1. Spreadsheet Problem Solving
fitting models to data
straight-line regression
multilinear regression
nonlinear regression
model building and selection
using
Data Analysis Regression tool
Trendline
Solver

2. Review of Straight-line Linear Regression
[ from Class #6 ] x y
y = ax + b y1
y11
e11
Model
x11
For each data point, there is an error between that
point and the model line. Fitting the model has to do
with minimizing these errors.

3. Finding the model parameters that give the best fit
For the straight-line model, the model parameters are
the slope (a) and the intercept (b).
The problem is then to find the values of a and b that
give the best fit. What is meant by the best fit?
The standard measure of goodness of fit is the sum
of squares of the errors:
So, the problem reduces to finding the minimum of
SSE by adjusting a and b.

4. Fitting a straight-line model to data
The minimization of SSE can be solved by calculus
to give formulas for the best values of a and b:
and Excel solves problems like this with either formulas
or built-in tools (Data Analysis Regression & Trendline).

5. Example: straight-line fit

6. Transfer the data to an Excel spreadsheet
and create a graph

7. Calculating the slope and intercept using Excel formulas

8. The formulas behind the numbers

9. Using the model straight-line equation to compute
the predictions:
and copy these
to the graph,
displaying as
a straight line

11. Using an alternate, shortcut approach Trendline
Start with a simple graph of the data
Select the data series by
clicking on it
Right-click on a
data point to get
context-sensitive
menu
Select
Add Trendline
option

12. The Add Trendline dialog box
Linear selected
by default
OK for this
problem
Click on
Options tab

13. Options tab
Set for
Display equation
on chart
Click OK

14. Initial form of graph with straight-line added
Fix up
equation
display

15. Looks just like before, but we got there quicker
But neither of these approaches gives us much information
about the model, how good it is, etc.

16. A 2nd alternate approach Data Analysis Regression tool
Tools Data Analysis
recall that, if Data Analysis
does not appear on the Tools
menu, you will need to check
Analysis Toolpak in the Add-ins
dialog box [if it’s not there, you
will have to go back to Microsoft
Office/Excel set-up]
Initial, empty
Regression
dialog box

17. Regression dialog box set up for our problem
checking Residuals
will give us also
model predictions

18. Initial (poorly formatted) Regression output display
[ on new worksheet ]
Format
Autoformat OK
and fix up
display for
appropriate
significant
figures

19. Final Display of Regression Output
[ tons of info, most of
which you will not
understand for a
couple years ]
used to judge
goodness of
fit
intercept
and slope
values
used to judge
whether terms
“belong” in the
model
add to data graph
for visual comparison
with model

20. Judging Goodness of Fit
correlation coefficient: if close
to +1 or –1, indicates strong
correlation between x and y
[something we already know
from the original graph!]
coefficient of determination:
%-age of the variability in y
that’s accounted for by the
model
adjustment to R2 that
penalizes the value for
using a model with too
many terms
gives an idea of how
far off the model
predictions will be
Adjusted R2 or Standard Error can be used to compare
different models and choose which fits best. The higher
the value of Adjusted R2 the better, the lower the value
of Standard Error the better.

21. Judging whether terms belong in the model
P-values estimate the probability
that the true value of the coefficient
could be zero
P-values that are quite small, like
these, indicate that there is little
question about the significance of
the term coefficients. In our case
here, that means that both the
intercept term and the slope term
belong in the model.
A P-value of 5%
(0.05) or greater
causes suspicion
that the coefficient
may not be
significant and that
the term should
probably be dropped
from the model

22. The Data Analysis Regression tool appears much more complicated and involved that the shortcut Trendline tool, so . . .
Why use Data Analysis Regression?
It provides more information that let’s us
judge the goodness of fit and significance
of model terms
2) It can handle model forms that cannot be
handled by Trendline
So, generally, when using Excel, we prefer
the Data Analysis Regression tool over Trendline
but Trendline is still quite good for “quick and dirty”
looks at the data
Learn to use both!

23. More complicated models
Note: it is called linear regression,
even when there are nonlinear
terms in x, because the terms are
linear in the model parameters,
a, b, c, etc.
More complicated models
Polynomial models
General linear models
Examples:
polynomial models above
Multilinear models
Examples:

24. Nonlinear models
Transformable to linear
Not transformable
straight-line
regression!
We can use the Data Analysis Regression tool for everything
except the nonlinear models that can’t be transformed into
linear. For those, we can use the Solver.

25. Example: polynomial regression
curvature evident

26. Setting up for polynomial fits
Select for quadratic model, etc

27. Data Analysis Regression tool
check Labels because
headings are included
in selections for Y and X
check
Residuals

28. Quadratic model regression results
model performance
adjR2
model coefficients
copy to graph

29. Quadratic model really doesn’t “capture” behavior of data

30. Continue with fits of cubic, 4th- & 5th-order polynomials
Summary of results
Looks like 5th-order offers best performance
but improvement is marginal over 4th-order.
Resulting model:

32. Precautions on polynomial fitting
Try to use the lowest-order model that gives a good fit.
Higher-order models will have “wiggles” between data
points that will cause prediction errors.
In fact, an (n-1)th-order polynomial will provide a perfect
fit to the n data points, but it will usually do bizarre things
in between the data points.

33. Example: multi-linear regression Model 1:
X-input range includes
two independent variables:
x1 and x2
High P value for intercept in
Model 1 suggests Model 2
without intercept, but there
is a significant loss in adjR2

34. Model performance isn’t that
great for either model, and
Model 1 doesn’t appear
dramatically better than Model 2
Note: for multi-linear models, we plot Predicted vs Measured y.
A perfect model would place points directly on the 45-degree line.

35. Nonlinear Regression
Fitting the parameters of the van der Waals’ equation of state
Data for SO2
Find the values of a and b
that give the best predictions
for P, when compared to the
measured values of P

36. Strategy for Nonlinear Regression
1) estimate initial values for a and b
2) compute predicted P’s using data for and T
3) compute errors between predicted P’s and measured P’s
4) sum the squares of these errors to compute SSE
5) have the Solver minimize SSE
by adjusting the values of a and b

37. - Basic data Calculated Pressure by both ideal gas law
and van der Waals
Sum of
squares
of this
column -

38. van der Waals Calculation
Ideal Gas
Calculation
Sum of Squares
Calculation
van der Waals Calculation
Error Calculation

39. Setting up Solver Parameters
SSE as Target Cell
Minimize
by adjusting a and b
with b>=0 constraint
Results

40. Results

41. Note departure of
ideal gas predictions
at higher pressures