https://doi.org/10.31449/inf.v46i4.3984 Informatica 46 (2022) 551–556 551 
 
Estimation of Parameters in Regression Analysis Based on QR 
Decomposition of Rectangular Matrices by Householder Reflections  
Oleksandr Dorokhov*
1
, Lyudmyla Malyarets
2
, Kadri Ukrainski
1
 and Dmytro Yevstrat
3 
1
Chair of Public Economics and Policy, University of Tarty Narva mnt 18, 51009, Tarty, Estonia 
2
Department of Higher Mathematics and Economic and Mathematical Methods, Simon Kuznets Kharkiv National 
University of Economics Nauka Ave 9A, 61166, Kharkiv, Ukraine 
3
Department of Information Systems, Simon Kuznets Kharkiv National University of Economics Nauka Ave 9A, 
61166, Kharkiv, Ukraine 
E-mail: oleksandr.dorokhov@ut.ee, malyarets@ukr.net,  kadri.ukrainski@ut.ee, dmitryyevstrat@gmail.com 
*Corresponding author 
 
Keywords: regression analysis, Householder reflection, QR decomposition  
Received: February 04, 2022 
An approach to eliminate multicollinearity problems in regression analysis using QR decomposition of 
rectangular matrices by Householder reflection has been proposed. The reliability of this computational 
procedure has been proved. 
Povzetek: V regresijski analizi je narejena ocena parametrov s pomočjo dekompozicije QR. 
 
1 Introduction 
In substantiating the decision taken in the management of 
various socio-economic systems, it is important to 
choose an analytical tool that analyzes the state of 
systems and predicts their further development.  
In the presence of various kinds of uncertainties and 
a large amount of various data, in the problems of 
economic modeling, two main approaches are most often 
used recently: fuzzy modeling [1-5] and various 
statistical methods [6-8]. 
Among the last, most often, multiple regression 
analysis is used as such a tool [9-13].  
Despite its prevalence in economics and duration in 
applications, many problems still need to be solved, since 
the existing algorithms for multiple regression analysis 
are far from perfect [14-17]. 
To solve many problems of mathematical methods in 
economics, the so-called QR decomposition of 
rectangular matrices is useful [18-21]. 
An n×m matrix X is decomposed into two factors X 
= QR, where Q is an n×n orthogonal matrix and R is an 
upper triangular n×m matrix with zero entries below the 
main diagonal. 
Recall that the determinant of the orthogonal matrix 
is equal to one |Q| = 1; its inverse matrix coincides with 
the transposed Q
–1
=Q  (i.e. QQ = Q Q = I), and 
orthogonal transformation of vectors a, b does not 
change their scalar products: (Qa,Qb) = (a,b); 
|Qa| = |a| = a; |Qb| = |b| = b.  
It can be argued that numerical algorithms with 
orthogonal transformations do not introduce additional 
errors into the solution of the problem. 
2 QR decomposition to solve the 
problem of multicollinearity 
Consider how QR decomposition helps to overcome the 
problem of multicollinearity in regression analysis. 
Suppose we need to find the best estimates of the 
parameters b of a linear three factor regression model 
from 6 observations (n = 6; m = 3+1 =4): 
Y = Xb + E, 
which in expanded form reduces to solving the overvalue 
value of a system of 6 linear equations with respect to 4 
unknowns (b 0 , b 1 , b 2 , b 3):  
y 1 = b 0 + b 1 x 11 + b 2 x 21 + b 3 x 31 + e 1 
y 2 = b 0 + b 1 x 12 + b 2 x 22 + b 3 x 32 + e 2 
y 3 = b 0 + b 1 x 13 + b 2 x 23 + b 3 x 33 + e 3 
y 4 = b 0 + b 1 x 14 + b 2 x 24 + b 3 x 34 + e 4 
y 5 = b 0 + b 1 x 15 + b 2 x 25 + b 3 x 35 + e 5 
y 6 = b 0 + b 1 x 16 + b 2 x 26 + b 3 x 36 + e 6 
 
Here Е – vector of errors (residuals e i ), which can be 
found after determining the estimates of the model 
parameters b 0 , b 1 , b 2 , b 3 . 
If the QR decomposition of the matrix Х = QR is 
known, then the above problem is solved as follows.  
Should multiply the matrix equation Y = QRb + E 
leftward to the orthogonal matrix Q  and we obtain an 
equivalent equation Z = Rb +  , where marked Z = Q Y, 
 = Q E.  

552 Informatica 46 (2022) 551–556 O. Dorokhov et al.  
 
 
In expanded form we have a system of linear equations 
with a triangular matrix R: 
 
z 1 = b 0 r 01 + b 1 r 11 + b 2 r 21 + b 3 r 31 +  1 
z 2 = b 1 r 12 + b 2 r 22 + b 3 r 32 +  2 
z 3 = b 2 r 23 + b 3 r 33 +  3 
z 4 = b 3 r 34 +  4 
z 5 =  5 
z 6 =  6 
 
Due to the orthogonality of the matrix 
transformation, the following relation is preserved 
||  || = ||E||, that is, the sum of the squares of the 
converted errors 

2
i
 (vector norm ) is always equal 
to the sum of the squares of the residuals of the original 
system of equations 

2
i
e (vector norm Е).  
Due to the successful determination of the model 
parameters b i  you can equate to zero the first few 
components of the vector   and obtain the minimum 
value of the sum of the squares of the residuals  
 
min e
i
→

2
.  
 
So it is possible to obtain estimates of the parameters 
of the model by the least squares method (but in a 
slightly different, non-standard computational way, 
without first drawing up a system of normal equations). 
If the last diagonal element of the triangular matrix R 
is nonzero r 34 ≠ 0, then you can equate to zero the 
maximum number (m) of the first components  i and 
find estimates of the model parameters from the 
triangular system of equations. 
This also determines the (minimum) sum of the 
squares of the residuals 
 
       
   
+ = + = = =
= = =
n
m i
i
n
m i
i
n
i
i
n
i
i
z e
1
2
1
2
1
2
1
2
  . 
 
If the last diagonal element of the triangular matrix R 
is exactly equal to zero r 34 = 0, then this means that the 
last variable х 4  is not independent, but it is a linear 
combination of other argument variables    
х 4 = а 0 + а 1 х 1 + а 2 х 2 + а 3 х 3 .  
In this case, the estimate b 4 = 0 should be equated to 
zero.  
That is, it is not necessary to include in the model a 
combination of already taken into account variables, and 
estimates of other parameters of the model should be 
obtained from the conditions of zero of the smaller 
number of the first components  i = 0.  
If the last diagonal element of the triangular matrix R 
is not exactly zero, but close to it |r 34|  0, this means that 
there is a multicollinear relationship between the source 
variables x i .  
In this case, it makes sense to equate the score b4 = 0 
to zero and remove the questionable term from the model 
(otherwise an unstable solution with large errors will be 
obtained). 
So, if we have a QR decomposition of the matrix X, 
the problem of multi-linearity is solved quite simply. 
At the same time, the strong side of the QR 
decomposition is that it allows you to calculate (find a 
numerical) solution to the least squares problem.  
As is well known, the classical, ordinary least 
squares method gives us a closed solution in the form of 
normal equations. But this solution is not always suitable 
for practical, specific applications.  
Therefore, if you need to find the actual numerical 
solution, the least squares method is not suitable, at the 
same time, the QR decomposition easily gives such 
numerical values. 
It should be noted that today the most well-known 
and used methods for obtaining an orthogonal matrix Q 
are the Gram-Schmidt process, the Householder 
transformation, and the Givens rotation. 
An orthogonal matrix Q (and a triangular matrix R) 
can be obtained by successive operations with matrices 
H k = I – 2  k  k , where I – single matrix,  k – normalized 
vector (  k   k=1), in which the first (k-1) components are 
zero. 
It's not hard to see that H k H k = H kH k = I, that is, the 
matrix H k is orthogonal. This matrix is also called as the 
Householder reflection matrix. 
Any vector can be represented as a = a 1  + a 2 , 
where ,  – orthonormal vectors (    = 0;    = 1; 
   = 1).  
Householder reflection Нa = –a 1  + a 2   changes 
the sign of the first component to the opposite. 
We have to take into account some important 
properties of the Householder matrix: it is Hermitian 
H=H* and unitary HH*=I, and therefore it is an 
involution H
2
=I. In this case, the transformation H u(x)  
displays (reflects) point x to point x-2(u, x)u.  
The Householder matrix has one eigenvalue equal to 
(-1), which corresponds to the eigenvector u, while all 
other eigenvalues are equal to (+1).  
In this case, the determinant of the Householder 
matrix is (-1), and the Householder transformation in the 
metric space preserves distances. 
3 Converting a rectangular matrix 
to upper triangular 
Consider the process of sequential transformation of a 
rectangular matrix А = [a 1 , a 2 , … , a m] to the upper 
triangular shape (possibly with permutations of 
columns). 
The first transformation with a matrix Н 1  must 
transform the first vector a 1  (the first column of the 
matrix А) to vector а 1 е 1 , where е 1 a coordinate vector 
in which only one first element is nonzero (this element 
is equal to 1, the other elements are zero); as а 1 is marked 

Estimation of Parameters in Regression Analysis Based on…         Informatica 46 (2022) 551–556   553 
length of the vector а 1 (because the orthogonal 
transformation does not change the length of the vector).  
Consider the approach of how to find a vector  1 , 
which defines the whole matrix Н 1 .  
To do this, consider that again (re-)reflection restores 
vector а 1 : 
а 1 = Н 1 ( а 1 е 1) = (I - 2  1  1 )( а 1 е 1) = а 1 е 1 +2а 1  11  1 , 
hence the vector  1  proportional to the vector а 1 , to the 
first component of which a value а 1 is added; sign (+ or 
–) should be selected by the sign of the element а 11 .  
So we get a (still normalized) vector  1  in the form 
 1 = a 1 + a 1 sgn(a 11) e 1 .  
The square of the length of this vector is equal to 
 1   1 = 2a 1 (a 1 + |a 11|).  
Thus the first Hausholder matrix is constructed 
𝐻 1
= 𝐼 −
𝛺 1
𝛺 1
′
𝑎 1
( 𝑎 1
+ | 𝑎 11
| )
.  
In the transformed matrix А in the first column will 
have only one non-zero element: H 1 a 1 = a 1 -  1 = 
= -a 1 sgn(a 11) e 1 .  
Other converted columns of matrix А (and the 
dependent variable column Y) have the form: H 1 a j = a j  –
  j  1 , where  j – numerical coefficients 𝜆 𝑗 =
( 𝑎 1
, 𝑎 𝑗 ) + 𝑎 1
⋅ 𝑠𝑔𝑛 ( 𝑎 11
) ⋅ 𝑎 𝑗 1
𝑎 1
( 𝑎 1
+ | 𝑎 11
| )
  
The first elements of the converted columns will no 
longer change in subsequent ones Householder 
reflections.  
Without these first elements of the norms of all 
vectors (columns of the transformed matrix А) are 
reduced on the value 
2
jk
a (by now k = 1).  
Let find the column a q  with the highest residual 
norm (relative to the original norm): 








−
=
−
2
2 2
2
2 2
max
j
jk j
q
qk q
a
a a
a
a a
.  
If the largest relative residual norm is less than some 
limit value (for instance, 0,01), further transformations 
are canceled, the process ends prematurely due to the 
detection of multicollinear connections. 
In the second stage we find the matrix Н 2 , which 
transform vector a q  (without first component) to vector 
(а q)*е 2 , where е 2  – the second coordinate vector; by 
(а q)* marked shortened vector а q length without its first 
component.  
Thus, after the second transformation, the vector а q 
will have two non-zero components. 
Then again we find the residual norms and determine 
the next vector а р , which (without the first two 
components) will be converted to (а р)*е 3 . 
The process will end after m iterations, or early if the 
residual norms become less than accepted limit level. 
Usually in regression analysis the first column of the 
matrix Х (matrix А) there is a column of ones х 0  1 (to 
take into account in the regression model the obligatory 
free term).  
Therefore, at the first stage (of Householder reflections), 
it is the first The order of selection of the following 
columns for conversion is determined by the values of 
their relative residual norms. 
4 An example of the implementation 
of the algorithm 
The described above algorithm was implemented in an 
electronic spreadsheet Excel in the macros form on the 
VBA language (Visual Basic for Application).  
Consider numerical example of the analysis of the 
regression dependence of the resultant feature on 5-
factors, the corresponding initial data for this are 
presented in Table.1. 
 
Table 1. Data for regression analysis 
№ у x 0 x 1 x 2 x 3 x 4 x 5 
1 1075,3 1 32,06 17,9 12,08 35,61 8,33 
2 1002,7 1 27,57 10,23 14,06 37,48 10,63 
3 995,6 1 27,88 10,29 11,26 37,77 12,72 
4 872,4 1 31,65 11,72 7,32 34,98 14,25 
5 909,3 1 34,81 12,64 1,68 39,04 11,75 
6 1009,9 1 29,47 10,87 1,31 46,14 12,15 
7 919,7 1 34,42 12,77 1,28 41,04 10,48 
8 876,2 1 32,76 12,26 1,06 42,53 11,32 
9 908,6 1 31,24 11,65 4,49 41,27 11,28 
10 935,8 1 30,4 11,33 6,88 40,07 11,26 
11 949,9 1 29,96 11,18 8,84 39,48 10,5 
12 927,4 1 30,49 11,41 7,73 39,55 10,78 
13 1003,9 1 29,71 11,05 13,08 29,46 16,68 
14 1017,6 1 29,02 10,79 14,34 29,06 16,77 
15 997,6 1 29,55 10,99 11,75 30,07 17,63 
16 958,2 1 30,79 11,44 7,94 31,29 18,54 
17 907,5 1 32,55 12,08 1,45 33,03 20,87 
18 928 1 33,27 12,35 1,41 32,32 20,63 
19 930,1 1 35,34 13,42 0,76 32,24 18,23 
20 892,6 1 33,71 12,79 0,78 33,58 19,13 
21 917,7 1 32,3 12,03 4,26 32,68 18,7 
22 947,2 1 31,32 11,64 6,99 31,65 18,38 
23 959,6 1 30,97 11,55 8,94 31,24 17,3 
24 943,4 1 31,52 11,7 7,63 31,64 17,5 
 
 
By using Householder reflections matrix Х was 
transformed into a triangular form (to matrix R = Q X).  
The process of transformation ended prematurely.  
Variables х 1, х 4 was not connected in the model since 
their residual norms decreased to values of about 0.1% of 
the original norms as shown in Table 2 and Table 3. 
Let move converted columns х 1 and х 4 (with small 
relative residual rates less than 0.01) to the left to the 
column Z.  
As result, we can obtain a system of 4 equations with 
3 columns of free terms (Z, х 1 and х 4 ) relative to 4 
parameters of the considered model b 0 , b 2 , b 3 , b 5  as 
shown in Table 4. 

554 Informatica 46 (2022) 551–556 O. Dorokhov et al.  
 
 
 
Table 2: Data after householder reflections 
№ z = Q' у  x0 x1 x2 x3 x4 x5 
1 -4651,2 -4,8 -153,6 -58,3 -32,1 -174,1 -72,6 
2 -168,7  7,6 1,4 -22,2 8,3 3,2 
3 -18,0  0,7 -1,4  -19,2 18,2 
4 62,9  4,5 6,8  -6,2  
5 -3,9  2,0   -2,5  
6 135,5  -2,4   3,4  
7 5,7  1,6   -2,1  
8 -24,3  0,1   -0,2  
9 -12,4  -0,1   0,1  
10 -1,6  -0,0   -0,0  
11 -4,2  0,2   -0,4  
12 -20,3  0,3   -0,5  
13 21,9  0,5   -0,7  
14 28,9  0,3   -0,4  
15 30,8  -0,1   0,1  
16 20,0  -0,3   0,5  
17 21,8  -1,1   1,7  
18 37,0  -0,6   0,8  
19 20,4  1,0   -1,6  
20 -3,4  -0,3   0,3  
21 3,0  -0,2   0,4  
22 13,8  -0,2   0,3  
23 7,9  0,1   -0,2  
24 1,4  0,2   -0,3  
 
Table 3: Norms 
Output 
21691,3 24 23707,1 3461,7 1524,9 30845,7 5617,9 
Residual 
24827,4 0 16,928 0 0 31,7126 0 
Relative 
0,00114 0 0,00071 0 0 0,00102 0 
 
Table 4: System of equations with a triangular 
matrix 
у x1 x4 x0 x2 x3 x5 
-4651,2 -153,6 -174,1 -4,8 -58,3 -32,1 -72,6 
-168,7 7,6 8,3  1,4 -22,2 3,2 
-18,0 0,7 -19,2  -1,4  18,2 
62,9 4,5 -6,2  6,8   
 
The solution to this system is given in Table 5. 
Explanatory variables are shown in the model x 2, x 3, x 5 . 
Other variables (including у) are expressed through these 
explanatory variables. 
 
Table 5: Inverse matrix and system solution 
Model parameters Inverse matrix 
у x1 x4 x0 x2 x3 x5 
790,78 24,03 67,07 -0,20 0,29 -0,86 -1,97 
9,14 0,65 -0,90    0,14 
8,14 -0,28 -0,60  -0,04 0,01 0,01 
-0,25 0,09 -1,13   0,05 0,01 
5 The results of calculations 
As a result, the following regression models can be 
obtained (together with the coefficients of 
determination): 
у = 790,78 + 9,14 х 2 + 8,14 х 3 – 0,25 х 5 ; R
2 
= 0,56 
х 1 = 24,03 + 0,65 х 2 – 0,28 х 3 + 0,09 х 5 ; R
2 
= 0,82 
х 4 = 67,07 – 0,90 х 2 – 0,60 х 3 – 1,13 х 5 ; R
2 
= 0,93 
 
Coefficients of determination R
2 
 can be calculated, 
for example, as follows: 
 
𝑅 2
= 1 −
| | 𝒆 | |
| | 𝒚 | | − 𝑁 ⋅ ( 𝑦 ̄ )
2
= 1 − 0 , 4312 = 0 , 5688 
So we have multicollinear relationships of variables 
х 1  and х 4  with explanatory variables х 2 , х 3 , х 5  and with 
multiple correlation coefficients 
9084 , 0 8251 , 0
1
= = R
 
and  9686 , 0 9382 , 0
4
= = R , which exceeds the 
closeness of the relationship of the explanatory variables 
with the resultant feature 
7542 , 0 5686 , 0 = =
y
R
.  
Note that the matrix of paired correlation coefficients 
r xy does not show (not demonstrate) any effect of 
multicollinearity at once. 
But it turns out that the determinant of this 
correlation matrix is almost zero |r xy| = 0,0000656, that is, 
all explanatory variables are interconnected by a precise 
multicollinear relationship as shown in Table 6.  
So, with the help of QR decomposition of Х 
rectangular matrix it is possible (without first compiling 
a system of normal equations) to obtain least square 
methods’ estimates of the parameters of the regression 
model together with all multicollinear connections. 
 
Table 6: Correlation matrix 
rxy x1 x2 x3 x4 X5 y 
x1 1 0,5817 -0,7796 -0,0065 0,2102 -0,5863 
x2 0,5817 1 -0,2002 -0,0160 -0,1664 0,1259 
x3 -0,7796 -0,2002 1 -0,3694 -0,1748 0,7031 
x4 -0,0065 -0,0160 -0,3694 1 -0,7743 -0,1736 
x5 0,2102 -0,1664 -0,1748 -0,7743 1 -0,1969 
6 Analysis of the stability of the 
obtained results 
We turn to consider a very important problem of the 
stability of the results obtained from some particular 
observations, which have an excessive effect on the 
values of the model parameters. Using a matrix Q you 
can detect all such questionable observations, such as 
observation №1 in the above tables.  
If you delete the observation №1, the values of the 
model parameters will change significantly: 
With observation №1:  
у = 790,78422 + 9,14172 х 2 + 8,14859 х 3 – 0,25244 х 5 ; 
Without observation №1:   
у = 1300,5080 – 34,4492 х 2 + 1,75898 х 3 + 2,25143 х 5 . 
This unpleasant effect is shown in the graphs of 
component effects in Figure 1 (with), Figure 2 (without). 
 

Estimation of Parameters in Regression Analysis Based on…         Informatica 46 (2022) 551–556   555 
 
 
 
Figure 1: Along with observation № 1. 
 
On the graphs empirical points are superimposed on 
theoretical regression lines (calculated values). It helps to 
find out which dependencies are significant and which 
are not. Indeed, it is always possible to choose such 
scales of coordinate axes that all theoretical lines will 
have the same slope in 45 .  
The presence of empirical points (or 95% 
confidence bands) will not allow incorrect conclusions in 
this case. 
However, the question arises, how to determine 
such empirical points so that only one variable varies on 
each graph, and the rest is fixed at the average levels? 
 
 
 
 
 
Figure 2: Without observation № 1. 
 
It turns out that this is a problem whose solution is 
insufficiently covered in the literature. Therefore, we 
propose such a solution. 
After determining the parameters of the model, you 
can find the deviation of each observation from the 
theoretical values (residuals): e = y – y p .  
Now, for each observation, you can write down the 
identity: 
i i i i i
e x x b x x b x x b y y + − + − + − + = ) ( ) ( ) (
5 5 5 3 3 3 2 2 2
Members ) (
k k k
x x b − are called "component effects". 
Sum y and the corresponding component effect is 
the equation of the theoretical dependence of the 
resultant feature y on one variable x k at the average values 
of the remaining explanatory variables. 
We will fix in identities all variables (except one) 
on average values and we will receive the corrected data 
as the sum of the total mean y , the corresponding 
component effect ) (
k k k
x x b − and the residual: 
. ) ( ) (
; ) ( ) (
; ) ( ) (
5 5 5 5 5
3 3 3 3 3
2 2 2 2 2
e x x b y e x Y Y
e x x b y e x Y Y
e x x b y e x Y Y
p
p
p
+ − + = + =
+ − + = + =
+ − + = + =
 
7 Conclusions and further research 
Since the regression analysis does not end only with 
the assessment of the parameters of the model, it is 
necessary to identify all the most influential observations 
and assess their negative contribution. 
Modern mathematical theory offers methods for 
identifying such components and assessing their 
acceptability in the data. These methods are based on the 
previous QR decomposition of the data matrix. 
The extra-large amount of computing work is no 
longer an obstacle in the presence of modern computers. 
800
900
1000
1100
10 12 14 16 18
Ур ( х 2 ) Y2
800
900
1000
1100
0 5 10 15
Ур ( х 3 ) Y3
800
900
1000
1100
5 10 15 20 25
Ур ( х 5 ) Y5
800
900
1000
1100
10 12 14 16 18
Ур ( х 2 ) Y2
800
900
1000
1100
0 5 10 15
Ур ( х 3 ) Y3
800
900
1000
1100
5 10 15 20 25
Ур ( х 5 ) Y5

556 Informatica 46 (2022) 551–556 O. Dorokhov et al.  
 
 
Comparing the results of calculations of multifactor 
regression analysis by the method of QR decomposition 
of rectangular matrices by House-Holder mappings with 
the calculations performed in the StatGraphics package, 
we obtained a 100% match. 
Checking the significance of the parameters of the 
equations by the student’s criterion shows that not all of 
them are significant:  
 
y = 899,649 + 7,59366*x 3; 
x 1 = 26,1328 + 0,60714*x 2 - 0,305855*x 3; 
x 4 = 67,0703 - 0,906909*x 2 - 0,600053*x 3 - 1,13156*x 5. 
 
In the following, we can consider how the 
procedures for estimating the parameters of the equation 
can be performed in the StatGraphics package. 
Summarizing all the above, as a conclusion, we can 
say that the use of QR decomposition matrices has 
significant advantages over the standard procedure of the 
least squares’ method in the presence of multicollinearity 
of data and is a reliable computational procedure. 
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