Mean error and root mean square error. The lower the values of these criteria, the greater the reliability of the predictive model.
linear correlation coefficient is determined by the formula
Root mean square error (standard deviation) for the S score and prediction confidence interval
In fact, the problem is reduced to estimating the average elasticity over a more or less long period of time. Let us analyze estimates of the elasticity of specific prices (joint elasticity) of different levels, i.e. species structure, for elevator grain, grain on the exchange and for flour. The obtained estimates are summarized in Table. 14.5 along with their standard mean square errors - estimation errors, or limits of confidence intervals of indicators of elasticity.
To check the significance of the correlation coefficients, we calculate the root-mean-square errors of the correlation coefficients r
The degree of closeness of the multiple statistical relationship and the standard error of the forecast (approximation) of one variable in the aggregate of others. Intuitively and from the meaning of the above characteristics of the degree of tightness of the statistical relationship, it is clear that the closer this relationship, the more information one variable contains relative to another, the more accurately it is possible to restore (predict, approximate) the unknown value of one variable from a given value of another.
Thus we again (as in B.5 and 1.1.1) arrive at the regression function f(X) = E(m] = X), this time as a function of the p variables (1, c (2), ..., x(p) most accurately (in the sense of root-mean-square error) reproducing the conditional value of the resulting indicator m] (X) under study for a given value of X explanatory variables.
The root-mean-square error of the combined forecast is respectively equal to
If the term standard deviation is used to describe the spread of a variable, then the term standard error is used to describe such a statistical parameter.
It is well known that the algorithm called the R. Kalman filter is optimal in terms of the minimum mean square error in estimating the state (current, past and future) of a dynamic system. All other accuracy estimation algorithms can only approach the estimation accuracy provided by the Kalman filter. Potentially possible estimation accuracy achieved by the specified filter is ensured due to the fact that the structure and parameters of the specified algorithm are preliminarily adjusted to the statistical portrait of the estimated dynamic system . That is why it is necessary to conduct preliminary statistical studies of the financial market in order to obtain a mathematical model adequate to the market in the form of a system of differential (difference) equations, and only then adjust the corresponding Kalman filter to the resulting mathematical model of the financial market.
Thus, the use of formulas (1.13)-(1.16) leads to a contradiction in determining the smoothing parameter with a decrease in a, the root-mean-square error decreases, but the error in the initial conditions increases, which in turn affects the accuracy of the forecast.
This fact makes it possible to use relations (1.81) to construct the predictive values of the analyzed time series for 1 clock cycle ahead. The theoretical basis of this approach to forecasting is provided by the well-known result, according to which the best (in the sense of root-mean-square error) linear forecast at time t with a lead of 1 is the conditional mathematical expectation of the random variable xt + i, calculated under the condition that all values of xm up to the moment time t. This result is a special case of the general theory of forecasting (see ).
With any division of a complete polynomial of a given degree into partial polynomials, the criterion for the minimum of the mean square error determined on the training sequence (the first criterion) makes it possible to uniquely determine the optimal estimates of all coefficients if the number of points in the training sequence is greater than the number of members of each of the partial polynomials by at least one .
For a given degree of a complete polynomial, there are many options for splitting it into partial polynomials. A complete search of all combinations by the criterion of the standard error, measured on a separate test sequence of data, allows you to find the only best separation.
Therefore, just as in the case of pairwise dependence, the variation (random scatter) of the resulting indicator t] consists of the variation of the regression function / (X) that we control (according to the value of the predictor variable X) and from the random scatter of values r (X ) (for a fixed X) with respect to the regression function / (X). It is this uncontrolled scatter (characterized by the value of o (X)) that simultaneously determines both the root-mean-square error of the forecast (or approximation) of the value of the resulting indicator r based on the values of the predictor variables X, and the degree of closeness of the relationship existing between the value of r, on the one hand, and values
X. Theil proposed in this case to use the standard error
This correlation does little to reduce uncertainty. Indeed, the standard error of the forecast is reduced by only 1%. Thus, although some weak signs of autocorrelation in the NASDAQ index have been found, they are of little use in practice. All other correlations are random and not statistically significant. Considering how many correlations we have analyzed to find only one more or less statistically significant, it can be argued with a high degree of probability that this single correlation is most likely a random result, similar to getting several heads in a row when a coin is tossed.
To assess the accuracy of any measurements means to determine, on the basis of the results obtained, comparable numerical (quantitative) characteristics that express the qualitative side of the measurements themselves and the conditions for their implementation. Quantitative characteristics of measurements or criteria for assessing the accuracy of measurements are established by the theory of probability and the theory of errors (in particular, by the method of least squares). According to these theories, the accuracy of measurement results is estimated only by random errors.
Measurement accuracy indicators can be:
Mean square measurement error;
Relative measurement error;
Limit measurement error.
The concept of root-mean-square error was introduced by Gauss, and it is currently accepted as the main characteristic of measurement accuracy in geodesy.
The root mean square error is the mean square value of the sum of the squared errors of the individual measurements. To calculate it, either true measurement errors or deviations of the measurement results from the arithmetic mean are used.
Let us denote the true value of the measured value through X, the measurement result through l i .
True measurement errors Δ i are called the differences between the measurement results and the true values, i.e.
In this case, the mean square error m of a single result is calculated by the formula:
where n is the number of equally accurate measurements.
However, in most cases of practice, except for rare cases of special studies, the true value of the measured quantity and, therefore, the true errors remain unknown. In these cases, to find the final value of the measured quantity and evaluate the accuracy of the measurement results, the principle of the arithmetic mean is used.
Let l 1 , l 2 , .... l n results n equal measurements of the same quantity. Then the quotient
is called the arithmetic mean of the measured values of this quantity.
The difference between each individual measurement result and the arithmetic mean is called the deviation of the measurement results from the arithmetic mean and is denoted by the letter v:
v i = l i - .
Example. A single angle is measured in four steps and the results are:
l 1= 74° 17"42"; l 2= 74° 17"46"; l 3= 74° 17"43"; l 4= 74° 17"47".
Then the arithmetic mean value of the angle will be = 74° 17 "44", 5, and the deviations of the measurement results from the arithmetic mean, respectively, will be v1= - 2",5; v2= +1",5; v 3= - 1",5 and v 4= +2",5.
Deviations of measurement results from the arithmetic mean have two important properties:
For any series of equally accurate measurements, the algebraic sum of deviations is equal to zero [ v] = 0;
For any series of equally accurate measurements, the sum of squared deviations is minimal, that is, less than the sum of squared deviations of individual measurements from any other value taken instead of the arithmetic mean, [ v2] = min.
The first property of deviations serves as a reliable control for calculating the arithmetic mean from the measurement results. The second property of deviations is used to evaluate the accuracy of the measurement results.
If the errors of individual measurements are calculated relative to the arithmetic mean of the measurement results, the standard error of the individual result is calculated by the formula
Example. Using the data from the previous example, we find the root mean square error of the angle measurement in one step:
When determining the mean square errors of measurements, it is necessary to follow the following rules:
1) the mean square error of the sum or difference of the measured values is equal to the square root of the sum of the squared mean square errors of the terms, i.e. for the expression A \u003d a + b - c + ... + q, the mean square error will be equal to
with equally accurate measurements, when m a = m b = m c = ... = m q:
2) the mean square error of the product of the measured value by a constant number is equal to the product of the mean square error of this value by the same number, i.e. for the expression L = kl;
3) the mean square error of the results of equal measurements is directly proportional to the mean square error of one measurement m and inversely proportional to the square root of the number of measurements, i.e.
or taking into account formula (12):
Examples: 1. The angle β is obtained as the difference between two directions, determined with errors m 1 = ± 3" and m 2 = ± 4".
By the first rule, we find .
2. The radius of the circle is measured with the root mean square error m R = ±5 cm.
According to the second rule, we find the root mean square error of the circumference
m 0 \u003d 2πm R \u003d 2 × 3.14 × 5 \u003d ± 31 cm.
3. The root-mean-square error of measuring the angle with one step is equal to m = ± 8 ". What is the accuracy of measuring the angle with four steps?
According to the third rule
.
4. The angle β was measured in five steps. At the same time, the deviations from the arithmetic mean were: - 2", + 3", - 4", +4" and -1". What is the accuracy of the final result?
According to the third rule
If the master action applied to the linear system (Fig. 7.2) is a random stationary function, then the controlled value and the system reproduction error are also random stationary functions. It is clear that, under these conditions, the accuracy of the system can be judged not by instantaneous, but only by some average values of the error. With the statistical method of analysis and synthesis, the dynamic accuracy of the system is determined by the root-mean-square value of its error, i.e., the square root of the mean value of the squared error:
Rice. 7.2. Block diagram of the ACS.
Rice. 7.3. On the concept of root-mean-square error.
which is used as a criterion that determines the accuracy or quality of the system in the presence of stationary random influences (the relationship between and is illustrated in Fig. 7.3).
If the correlation function or the spectral density of the error is known, then, in accordance with expression (7.11), the error variance can be calculated by the formula
The optimal transfer function when using the RMS criterion is such a transfer function of the system for which the root mean square error has a minimum.
Let us note the advantages and disadvantages of estimating the accuracy of the system using RMS. When the standard deviation is taken as a criterion for accuracy, the analysis and synthesis of the system turns out to be relatively simple. With the help of standard deviation (or variance) it is possible to estimate from above the probability of occurrence of any error. So, for example, with a normal distribution of errors, the probability that the error (deviation from the average value) will exceed is very small (less than 0.003). According to the RMS criterion, the undesirability of an error increases with its magnitude.
There is a large class of systems for which the RMS criterion is effective. However, the RMS criterion, like any other criterion, is not universal. It provides a small value of only the average, and not the instantaneous error, therefore, in those systems where large, albeit short-term errors are unacceptable, it is desirable to use another criterion. This shortcoming of the standard deviation criterion is especially evident in the calculation of ACS with feedback. The expressions for the correlation function, spectral density, and root-mean-square error are valid only for long time intervals. Therefore, system errors associated with relatively short-term transient processes in it have practically no effect on the root-mean-square value of the error, i.e., the error averaged over an infinitely long period of time. In practice, there are often systems that operate in a limited period of time, when the errors associated with the transient process cannot be neglected. As a rule, if the system parameters are chosen from the condition of obtaining a minimum RMS when operating over a long period of time, then the closed system has a weakly damped transient. Therefore, in practice, the problem of the rational choice of the transfer function of the system
Arithmetic mean value of a series of measurements
As n increases, the average value
The Gauss formula can be derived from the following assumptions:
- measurement errors can take a continuous series of values;
- with a large number of observations, errors of the same magnitude but different signs occur equally often;
- the probability, i.e. the relative frequency of occurrence of errors, decreases as the magnitude of the error increases. In other words, large errors are less common than small ones.
The normal distribution law is described by the following function:
where σ is the root mean square error; σ2 is the measurement variance; X ist - the true value of the measured value.
Analysis of formula (1.13) shows that the normal distribution function is symmetric with respect to the straight line X = X true and has a maximum at X = X true. We find the value of the ordinate of this maximum by putting in the right side of equation (1.13) X ist instead of X. We obtain
,
whence it follows that as σ decreases, y(X) increases. Area under the curve
must remain constant and equal to 1, since the probability that the measured value of X will be in the range from -∞ to +∞ is equal to 1 (this property is called the probability normalization condition).
On fig. 1.1 shows graphs of three normal distribution functions for three values of σ (σ 3 > σ 2 > σ 1) and one X ist. The normal distribution is characterized by two parameters: the mean value of a random variable, which, for an infinitely large number of measurements (n → ∞), coincides with its true value, and the variance σ. The value σ characterizes the spread of errors relative to the average value taken as true. At small values of σ, the curves go more steeply and large values of ΔХ are less likely, that is, the deviation of the measurement results from the true value of the value is smaller in this case.
There are several ways to estimate the magnitude of a random measurement error. The most common estimation is by means of the standard or root mean square error. Sometimes the mean arithmetic error is used.
The standard error (root mean square) of the mean over a series of n measurements is given by:
If the number of observations is very large, then the quantity Sn subject to random random fluctuations tends to some constant value σ, which is called the statistical limit Sn:
It is this limit that is called the root mean square error. As noted above, the square of this quantity is called the measurement variance, which is included in the Gauss formula (1.13).
The value of σ is of great practical importance. Let, as a result of measurements of a certain physical quantity, we find the arithmetic mean<Х>and some error ΔX. If the measured quantity is subject to random error, then it cannot be unconditionally assumed that the true value of the measured quantity lies in the interval (<Х>– ΔX,<Х>+ ΔХ) or (<Х>– ΔX)< Х < (<Х>+ ΔХ)). There is always some probability that the true value lies outside this interval.
The confidence interval is the range of values (<Х>– ΔX,<Х>+ ΔХ) of the value X, in which, by definition, its true value X sr falls with a given probability.
The reliability of the result of a series of measurements is the probability that the true value of the measured quantity falls within a given confidence interval. The reliability of the measurement result or the confidence level is expressed as a fraction of a unit or a percentage.
Let α denote the probability that the measurement result differs from the true value by an amount not greater than ΔX. This is usually written as:
R((<Х>– ΔX)< Х < (<Х>+ ΔХ)) = α
Expression (1.16) means that with a probability equal to α, the measurement result does not go beyond the confidence interval from<Х>– ΔХ up to<Х>+ ΔX. The larger the confidence interval, that is, the larger the specified error of the measurement result ΔX, the more reliable the sought-for value X falls into this interval. Naturally, the value of α depends on the number n of measurements. as well as on the specified error ΔХ.
Thus, to characterize the magnitude of a random error, it is necessary to set two numbers, namely:
- the magnitude of the error itself (or confidence interval);
- the value of confidence probability (reliability).
Specifying only the magnitude of the error without specifying the corresponding confidence probability is largely meaningless, since in this case we do not know how reliable our data are. Knowing the confidence level allows you to assess the degree of reliability of the result.
The required degree of reliability is given by the nature of the changes being made. The mean square error S n corresponds to a confidence probability of 0.68, doubled mean square error (2σ) corresponds to a confidence probability of 0.95, and triple (3σ) corresponds to 0.997.
If the interval (X - σ, X + σ) is chosen as the confidence interval, then we can say that out of a hundred measurement results, 68 will necessarily be within this interval (Fig. 1.2). If during the measurement the absolute error ∆Х > 3σ, then this measurement should be attributed to gross errors or a miss. The value of 3σ is usually taken as the limiting absolute error of a single measurement (sometimes, instead of 3σ, the absolute error of the measuring device is taken).
For any value of the confidence interval, the corresponding confidence probability can be calculated using the Gauss formula. These calculations were carried out and their results are summarized in Table. 1.1.
Confidence probabilities α for a confidence interval expressed as a fraction of the root mean square error ε = ΔX/σ.
Vidutinė kvadratinė paklaida statusas T sritis automatika atitikmenys: engl. mean square error vok. mittlerer quadratischer Fehler, m rus. root mean square error, fpranc. écart quadratique moyen, m; erreur quadratique moyenne, f … Automatikos terminų žodynas
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