# Re: Calculate slope of a data set or trend line

kux wrote:
Hello everyone,
I hope someone is out here who can help me with a simple calculation...

I have a sales data base in access with monthly sales history by
product. to make future predictions I need the slope (representing
growth) of monthly sales.
the data is setup in a crosstab with the months in different collums,
say column 1 to 12 for last year. I want to calculate the slope for
every row (on the same query or another if needed)

Does anyone have an idea how to calculate the slope of the trendline
for those values? In excel I can just use =slope(known y's, known x's).
the slope i need is from the trendline in common statistics (I believe
it is calcuated by minimizing the sum of the squared deviation - OLS)

I cannot use an ordinary excel export (too many lines).

Anyone an idea how to work around that?

Help is much appreciated,

Thanks
Kai

Data points: (X1,Y1), ..., (XN, YN)

Best Fit Line: Y = AX + B

Error from the line: Ei = Yi - AXi - B

Square of each error:

Ei^2 = Yi^2 + A^2 Xi^2 + B^2 - 2A XiYi - 2B Yi + 2AB Xi

Sum of the squares of the errors:

Sum(Ei^2) = Sum(Yi^2) + A^2 Sum(Xi^2) + B^2 Sum(1) - 2A Sum(XiYi) - 2B
Sum(Yi) + 2AB Sum(Xi), i = 1,...,N

Taking the partial derivatives with respect to A and B and setting them
to 0,

w.r.t. A: 0 = 2A Sum(Xi^2) - 2 Sum(XiYi) + 2B Sum(Xi)
w.r.t. B: 0 = 2BN + 2A Sum(Xi) -2Sum(Yi)

Rearranging so that A and B are the variables:

Sum(Xi^2) A + Sum(Xi) B = Sum(XiYi)
Sum(Xi) A + N B = Sum(Yi)

as a Matrix Equation:

- - - - - -
| Sum(Xi^2) Sum(Xi) | * | A | = | Sum(XiYi) |
| Sum(Xi) N | | B | | Sum(Yi) |
- - - - - -

As long as Abs(N Sum(Xi^2) - [Sum(Xi)]^2) is not miniscule, Cramer's
rule will work without numerical instabilities:

A = [N Sum(XiYi) - Sum(Xi) Sum(Yi)] / [N Sum(Xi^2) - Sum(Xi) ^ 2]
B = [Sum(Xi) ^ 2 Sum(Yi) - Sum(Xi) Sum(XiYi)] / [N Sum(Xi^2) - Sum(Xi)
^ 2]

where each sum is performed on i = 1,...,N

So if tblData has N records of (X, Y) values:

SELECT SUM(1) * SUM(X*X) - SUM(X) * SUM(X) AS Denom, (SUM(1) * SUM(X *
Y) - SUM(X) * SUM(Y)) / Denom AS A, (SUM(X * X) * SUM(Y) - SUM(X) *
SUM(X * Y)) / Denom AS B FROM tblData;

should return unrounded values for A and B.

I didn't check the equations against a book but they seem to be giving
the correct results for a couple of simple data sets. Least squares
regression of physical data often results in miniscule Denom values.
You'll know that numerical instabilities are creeping in when the
answers you get no longer solve the 2 X 2 matrix equation. In those
cases, alternate matrix solutions such as partial or full (matrix)
pivoting during gaussian elimination, may be required. A new wrinkle
is added when the numbers are in rows. Perhaps create a public
function with 12 TotalMonthlySales arguments, that returns A (the
slope) for an N value of 12.

I hope this helps,

James A. Fortune
CDMAPoster@xxxxxxxxxxxxxxxx

.

• References:

## Relevant Pages

• Re: Calculate slope of a data set or trend line
... to make future predictions I need the slope (representing ... it is calcuated by minimizing the sum of the squared deviation - OLS) ... Sum of the squares of the errors: ... answers you get no longer solve the 2 X 2 matrix equation. ...
(comp.databases.ms-access)
• Re: Americans finally see the global warming scam
... slope of the least squares line is positive. ... They both have a positive slope. ... LMFAO! ... You said you knew I was lying, and that I couldn't produce a reference ...
(rec.gambling.poker)
• Re: linear trend line on a scatter graph
... This is general least squares theory. ... I have scatter-plotted two matrices (matA & matB) onto one single ... how do I plot three linear trend lines on the data points ... I would also like to find the slope of these trend lines at ...
(comp.soft-sys.matlab)
• Re: Linear regression vs. vectors from principal components analysis
... origin the mean of both vars) might be construed as a prediction line? ... the sum of the squared distances from the points to the line. ... least squares", or "error in variables" if they want to follow up. ... minimization to find a closed-form solution for the slope, ...
(sci.stat.consult)
• Re: MLE of a restricted RV
... vector of length 1 and angle theta in the complex plane (you can cast the ... doing my stats on the slope of a zero-intercept orthogonal least squares ... This approach uses an ordinary least squares approach, ... more detail-- just remember to reverse the first part of my email addy. ...
(sci.stat.edu)