(45) Straight-Line Fitting

[SlideRule]

  The Hewlett-Packard HP-45 pocket calculator includes a key ∑+ that accumulates and stores the sums ∑1(=n), ∑x, ∑x², and ∑y, with another key that directs the calculator to compute the mean of x and its standard deviation. An independent y entry can be summed simultaneously.
  Another problem occurring often in data reduction is to find the least-squares slope and intercept of a straight line and their standard deviations. For this case the sums ∑xy and ∑y² are needed, in addition to those provided by the ∑+ key. An efficient (though not unique) procedure for accumulating the necessary sums with the HP-45 is the following: In succession, key in the data pairs (xi, yi) together with keyboard steps in the order: xi, ENTER↑, ENTER↑, yi, ENTER↑, STO + 3, X², STO + 4, R↓, ×, x⇄y, ∑+. After processing all data pairs in this manner the sums can be recalled from storage as needed, with ∑y in R3, ∑y² in R4, n in R5, ∑x² in R6, ∑x in R7 and ∑xy in R8. It is then a simple task to calculate the slope, intercept, and standard deviations using the well-known least-squares formulas. If the standard deviations are not required, the sum ∑y² can be omitted, with a saving of six keyboard steps (steps 5, 9, 10, 11, 12, 13 in the previous list).

source: AJP Volume 42, COMPUTER NOTES, Straight-Line Fitting with an HP-45 Calculator, W. C. ELMORE (Swarthmore College), April 1974, pg. 253

BEST!
SlideRule

[Thomas Klemm]

(01-31-2020 12:34 PM)SlideRule Wrote:  If the standard deviations are not required, the sum ∑y² can be omitted, with a saving of six keyboard steps (steps 5, 9, 10, 11, 12, 13 in the previous list).

This leaves us with:

1. \(x_i\)
2. ENTER
3. ENTER
4. \(y_i\)
5. STO
6. +
7. 3
8. ×
9. x<>y
10. Σ+
    

But we can do slightly better with:

1. \(y_i\)
2. STO
3. +
4. 3
5. \(x_i\)
6. ×
7. LASTx
8. Σ+
   

Registers

These registers are used:
\(
\begin{array}{|c|c|}
\hline
\text{Register} & \text{Value} \\
\hline
1 & a \\
2 & b \\
3 & \sum{y} \\
5 & n \\
6 & \sum{x^2} \\
7 & \sum{x} \\
8 & \sum{xy} \\
\hline
\end{array}
\)

Formula

(01-31-2020 12:34 PM)SlideRule Wrote:  It is then a simple task to calculate the slope, intercept, and standard deviations using the well-known least-squares formulas.

The following formulas are used to calculate slope \(a\) and intercept \(b\):

\(
\begin{align}
a &= \frac{\sum{xy} - \frac{\sum{x} \sum{y}}{n}}{\sum{x^2} - \frac{\sum{x}^2}{n}} \\
\\
b &= \frac{\sum{y} - a \cdot \sum{x}}{n} \\
\end{align}
\)

Program

These steps calculate both \(a\) and \(b\):

Code:

# calculate a
RCL Σ
RCL 3
×
RCL 5
÷
−
RCL 6
RCL 7
x²
RCL 5
÷
−
÷
STO 1
# calculate b
RCL 7
×
RCL 3
x<>y
−
RCL 5
÷
STO 2

Example

\(
\begin{array}{|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
C & 40.5 & 38.6 & 37.9 & 36.2 & 35.1 & 34.6 \\
\hline
F & 104.5 & 102 & 100 & 97.5 & 95.5 & 94 \\
\hline
\end{array}
\)

Initialise the Registers

CLEAR
STO 3


Enter the Data

104.5 STO + 3
40.5 × LASTx Σ+

102 STO + 3
38.6 × LASTx Σ+

100 STO + 3
37.9 × LASTx Σ+

97.5 STO + 3
36.2 × LASTx Σ+

95.5 STO + 3
35.1 × LASTx Σ+

94 STO + 3
34.6 × LASTx Σ+


Result

RCL 1
1.76

RCL 2
33.53


HP-25

The statistics functions also store \(\sum{xy}\).
In addition to that it is programmable.
Both makes entry of the data and calculating the best fit much easier.

Initialise the Registers

CLEAR REG

Enter the Data

104.5 ENTER 40.5 Σ+
102 ENTER 38.6 Σ+
100 ENTER 37.9 Σ+
97.5 ENTER 36.2 Σ+
95.5 ENTER 35.1 Σ+
94 ENTER 34.6 Σ+


Program

Code:

01: 24 04    : RCL 4
02: 24 03    : RCL 3
03: 71       : /
04: 24 04    : RCL 4
05: 14 21    : f mean
06: 61       : *
07: 24 05    : RCL 5
08: 41       : -
09: 24 07    : RCL 7
10: 14 21    : f mean
11: 61       : *
12: 24 06    : RCL 6
13: 41       : -
14: 71       : /
15: 23 01    : STO 1
16: 14 21    : f mean
17: 61       : *
18: 41       : -
19: 23 02    : STO 2

Data

Code:

DATA
8
00: 0
01: 1.760149049
02: 33.5271295
03: 6
04: 593.5
05: 22093.4
06: 8306.23
07: 222.9

Python

Just in case you want to compare the results:

Code:

C = [40.5, 38.6, 37.9, 36.2, 35.1, 34.6]
F = [104.5, 102, 100, 97.5, 95.5, 94]

Σx = sum(C)
Σy = sum(F)
Σx2 = sum([x**2 for x in C])
Σxy = sum([x*y for x, y in zip(C, F)])
n = len(C)

a = (Σxy - Σx*Σy/n) / (Σx2 - Σx**2/n)
b = (Σy - a*Σx) / n

a, b, Σx, Σy, Σx2, Σxy, n

(1.7601490488333176, 33.52712950250892, 222.9, 593.5, 8306.23, 22093.4, 6)

References

• HP-45 Emulator
• HP-45 Applications Book: Curve Fitting pp. 77
• HP-25 Emulator
• HP-25 Applications Programs: Curve Fitting - Linear Regression pp. 87
• HP Calculators with native C-F temperature conversion
• Straight-Line Fitting with an HP-45 Calculator
  

[Namir]

And that is why I bought the HP-55 in 1975 as my first HP calculator because it had built-in linear regresssion! Earlier that year, I had done linear regression (ln(y) vs 1/(x+constant)) calculations on a Sharp scientific calculator with one memory register ... and I swore it was the first and last time I do linear regression in that painfull way!!!

Namir