ON THE JUSTIFICATION OF THE LEAST SQUARES METHOD

V. S. Nuvaryev

The modern method of reconciling numerous observations was first published by Legendre in a short study in 1805. This method, called by Legendre the method of least squares, was proposed by him without proof.

Gauss, who had used this method since 1795, first gave a justification of this method in 1809. However, his first proof had the shortcoming that it was based on the requirement of the greatest probability of the system of corrections, and not of each of them.

Entirely different principles were laid by Laplace at the basis of his proof, proposed by him in 1811. The general principle of Laplace’s proof, later used by Gauss in his second proof of the method of least squares, consists in the following: suppose that from a very large number of equations of errors of the form

\[ \varepsilon_i=-l_i+a_iX+b_iY+c_iZ+\ldots\quad (i=1,2\ldots n) \tag{a} \]

it is required to determine the most advantageous estimates \(m\) (\(m<n\)) of the unknowns \(x, y, z\ldots\).

Let us multiply each equation of errors by certain positive or negative numbers \(z_i\), and form the sum:

\[ [z\varepsilon]=-[zl]+[za]X+[zb]Y+[zc]Z+\ldots \]

and subject the \(n\) multipliers \(z_i\) to \(m\) conditional equations:

\[ [za]=1,\quad [zb]=0,\quad [zc]=0\ldots \tag{b} \]

Then the true value is equal to

\[ X=[zl]+[z\varepsilon], \tag{c} \]

where \(l_i\) are the measured values \(L_i\), and \(\varepsilon_i\) are the true errors of these measurements.

If \([z\varepsilon]\) is set equal to zero, then the approximate value of the unknown will be determined from the equation

\[ x=[zl] \tag{d} \]

with error

\[ u=[z\varepsilon]. \]

Of all possible determinations, the best will be that value for which \(u^2\) is the least (\(|u|=\min.\), according to Laplace). These requirements lead, for observations of unequal weight, to the condition:

\[ [z^2m^2]=\mathrm{minimum}; \]

and for observations of equal weight—

\[ [z^2]=\mathrm{minimum}, \]

The distribution of errors in the given series of measurements is not known to us, i.e., the values \(\varepsilon_i\) in equations \((a)\) and \((c)\) are also unknown. Hence we must conclude that the values \(a_i\), determined from the requirement \(u^2=\min\) and from equations \((b)\), do not depend on the errors of the given series of observations, i.e., the mechanism of the method of least squares is defined for some average series; the given series of measurements is used by it independently of the distribution of errors in this series.

From equation \((c)\) it follows that, as \([a\varepsilon]\) decreases, \([a l]\) tends to the true value \(X\). However, if one turns to the given series of measurements, it is not sufficiently evident that \([a l]\) tends to the true value \(X\), i.e., we cannot set the problem \([a\varepsilon]^2=\min\) for the given series of measurements. Therefore we consider it more intuitive to represent equation \((d)\) in the following form:

\[ x = [aL] - [a\varepsilon]. \]

Since \([aL]\) represents the true value \(X\) for any system \(a\) satisfying the system of equations \((b)\), a decrease in the absolute value of the sum \([a\varepsilon]\) (or \([a\varepsilon]^2\)) brings \(x\) closer to the true value \(X\).

In studying the results obtained by the method of least squares, various authors assert that these results are in complete agreement with the arithmetic mean, i.e., that it lies at the basis of the method of least squares. As proof of this assertion, they give an investigation of the simplest example with one conditional equation. However, these investigations do not answer the question of whether the principle of the method of least squares and the principle of the arithmetic mean are always in agreement, or whether there are cases in which one of them must be given preference.

In investigating this question, we were compelled to construct new proofs of the method of least squares, based on the principle of keeping the sought value in the smallest possible neighborhood of its exact value. In doing so, we proceeded from the formula, proved by Laplace, Gauss, and later authors, for the mean square error of a compound deviation that is a linear function of individual deviations obeying the normal law of distribution. Thus, if

\[ u = F(x,y,z\ldots), \]

then

\[ m_u^2 = F_x^2 m_x^2 + F_y^2 m_y^2 + F_z^2 m_z^2 + \ldots, \tag{e} \]

where \(F_x, F_y\ldots\) are the partial derivatives of \(u\) with respect to \(x,y\ldots\), and \(m_x,m_y\ldots\) are the mean square errors of these quantities.

The proofs of the method of least squares developed by us proceed from the following consideration: the equation value of the sought quantity, irrespective of the form of the function connecting it with the observations, must be equal to its true value if the observations contain no errors.

Let there be \(n\) linear equations of the form:

\[ 0 = -L_i + a_i X + b_i Y - c_i Z + \ldots, \tag{f} \]

where \(m\) \((m<n)\) quantities \(X,Y,Z\ldots\) are to be determined from \(n\) measured quantities, whose true value is equal to \(L_i\).

Let us find \(X\) from the algebraic solution, say, of the first \(m\) equations. Denote the true value of \(X\) from the first solution by

\[ X_1 = f_i(L_i), \quad (i=1,2\ldots m), \]

and by \(X_2, X_3,\ldots\) the true values of \(X\), found from the algebraic solution of combinations of equations \((f)\), chosen so that each subsequent combination would include each time an equation not used earlier.\(^1\)

If \(X_i\) are multiplied by undetermined multipliers \(p_i\), satisfying the condition:

\[ [p]=1, \tag{1} \]

then the true value may be represented in the form:

\[ X=[X_i p_i]. \qquad (i=1,2\ldots,n-m+1) \tag{2} \]

If in equation (2) the measured values \(l_i\) are substituted instead of \(X_i\), then we obtain an approximate value \(x\) with mean error \(m_x\), depending on \(m_{l_i}\) and \(p_i\).

Let us find the system of multipliers \(p_i\) under the condition:

\[ m^2=\mathrm{minimum}. \tag{3} \]

The solution of this problem presents no difficulty.

Let us apply what has been said to a system of three equations with two unknowns. Let there be a system of equations:\(^2\)

\[ \begin{aligned} a_1X+b_1Y-l_1&=0\\ a_2X+b_2Y-l_2&=0\\ a_3X+b_3Y-L_3&=0 \end{aligned} \tag{4} \]

Then from the solution of the first two equations we have:

\[ X_{12}=\frac{1}{D_{12}}(b_2L_1-b_1L_2). \]

From the second and third we obtain:

\[ X_{23}=\frac{1}{D_{23}}(b_3L_2-b_2L_3), \]

where \(D_{12}\) is the determinant of the system from the first and second equations, and \(D_{23}\) from the second and third equations.

Then

\[ X=\frac{1}{D_{12}}(b_2L_1-b_1L_2)p_1+ \frac{1}{D_{23}}(b_3L_2-b_2L_3)p_2, \tag{5} \]

where

\[ p_1+p_2=1. \]

Suppose that, from the measurements, equally weighted values \(l_1,l_2,l_3\) have been found;\(^3\)

\[ m_x^2= \frac{[b^2]_3}{D_{12}^2}p_1^2+ \frac{[b^2]_1}{D_{23}^2}p_2^2- 2\frac{b_1b_3}{D_{12}D_{23}}p_1p_2, \tag{6} \]

where \([b^2]_3\) represents the sum of \(b^2\) without \(b_3^2\), and so on.

\(^1\) The thought may arise (following Jacobi) to use solutions in which all possible combinations of \(L_i\) will participate, but then the determinant of the system in determining \(p_i\) from system (7) will be equal to zero, \(D_0=0\), even when the number of combinations is \(n-m+2\).

\(^2\) More than one \(L\) may enter into each equation.

\(^3\) If the \(l_i\) are taken to be of unequal weight, the essence of the question does not change.

From (1) and (6) we find:

\[ \left. \begin{aligned} \frac{\partial (m_x^2)}{\partial p_1} &= \frac{[b^2]_3}{D_{12}^2}p_1 - \frac{b_1 b_3}{D_{12}D_{23}}p_2 = k,\\ \frac{\partial (m_x^2)}{\partial p_2} &= - \frac{b_1 b_3}{D_{12}D_{23}}p_1 + \frac{[b^2]_1}{D_{23}^2}p_2 = k . \end{aligned} \right\} \tag{7} \]

The solution of this system gives:

\[ \left. \begin{aligned} p_1 &= \frac{A_1}{A_1 + A_2};\quad p_2 = \frac{A_2}{A_1 + A_2};\\ A_1 &= \frac{[b^2]_1}{D_{23}^2} + \frac{b_1 b_3}{D_{12}D_{23}};\quad A_2 = \frac{[b^2]_3}{D_{12}^2} + \frac{b_1 b_3}{D_{12}D_{23}} . \end{aligned} \right\} \tag{8} \]

Then, on the basis of (8) and (5), the adjusted value \(x^\circ\) will be equal to:

\[ x^\circ = \frac{1}{D_{12}}(b_2 l_1 - b_1 l_2)p_1 + \frac{1}{D_{23}}(b_3 l_2 - b_2 l_3)p_2 \tag{9} \]

If we had applied to system (4) the requirement \([v^2] = \min\), i.e., had solved the normal equations

\[ \begin{aligned} [aa]x + [ab]y - [al] &= 0\\ [ab]x + [bb]y - [bl] &= 0, \end{aligned} \tag{10} \]

then

\[ x = \frac{1}{D}\bigl([bb][al] - [ab][bl]\bigr), \tag{11} \]

where \(D\) is the determinant of system (10).

It can be shown in general that the coefficients of the \(l_i\) in equations (9) and (11) are identical, i.e., the requirements \(m_x^2 = \min\) and \([v^2] = \min\) are equivalent.

Since the size of this article does not allow us to present the indicated investigation, we shall apply the formulas derived to the solution of a particular example.

Take the system of equations

\[ \begin{aligned} 2X - y - L_1 &= 0\\ X + y - L_2 &= 0\\ X - y - L_3 &= 0 \end{aligned} \tag{12} \]

Let the measured values be

\[ l_1 = 4,\quad l_2 = 5,\quad l_3 = 2. \]

Then by the method of least squares we find:

\[ x^\circ = 3.072;\quad y^\circ = 1.714. \]

Let us solve system (12), using formulas (8) and (9), for which we compute the determinants

Then

\[ x_{12}=\frac{1}{D_{12}}(b_2l_1-b_1l_2)=3; \]

\[ x_{23}=\frac{1}{D_{23}}(b_3l_2-b_2l_3)=3.5 \]

\[ A_1=\frac{1}{3};\quad A_2=\frac{1}{18};\quad p_1=\frac{6}{7};\quad p_2=\frac{1}{7}; \]

\[ x=3.072. \]

In deriving formulas (8) and (9) we did not restrict the choice of the values \(x_{ik}\), and therefore we obtain this same result by choosing another system of values of \(x\), for example:

\[ x_{13}=2.0;\quad x_{23}=3.5.\quad \text{Then } p_1=0.286;\quad p_2=0.714; \]

\[ x^\circ=3.071. \]

Let us apply the method described above to conditional equations in which the number of unknowns is greater than the number of equations. Suppose there are two equations with four unknowns:

\[ \left. \begin{array}{l} a_1X_1+a_2X_2+a_3X_3+a_4X_4+a_0=0\\ b_1X_1+b_2X_2+b_3X_3+b_4X_4+b_0=0 \end{array} \right\} \tag{13} \]

Solving this system with respect to \(X_1\) and \(X_2\), \(X_1\) and \(X_3\), we find:

\[ X_{12}=\frac{1}{D_{12}}\left\{(a_2b_3-b_2a_3)X_3+(a_2b_4-b_2a_4)X_4+(a_2b_0-b_2a_0)\right\}. \]

Denoting the determinants standing in the braces by \(D_{23}\), \(D_{24}\), and \(D_{20}\), and also solving system (13) with respect to \(X_1\) and \(X_3\), we shall write \(X\) as a function of all the quantities subject to measurement:¹

\[ \left. \begin{array}{l} X_1=X_1\\[4pt] X_{12}=\dfrac{1}{D_{12}}\left(D_{23}X_3+D_{24}X_4+D_{20}\right)\\[6pt] X_{13}=\dfrac{1}{D_{13}}\left(-D_{23}X_2+D_{34}X_4+D_{30}\right). \end{array} \right\} \tag{14} \]

If the expressions (14) are multiplied by factors \(p\) under the condition

\[ [p]=1, \tag{15} \]

the true value of \(X_1\) will have the form:

\[ X_1=p_1X_1-\frac{D_{23}}{D_{13}}p_3X_2+\frac{D_{23}}{D_{12}}p_2X_3+ \]

\[ +\left(\frac{D_{24}}{D_{12}}p_2+\frac{D_{34}}{D_{13}}p_3\right)X_4+p_2\frac{D_{20}}{D_{12}}+p_3\frac{D_{30}}{D_{13}}. \tag{16} \]

¹ It can be proved that, instead of the values \(X_{12}\), \(X_{13}\), one may use \(r\) values of \(X\) obtained from each equation of system (13),

\[ X_1'=-\frac{1}{a_1}\{[aX]_1+a_0\}; \]

\[ X_1''=-\frac{1}{b_1}\{[bX]_1+b_0\}\ldots \quad \text{or} \quad X_1'=l_1-\frac{W_1}{a_1};\quad X_1''=l_1-\frac{W_2}{b_1}\ldots \]

If the measured values are substituted into the right-hand side of equation (16), then the equation will determine the approximate value \(x_1\) with mean error

\[ m_{x_1}^{2}=p_{1}^{2}+\frac{D_{23}^{2}+D_{24}^{2}}{D_{12}^{2}}p_{2}^{2} +\frac{D_{23}^{2}+D_{34}^{2}}{D_{13}^{2}}p_{3}^{2} +2\frac{D_{24}D_{34}}{D_{12}D_{13}}p_{2}p_{3}. \tag{17} \]

Let us find \(p_i\) under condition (3). Then, multiplying equation (15) by the factor \(2k\) and equating the partial derivatives of this equation with respect to the variables \(p_i\) to the partial derivatives of equation (17) with respect to the same variables, we obtain:

\[ \left\{ \begin{aligned} p_1&=k,\\ \frac{1}{D_{12}^{2}}(D_{23}^{2}+D_{24}^{2})p_2 +\frac{D_{24}D_{34}}{D_{12}D_{13}}p_3&=k,\\ \frac{D_{24}D_{34}}{D_{12}D_{13}}p_2 +\frac{1}{D_{13}^{2}}(D_{23}^{2}+D_{34}^{2})p_3&=k. \end{aligned} \right. \tag{18} \]

The solution of the last two equations gives:

\[ p_2=\frac{1}{D_0}\left\{ \frac{1}{D_{13}^{2}}(D_{23}^{2}+D_{34}^{2}) -\frac{D_{24}D_{34}}{D_{12}D_{13}} \right\}k \]

\[ p_3=\frac{1}{D_0}\left\{ \frac{1}{D_{12}^{2}}(D_{23}^{2}+D_{24}^{2}) -\frac{D_{24}D_{34}}{D_{12}D_{13}} \right\}k, \]

where \(D_0\) is the determinant of the system (18).

Substituting the value of \(p\) into equation (15), we find:

\[ k=\frac{D_0}{D_0+D_2+D_3}; \]

therefore

\[ p_1=\frac{D_0}{N};\qquad p_2=\frac{1}{N}\left\{ \frac{1}{D_{13}^{2}}(D_{23}^{2}+D_{34}^{2}) -\frac{D_{24}D_{34}}{D_{12}D_{13}} \right\} \]

\[ p_3=\frac{1}{N}\left\{ \frac{1}{D_{12}^{2}}(D_{23}^{2}+D_{24}^{2}) -\frac{D_{24}D_{34}}{D_{12}D_{13}} \right\}, \]

where

\[ N=D_0+D_2+D_3. \]

Let us find the values of \(D_0\) and \(N\),

\[ D_0=\frac{1}{D_{12}^{2}D_{13}^{2}}(D_{23}^{2}+D_{24}^{2})(D_{23}^{2}+D_{34}^{2}) -\frac{D_{24}^{2}D_{34}^{2}}{D_{12}^{2}D_{13}^{2}}, \]

\[ D_0=\frac{D_{23}^{2}}{D_{12}^{2}D_{13}^{2}} (D_{23}^{2}+D_{24}^{2}+D_{34}^{2}). \]

\[ N=\frac{1}{D_{12}^{2}D_{13}^{2}} \left\{(D_{12}^{2}+D_{13}^{2}+D_{23}^{2}+D_{24}^{2}+D_{34}^{2})D_{23}^{2} +(D_{12}D_{34}-D_{13}D_{24})^{2}\right\} \]

but

\[ D_{12}D_{34}-D_{13}D_{24}=-D_{23}D_{14}, \]

i.e.

\[ N=\frac{D_{23}^{2}D}{D_{12}^{2}D_{13}^{2}}, \quad\text{where}\quad D=D_{12}^{2}+D_{13}^{2}+D_{14}^{2}+D_{23}^{2}+D_{24}^{2}+D_{34}^{2}. \]

It can be shown that \(D\) is the determinant of the system of normal equations corresponding to the system of conditional equations (13).

Using the values \(D_0\) and \(N\), we find the final expressions for the multipliers \(p\) in equation (16)

\[ \left. \begin{aligned} p_1&=-\frac{1}{D}\left(D_{23}^{2}+D_{24}^{2}+D_{34}^{2}\right),\\[4pt] p_2&=\frac{D_{12}}{DD_{23}}\left(D_{12}D_{23}-D_{14}D_{34}\right),\\[4pt] p_3&=\frac{D_{13}}{DD_{23}}\left(D_{13}D_{23}+D_{14}D_{24}\right). \end{aligned} \right\} \tag{19} \]

\[ x_1^{\circ}=lp_1+x_{12}p_2+x_{13}p_3 \tag{20} \]

Formulas (19) solve the problem of finding the multipliers, having the meaning of weights, with which the directly measured value \(l_1\) and the values \(x_{12}\) and \(x_{13}\), obtained from the algebraic solution of the conditional equations, enter into the arithmetic mean (20).

It can be shown in general form that the solution of the system of equations (13) under the condition \([v^2]=\min\) and under the condition \(m_x^2=\min\) lead to one and the same result. For this it is necessary to prove that the expression, made equal under the condition \([v^2]=\min\), for the value \(x_1\)

\[ x_1^{\circ}=(1+a_1r_1+b_1r_2+\ldots)l_1+(0+a_2r_1+b_2r_2+\ldots)l_2+\ldots+ \]

\[ {}+a_0r_1+b_0r_2, \]

where \(r\) are transition coefficients, can be reduced to equation (16), if in it \(X_i\) is replaced by \(l_i\).

Let us give an example of the application of formulas (14), (19), and (20). Suppose there is a system of conditional equations:

\[ \left. \begin{aligned} -4X_1+X_2+2X_3+X_4+a_0&=0,\\ 5X_1-2X_2+X_3+2X_4+b_0&=0. \end{aligned} \right\} \tag{21} \]

Assume that the values of \(X\) are:

Then

\[ a_0=-1;\qquad b_0=-36;\qquad W_1=[aa]+a_0=5;\qquad W_2=4. \]

Solving system (21) by the method of least squares, we find:

\[ v_1=+0.45, \]

i.e. the adjusted value

\[ x_1^{\circ}=5.45.\qquad m_x^2=0.118 \]

Let us apply to the solution of system (21) the formulas (14), (16), and (19) derived by us, for which purpose we compute the determinants

Then, calculating by formulas (14) and (19) the values of \(x\) and \(p\), taking (20) into account, we find

Substituting \(p\) in (17), we find \(m_{x_1}^2 = 0.118\).

A New Interpretation of the Results of Adjustment by the Method of Least Squares

Above we found that the value \(x\), adjusted by the method of least squares, can be represented in the form

\[ x_1^0 = [xp] \tag{22} \]

where \(x\) are the results of the algebraic solution of systems of equations, and \(p\) are found under the condition

\[ [p] = 1, \qquad m_x^2 = \text{minimum}. \]

From consideration of expressions (14) and (16) it follows that the value \(x\) adjusted by the method of least squares is the arithmetic mean of dependent quantities, since the same measured quantity \(l_4\) enters into the determination of the values \(x_{12}\) and \(x_{13}\). The same conclusion can be drawn from formula (9) for the method of indirect observations. On this basis we give a new interpretation of the results of adjustment by the method of least squares, consisting in the fact that the sought value, always adjusted by the method of least squares, is the arithmetic mean of the results of direct measurement and dependent values found from the algebraic solution of systems of equations, with weights satisfying the condition of greatest weight of the sought quantity. Such an interpretation of the results of adjustment by the method of least squares, as we shall see below, expands the field of possible investigations and applications of the method of least squares, but does not determine the basic properties of the results of adjustment. If by the arithmetic mean one understands the mean of independent values, then it is a particular case of the principle of greatest weight of the sought quantity that underlies the method of least squares. For the method of conditional observations, independence of the values is fulfilled if there is only one conditional equation, or if the number of unknowns \(n\) and the number of equations are connected by the condition \(r = n - 1\).

Only in this case will formulas (14) give independent values \(x\), and the \(p_i\) from formulas (19) will coincide with the weights calculated by the basic formula \((e)\).

Examples of Application of the New Interpretation

1. Adjustment of angles and directions observed by Schreiber’s method. Let at point \(P\) there have been four directions, between which angles were measured by Schreiber’s method. The value of the angle \(x\) by the method of least squares will be determined from the expression

\[ x = \frac{2(12) + [(13) - (23)] + [(14) - (24)]}{4}. \tag{a} \]

In expression $(a)$ the value of the angle $(3,4)$ is entirely absent, i.e., the error in this angle has no effect whatsoever on the adjusted value of the angle $x$.

Turning to the error equations

\[ \begin{array}{lll} x-(12)=v_{12} & \quad & y-x-(23)=v_{23}\\ y-(13)=v_{13} & \quad & z-x-(24)=v_{24}\\ z-(14)=v_{14} & \quad & z-y-(34)=v_{34} \end{array} \tag{6} \]

we observe that the angle $(34)$ does not enter into the equations containing the angle $x$; therefore one may suppose that the result $(a)$ will change if the angle $(34)$ is introduced into the determination of $x$.

It was established above that the results of adjustment by the method of least squares can be represented in the form of an arithmetic mean of dependent values with weights satisfying the requirement of greatest weight. Formulas (14), (19), and (20) give us the possibility of arriving at the determination of the most probable value of $x$ by using all the values of the observed angles.

Let us write the expressions for the angle $x$, imposing the condition that the value

\[ x=[p_i x_i]\quad \text{when } [p]=1 \]

be a function of all the measured values of the angles; then

\[ x=(12)p_1+\{(13)-(23)\}p_2+\{(14)-(24)\}p_3+\{(14)-(23)-(34)\}p_4; \tag{c} \]

\[ x=p_1(12)+p_2(13)+(p_3+p_4)(14)-(p_2+p_4)(23)-p_3(24)-p_4(34) \]

\[ m_x^2=p_1^2+2p_2^2+2p_3^2+3p_4^2+2p_2p_4+2p_3p_4 \tag{d} \]

We shall seek the minimum of the function $(d)$ under the condition that

\[ [p]=1 \tag{e} \]

Then, reducing the right-hand sides by 2, we obtain:

\[ \frac{\partial(m_x^2)}{\partial p_1}=p_1=k; \qquad \frac{\partial(m_x^2)}{\partial p_3}=2p_3+p_4=k; \]

\[ \frac{\partial(m_x^2)}{\partial p_2}=2p_2+p_4=k; \qquad \frac{\partial(m_x^2)}{\partial p_4}=p_2+p_3+3p_4=k. \]

Whence it follows:

\[ p_2=p_3;\qquad p_3=\frac{k}{D} \begin{vmatrix} 1 & 1\\ 1 & 3 \end{vmatrix} =\frac{k}{2}; \qquad p_4=\frac{1}{D} \begin{vmatrix} 2 & k\\ 2 & k \end{vmatrix} =0. \]

Substituting the found values of $p$ into equation $(e)$, we find:

\[ k=0.5;\qquad p_1=0.5;\qquad p_2=p_3=0.25. \]

Then formula $(c)$ may be reduced to the usual form:

\[ x=(12)p_1+\{(13)-(23)\}p_2+\{(14)-(24)\}p_3. \]

According to $(d)$

\[ m_x^2=\frac{1}{2};\qquad P_x=2=\frac{n}{2}, \]

i.e., we have obtained a result coinciding with the solution by formula $(a)$, if the weight of the observed angle is taken equal to unity.

In this example we have shown that the proposition we have put forward and justified, concerning the possibility of using dependent values with weights satisfying the requirement of greatest weight, gives us greater scope in investigations. In this same example we shall show that the proposition we have put forward makes it possible to solve problems that cannot be solved by the usual devices of the method of least squares. Thus, in the preceding example we considered the angles as directly observed quantities. It is therefore very important, both theoretically and practically, to show that we obtain the same result even without the indicated assumption. For convenience of notation let us number the angles as follows:

\[ \tag{f} \]

Let \(a_i, b_i, c_i, d_i\) denote the directions read in set \(k\) when measuring angle \(i\).

Let us write the expression for the required angle \(x\) as a function of all the observed directions and the factors \(p\).

\[ \begin{aligned} x={}&(b_1-a_1)p_1+\bigl[(c_2-a_2)-(c_4-b_4)\bigr]p_2 \\ &+\bigl[(d_3-a_3)-(d_5-b_5)\bigr]p_3 \\ &+\bigl[(c_2-a_2)+(d_6-c_6)-(d_5-b_5)\bigr]p_4; \end{aligned} \tag{g} \]

\[ \begin{aligned} x={}&p_1b_1-p_1a_1+(p_2+p_4)c_2-(p_2+p_4)a_2-p_2c_4+p_2b_4 \\ &+p_3d_3-p_3a_3-(p_3+p_4)d_5+(p_3+p_4)b_5+p_4d_6-p_4c_6. \end{aligned} \]

If the mean square error of a direction is taken equal to unity,

\[ m_x^2=2p_1^2+4p_2^2+4p_3^2+6p_4^2+4p_2p_4+4p_3p_4 . \tag{h} \]

We shall seek the roots of the equation \([p]=1\) under the condition \(m_x^2=\min\); then, canceling \(2\), we obtain:

\[ 2p_1=k;\qquad 4p_2+2p_4=k \]

\[ 4p_3+2p_4=k \]

\[ 2p_2+2p_3+6p_4=k \]

whence it follows:

\[ p_2=p_3;\quad p_2=\frac{k}{D} \begin{vmatrix} 1&0&2\\ 1&4&2\\ 1&2&6 \end{vmatrix} =\frac{k}{4}; \quad p_4=0. \]

Substituting the found values of \(p\) into \([p]=1\), we find: \(k=1\), and

\[ p_1=\frac{1}{2};\quad p_2=p_3=\frac{1}{4};\quad p_4=0. \]

Taking into account the numbering of the angles \((f)\), expression \((g)\), if all the sets at the given station are included in the calculation, leads to the value of \(x\) determined by expression \((a)\). Substituting the values of \(p\) into \((h)\), we obtain:

\[ m_x^2=1;\qquad P_x=1. \]

If, however, the weight of a direction is taken to be equal to 2, then we obtain the previous result

\[ P_x = 2 = \frac{n}{2}. \]

Thus, on the basis of the investigation carried out, we have come to the conclusion that formula \((a)\), in whose derivation the angles, and not the directions, were taken as the observed quantities, gives results that are rigorous from the point of view of the method of least squares. From this same investigation it is clear that the formulas given by us can be used in studying questions of extracting the most probable results in cases where the application of the usual procedures of the method of least squares is either difficult or impossible.

2. Adjustment of directions observed by the method of circular sets

Suppose that at some station equal-precision observations of \(n\) directions have been made by \(m\) complete sets from one initial direction.

Let in set \(i\) the following readings have been obtained: \(a_i, b_i \ldots n_i\), corresponding to the directions \(A, B, \ldots N\).

On the basis of the new interpretation, the most probable value of the angle \(X\) is written as:

\[ x^\circ = [(b_i-a_i)p_i], \quad (i=1.2\ldots m) \tag{23} \]

where the \(p_i\) are subject to the condition \(m_x^2=\min\) and

\[ |p| = 1 \tag{a} \]

If we put \(m_{a_i}^2=m_{b_i}=\ldots=1\), then

\[ m_x^2 = 2[p^2] \tag{24} \]

Multiplying \([p]\) by \(4k\), differentiating \(4k[p]\) and (24) with respect to the variables \(p\), and equating the partial derivatives, we obtain:

\[ p_i = k \]

Substituting \(p\) in \((a)\), we shall have: \(mk=1\);

\[ p_i=\frac{1}{m} \tag{b} \]

Then the most probable result for the angle \(x\) is written in the form:

\[ x^\circ=\left[(b_i-a_i)\frac{1}{m}\right] =\frac{1}{m}\bigl([b]-[a]\bigr) \tag{25} \]

In exactly the same way we could obtain the values \(y^\circ, z^\circ \ldots\).

From the investigation presented it is clear that the application of the new interpretation has enabled us to obtain the well-known result by an extremely simple route.

The most probable value of the angle \(\omega\) in set \(i\) can also be represented in the form:

\[ \omega_i=a_i p_1+(b_i-x^\circ)p_2+(c_i-y^\circ)p_3+\ldots+(n_i-N^\circ)p_n \tag{26} \]

where by \(x^\circ, y^\circ\) are meant the most probable values of the angles, or

\[ \omega_i=a_i p_1+\left\{b_i-\frac{1}{m}\bigl(|b|-|a|\bigr)\right\}p_2+\cdots+ \left\{n_i-\frac{1}{m}\bigl(|n|-|a|\bigr)\right\}p_n. \]

Writing the expression \(m^2\omega_i\), after simple algebraic transformations we obtain:

\[ \omega_i=\frac{a_i+b_i+\cdots+n_i}{n}-\frac{x^\circ+y^\circ+\cdots+N^\circ}{n}, \]

which agrees with the result obtained from the equations of errors. It should be noted that the most probable value of the angle \(\omega_i\), i.e. the most probable value of the reading on object \(A\) in the set with number \(i\), is obtained more simply by the usual route. We have given this investigation in order to show that the sphere of application of the new interpretation can be extended, since in expression (26) the values \((b_i-x^\circ)\), \((c_i-y^\circ)\ldots\) are not the observed values \(\omega_i\), because \(x^\circ, y^\circ\ldots\) are the adjusted values of these angles.

Conclusions.

The results of the investigation may be formulated as follows:

1. The method of least squares is defined for a certain mean series of observations; the given series of observations is used by it regardless of the distribution of the errors in this series.

2. Theorem 1. The most probable value of each of \(n\) observed quantities connected by \(r\) exact conditions \((n>r)\) can be represented in the form of an arithmetic mean of the result of the direct measurement of the required quantity and of dependent values of it delivered by each conditional equation (or obtained from \(r\) simultaneous solutions of equations), with weights satisfying the requirement of the greatest weight of the quantity sought.

3. Theorem 2. The most probable value of each of \(m\) functions of direct observations, connected with the direct observations by \(n\) equations \((n>m)\), can be represented in the form of an arithmetic mean of dependent values obtained from the solution of the necessary and sufficient \((n-m+1)\) number of combinations of equations, with weights satisfying the requirement of the greatest weight of the quantity sought.

4. These theorems have theoretical and practical consequences and broaden the field of applications of the method of least squares.

5. The weights with which the values of the required quantity enter into the arithmetic mean make it possible also to compute the weight of its adjusted value.

6. The arithmetic mean of independent values is a special case of the principle underlying the method of least squares: the principle of the greatest weight of the quantity sought.