkins). Since the construction of similar models and obtaining on their basis of forecasts is sometimes called Box-Jenkins methods. For more information hierarchy of algorithms for the Box-Jenkins, we consider below, in the meantime, we note that in this family includes several algorithms, the most known and used of them is the algorithm ARIMA. It is built in almost any special package for forecasting. In the classic form ARIMA not used independent variables. The models are based only on information contained in the prehistory of the projected series, limiting the ability of the algorithm. Currently, the scientific literature often refers to variants of models ARIMA, to take into account the independent variables. Unlike previously discussed methods of forecasting time series in ARIMA methodology is not anticipated any clear model for the prediction of the time series. Set a general class of models describing the time series and allow somehow express the current value of the variable in terms of its previous value. Then the algorithm, adjusting the internal parameters, selects the most appropriate forecasting model. As noted above, there is a whole hierarchy of models of Box-Jenkins. Logically, it can be defined as
AR(p)+MA(q)->ARMA(p,q)->ARMA(p,q)(P,Q)->ARIMA(p,q,r)(P,Q,R)->... AR (p) autoregressive model of order p.
The model has the form
Y(t)=f_0+f_1*Y(t-1)+f_2*Y(t-2)+...+f_p*Y(t-p)+E(t)
Where: Y (t)-dependent variable at time t. f_0, f_1, f_2, ..., f_p - estimated parameters. E (t) - a mistake from the influence of variables that are not included in this model. The challenge is to determine the f_0, f_1, f_2, ..., f_p. They can evaluate a variety of ways. Correct to search for them via the Yule-Walker equations, for the preparation of this system would require calculation of the autocorrelation function. You can do a simpler way - to calculate their method of least squares.

MA (q)-model with a moving average of order q.
The model has the form:
Y(t)=m+e(t)-w_1*e(t-1)-w_2*e(t-2)-...-w_p*e(t-p)
Where Y (t) the dependent variable at time t. w_0, w_1, w_2, ..., w_p estimated parameters.

    3.2.6 Neural network prediction model

Currently, in my opinion, the most promising quantitative method of forecasting is the use of neural networks. We can mention many advantages over other neural network algorithms, here are two basic.

Using neural networks can easily explore the dependence of the predicted value of the independent variables.

Another major advantage of neural networks is that the expert is not held hostage to the choice of mathematical models of behavior of time series. Construction of neural network model is adaptively during training, without the participation of an expert. In this neural network are presented examples from the database and she adapts to the data.

Disadvantage of neural networks is their non-determinism. Refers to the fact that after learning there is a "black box" that somehow works, but the logic of decision-making neural network completely hidden from the expert. In principle, there are algorithms extract knowledge from neural networks ", which formalize the training of the neural network to the list of logical rules, thus creating a network of expert system. Unfortunately, these algorithms are not embedded in the neural network packages to the same set of rules that are generated by such algorithms can be voluminous.

Nevertheless, for people who know how to work with neural networks and knowing the nuances of settings, training and application in practical problems, the opacity of neural networks is not any serious drawback.

The simplest version of the application of artificial neural networks in forecasting problems - the use of conventional perceptron with one, two, or (at most) three hidden layers. In this case the inputs of the neural network is usually served a set of parameters, on the basis of which (the expert's opinion) can be successfully predicted. The solution is usually the weather network in the next time.

Another frequently used neural network architecture used in the prediction of a neural network with general regression. Despite the fact that the principle of learning and application of such networks is fundamentally different from conventional perceptrons external network is used in the same manner as a conventional perceptron. In other words, it is compatible architecture in the sense that a running system, forecasting can be replaced by working on the perceptron network with the overall regression and everything will work. No need to carry out any further manipulation of the data.