HP 9g Handbuch

hp calculators

HP 9g Statistics – Non-Linear Regression

Non-Linear Regression

Practice Solving Non-Linear Regression Problems

hp calculators

HP 9g Statistics –Non-Linear Regression

Non-linear regression

In addition to the linear regression (described in the HP 9g learning module Statistics – Linear Regression) the HP 9g

provides five more types of regression of a variable yon a variable x, namely:

♦Logarithmic model: .xlnbay +=

The logarithmic curve-fitting model requires positive x-values. An example of this model is the

psychophysics law of Weber-Fechner.

♦Exponential model: .

eay ⋅=

This model requires positive y-values. Examples of this model are the number of undecayed

nuclei of radioactive materials after a period x of time, the discharging process in Capacitor-

Resistance circuits, etc.

♦Power model: .

xay ⋅=

The power model requires positive x-values and positive y-values, otherwise the DOMAIN error

occurs when attempting to display the STATVAR menu. Newton’s law of gravitation is a classic

example.

♦Inverse model: x

ay += .

Zero cannot be one of the x-values, since it will trigger the DIVIDE BY 0. Examples are Curie-

Weiss’ law, the reactance of a capacitor, relation between potential energy and distance, etc.

♦Quadratic model: .

cxbxay ++=

Bear in mind that this model needs at least three points. An example is the relation between

kinetic energy and speed.

Regression curves can be found by following the same procedure used with linear regressions. That is, press ` 1,

select REG, and press =. The REG menu is now displayed showing all the types of regression available. Select the

desired one then press =. Once the data has been entered, regression results are shown in the STATVAR menu

(~e), which displays the values of a and b (and c) and also the correlation coefficient r (except in the case of

quadratic regression). Also in this menu are the functions and , which are used for predicting values. Let’s illustrate

ˆx

all this with some examples.

hp calculators - 2 - HP 9g Statistics – Non-Linear Regression - Version 1.0

hp calculators

HP 9g Statistics –Non-Linear Regression

Practice solving non-linear regression problems

Example 1: Enter the following points into your HP 9g as statistical data and plot them. Find the equation that best fits

the points.

x 10 25 40 80 160 250

y 106 46 31 18.5 12.25 10

Solution: Let’s first clear any previous data by pressing `1, then select D-CL and press =.

To plot the scatter graph (or scatterplot) we must set the 2-VAR mode, so press: `1, select 2-VAR

and press =.Let’s now enter the data: E, select DATA-INPUT and =, then press:

10‡106‡

25‡46‡

40‡31‡

80‡18.5‡

160‡12.25‡

250‡10‡

To plot the graph press: ~e P. Figure 1 shows the resulting

graph. These points seem to belong to a rectangular hyperbola. Let’s try to

verify it by finding the inverse regression equation. Press first: `1,

select REG, press =, select INV and press =again. Figure 1

Regression variables are displayed in the ~e menu, where we can

read a= 6, b = 1000 and r = 1, which means that the fit is perfect. Without

leaving the currently displayed menu, the curve can be plotted by pressing

a single key: P. Figures 2 shows the displayed graph (notice the

asymptote in the y axis). Figure 2

Answer: The equation that best fits these points is x

y1000

6+= .

Example 2: Using the data of the previous example, what are the predicted values of y for x = 1000 and for x = 95?

Which prediction is more reliable? And what is the predicted value of x for y = 35?

Solution: As to the first question, we have to evaluate the regression line at x = 1000, and 95, that is to say: y(1000)

and y(95). Actually, y and x are often written as and , respectively, when we are referring to predicted

values. On the HP 9g their symbols are y’ and x’ and are displayed in the STATVAR menu. To calculate

(1000) and (95), press:

ˆx

ˆy

~e , select y’, =1000= and ††55:= respectively.

Likewise, to find (35), press:

~e , select x’, =35=

hp calculators - 3 - HP 9g Statistics – Non-Linear Regression - Version 1.0

hp calculators

HP 9g Statistics –Non-Linear Regression

Answer: (95) = 16.52, (1000) = 7 and (35) = 34.48 (to two decimal digits). (95) is more reliable than

(1000) because the former is the result of an interpolation whereas the latter is an extrapolation.

ˆy

ˆx

ˆy

ˆ1Also,

95 is much closer to x=94.17 than 1000 (xbeing the arithmetic mean of the x values, and can be

obtained by this sequence of keystrokes: ~` , select 2-VAR, ~e †), which means that any

variability in the former value will be far less significant than in the latter.

Example 3: The following data shows the number of AIDS cases diagnosed in the United Sates from theyears 1981

to 1994. (From The Spread of AIDS in the United Sates, a data analysis activity for the HP 38G by Michael

Grasse)

Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Cases 315 1184 3119 6276 11894 19195 28833 35662 42700 48092 58549 75908 73774 58225

Verify that from 1981 to 1987 the data fits an exponential model quite well. Considering all the years, what

model seems to provide the best fit?

Solution: First, clear the previous data by pressing `1, then selecting D-CL and finally pressing =. Let’s

now enter the data, first press: E, select DATA-INPUT and =. (The 2-VAR mode is assumed to be

set from the previous example). We’ll enter the years 198x as xand 199x as 1x, but we’ll enter up to 1987

for now:

1‡315‡2‡1184‡3‡3119‡4‡6276‡

5‡11894‡6‡19195‡7‡28833‡

Let’s find the correlation coefficient corresponding to an exponential relation:

`1 select REG =select “e^” =~e ††

Rounding to four decimal digits r = 0.9793, which indicates that the exponential equation

fits the data well unfortunately.

e.y 73070

8619250=

We’ll enter now the rest of the data. Note that you don’t need to be in the 2_VAR mode now, just press:

E, select DATA-INPUT =, and then:

……

8‡35662‡

9‡42700‡

10‡48092‡

11‡58549‡

12‡75908‡

13‡73774‡

1458255‡

1To understand the lack of validity in extrapolation, consider the number of AIDS diagnoses that the regression models in Example 3 predict for the

year 1974, a year in which AIDS as a disease was unknown!

hp calculators - 4 - HP 9g Statistics – Non-Linear Regression - Version 1.0

hp calculators

HP 9g Statistics –Non-Linear Regression

The correlation coefficient is now: (~e ††) 0.9097, which means that data (from 1981 to1994)

begins to diverge from an exponential model. In order to know what model best fits the data, we’ll find the

correlation coefficient associated with each of the models available on the HP 9g (except for the quadratic

model because no correlation coefficient is provided. You can plot the resulting quadratic, however, and

see how well it fits the data):

Linear fitting: `1REG =LIN =~e †† r = 0.9647

Logarithmic fitting: `1REG =LOG =~e †† r = 0.8795

Power fitting: `1REG =PWR =~e †† r = 0.9927

Inverse fitting: `1REG =INV =~e †† r = – 0.6694

Solution: Of the considered models, the power curve is the line that best fits the data.

1682

91243219 .

x.y ⋅=

Notice that when taking all the years into account, the linear model is better than the exponential model.

hp calculators - 5 - HP 9g Statistics – Non-Linear Regression - Version 1.0

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