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14.8. Smoothing of experimental dependencies by the least squares method

Lecture



The issues related to the processing of the experiments discussed in this chapter are closely related to the question of smoothing out experimental dependencies.

Let an experiment be made, the purpose of which is to study the dependence of a certain physical quantity   14.8.  Smoothing of experimental dependencies by the least squares method from physical quantity   14.8.  Smoothing of experimental dependencies by the least squares method (for example, the dependence of the path traveled by the body, on time; the initial velocity of the projectile on the temperature of the charge; lift force on the angle of attack, etc.). It is assumed that the values   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method linked by functional dependence:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.1)

The type of this dependence is required to be determined from experience.

Suppose that as a result of the experiment we obtained a number of experimental points and plotted the dependence   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method (fig. 14.8.1). Usually, the experimental points on such a graph are not arranged in exactly the right way - they give some “scatter”, that is, they detect random deviations from a visible general pattern.

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.1.

These deviations are associated with measurement errors that are inevitable with every experience.

The question arises of how best to reproduce the dependence on these experimental data.   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method ?

It is known that through any   14.8.  Smoothing of experimental dependencies by the least squares method points with coordinates   14.8.  Smoothing of experimental dependencies by the least squares method you can always draw a curve expressed analytically by a polynomial of degree   14.8.  Smoothing of experimental dependencies by the least squares method so that it passes exactly through each of the points (fig. 14.8.2).

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.2.

However, such a solution to the problem is usually not satisfactory: as a rule, the irregular behavior of experimental points, similar to that shown in Fig. 14.8.1 and 14.8.2, is not related to the objective nature of dependence   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method , but only with measurement errors. This is easy to detect by comparing observed deviations (spread of points) with roughly known errors in instrumentation.

Then there is a very typical practice problem of smoothing the experimental dependence. It is desirable to process the experimental data so as to reflect the general trend of dependence as accurately as possible.   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method but at the same time smooth irregular, random deviations due to the inevitable errors of the observation itself.

To solve such problems, a computational method commonly known as the “least squares method” is usually used. This method allows for a given type of dependency   14.8.  Smoothing of experimental dependencies by the least squares method so choose its numeric parameters to curve   14.8.  Smoothing of experimental dependencies by the least squares method In a sense, the best way to display the experimental data.

Let us say a few words about the reasons for the type of curve   14.8.  Smoothing of experimental dependencies by the least squares method . Often this issue is solved directly by the appearance of experimental dependence. For example, the experimental points shown in Fig. 14.8.3, clearly suggest a straight line dependence   14.8.  Smoothing of experimental dependencies by the least squares method .

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.3.

The dependence shown in Fig. 14.8.4, well can be represented by a polynomial of the second degree   14.8.  Smoothing of experimental dependencies by the least squares method .

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.4.

If we are talking about a periodic function, it is often possible to choose several harmonics of a trigonometric series for its image, etc.

It often happens that the type of dependence (linear, quadratic, exponential, etc.) is known from physical considerations related to the essence of the problem to be solved, and from experience only some parameters of this dependence need to be established.

The problem of the rational choice of such numerical parameters for this type of dependence we will solve in the present   14.8.  Smoothing of experimental dependencies by the least squares method .

Let there be results   14.8.  Smoothing of experimental dependencies by the least squares method independent experiments, designed in the form of a simple statistical table (table. 14.8.1), where   14.8.  Smoothing of experimental dependencies by the least squares method - number of experience;   14.8.  Smoothing of experimental dependencies by the least squares method - the value of the argument;   14.8.  Smoothing of experimental dependencies by the least squares method - the corresponding value of the function.

Table 14.8.1

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

Points   14.8.  Smoothing of experimental dependencies by the least squares method plotted on the chart (Fig. 14.8.5).

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.5.

From theoretical or other considerations, the principle form of dependence is chosen.   14.8.  Smoothing of experimental dependencies by the least squares method . Function   14.8.  Smoothing of experimental dependencies by the least squares method contains a series of numeric parameters   14.8.  Smoothing of experimental dependencies by the least squares method . It is required to select these parameters so that the curve   14.8.  Smoothing of experimental dependencies by the least squares method in a sense, it best depicted the dependence obtained in the experiment.

The solution to this problem, like any leveling or smoothing problem, depends on what we agree to be considered the “best”. You can, for example, consider the “best” such mutual arrangement of the curve and experimental points, at which the maximum distance between them turns to a minimum; it is possible to require that the sum of absolute values ​​of deviations of points from a curve, etc., be addressed to the minimum, etc. With each of these requirements we will get our solution to the problem, our values ​​of the parameters   14.8.  Smoothing of experimental dependencies by the least squares method .

However, the so-called least squares method, which requires the best fit of a curve, is generally accepted when solving such problems.   14.8.  Smoothing of experimental dependencies by the least squares method and experimental points is reduced to ensure that the sum of the squares of the deviations of the experimental points from the smoothing curve turns to a minimum. The method of least squares has significant advantages over other smoothing methods: firstly. it leads to a relatively simple mathematical method for determining parameters   14.8.  Smoothing of experimental dependencies by the least squares method ; secondly, he admits a rather weighty theoretical substantiation from a probabilistic point of view.

We present this rationale. Suppose true dependency   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method exactly expressed by the formula   14.8.  Smoothing of experimental dependencies by the least squares method ; experimental points shy away from this dependence due to unavoidable measurement errors. We have already mentioned that measurement errors, as a rule, are subject to normal law. Assume that it is. Consider some argument value   14.8.  Smoothing of experimental dependencies by the least squares method . The result of the experience is a random variable.   14.8.  Smoothing of experimental dependencies by the least squares method distributed according to the normal law with the expectation   14.8.  Smoothing of experimental dependencies by the least squares method and with standard deviation   14.8.  Smoothing of experimental dependencies by the least squares method characterizing the measurement error. Suppose that the measurement accuracy at all points is the same:

  14.8.  Smoothing of experimental dependencies by the least squares method .

Then the normal law on which the value is distributed   14.8.  Smoothing of experimental dependencies by the least squares method can be written in the form:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.2)

As a result of our experience - a series of measurements - the following event occurred: random variables   14.8.  Smoothing of experimental dependencies by the least squares method took a set of values   14.8.  Smoothing of experimental dependencies by the least squares method . We set the task: so pick up the mathematical expectations   14.8.  Smoothing of experimental dependencies by the least squares method so that the probability of this event is maximum.

Strictly speaking, the probability of any of the events   14.8.  Smoothing of experimental dependencies by the least squares method , as well as their combination, is equal to zero, since the values   14.8.  Smoothing of experimental dependencies by the least squares method continuous; so we will use no probabilities of events   14.8.  Smoothing of experimental dependencies by the least squares method , and the corresponding elements of probability:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.3)

Find the probability that the system of random variables   14.8.  Smoothing of experimental dependencies by the least squares method will take a set of values ​​lying within

  14.8.  Smoothing of experimental dependencies by the least squares method   14.8.  Smoothing of experimental dependencies by the least squares method .

Since the experiments are independent, this probability is equal to the product of probability elements (14.8.3) for all values   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.4)

Where   14.8.  Smoothing of experimental dependencies by the least squares method - some coefficient not depending on   14.8.  Smoothing of experimental dependencies by the least squares method .

It is required to choose the mathematical expectations   14.8.  Smoothing of experimental dependencies by the least squares method so that the expression (14.8.4) turns to a maximum.

Magnitude

  14.8.  Smoothing of experimental dependencies by the least squares method

always less than one; obviously, it has the greatest value when the exponent in absolute magnitude is minimal:

  14.8.  Smoothing of experimental dependencies by the least squares method .

From here, rejecting the constant multiplier   14.8.  Smoothing of experimental dependencies by the least squares method , we obtain the requirement of the method of least squares: in order for a given set of observed values

  14.8.  Smoothing of experimental dependencies by the least squares method

was the most probable, you need to select a function   14.8.  Smoothing of experimental dependencies by the least squares method so that the sum of the squares of the deviations of the observed values   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method was minimal:

  14.8.  Smoothing of experimental dependencies by the least squares method .

Thus, the least squares method is justified on the basis of the normal law of measurement errors and the requirement of the maximum probability of a given set of errors.

Let us turn to the problem of determining the parameters   14.8.  Smoothing of experimental dependencies by the least squares method based on the principle of least squares. Let there be a table of experimental data (Table 14.8.1) and let some considerations (related to the essence of the phenomenon or simply to the appearance of the observed dependence) be selected   14.8.  Smoothing of experimental dependencies by the least squares method depending on several numeric parameters   14.8.  Smoothing of experimental dependencies by the least squares method ; It is these parameters that need to be chosen according to the least squares method so that the sum of the squares of the deviations   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method was minimal. Write   14.8.  Smoothing of experimental dependencies by the least squares method as a function of not only an argument   14.8.  Smoothing of experimental dependencies by the least squares method but also parameters   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method (14 8.5)

Required to choose   14.8.  Smoothing of experimental dependencies by the least squares method так, чтобы выполнялось условие:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.6)

Найдем значения   14.8.  Smoothing of experimental dependencies by the least squares method , обращающие левую часть выражения (14.8.6) в минимум. Для этого продифференцируем ее по   14.8.  Smoothing of experimental dependencies by the least squares method и приравняем производные нулю:

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.7)

Where   14.8.  Smoothing of experimental dependencies by the least squares method - значение частной производной функции   14.8.  Smoothing of experimental dependencies by the least squares method по параметру   14.8.  Smoothing of experimental dependencies by the least squares method at the point   14.8.  Smoothing of experimental dependencies by the least squares method ,   14.8.  Smoothing of experimental dependencies by the least squares method - аналогично.

Система уравнений (14.8.7) содержит столько же уравнений, сколько неизвестных   14.8.  Smoothing of experimental dependencies by the least squares method .

Решить систему (14.8.7) в общем виде нельзя; для этого необходимо задаться конкретным видом функции   14.8.  Smoothing of experimental dependencies by the least squares method .

Рассмотрим два часто встречающихся на практике случая: когда функция   14.8.  Smoothing of experimental dependencies by the least squares method линейна и когда она выражается полиномом второй степени (параболой).

1. Подбор параметров линейной функции методом наименьших квадратов

В опыте зарегистрирована совокупность значений   14.8.  Smoothing of experimental dependencies by the least squares method   14.8.  Smoothing of experimental dependencies by the least squares method (см. рис. 14.8.6).

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.6.

Требуется подобрать по методу наименьших квадратов параметры   14.8.  Smoothing of experimental dependencies by the least squares method линейной функции

  14.8.  Smoothing of experimental dependencies by the least squares method ,

изображающей данную экспериментальную зависимость.

Decision. We have:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.8)

Дифференцируя выражение (14.8.8) по   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method , we have:

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

Подставляя в формулы (14.8.7), получим два уравнения для определения   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

или, раскрывая скобки и производя суммирование,

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.9)

Разделим оба уравнения (14.8.9) на   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.10)

Суммы, входящие в уравнения (14.8.10), представляют собой не что иное, как уже знакомые нам статистические моменты:

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

Подставляя эти выражения в систему (14.8.10), получим:

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.11)

Выразим   14.8.  Smoothing of experimental dependencies by the least squares method из второго уравнения (14.8.11) и подставим в первое:

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method .

Решая последнее уравнение относительно   14.8.  Smoothing of experimental dependencies by the least squares method , we have:

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.12)

Выражение (14.8.12) можно упростить, если ввести в него не начальные, а центральные моменты. Really,

  14.8.  Smoothing of experimental dependencies by the least squares method ,   14.8.  Smoothing of experimental dependencies by the least squares method ,

from where

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method , (14.8.13)

Where

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.14)

Таким образом, поставленная задача решена, и линейная зависимость, связывающая   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method , имеет вид:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

или, перенося   14.8.  Smoothing of experimental dependencies by the least squares method в левую часть,

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.15)

Мы выразили коэффициенты линейной зависимости через центральные, а не через начальные вторые моменты только потому, что в таком видe формулы имеют более компактный вид. При практическом применении выведенных формул может оказаться удобнее вычислять моменты   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method не по формулам (14.8.14), а через вторые начальные моменты:

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.16)

Для того чтобы формулы (14.8.16) не приводили к разностям близких чисел, рекомендуется перенести начало отсчета в точку, не слишком далекую от математических ожиданий   14.8.  Smoothing of experimental dependencies by the least squares method ,   14.8.  Smoothing of experimental dependencies by the least squares method .

2. Подбор параметров параболы второго порядка методом наименьших квадратов

В опыте зарегистрированы значения   14.8.  Smoothing of experimental dependencies by the least squares method   14.8.  Smoothing of experimental dependencies by the least squares method (см. рис. 14.8.7).

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.7.

Требуется методом наименьших квадратов подобрать параметры квадратичной функции - параболы второго порядка:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

соответствующей наблюденной экспериментальной зависимости. We have:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method .

Подставляя в уравнения (14.8.7), имеем:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method

или, раскрывая скобки, производя суммирование и деля на   14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.17)

Коэффициенты этой системы также представляют собой статистические моменты системы двух величии   14.8.  Smoothing of experimental dependencies by the least squares method , а именно:

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method .

Пользуясь этими выражениями для коэффициентов через начальные моменты одной случайной величины и системы двух величин, можно придать системе уравнений (14.8.7) достаточно компактный вид. Действительно, учитывая, что   14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method и перенося члены, не содержащие неизвестных, в правые части, приведем систему (14.8.17) к виду:

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.18)

Закон образования коэффициентов в уравнениях (14.8.18) нетрудно подметить: в левой части фигурируют только моменты величины   14.8.  Smoothing of experimental dependencies by the least squares method в убывающем порядке; в правой части стоят моменты системы   14.8.  Smoothing of experimental dependencies by the least squares method , причем порядок момента по   14.8.  Smoothing of experimental dependencies by the least squares method убывает от уравнения к уравнению, а порядок по   14.8.  Smoothing of experimental dependencies by the least squares method всегда остается первым.

Аналогичными по структуре уравнениями будут определяться коэффициенты параболы любого порядка.

Мы видим, что в случае, когда экспериментальная зависимость выравнивается по методу наименьших квадратов полиномом некоторой степени, то коэффициенты этого полинома находятся решением системы линейных уравнений. Коэффициенты этой системы линейных уравнений представляют собой статистические моменты различных порядков, характеризующие систему величии   14.8.  Smoothing of experimental dependencies by the least squares method , если ее рассмотреть как систему случайных величин.

Почти так же просто решается задача сглаживания экспериментальной зависимости методом наименьших квадратов в случае, когда сглаживающая функция представляет собой не полином, а сумму произвольных заданных функций   14.8.  Smoothing of experimental dependencies by the least squares method с коэффициентами   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method , (14.8.19)

и когда требуется определить коэффициенты   14.8.  Smoothing of experimental dependencies by the least squares method .

Например, экспериментальную зависимость можно сглаживать тригонометрическим полиномом

  14.8.  Smoothing of experimental dependencies by the least squares method

or linear combination of exponential functions

  14.8.  Smoothing of experimental dependencies by the least squares method ,

etc.

If the function is given by an expression of the type (14.8.19), the coefficients   14.8.  Smoothing of experimental dependencies by the least squares method are found by solving a system of   14.8.  Smoothing of experimental dependencies by the least squares method linear equations of the form:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method .

Performing term-by-word summation, we have:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

or shorter

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.20)

The system of linear equations (14.8.20) can always be solved and thus the coefficients   14.8.  Smoothing of experimental dependencies by the least squares method .

The least squares smoothing problem is more difficult to solve if the   14.8.  Smoothing of experimental dependencies by the least squares method numeric parameters   14.8.  Smoothing of experimental dependencies by the least squares method enter the function expression nonlinearly. Then the solution of system (14.8.7) can be difficult and time consuming. However, in this case, it is often possible to obtain a solution to the problem using relatively simple techniques.

We illustrate the idea of ​​these techniques on the simplest example of a function that depends nonlinearly on only one parameter.   14.8.  Smoothing of experimental dependencies by the least squares method (eg,   14.8.  Smoothing of experimental dependencies by the least squares method or   14.8.  Smoothing of experimental dependencies by the least squares method , or   14.8.  Smoothing of experimental dependencies by the least squares method ). We have:

  14.8.  Smoothing of experimental dependencies by the least squares method (14.8.21)

Where   14.8.  Smoothing of experimental dependencies by the least squares method - параметр, подлежащий подбору методом наименьших квадратов для наилучшего сглаживания заданной экспериментальной зависимости (рис. 14.8.8).

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.8.

Будем решать задачу следующим образом. Зададимся рядом значений параметра   14.8.  Smoothing of experimental dependencies by the least squares method и для каждого из них найдем сумму квадратов уклонений   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method . Эта сумма квадратов есть некоторая функция   14.8.  Smoothing of experimental dependencies by the least squares method ; обозначим ее   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method .

Нанесем значение   14.8.  Smoothing of experimental dependencies by the least squares method на график (рис. 14.8.9).

То значение   14.8.  Smoothing of experimental dependencies by the least squares method , для которого кривая   14.8.  Smoothing of experimental dependencies by the least squares method имеет минимум, и выбирается как подходящее значение параметра   14.8.  Smoothing of experimental dependencies by the least squares method в выражении (14.8.21).

Совершенно так же, в принципе, можно, не решая уравнений (14.8.7), подобрать совокупность двух параметров   14.8.  Smoothing of experimental dependencies by the least squares method , удовлетворяющую принципу наименьших квадратов; работа при этом лишь незначительно усложнится и сведется к построению не одного, а нескольких графиков (рис. 14.8.10); при этом придется искать совокупность значений   14.8.  Smoothing of experimental dependencies by the least squares method , обеспечивающую минимум минимального значения суммы квадратов отклонений   14.8.  Smoothing of experimental dependencies by the least squares method .

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.9.

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.10.

Пример1. В опыте исследована зависимость глубины проникания   14.8.  Smoothing of experimental dependencies by the least squares method тела в преграду от удельном энергии   14.8.  Smoothing of experimental dependencies by the least squares method (энергии, приходящейся на квадратный сантиметр площади соударения). Экспериментальные данные приведены в таблице 14.8.2 и на графике (рис. 14.8.11).

Таблица 14.8.2.

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

one

41

four

2

50

eight

3

81

ten

four

104

14

five

120

sixteen

6

139

20

7

154

nineteen

eight

180

23

9

208

26

ten

241

thirty

eleven

250

31

12

269

36

13

301

37

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.11.

Требуется по методу наименьших квадратов подобрать и построить прямую, изображающую зависимость   14.8.  Smoothing of experimental dependencies by the least squares method from   14.8.  Smoothing of experimental dependencies by the least squares method .

Decision. We have:

  14.8.  Smoothing of experimental dependencies by the least squares method ,   14.8.  Smoothing of experimental dependencies by the least squares method .

Для обработки по начальным моментам переносим начало координат в близкую к средней точку:

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method .

Получаем новую таблицу значений величин:

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method

(табл. 14.8.3).

Таблица 14.8.3.

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

one

-109

-sixteen

2

-100

-12

3

-69

-ten

four

-46

-6

five

-thirty

-four

6

-eleven

0

7

four

-one

eight

thirty

3

9

58

6

ten

91

ten

eleven

100

eleven

12

119

sixteen

13

151

17

Определяем моменты:

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method .

Уравнение прямой имеет вид:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

or

  14.8.  Smoothing of experimental dependencies by the least squares method . (14.8.22)

Прямая (14.8.22) показана на рис. 14.8.11.

Пример 2. Произведен ряд опытов по измерению перегрузки авиационной бомбы, проникающей в грунт, при различных скоростях встречи. Полученные значения перегрузки   14.8.  Smoothing of experimental dependencies by the least squares method в зависимости от скорости   14.8.  Smoothing of experimental dependencies by the least squares method приведены в таблице 14.8.4.

Таблица 14.8.4.

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

one

120

540

2

131

590

3

140

670

four

161

760

five

174

850

6

180

970

7

200

1070

eight

214

1180

9

219

1270

ten

241

1390

eleven

250

1530

12

268

1600

13

281

1780

14

300

2030

Построить по методу наименьших квадратов квадратичную зависимость вида:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

наилучшим образом согласующуюся с экспериментальными данными.

Decision. For convenience of processing, it is convenient to change the units of measurement so as not to deal with multi-digit numbers; for this you can value   14.8.  Smoothing of experimental dependencies by the least squares method Express in hundreds of meters per second (multiply by   14.8.  Smoothing of experimental dependencies by the least squares method ), but   14.8.  Smoothing of experimental dependencies by the least squares method - in thousands of units (multiplied by   14.8.  Smoothing of experimental dependencies by the least squares method ) and all processing is carried out in these conventional units.

Find the coefficients of equations (14.8.18).

In the accepted conventional units:

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method ;

  14.8.  Smoothing of experimental dependencies by the least squares method .

The system of equations (14.8.18) has the form:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method .

Solving this system, we find:

  14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method ;   14.8.  Smoothing of experimental dependencies by the least squares method .

In fig. 14.8.12 plotted experimental points and dependence

  14.8.  Smoothing of experimental dependencies by the least squares method ,

least squares method.

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.12.

Note. In some cases, it may be necessary to draw a curve so that it accurately passes through some predetermined points. Then some of the numeric parameters   14.8.  Smoothing of experimental dependencies by the least squares method included in the function   14.8.  Smoothing of experimental dependencies by the least squares method , can be determined from these conditions.

For example, in the conditions of example 2, it may be necessary to extrapolate the dependence   14.8.  Smoothing of experimental dependencies by the least squares method for small values   14.8.  Smoothing of experimental dependencies by the least squares method , it is natural to hold a second order parabola so that it passes through the origin (i.e., zero meeting speed corresponds to zero meeting speed). Then naturally   14.8.  Smoothing of experimental dependencies by the least squares method and addiction   14.8.  Smoothing of experimental dependencies by the least squares method takes the form:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

and the system of equations to determine   14.8.  Smoothing of experimental dependencies by the least squares method and   14.8.  Smoothing of experimental dependencies by the least squares method will look like:

  14.8.  Smoothing of experimental dependencies by the least squares method ,

  14.8.  Smoothing of experimental dependencies by the least squares method .

Example 3. Condenser charged to voltage   14.8.  Smoothing of experimental dependencies by the least squares method volt, discharged through some resistance. Voltage dependence   14.8.  Smoothing of experimental dependencies by the least squares method between capacitor plates from time to time   14.8.  Smoothing of experimental dependencies by the least squares method registered on the time interval of 10 seconds with an interval of 1 second. Voltage is measured to an accuracy of 5 volts. The measurement results are shown in table 14.8.5.

Table 14.8.5.

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

one

0

100

7

6

15

2

one

75

eight

7

ten

3

2

55

9

eight

ten

four

3

40

ten

9

five

five

four

thirty

eleven

ten

five

6

five

20

According to theoretical data, the dependence of voltage on time should be of the form:

  14.8.  Smoothing of experimental dependencies by the least squares method .

Based on the experimental data, select the parameter value using the least squares method   14.8.  Smoothing of experimental dependencies by the least squares method .

Decision. By function tables   14.8.  Smoothing of experimental dependencies by the least squares method make sure that   14.8.  Smoothing of experimental dependencies by the least squares method comes to about 0.05 at   14.8.  Smoothing of experimental dependencies by the least squares method ; hence the coefficient   14.8.  Smoothing of experimental dependencies by the least squares method must be of order 0.3. Ask in the area   14.8.  Smoothing of experimental dependencies by the least squares method multiple meanings   14.8.  Smoothing of experimental dependencies by the least squares method :

  14.8.  Smoothing of experimental dependencies by the least squares method

and calculate for them the function values

  14.8.  Smoothing of experimental dependencies by the least squares method

in points   14.8.  Smoothing of experimental dependencies by the least squares method (tab. 14.8.6). The bottom line of table 14.8.6 contains the values ​​of the sum of squared deviations.   14.8.  Smoothing of experimental dependencies by the least squares method depending on the   14.8.  Smoothing of experimental dependencies by the least squares method .

Table 14.8.6.

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

  14.8.  Smoothing of experimental dependencies by the least squares method

one

0

100.0

100.0

100.0

100.0

100.0

100.0

2

one

75.5

74.8

74.1

73.3

72.6

71.9

3

2

57.1

56.0

54.9

53,8

52.7

51.7

four

3

43.2

41.9

40.7

39.5

38.3

37.2

five

four

32.6

31.3

30.1

28.9

27.8

26.7

6

five

24.6

23.5

22.3

21.2

20.2

19.2

7

6

18.6

17.6

16.5

15.6

14.7

13.8

eight

7

14.1

13.1

12.2

11.4

10.6

9.9

9

eight

10.7

9.8

9.1

8.4

7.7

7.1

ten

9

8.0

7.4

6.7

6.1

5.6

5.1

eleven

ten

6.1

5.5

5.0

4.5

4.1

3.7

  14.8.  Smoothing of experimental dependencies by the least squares method

83.3

40.3

17.4

13.6

25.7

51.4

Function graph   14.8.  Smoothing of experimental dependencies by the least squares method is shown in fig. 14.8.13.

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.13.

From the graph it is clear that the value   14.8.  Smoothing of experimental dependencies by the least squares method corresponding to the minimum is approximately 0.307. Thus, using the least squares method, the best approximation to the experimental data would be the function   14.8.  Smoothing of experimental dependencies by the least squares method . The graph of this function together with the experimental points is given in fig. 14.8.14.

  14.8.  Smoothing of experimental dependencies by the least squares method

Fig. 14.8.14.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis