Looking at the quality of the regression equation, two main indicators should be distinguished: accuracy and reliability. The precision of the regression equation is characterized by the mean square deviation (standard deviation) of the table data from the regression equation values at the corresponding points - the smaller the deviation, the higher the accuracy. By increasing the number of coefficients of the regression equation and thus making the equation more complicated, it can unlimitedly reduce the deviation of the output data from the equation. In a boundary case, when the number of coefficients of the equation coincides with the number of output data, it is possible to achieve a complete match of the output data with the values calculated using this regression equation. This regression equation is unlikely to have any meaning because it is unlikely to have good predictive properties: the values calculated in the intersections of the parameter area of the object from actual ones can vary unforgivably.
The reliability of the regression equation shall be characterized by the extent to which the deviations calculated at starting-points correspond to the inter-point deviations of the object's parameter room. It becomes understood that the less the number of regression equation coefficients, the higher the credibility of the equation. The good compliance of the regression equation with the output data at a small number of regression coefficients (relative to the set points) indicates that this harmony was achieved due to the structure of the regression equation itself, but not to the recovery of coefficients. Thus, the key indicators of the regression equation are contradictory: one improvement leads to a deterioration of the other.
Synthesis of the regression equation
In practice, the following task appears quite often: information about the object is given in the form of a table. Assuming that there is some relationship between the object's parameters and response (feedback), it is necessary to express this relationship mathematically - it is to create an object regression model based on the table data. Existing regression analysis methods, as a rule, require the existence of a regression equation with accuracy up to coefficients, the determination of which also constitutes the main computational work. However, in most cases the structure of the regression equation is unknown a priori. In this case, the use of regression analysis methods is difficult, especially the determined links between the parameters and feedback of the object. The proposed regression synthesis method does not require a priori knowledge of the regression equation structure. As previously mentioned, the approximation function is not predefined, but is synthesized in the process of processing information. The mathematical expression has the following form:
m
Y = ⅀ [Bi * F(X)]
i=1
where,
m - number of expression members;
Bi - coefficients;
F - elementary function;
X - vector of factors.
The elementary function is constructed as follows:
NX
F(X)= П[Aj+Bj*Xj]*Lk,j
J=1
where,
NX - number of factors;
Aj, Bj - coefficients;
Lk,j - integers that can accept both positive and negative values, including zero.
The algorithm provides both an automatic selection of the elementary function depending on the number of factors and the results of the information analysis, as well as the analysis of the influence of individual factors and the exclusion of non-essential factors. For more detailed information, please see [10, 11].
"USA Temperature" data example file "RES.DAT" listing below:
;OS RSSSP/101E R E S I N T Version SE 4.44(20210917v0) (C)Copyright 1994-2021 RANDEC Ltd
;F:\RSSSP\RESINT\DAT\RES.DAT Sun 19-SEP-2021 13:59:57.633
;
; You must prepare such 8 lines above data table:
;
; 1. "OBN/OBJ"-Object(s) name(s) (up to 16 chars each) separated by "|"(#124=$7C)
; character or horizontal tabulator (#9=$09)
;
; 2. Miscellaneous parameters:
;
; 1-"YNU" : Number of functions (up to 30..39)
; 2-"XNU" : Number of variables (up to 10)
; 3-"MNU" : Magic Number (0-without MGN: "MNU"="PNU")
; 4-"ELN" : Maximum length of equation (up to 20 members, 0-all)
; 5-"VNU" : Number of variants for printing (0-all)
; 6-"DVN" : Number of deviations for printing (0-all, if "DVN" < 0 then in
; addition to deviations printing, writes actual function and data
; values at the beginning of result file, e.g. for testing purposes)
; 7-"CLN" : Number of required values (columns) in data line for reading
; (0-all: based on first data line as main one)
; 8-"PNU" : Number of points (data lines, 4 and more rows) in file for reading (0-all)
; 9-"DGN" : Number of digits for output floating-point values (0-all)
;
; 3. "FLG"-Flags (0-inactive/false/disabled/"NO", 1-active/true/enabled/"YES"):
;
; 1-"ASC" : Create unformatted ASCIIZ output file
; 2-"FMT" : Use large format for output diagrams
; 3-"STA" : Print table of statistics
; 4-"ELM" : Print diagram of eliminations
; 5-"ERA" : Print eliminated (erased) functions
; 6-"DEV" : Print distribution of deviations
; 7-"DRW" : Use MS-DOS line-draw characters (flg exception: "2"-use the "Seventies" ones)
;
; 4. "NLY"-Nonlinear deformation code(s) for Y column(s):
;
; 0 - Without deformation : Y=Y
; 1 - Natural logarithm : Y=LN(Y)
; 2 - Square root : Y=SQRT(Y)
; 3 - Decimal logarithm : Y=LOG10(Y)
; 4 - Double square root : Y=SQRT(SQRT(Y))
; 5 - Exponent : Y=EXP(Y)
;
; 5. "NLX"-Nonlinear deformation code(s) for X column(s):
;
; 0 - Without deformation : X=X
; 1 - Natural logarithm : X=LN(X)
; 2 - Square root : X=SQRT(X)
; 3 - Decimal logarithm : X=LOG10(X)
; 4 - Double square root : X=SQRT(SQRT(X))
; 5 - Exponent : X=EXP(X)
;
; WARNING: Deformation function for required data value(s) is NOT used
; if this value is < = 0 or the 5678 is exceeded for exponent
; function
;
; 6. "NRM/NOR"-Coefficient(s) of normalization for X column(s):
;
; -2 - Without normalization (inverse functions disabled)
; -1 - Without normalization (inverse functions enabled)
; 0 - Normalization within +0.5 through +1.5
; 1 - Normalization within +0.0 through +1.0
; 2 - Normalization within -1.0 through +1.0
;
; 7. "KYC"-Y column(s) number(s) (1 through 40, equal ones possible)
;
; 8. "KXC"-X column(s) number(s) (1 through 40)
;
; Then follows data table
;
; Maximum data file line length : 254 characters
; Maximum number of columns ("CLN"): 40 numbers in line
; Floating-point numbers precision : 2 through 18 decimal digits
; Calculations results file(s) : RESINT01.REZ through RESINT39.REZ
; Unformatted ASCIIZ output file(s): ASCIIZ01.REZ through ASCIIZ39.REZ
;
; Please use ";" character for the commentary lines or to disable data row(s)
;
USA Temperature ExpY=F(X1,X2,X3)
2 3 62 12 3 3 6 62 7
0 0 1 1 1 1 2
0 5
0 0 0
0 0 0
2 3
4 5 6
;
;Temperature [Y] = F(Latitude [X1],
; Longitude [X2],
; Altitude [X3])
;==============================================
;NUM. TEMP. TEMP. LAT. LONG. ALT.
;[-] [Y1] [Y2] [X1] [X2] [X3]
;1st 2nd 3rd 4th 5th 6th
;==============================================
1. 61 61 30 88 5
2. 59 59 32 86 160
3. 30 30 58 134 50
4. 64 64 33 112 1090
5. 51 51 34 92 286
6. 65 65 34 118 340
7. 55 55 37 112 65
8. 42 42 39 104 5280
9. 34 34 41 72 40
10. 41 41 39 75 135
11. 44 44 38 77 25
;******** Data line below is wrong ************
12. 67 67 34 81 20
;********* Data line below is correct *********
;12. 67 67 29 81 20
;**********************************************
13. 74 74 24 81 5
14. 76 76 25 80 10
15. 52 52 33 84 1050
16. 79 79 21 157 21
17. 36 36 43 116 2704
18. 33 33 41 87 595
19. 37 37 39 86 710
20. 29 29 41 93 805
21. 27 27 42 90 620
22. 42 42 37 97 1290
23. 44 44 38 85 450
24. 64 64 29 90 5
25. 32 32 43 70 25
26. 44 44 39 76 20
27. 37 37 42 71 21
28. 33 33 42 83 585
29. 23 23 46 84 650
30. 22 22 44 93 815
31. 40 40 38 90 455
32. 29 29 46 112 4155
33. 32 32 41 95 1040
34. 32 32 43 71 290
35. 43 43 39 74 10
36. 46 46 35 106 4945
37. 31 31 42 73 20
38. 40 40 40 73 55
39. 51 51 35 80 720
40. 52 52 35 78 365
41. 20 20 46 100 1674
42. 41 41 39 84 550
43. 35 35 41 81 660
44. 46 46 35 97 1195
45. 44 44 45 122 77
46. 39 39 40 76 365
47. 40 40 39 75 100
48. 61 61 32 79 9
49. 34 34 44 103 3230
50. 49 49 36 86 450
51. 50 50 35 101 3685
52. 61 61 29 94 5
53. 64 64 29 95 40
54. 37 37 40 111 4390
55. 25 25 44 73 110
56. 50 50 36 76 10
57. 44 44 47 122 10
58. 31 31 47 117 1890
59. 26 26 43 89 860
60. 28 28 43 87 635
61. 37 37 41 104 6100
62. 81 81 18 66 35
"USA Temperature" calculations results example file "RESINT01.REZ" listing below:
OS RSSSP/101E RESINT SE v4.44(20210917v0) Copyright(C)1994-2021 RANDEC Ltd. Sun 19-SEP-2021 14:00.06
F:\RSSSP\RESINT\DAT\RES.DAT=>F:\RSSSP\RESINT\REZ\RESINT01.REZ
"Matiss Paraudzens" + 80686(3000MHz)"Intel(R) Xeon(R) CPU X5450 @ 3.00GHz"
"Microsoft Windows 7 Professional (6.1.7601) 32-bit_x86"
YNU(1)
XNU = 3 ELN = 12 VNU = 3 MNU = 62 NRM: 0 0 0
DGN = 7 CLN = 6 PNU = 62 NLX: 0 0 0
DVN = 3 NLY: 0
NORMALIZATION COEFFICIENTS
KXC A B XMin XMax
1 4 5.000000E-0002 2.500000E-0002 18. 58.
2 5 -2.252747E-0001 1.098901E-0002 66. 157.
3 6 4.991797E-0001 1.640689E-0004 5. 6100.
Y0 = 44.12903 SIGMA = 14.74425 KYC = 2
USA Temperature CORRELATION: 79.77 % SIGMA: 2.982067 12
+------------------+-----------------+------------------+-----------------+
! COEFFICIENT ! FUNCTION CODE ! COEFFICIENT ! FUNCTION CODE !
+------------------+-----------------+------------------+-----------------+
! 176.4337 ! 0 0 0 0 0 ! -351.5554 ! 1 0 0 0 0 !
! 177.4008 ! 2 0 0 0 0 ! 45.56233 ! -3 0 0 0 0 !
! -76.31198 ! 3 0 0 0 0 ! -66.43067 ! -1 2 0 0 0 !
! 8.797121 ! -2 -2 0 0 0 ! -26.37517 ! -1 -1 0 0 0 !
! 18.36291 ! 1 2 0 0 0 ! 81.62391 ! -2 3 0 0 0 !
! -32.31595 ! -3 1 0 0 0 ! 88.25351 ! 1 1 0 0 0 !
+------------------+-----------------+------------------+-----------------+
CORRELATION FOR EACH EXPRESSION, %:
1- 79.77 2- 79.96 3- 78.74 4- 78.61 5- 78.29 6- 75.97 7- 69.05
8- 68.36 9- 60.81 10- 59.13 11- 54.97
USA Temperature CORRELATION: 54.97 % SIGMA: 6.639196 11
+------------------+-----------------+------------------+-----------------+
! COEFFICIENT ! FUNCTION CODE ! COEFFICIENT ! FUNCTION CODE !
+------------------+-----------------+------------------+-----------------+
! 122.7567 ! 0 0 0 0 0 ! -78.46954 ! 4 0 0 0 0 !
+------------------+-----------------+------------------+-----------------+
DISTRIBUTION OF DEVIATIONS:
I+ + + ++ + + + ++ + +++++O + ++ +++ + + + + + +I
3/ -24.94753 Y = 30. 30/ 10.51680 Y = 22.
** WARNING: Deviation more than 3*SIGMA for 3 line. PROGNOSIS: Y = 5.052469
57/ -17.36841 Y = 44. 21/ 9.440283 Y = 27.
12/ -14.86581 Y = 67. 20/ 9.402022 Y = 29.
USA Temperature CORRELATION: 78.29 % SIGMA: 3.200375 5
+------------------+-----------------+------------------+-----------------+
! COEFFICIENT ! FUNCTION CODE ! COEFFICIENT ! FUNCTION CODE !
+------------------+-----------------+------------------+-----------------+
! 269.1349 ! 0 0 0 0 0 ! -405.5915 ! 4 0 0 0 0 !
! 144.0693 ! 5 0 0 0 0 ! 5.565434 ! -6 0 0 0 0 !
! -59.79430 ! -4 5 0 0 0 ! 14.53710 ! -5 -5 0 0 0 !
! -22.87760 ! -4 -4 0 0 0 ! 102.2578 ! 4 4 0 0 0 !
+------------------+-----------------+------------------+-----------------+
DISTRIBUTION OF DEVIATIONS:
I+ + + + +++ + ++++++ O++++++++ ++ +++++ ++I
12/ -12.73518 Y = 67. 20/ 4.683998 Y = 29.
** WARNING: Deviation more than 3*SIGMA for 12 line. PROGNOSIS: Y = 54.26482
49/ -6.789119 Y = 34. 52/ 4.592093 Y = 61.
57/ -4.462693 Y = 44. 54/ 4.158341 Y = 37.
USA Temperature CORRELATION: 79.96 % SIGMA: 2.955190 2
+------------------+-----------------+------------------+-----------------+
! COEFFICIENT ! FUNCTION CODE ! COEFFICIENT ! FUNCTION CODE !
+------------------+-----------------+------------------+-----------------+
! 180.7910 ! 0 0 0 0 0 ! -373.0410 ! 4 0 0 0 0 !
! 211.2258 ! 5 0 0 0 0 ! 46.83002 ! -6 0 0 0 0 !
! -79.32046 ! 6 0 0 0 0 ! -78.96880 ! -4 5 0 0 0 !
! 8.744062 ! -5 -5 0 0 0 ! -26.26420 ! -4 -4 0 0 0 !
! 84.59887 ! -5 6 0 0 0 ! -33.41774 ! -6 4 0 0 0 !
! 102.2244 ! 4 4 0 0 0 ! ! !
+------------------+-----------------+------------------+-----------------+
DISTRIBUTION OF DEVIATIONS:
I+ + + + +++++ + + ++++O ++++++++ ++++ ++ +I
12/ -12.26501 Y = 67. 52/ 4.636819 Y = 61.
** WARNING: Deviation more than 3*SIGMA for 12 line. PROGNOSIS: Y = 54.73499
4/ -6.161225 Y = 64. 20/ 3.948468 Y = 29.
29/ -4.962709 Y = 23. 9/ 3.495374 Y = 34.
USA Temperature STATISTICS
+-------------+------+------------------+------------------+-----------------+
! CORRELATION ! EL ! SIGMA ! COEFFICIENT ! ELIMIN.FUNCTION !
+-------------+------+------------------+------------------+-----------------+
! 1- 79.77 ! 12 ! 2.982067 ! 18.36291 ! 4 5 0 0 0 !
! 2- 79.96 ! 11 ! 2.955190 ! -33.41774 ! -6 4 0 0 0 !
! 3- 78.74 ! 10 ! 3.135293 ! -30.08354 ! 6 0 0 0 0 !
! 4- 78.61 ! 9 ! 3.153956 ! 11.01566 ! -5 6 0 0 0 !
! 5- 78.29 ! 8 ! 3.200375 ! 5.565434 ! -6 0 0 0 0 !
! 6- 75.97 ! 7 ! 3.543191 ! 145.4588 ! 4 4 0 0 0 !
! 7- 69.05 ! 6 ! 4.563275 ! -5.868199 ! -4 -4 0 0 0 !
! 8- 68.36 ! 5 ! 4.665556 ! 14.05896 ! -5 -5 0 0 0 !
! 9- 60.81 ! 4 ! 5.778814 ! -23.87004 ! -4 5 0 0 0 !
! 10- 59.13 ! 3 ! 6.025672 ! 14.92430 ! 5 0 0 0 0 !
! 11- 54.97 ! 2 ! 6.639196 ! -78.46954 ! 4 0 0 0 0 !
+-------------+------+------------------+------------------+-----------------+
DIAGRAM OF ELIMINATIONS:
N 10 20 30 40 50 60 70 80 90 100 ELIMINATED FUNCTION
12!------!------!------!------!------!------!------!-----O!------!------! 4 5 0 0 0
11!------!------!------!------!------!------!------!-----O!------!------! -6 4 0 0 0
10!------!------!------!------!------!------!------!-----O!------!------! 6 0 0 0 0
9!------!------!------!------!------!------!------!-----O!------!------! -5 6 0 0 0
8!------!------!------!------!------!------!------!----O-!------!------! -6 0 0 0 0
7!------!------!------!------!------!------!------!---O--!------!------! 4 4 0 0 0
6!------!------!------!------!------!------!-----O!------!------!------! -4 -4 0 0 0
5!------!------!------!------!------!------!----O-!------!------!------! -5 -5 0 0 0
4!------!------!------!------!------!------O------!------!------!------! -4 5 0 0 0
3!------!------!------!------!------!-----O!------!------!------!------! 5 0 0 0 0
2!------!------!------!------!------!--O---!------!------!------!------! 4 0 0 0 0
0 10 20 30 40 50 60 70 80 90 100
"USA Temperature" elapsed time 64ms997us257ns (Epsilon_64: 5.42101086242752217E-0020), thanks!
The latest available "RESINT" version will be included in the software CD
Parallel port LPT (Line Print Terminal) is required to attach the Aladdin HASP (Hardware Against Software Piracy) Security Key. For portable computers, you can add an LPT port using Express Card or PCMCIA CardBus PC cards (Type II card) for older laptops
1. https://lv.wikipedia.org/wiki/Vilnis_Egl%C4%81js
2. https://en.wikipedia.org/wiki/PDP-11
3. Vanags J. J., Rikmanis M. A., Ushkans E. J., Viesturs U. E. (1990). "Stirring Characteristics in Bioreactors". American Institute of Chemical Engineers (AIChE) Journal, 1361-1369.
4. Rikmanis M. A., Vanags J. J., Ushkans E. J., Viesturs U. E. (1987). "Distribution of Energy Introduced into Bioreactors with Various Constructions of Stirrers and Rheological Properties of the Liquid". Abstr., Cong. on Biotechnol., Amsterdam, 110.
5. Ruklisha M. P., Vanags J. J., Rikmanis M. A., Toma M. K., Viesturs U. E. "Biochemical Reactions of Brevibacterium flavum Depending on Medium Stirring Intensity and Flow Structure". Acta Biotechnologica 9 (1989) 6, 565-575, Akademie-Verlag Berlin.
6. Smite I. A., Eglajs, V. O., Ruklisa M. P., Viesturs U. E. "Biosynthesis of Polyribonucleotide phosphorylase and Polyribonucleotides by E. coli". Acta Biotechnologica 2 (1982) 4, 359-368.
7. M. Rikmanis, J. Vanags, E. Ushkans and U. Viesturs: “Stirring characteristics in bioreactors”, In: Proc. Congress CHISA’ 87, 1987, paper E9-137.
8. J. Vanags, M. Rikmanis, E. Uschkans, J. Grants and U. Viesturs: “Entwicklung eines Gerätes zur Messung der Vermichungsintensität in Bioreaktoren”, 4. Heiligenstädter Kolloquium Wissenschaftliche Geräte für die Biotechnologie, DDR, Heiligenstadt, 1988, pp. 282–287 (in German).
9. M. Rikmanis, J. Vanags, J. Grants and E. Ushkans: “The optimum stirring regime during microorganism cultivation”, In: Proc. 10th Congress CHISA’ 90, 1990, paper J4-3.
10. Eglajs, V. (1981). "Approximation of table data by multidimensional regression equation" (Russian). Problems of Dynamics and Strengths. 39 (Riga: Zinatne Publishing House): 120-125.
11. Eglajs, V. O. (1980). "Synthesis of a Regression Model of an Object on the Basis of Table Data" (Russian). Izv. AN LatvSSR, Ser. Phiz. i Tekhn. Nauk, 4, 107.
* Note: The original "RESINT" version on a CM-4 computer in the RSX-11M operating system was used for calculations