DETERMINATION OF CORRELATION CHARACTERISTICS USING THE MINSK-1 ELECTRONIC COMPUTER

G. L. Kalinichenko, L. V. Baklanova, S. S. Sulakshin

(Submitted by the Department of Engineering and Computational Mathematics)

All calculations connected with determining the correlation coefficient and the correlation ratio can be programmed and carried out on the Minsk-1 electronic digital computer.

The preparation of a program for each separately taken problem and the preparation for direct computation on the machine at first require considerable time. In this connection, programming for establishing the form and degree of dependence between two random variables should expediently be carried out in such a way that the compiled program can be used for any computations of this type without changes, except for the initial data.

Thus, in the present case this should be a program with the largest number of initial data, which are elements of a matrix of the form

\[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots\\ a_{m1} & a_{m2} & \cdots & a_{mn}, \end{matrix} \]

where \(m\) is the number of rows,

\[ n \]

is the number of columns.

When compiling a program with the greatest number of initial data (without using external memory), one may take a matrix of type

\[ m \times n = 25 \times 20. \]

If the initial data constitute a matrix of type \(q \times l\), where \(q \times l \le 500\), then the program will be the same; only the auxiliary numbers selected for comparison will be changed in it.

The initial data (matrix elements) are placed in the following manner (Table 1). The matrix element located in the \(k\)-th row and \(j\)-th column of correlation Table 1 is placed in cell \(a\)

\[ a + (k-1)n + j, \]

where \(n\) is the number of columns in the matrix,

\[ a \]

is a certain constant determining the initial address of the array.

Thus a matrix can be obtained whose elements will be the numbers of the cells containing the initial data placed in them. The matrix with the largest number of initial data, \(m \times n = 25 \times 20\), will have the following form (Table 1). Here \(p, p+1, \ldots, p+763\) are cell numbers taken in the octal number system.

Table 1

The possible values of the random variables \(x_q, y_p\) are the components (coordinates) of the column vector \(Y\) and the row vector \(X\) of the correlation table [1]. Therefore their possible values are placed in cells having consecutive numbers:

for the column vector \(Y\):
\[ l,\quad l+1,\quad l+2,\ldots,\quad l+23 \]

for the row \(X\):
\[ q,\quad q+1,\quad q+1,\ldots,\quad q+30. \]

When this program (Table 1) is used to compute the relationship between other random variables, only the contents of all the above-listed cells will change.

To compile the complete correlation table [1], the \(j+1\)-th column and the \(k+1\)-th row are filled. In other words, it is necessary to obtain a column vector and a row vector.

The column vector \(Y_{k+1}\), for \(m \times n = 25 \times 20\), is the sum of the column vectors

\[ \sum_{k=1}^{k=20} y_k, \]

and the row vector

\[ X=\sum_{q=1}^{q=25} x_q, \]

where the column vector \(y_k\) is any \(j\)-th column of the correlation table containing the frequencies \(n_{zq} y_k\) corresponding to a particular value of the attribute \(y_k\), for different values of the attribute \(x_q\). The computation of the \(y_{k+1}\) column is carried out by means of a program containing the following commands (the first section of the program, Table 2).

where cells $m$, $m+1$, $m+2$ contain auxiliary numbers needed for organizing the cycle; in cells $r$, $r+1$, $r+2,\ldots,r+30$ the components of the column vector are obtained. The operation of this program is characterized in more detail as follows.

Command $k$ clears cell $r$, i.e., sends 0 into cell $r$. The following four commands form a cyclic program, which obtains one of the components of the column vector in the following way.

Command $k+1$ adds the contents of cell $p$ to the contents of cell $r$ and sends the result to cell $r$, which is thus selected so that the sum of the components is accumulated in it.

Command $k+2$ performs redirection of command $k+1$, i.e., increases the first address of the command by 1; the redirection constant is in cell 0011 and has the form $+00\ 0001\ 0000$.

Command $k+3$ forms the condition, depending on which command $k+4$ transfers control to command $k+1$ to continue the cycle or to $k+5$.

In order to use this same cycle to obtain the second and subsequent components, it is necessary to redirect commands $k$ and $k+1$, i.e., to increase their second addresses by 1 and, in the constant $m$, to increase the first address by 24 units. This is carried out by means of three commands: $k+5$, $k+6$, $k+7$.

Command $k+10$ checks the condition, depending on which it transfers control to command $k$ for the start of the outer cycle or to $k+12$ to continue the computation.

Thus, this section of the program performs the addition of rows (finds the frequencies) and places them in successive cells $r$, $r+1$, $r+2,\ldots,r+30\ldots$

The second section of the program, from command $k+12$ through $k+25$, fills column $k+4$ of the correlation table and sends them into cells. This section of the program is constructed analogously to the first, but with the sole difference that one command, $k+13$, is added, which obtains the product $n_{xy}\cdot y_k$.

The third section of the program finds the frequencies and places them in cells $a$, $a+1$, $a+2,\ldots,a+23$. This section of the program is constructed analogously to the first section; only in this case the redirection of the variable command $k+27$ increases the first address not by 1, but by 24.

The fourth section of the program, from command $k+41$ through $k+56$, obtains row $q+4$ of the correlation table and places it in cells $b$, $b+1$, $b+2,\ldots,b+23$.

The fifth section of the program finds $N=\Sigma n_{xy}$ and sends the number found into cell $\alpha$.

The sixth and seventh sections of the program give the totals of column $j+4$ and row $i+4$ and send them respectively into cells $\alpha+1$ and $\alpha+2$.

The eighth section of the program obtains the total of row $i+3$ of the correlation table. This computation is carried out by commands from $k+76$ through $k+105$, where $k+76$ clears cell $\alpha+3$, in which the total of row $i+3$ will be accumulated. $k+77$ obtains $y^2$, $k+100$—$y^2 n_y$ (the total of row $i+3$), $k+101$ adds the result of the preceding operation to the contents of cell $\alpha+3$ and places the sum in $\alpha+3$. $k+102$ and $k+103$ are commands

readdressing, \(k+104\) forms a condition, depending on which \(k+105\) transfers control to \(k+77\) or to \(k+106\).

The ninth section of the program obtains the total of the \(i+7\)-th row and sends it to cell \(a+4\).

The tenth section of the program fills the \(i+8\)-th row of the correlation table and prints it, i.e., gives the coordinates of the empirical regression line of \(x\) on \(y\).

The eleventh section of the program obtains the total of the \(j+3\)-rd column and sends it to cell \(a+5\). The twelfth section of the program gives the total of the \(j+5\)-th column of the table and sends it to cell \(a+6\).

The thirteenth section obtains the total of the \(j+7\)-th column. The fourteenth section of the program fills the \(j+8\)-th column and prints it, i.e., gives the coordinates of the empirical regression line of \(y\) on \(x\).

Program sections 15 through 21 respectively find

\[ \bar{x},\ \bar{y},\ \sigma_x,\ \sigma_y,\ r,\ \eta_{y/x},\ \eta_{x/y}. \]

When computing \(\sigma_y\), the commands \(k+163\), \(k+164\), and when computing \(\sigma\), the commands \(k+172\), \(k+173\), are commands for calling the standard square-root computation program.

The program is composed for computation with a matrix of size \(20 \cdot 25\) for a fixed-point machine, for which all numbers must be less than 1 in absolute value. Therefore, for all initial data \(x_i,\ y_k,\ n_{xy}\), a scale must be chosen.

Table 2

Program for computing correlation characteristics using the Minsk-1 electronic computer in conditional addresses

Continuation of Table 2

Continuation of Table 2

Continuation of Table 2

Auxiliary numbers