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This program is Copyright © 1976 by Hewlett-Packard and is used here by permission. This program was originally published in the HP-67 Stat Pac 1.
This program is supplied without representation or warranty of any kind. Hewlett-Packard Company and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
| Multiple Linear Regression | |||||
| Shift | →R 2 | x↑y→est z | →Σxi... | →Σxi2... | →Σxiyi:... |
| Label | Start | P? | xi↑...(Σ+) | xk↑...(Σ-) | →a:b:c |
| Key | A | B | C | D | E |
For a set of data points {(xi, yi, zi,), i = 1, 2, ..., n} this program fits a linear equation of the form
z = a + bx + cy
by the least squares method. Regression coefficients a, b, c can be found by solving the normal equations:
Σzi = an + bΣxi + cΣyi
Σxizi = aΣxi + bΣxi2 + cΣxiyi
Σyizi = aΣyi + bΣxiyi + cΣyi2
(i = 1, 2, ..., n)
c = (A-B)/[nΣxi2 - (Σxi)2][nΣyi2 - (Σyi)2] - [nΣxiyi - (Σxi)(Σyi)]2
where
A = [nΣxi2 - (Σxi)2] [nΣ yizi - (Σyi)(Σxi)]
B = [nΣxiyi - (Σxi)(Σyi)][nΣxizi - (Σxi)(Σzi)]
b = ([nΣxizi - (Σxi) (Σzi)] - c[nΣxiyi - (Σxi)(Σyi)])/(nΣxi2 - (Σxi)2)
a = (Σzi - cΣyi - bΣxi)/n
R2 = [aΣzi + bΣxizi + cΣyizi - (Σzi)2/n]/[Σzi2 -(Σzi)2/n]
Reference: Introduction to the Theory of Statistics Mood and Graybill McGraw-Hill 1963
| Step | Instructions | Input Data/Units | Keys | Output Data/Units |
| 1 | Load side 1 and side 2. | |||
| 2 | Initialize | A | 0.00 | |
| 3 | To set print mode* | B | 1.00 | |
| 4 | Do 5~6 for i = 1,2,...,n | |||
| 5 | Input xi | xi | ENTER↑ | |
| yi | yi | ENTER↑ | ||
| zi | zi | C | i | |
| 6 | If you made a mistake in inputting xk, yk, and zk, then correct by → | xk | ENTER↑ | |
| yk | yk | ENTER↑ | ||
| zk | zk | D | i-1 | |
| 7 | Calculate coefficients: a | E | a | |
| b | R/S | b | ||
| c | R/S | c | ||
| 8 | Calculate the square of multiple correlation coefficient R2 | f A | R2 | |
| 9 | Calculate estimated z from regression, input: x | x | ENTER↑ | |
| y | y | f B | est. z | |
| 10 | Repeat step 9 for different (x,y)'s | |||
| 11 | Recall sums: Σxi | f C | Σxi | |
| Σyi | R/S | Σyi | ||
| Σzi | R/S | Σzi | ||
| 12 | Recall sums of squares: Σxi2 | f D | Σxi2 | |
| Σyi | R/S | Σyi2 | ||
| Σzi | R/S | Σzi2 | ||
| 13 | Recall sums of cross products: Σxiyi | f E | Σxiyi | |
| Σxizi | R/S | Σxizi | ||
| Σyizi | R/S | Σyizi | ||
| For a new case, go to 2 | ||||
| *Note: To clear the print mode press → | CLF 0 |
A set of data points are given as the following
1 2 3 4 xi 1.5 0.45 1.8 2.8 yi 0.7 2.3 1.6 4.5 zi 2.1 4.0 4.1 9.4
Find the regression line, coefficients a, b, c, R2, est. z, sums, sums of squares, and sums of products.
Keystrokes Outputs
A 0.00
B 1.00
1.5 ENTER↑ 0.7 ENTER↑ 2.1 C 1.50 ***
0.70 ***
2.10 ***
1.00 ***
9 ENTER↑ 9 ENTER↑ 9 C 9.00 ***
9.00 *** (error)
9.00 ***
2.00 ***
9 ENTER↑ 9 ENTER↑ 9 D 9.00 ***
9.00 *** (correction)
9.00 ***
1.00 ***
0.45 ENTER↑ 2.3 ENTER↑ 4 C 0.45 ***
2.30 ***
4.00 ***
2.00 ***
1.8 ENTER↑ 1.6 ENTER↑ 4.1 C 1.80 ***
1.60 ***
4.10 ***
3.00 ***
2.8 ENTER↑ 4.5 ENTER↑ 9.4 C 2.80 ***
4.50 ***
9.40 ***
4.00 ***
E -0.10 *** (a)
R/S 0.79 *** (b)
R/S 1.63 *** (c)
f A 1.00 *** (R2)
DSP 9 PRINTx 0.998411259 ***
DSP 2 2 ENTER↑ 3 f B 2.00 ***
3.00 ***
6.37 *** (est. z)
f C 6.55 *** (Σxi)
R/S 9.10 *** (Σyi)
R/S 19.60 *** (Σzi)
f D 13.53 *** (Σxi2)
R/S 28.59 *** (Σyi2)
R/S 125.58 *** (Σzi2)
f E 17.57 *** (Σxiyi)
R/S 38.65 *** (Σxizi)
R/S 59.53 *** (Σyizi)
Regression line is
z = -0.10 + 0.79x + 1.63y
For x = 2 and y = 3, est. z = 6.37
LINE KEYS 001 *LBL A Initialize 002 CL REG 003 CF 0 004 CF 1 005 0 006 RTN 007 *LBL C 008 STO C 009 R↓ 010 STO B 011 R↓ 012 STO A 013 GSB 7 014 7 015 STO I 016 R↓ Input Xi, Yi, Zi. 017 GSB 1 018 8 019 STO I 020 RCL B Compute ΣXi, ΣYi, ΣZi, ΣXi2, ΣYi2 021 GSB 9 ΣZi2, ΣXiYi, ΣYiZi, ΣZiXi. 022 GSB 1 023 9 024 STO I 025 RCL C 026 GSB 9 027 GSB 1 028 RCL A 029 RCL B 030 × 031 GSB 2 032 STO + 1 033 RCL A 034 RCL C 035 × 036 GSB 2 037 STO + 2 038 RCL B 039 RCL C 040 × 041 GSB 2 042 STO + 3 043 1 044 GSB 2 045 STO + 0 046 RCL 0 047 GSB 9 048 RTN 049 *LBL 1 050 GSB 2 Subroutine for ΣXi, ΣXi2, ... 051 STO + (i) 052 RCL I 053 3 054 - 055 STO I 056 R↓ 057 X2 058 GSB 2 059 STO + (i) 060 RTN 061 *LBL E 062 RCL 0 063 RCL 4 064 × 065 RCL 7 066 X2 067 - 068 STO D 069 RCL 0 070 RCL 3 071 × 072 RCL 8 073 RCL 9 074 × 075 - 076 × 077 STO C 078 RCL 0 Calculating a, b, c 079 RCL 1 080 × 081 RCL 7 082 RCL 8 083 × 084 - 085 STO A 086 RCL 0 087 RCL 2 088 × 089 RCL 7 090 RCL 9 091 × 092 - 093 STO B 094 × 095 RCL C 096 X⇔Y 097 - 098 RCL D 099 RCL 0 100 RCL 5 101 × 102 RCL 8 103 X2 104 - 105 × 106 RCL A 107 X2 108 - 109 ÷ 110 STO C 111 RCL B 112 RCL A 113 RCL C 114 × 115 - 116 RCL D 117 ÷ 118 STO B 119 RCL 9 120 RCL C 121 RCL 8 122 × 123 - 124 RCL B 125 RCL 7 126 × 127 - 128 RCL 0 129 ÷ a 130 STO A 131 GSB 7 132 R/S b 133 RCL B 134 GSB 9 135 R/S c 136 RCL C 137 GSB 9 138 R/S 139 *LBL a 140 RCL A 141 RCL 9 142 × 143 RCL B 144 RCL 2 145 × 146 + 147 RCL C 148 RCL 3 Calculating R2 149 × 150 + 151 RCL 9 152 X2 153 RCL 0 154 ÷ 155 - 156 RCL 6 157 RCL 9 158 X2 159 RCL 0 160 ÷ 161 - 162 ÷ 163 GSB 7 164 RTN 165 *LBL b 166 X⇔Y 167 GSB 7 168 X⇔Y 169 GSB 9 170 ENTER↑ Calculating approximated z value for 171 RCL C given x,y. 172 × 173 X⇔Y 174 RCL B 175 × 176 + 177 RCL A 178 + 179 GSB 9 180 RTN 181 *LBL c 182 F0? 183 PRT SPC ΣXi, ΣYi, ΣZi. 184 6 185 STO I 186 GTO 8 187 RTN 188 *LBL d ΣXi2, ΣYi2, ΣZi2. 189 3 190 STO I 191 GSB 8 192 RTN 193 *LBL e ΣXiYi, ΣXiZi, ΣYiZi 194 0 195 STO I 196 GSB 8 197 RTN 198 *LBL 8 199 ISZ I 200 RCL (i) 201 GSB 9 Print for LBL a 202 R/S LBL b 203 GTO 8 LBL c 204 RTN 205 *LBL D 206 SF 1 207 GSB C 208 CF 1 For correction of Xk,Yk,Zk. 209 RTN 210 *LBL B 211 SF 0 212 1 213 RTN 214 *LBL 7 Set flag 0 for print. 215 F0? 216 PRT SPC 217 *LBL 9 218 F0? Print Xi, Yi, Zi. 219 PRTX 220 RTN 221 *LBL 2 222 F1? Change sign for correction. 223 CHS 224 RTN
R0 n R1 ΣXiYi R2 ΣXiZi R3 ΣYiZi R4 ΣXi2 R5 ΣYi2 R6 ΣZi2 R7 ΣXi R8 ΣYi R9 ΣZi A used, a B used, b C Zi, B in b, c D n*ΣXj2-(ΣXj)2 E n*ΣXi2 - (Σij)2
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