Regression
a program for the Texas Instruments TI-83
and TI-84 (Plus) graphing calculators

 

Summary

Every measurement is influenced by many uncertainties. Measurement uncertainties can never be completely eliminated, so the true value for any quantity is never known exactly.
If two physical quantities are known to have a linear relationship, experimental data, when plotted on a graph, are expected to fit a straight line in a more or less accurate and precise way. Due to errors in the measuring process, not all the data fall exactly on the drawn line.

   

Linear regression is a statistical tool that helps you to determine the "best-fitting" straight line for any given set of experimental data. In the relationship  y = ax + b  the constants a and b are being determined. The obtained function hands over which value is to be expected on an average for the one variable (y) at a given value of the other variable (x).

Quadratic regression produces the "best" parabola for a set of experimental data by calculating the constants a, b, and c of the quadratic equation  y = ax + bx + c.

Other regression models available in the program:
Cubic  y = ax + bx + cx + d
Quartic  y = ax4 + bx + cx + dx + e
Logarithm  y = a + bln(x)
Exponential  y = a * b^x
Power  y = a * x^b
Logistic  y = c / (1 + ae^(–bx))
Sine  y = a * sin(bx + c) + d

 

Program Listing

Help
PROGRAM:REGRESS R : ALPHA, R keys
ClrHome > PRGM > I/O > ClrHome
RectGC:CoordOn > 2nd, FORMAT > ...
: : ALPHA, : keys
GridOn:AxesOn  
LabelOn:ExprOn  
PlotsOff > 2nd, STAT PLOT > PlotsOff
FnOff > VARS > Y-VARS > On/Off... > ...
GraphStyle(1,1) > PRGM > CTL > GraphStyle
, : comma key
1 → Xres "→" Symbolizes the STO> key
> VARS > Window... > Xres
Disp "VALUES IN L1,L2" > PRGM > I/O > Disp
Space: ALPHA, .. (above 0 key)
L1 : 2nd, L1
0 → Z : While Z<40 > PRGM > CTL > While
> 2nd, TEST > "<"
Z+1 → Z:End > PRGM > CTL > End
(max(L1)–min(L1))/5 → U > MATH > NUM > max(
10^(int(log(U))) → Z 10^( : 2nd, 10x keys
> MATH > NUM > int(
U/Z → U  
(int(√(U–1))+1)Z → Xscl √( : 2nd, √ keys (square root)
: x key
> VARS > Window... > Xscl
(max(L2)–min(L2))/5 → V  
10^(int(log(V))) → Z  
V/Z → V  
(int(√(V–1))+1)Z → Yscl √( : square root
Xscl and Yscl will be set to 1, 2, or 5 times a power of ten.
Plot1(Scatter,L1,L2, ) > 2nd, STAT PLOT > Plot1(
> 2nd, STAT PLOT > TYPE > Scatter
> 2nd, STAT PLOT > MARK > (small square)
ZoomStat > ZOOM > ZoomStat
ZoomSto > ZOOM > MEMORY > ...
Lbl 0 : ClrHome > PRGM > CTL > Lbl
> PRGM > I/O > ClrHome
Menu("","VIEW L1,L2",1, "LinReg(ax+b) ", 2, "QuadReg ",3, "CubicReg ",4, "MORE REG",50, "ZOOM,Trace", 60, "Stop",7) > PRGM > CTL > Menu(
> STAT > CALC > LinReg(ax+b)
> TRACE
> PRGM > CTL > Stop
Lbl 50  
Menu("","QuartReg ", 51, "LnReg ",52, "ExpReg ",53, "PwrReg ",54, "Logistic ",55, "SinReg ",56, "HOME",57) > STAT > CALC > CubicReg
etc.
Lbl 60  
Menu("","Zoom Out",61, "Zoom In",62, "RESET ZOOM",63, "Trace",64, "HOME",65)  
Lbl 1  
Output(1,1,L1) > PRGM > I/O > ...
Pause :ClrHome > PRGM > CTL > Pause
Output(1,1,L2)  
Pause  
Goto 0 > PRGM > CTL > ...
Lbl 2  
Disp "LinReg(ax+b) ", "a,b,r,r" > STAT > CALC > LinReg(ax+b)
> VARS > Statistics... > EQ > a (and b,r,r)
LinReg(ax+b) Y1 > VARS > Y-VARS > Function... > Y1
Disp a,b,r,r > VARS > ...
Lbl 21  
Pause  
DispGraph > PRGM > I/O > DispGraph
Pause  
Goto 0  
Lbl 3  
Disp "QuadReg ", "Y=aX+bX+c", "a,b,c,R" X : X key or ALPHA, X keys
> VARS > Statistics... > EQ > R
QuadReg Y1 It needs 3 points at least (Reg-equation 4 at least).
Disp a,b,c,R > VARS > ...
Goto 21  
Lbl 4  
Disp "CubicReg Y=aX+..", "a,b,c,d,R" Y=aX+bX+cX+d
CubicReg Y1 It needs 4 points at least (Reg-equation 5 at least).
Disp a,b,c,d,R  
Goto 21  
Lbl 51  
Disp "QuartReg a..e,R" Y=aX4+bX+cX+dX+e
QuartReg Y1 It needs 5 points at least (Reg-equation 6 at least).
Disp a,b,c,d,e,R  
Goto 21  
Lbl 52  
Disp "LnReg ", "Y=a+bln(X)", "a,b,r,r"  
LnReg Y1  
Disp a,b,r,r  
Goto 21  
Lbl 53  
Disp "ExpReg ", "Y=a*b^X", "a,b,r,r"  
ExpReg Y1  
Disp a,b,r,r  
Goto 21  
Lbl 54  
Disp "PwrReg ", "Y=a*X^b", "a,b,r,r"  
PwrReg Y1  
Disp a,b,r,r  
Goto 21  
Lbl 55  
Disp "Logistic ", "Y=c/(1+ae^(–bX))", "a,b,c"  
Logistic Y1  
Disp a,b,c  
Goto 21  
Lbl 56  
Disp "ESTIMATE PERIOD"  
Input T  
ClrHome  
Disp "SinReg ", "Y=a*sin(bX+ c)+d", "a,b,c,d"  
SinReg 16,L1,L2,T,Y1 It needs 4 points at least; 2 points per period.
Disp a,b,c,d The results bX and c are always expressed in radians.
2π/abs(b) → T π : 2nd, π keys (pi)
Pause  
Disp "PERIOD, FREQ.",T,1/T Once the period is calculated, its value may be used to iterate once more, especially when b is a negative.
Goto 21  
Lbl 57:Goto 0  
Lbl 61  
√(2) → Z : Z → XFact (square root)
Z → YFact > VARS > Window... > X/Y > Xfact
10^(int(log((Xmax–Xmin)/Z))) → Xscl  
10^(int(log((Ymax–Ymin)/Z))) → Yscl  
Zoom Out > ZOOM > Zoom Out
Pause  
Goto 60  
Lbl 62  
Zoom In:Pause  
Goto 60  
Lbl 63  
ZoomRcl:Pause > ZOOM > MEMORY > ZoomRcl
Goto 60  
Lbl 64:Trace > TRACE
Goto 60  
Lbl 65:Goto 0  
Lbl 7  
Xscl → ΔTbl:Stop > VARS > Table... > ΔTbl (Delta-Tbl)
> PRGM > CTL > Stop

 

An Application

After putting in the program listing, you might like to see the program working. Try this example.

 

The Experiment

A long tube is placed vertically and filled with a viscous liquid. (Which liquid is not?) When a bullet, being made up of steel, is let loose in the liquid, the sinking speed soon becomes constant. To determine this constant velocity, the position x is measured as a function of time t. The measuring outcome is:

t
(s)
x
(cm)
0,0
1,3
2,7
4,2
5,5
7,0
8,1
10
20
30
40
50
60
70

 

Entering the Values

The time t is considered the independent variable and is measured along the horizontal axis of a graph. The position x, the dependent variable, is measured along the vertical axis. List 1 (L1) must contain values of the variable, which is to be plotted along the horizontal axis.
So, store the time values in List 1 and the position values in List 2 (L2). Here's how you can get access to these lists: > STAT > EDIT > Edit... > ...

 

Getting the Linear Equation and the Coefficient of Correlation

After preparing L1 and L2, run the program. A menu appears.
Select option "2:LinReg(ax+b)".
A fysical equivalent to the shown equation  y = ax + b  is the position equation
x(t) = v t  +  x(0).
The displayed value of the constant a in  y = ax + b  equals 7.26571..., meaning the constant velocity v amounts to 7,27 cm/s.
The position at the beginning of the measurements is
x(0) = = 10,1 cm.
Substituting the numerical values in the position equation leads to the linear equation in demand:
x(t) = 7,27 t  +  10,1  with x expressed in cm.

   

The produced value of the coefficient of correlation (r = 0,99952) indicates that the found linear equation is only a fair representation of the relationship between the measured t and x values. Counting the nines, that directly follow the comma in the coefficient of correlation, is a convenient way to find out the usefulness of the equation. In chemical and physical determinations the number of nines may be interpreted in this way: five nines mean an excellent representation, four nines a good one, three nines a fair one, two nines a poor one, and one nine means a very poor representation. With no nines directly behind the comma the equation is "unusable".

 

Viewing the Plotted Points and the Straight Line

Press ENTER.

 

Tracing

Once the graph has been drawn, press ENTER, select option "6:ZOOM,Trace" in the menu, and "4:Trace" in the following sub-menu. Next, use the left and right arrow keys to trace the values of the plotted measuring points.

   

Press the up arrow key once to enable tracing the plotted line. Again, use the left and right arrow keys to explore the coordinates of some points of the line.

Using the TABLE function is another way to get these values. Stop the execution of the program by selecting "5:HOME" in the sub-menu and "7:Stop" in the main menu. Then call on the TABLE function. You may have to adjust the table settings using the TBLSET function.

   

 

Other Applications

In other applications more regression methods may be needed. Selecting "5:MORE REG" in the main menu allows access to other regression functions in the following sub-menu.

 

Hein ten Horn
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