Curve Fitting: Adjusting Data Points (feat. HP 14B and HP Prime)

 Curve Fitting: Adjusting Data Points

Graphs are generated with the HP Prime emulator software.

A Curve Fitting Problem

Problem:

We have an investment project, which at the beginning will cost us 50,000 (insert the currency of your choice). Through analysis, the projected cash flows at the beginning of each term are shown here:

Period (beginning of)	Cash Flow
1	-50000
2	-3000
3	9000
4	16000
5	18000
6	21000

What is the projected cash flow at the beginning of period 7?

We could easily use linear regression, but the problem is, the data itself does not easily fit a line. So we can just use another regression model and that’s simple, right? Not quite, since two of the cash flows are negative.

Approach

Most calculators have at least the following curve fitting regressions:

	Model	Transformed	X’ =	Y’ =	True B =
Linear	Y = B + M * X	Y = B + M * X	X	Y	B
Exponential	Y = B * exp(M *X)	ln(Y) = ln(B) + M * X	X	ln(Y)	exp(B)
Logarithmic	Y = B + M * ln(X)	Y = B + M * ln(X)	ln(X)	Y	B
Power	Y = B * X^M	ln(Y) = ln(B) + M * ln(X)	ln(X)	ln(Y)	exp(B)

A lot of them rely on a transition of data to use a translated linear regression model. After the parameters slope (M) and y-intercept (B) are calculated, any adjustments are made to find the correct curve fitting parameters.

The statistical calculations operate on real numbers. If we try to fit an exponential, logarithmic, or power fit on data with non-positive numbers, we are going to have a problem because the natural logarithmic function returns non-real numbers in the transition process.

Still the data does not fit a linear regression line as good as the other regression curves. This calls for an adjustment of data. In essence, we are going to subtract out the minimum value of either the x or y data, or both “minus one”. Conceptually:

If min(X) ≤ 0, then:

P = min(X) -1

X_adj = X – P

If min(Y) ≤ 0, then:

Q = min(Y) -1

Y_adj = Y – Q

Why the “minus 1”?

If we only subtract the minimum, the lowest adjust amount would be 0. But ln(0) approaches negative infinity, which most calculators would not work with.

If we subtract the minimum minus one, the lowest adjusted amount would be 1. ln(1) =0. Beautiful!

Example: Adjusting the Raw Data

Raw data:

X	Y
1	-50000
2	-3000
3	9000
4	16000
5	18000
6	21000

min(X) = 1 > 0. No adjustment is needed.

min(Y) = -50000 ≤ 0. An adjustment is needed.

Let Q = min(Y) – 1 = -50000 – 1 = -50001

Adjusted data:

X	Y	Y’ = Y – Q	ln(Y’) (fix 3)
1	-50000	1	0
2	-3000	47001	10.758
3	9000	59001	10.985
4	16000	66001	11.097
5	18000	68001	11.127
6	21000	71001	11.170

Let’s fit the adjusted curve.

Keep in mind that:

Y’ = Y – Q

Y’ = Y - -50001

Y’ = Y + 50001

Y = Y’ – 50001

Fitting the Curve

Determine which one of the four regressions: linear, exponential, logarithmic, or power bets fits the adjusted data points. Predict the actual revenue (Y) for period 7.

Here is the data. We are going to use the data sets X and Y’.

X	Y	Y’ = Y – Q , (Q = -50001)
1	-50000	1
2	-3000	47001
3	9000	59001
4	16000	66001
5	18000	68001
6	21000	71001

Correlation Coefficients:

Model	Correlation (fix 4)
Linear	0.8477
Exponential	0.6772
Logarithmic	0.9520
Power	0.8290

Calculations were made with a HP 14B calculator. The best fit has absolute value of the correlation closest to 1. Correlations near -1 or +1 are better than correlations near 0.

The best fit is the logarithmic fit. Hence the curve fit is:

Y’ ≈ 9618.6378 + 38498.9034 * ln(X)

Remember that Y’ = Y + 50001. Then:

Y + 50001 ≈ 9618.6378 + 38498.9034 * ln(X)

Y ≈ -40382.3262 + 38498.9034 * ln(X)

To predict for X = 7:

Y ≈ -40382.3262 + 38498.9034 * ln(7)

Y ≈ 34533.0807

The beginning of period 7 will have the project cash flow of about $34,533.

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-30Xa Algorithms: Linear Regression

 TI-30Xa Algorithms: Linear Regression

Welcome to the March 2024 installment of TI-30Xa Algorithms.

Linear Regression with a TI-30Xa?

Today’s task is to fit bi-variate data to the line:

y = m * x + b

with the TI-30Xa. What? This calculator does not have a linear regression mode. Yes, with it is possible. We are going to use the technique shown in by the TI-36 Solar Guidebook from 1985 (see Source below). The TI-36 Solar was part of Texas Instruments’ line in the late 1980s and the early 1990s.

You can read my review on the TI-36 Solar from September 26, 2020 here:

https://edspi31415.blogspot.com/2020/09/retro-review-ti-36-solar.html

Where the TI-36 Solar only had one memory register, the TI-30Xa has three memory registers, and they are going to come in handy here.

Procedure

Caution: Be sure the calculator is on the entire time. Turning off the calculator will clear out statistics mode.

1. Enter the y data. Store the mean (y-bar) into memory register 2. Store the population deviation (σy) into memory register 3.

Keystrokes:

Mean of y-data: [ 2^nd ] [ x^2 ] {x-bar} [ STO ] [ 2 ]

Population of y-data: [ 2^nd ] [ ÷ ] { σxn } [ STO ] [ 3 ]

2. Clear the stat registers and enter the x data. As we are entering the x data, use memory register 1 to calculate Σxy.

Keystrokes:

Clear Stat Registers: [ 2^nd ] [ 7 ] {CSR}

1^st Point:

x1 [ × ] y1 [ = ] [ STO ] [ 1 ]

x1 [ Σ+ ]

Every point there after:

x_i [ × ] y_i [ = ] [ 2^nd ] [ RCL ] {SUM} [ 1 ]

x_i [ Σ+ ]

3. Calculate the slope, and replace Σxy with the slope.

m = ( y-bar * Σx – Σxy) / ( x-bar * Σx - Σx^2)

Keystrokes:

[ ( ] [ RCL ] [ 2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ RCL ] [ 1 ] [ ) ]

[ ÷ ] [ ( ] [ 2^nd ] [ x^2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ 2^nd ] [ ) ] [ = ] [ STO ] [ 1 ]

4. Calculate the y-intercept, replace y-bar with the y-intercept.

b = -m * x-bar + y-bar

Keystrokes:

[ RCL ] [ 1 ] [ +/- ] [ × ] [ 2^nd ] [ x^2 ] [ + ] [ RCL ] [ 2 ] [ = ] [ STO ] [ 2 ]

5. Calculate the correlation. If the correlation is close to -1 or +1, the linear fit will be excellent.

r = m * σx / σy

Keystrokes:

[ RCL ] [ 1 ] [ × ] [ 2^nd ] [ ÷ ] [ ÷ ] [ RCL ] [ 3 ] [ = ]

6. Use slope (m) and intercept (b) to predict x and y values:

y’ = m * x0 + b

Keystrokes: [ RCL ] [ 1 ] [ × ] x0 [ + ] [ RCL ] [ 2 ] [ = ]

x’ = (y0 – b) / m

Keystrokes: [ ( ] y0 [ - ] [ RCL ] [ 2 ] [ ) ] [ ÷ ] [ RCL ] [ 1 ] [ = ]

Key Map

[ RCL ] [ 1 ]: first Σxy, then m

[ RCL ] [ 2 ]: first y-bar, then b

[ RCL ] [ 3 ]: σy

[ 2^nd ] [ x^2 ]: x-bar

[ 2^nd ] [ ÷ ]: σxn

Example

Fit a line to the data:

X	Y
12	100
13	98.7
14	97.1
16	94.9
18	92.6
20	90

1. Enter the y data. Store the mean (y-bar) into memory register 2. Store the population deviation (σy) into memory register 3.

100 [ Σ+ ]

98.7 [ Σ+ ]

97.1 [ Σ+ ]

94.9 [ Σ+ ]

92.6 [ Σ+ ]

90 [ Σ+ ] (display n = 6)

Mean of y-data: [ 2^nd ] [ x^2 ] [ STO ] [ 2 ] (y-bar = 95.55)

Population of y-data: [ 2^nd ] [ ÷ ] [ STO ] [ 3 ] (σy = 3.465424457)

2. Clear the stat registers and enter the x data. As we are entering the x data, use memory register 1 to calculate Σxy.

Keystrokes:

Clear Stat Registers: [ 2^nd ] [ 7 ] {CSR}

12 [ × ] 100 [ = ] [ STO ] [ 1 ]

12 [ Σ+ ]

13 [ × ] 98.7 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

13 [ Σ+ ]

14 [ × ] 97.1 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

14 [ Σ+ ]

16 [ × ] 94.9 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

16 [ Σ+ ]

18 [ × ] 92.6 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

18 [ Σ+ ]

20 [ × ] 90 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

20 [ Σ+ ]

RCL 1: Σxy = 8827.7

[ 2^nd ] [ x^2 ]: x-bar = 15.5

3. Calculate the slope, and replace Σxy with the slope.

[ ( ] [ RCL ] [ 2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ RCL ] [ 1 ] [ ) ]

[ ÷ ] [ ( ] [ 2^nd ] [ x^2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ 2^nd ] [ ) ] [ = ] [ STO ] [ 1 ]

Slope: m = -1.230526316

4. Calculate the y-intercept, replace y-bar with the y-intercept.

[ RCL ] [ 1 ] [ +/- ] [ × ] [ 2^nd ] [ x^2 ] [ + ] [ RCL ] [ 2 ] [ = ] [ STO ] [ 2 ]

Intercept: b = 114.6231579

5. Calculate the correlation.

[ RCL ] [ 1 ] [ × ] [ 2^nd ] [ ÷ ] [ ÷ ] [ RCL ] [ 3 ] [ = ]

Correlation: r = -0.999092386

The line is:

y = -1.230526316 * x + 114.6231579

6. Predict values.

If x = 15, predict the y value (y’):

[ RCL ] [ 1 ] [ × ] x0 [ + ] [ RCL ] [ 2 ] [ = ]

y’ = 96.16526316

If y = 95, predict the x value (x’):

[ ( ] y0 [ - ] [ RCL ] [ 2 ] [ ) ] [ ÷ ] [ RCL ] [ 1 ] [ = ]

x’ = 15.94696322

Source

Alley, Chris M., Brenda M. Cornitius, et al. TI-36 Solar Guidebook Texas Instruments Incorporated. Dallas, TX. 1985, 1986, 1987. pp. 4.6 – 4.13

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Casio FX-702P and Swiss Micros DM42: Spearman’s Coefficient of Rank Correlation

Casio FX-702P and Swiss Micros DM42: Spearman’s Coefficient of Rank Correlation

Introduction: Correlation of Rankings Between Two Events

The Spearman’s coefficient rank correlation tests the relationship between two variables according to the rank of the two variables. Ranks are assumed to be positive integers; 1 for 1^st, 2 for 2^nd, 3 for 3^rd, etc. The Spearman’s correlation is said to be non-parametric, eliminating the requirement that the data is normally distributed.

The correlation has a range between -1 and +1. Two variables have a strong statistical relationship if the correlation is close to -1 or +1. Conversely, these variables have no relationship if the correlation is equal to 0.

Let the variables A and B be the two variables. Again, A and B are ranks.

Assuming that there are no ties in ranking in either A or B, the correlation is calculated as:

r = 1 - ( 6 × C)^2 ) ÷ (n^3 – n)

where:

C = Σ( (A_i – B_i)^2, k = 1 to n)

n = number of ranks

Casio FX-702P Code

This is a port of the Spearman’s Coefficient of Rank Correlation program set from Casio for the fx-702P. The original BASIC programs are listed here (Casio, pg. 89, Program Library FX-702P, see sources):

P0 (rank input):

10 C=0:N=1

20 PRT “A”;N;:INP A

30 PRT “B”;N;:INP B

40 C=C+(A-B)↑2

50 N=N+1:GOTO 20

P1 (calculate the correlation):

10 N=N-1

20 R=1-6*C/(N↑3-N)

30 PRT “R=”;R

40 END

Note: PRT and INP are shortcut commands for PRINT and INPUT, respectively. The three letter shortcuts are allowed (and encouraged) on the fx-702P.

Swiss Micros DM 42 Code: SPEAR

00 { 119-Byte Prgm }
01▸LBL "SPEAR"
02 "SPEARMAN CORR"
03 AVIEW
04 PSE
05 0
06 STO 03
07 1
08 STO 04
09 "ENTER RANK #S"
10 AVIEW
11 PSE
12▸LBL 00
13 "DONE? XEQ 01"
14 AVIEW
15 PSE
16 "A"
17 ARCL 04
18 ├"?"
19 PROMPT
20 "B"
21 ARCL 04
22 ├"?"
23 PROMPT
24 -
25 X↑2
26 STO+ 03
27 1
28 STO+ 04
29 GTO 00
30▸LBL 01
31 1
32 STO- 04
33 RCL 04
34 3
35 Y↑X
36 RCL- 04
37 1/X
38 RCL× 03
39 6
40 ×
41 +/-
42 1
43 +
44 STO 05
45 "R="
46 ARCL 05
47 AVIEW
48 .END.

Note: ├ is the append string character, call it up by pressing [ (shift) ] { ALPHA } [ ENTER ].

Instructions

1. Sort event A and B to their relative rankings. We are not analyzing data point by their values, but their rankings.

2. Press [ XEQ ] {SPEAR}

3. For each rank of event A, type the rank and press [ R/S ]. For each rank of event B, type the rank and press [ R/S ]. There are prompts to guide rank entry.

4. When done, execute label 01 by pressing [ XEQ ] 01.

Example

Two algorithms A and B are analyzed. The time between run time with algorithms A and B are:

A: 48 seconds, B: 63 seconds

A: 50 seconds, B: 64 seconds

A: 53 seconds, B: 61 seconds

A: 55 seconds, B: 66 seconds

A: 57 seconds, B: 69 seconds

Ranking the times relative to algorithm A are:

A: 48 seconds (rank 1), B: 63 seconds (rank 2)

A: 50 seconds (rank 2), B: 64 seconds (rank 3)

A: 53 seconds (rank 3), B: 61 seconds (rank 1)

A: 55 seconds (rank 4), B: 66 seconds (rank 4)

A: 57 seconds (rank 5), B: 69 seconds (rank 5)

Calculate the Spearman correlation:

XEQ SPEAR

A1? 1 R/S

B1? 2 R/S

A2? 2 R/S

B2? 3 R/S

A3? 3 R/S

B3? 1 R/S

A4? 4 R/S

B4? 4 R/S

A5? 5 R/S

B5? 5 R/S

XEQ 01

Result: R = 0.7000

The Spearman correlation is 0.7.

Sources

“Spearman’s rank correlation coefficient” DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL: https://datatab.net Retrieved January 17, 2024.

“Spearman’s rank correlation coefficient” Wikipedia. https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient Edited January 17, 2024. Retrieved January 17, 2024.

Casio. Program Library: FX-702P pg. 89 (English)

(The Casio Ledudu web page, https://casio.ledudu.com/pockets.asp?lg=eng&type=53, has the library in French).

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.