Robbie,
Here's what I *think* you are trying to do:
1.
y is a cubic function of x:
y = b1*x + b2*x^2 + b3*x^3
2.
s is the cumsum of y:
s_i = y_1 + ... + y_i
3.
Given a subset of x = 1:n and the corresponding
values of s, estimate the coefficients of the cubic.
If that is the correct understanding, then you should
be able to estimate the coefficients as follows:
a) since s_i = b1 * sum of x_k for k=1, ..., i
+ b2 * sum of (x_k)^2 for k=1, ..., i
+ b3 * sum of (x_k)^3 for k=1, ..., i
we can regress s on the cumsums of x, x^2 and x^3:
using your sample data:
d <- data.frame(x = c(1, 4, 9, 12),
s = c(109, 1200, 5325, 8216))
e <- data.frame(x = 1:12)
e <- merge(e, d, all.x = T)
e <- within(e,
{z3 <- cumsum(x^3)
z2 <- cumsum(x^2)
z1 <- cumsum(x)})
coef(lm(s ~ 0 + z1 + z2 + z3, data = e))
# z1 z2 z3
# 100 10 -1
Peter Ehlers
On 2012-05-22 09:43, Robbie Edwards wrote:
I don't think I can.
For the sample data
d<- data.frame(x=c(1, 4, 9, 12), s=c(109, 1200, 5325, 8216))
when x = 4, s = 1200. However, that s4 is sum of y1 + y2 + y3 + y4.
Wouldn't I have to know the y for x = 2 and x = 3 to get the value of y
for x = 4?
In the previous message, I created two sample data frames. d is what I'm
trying to use to create df. I only know what's in d, df is just used to
illustrate what I'm trying to get from d.
robbie
On Tue, May 22, 2012 at 12:30 PM, R. Michael Weylandt<
michael.weyla...@gmail.com> wrote:
But if I understand your problem correctly, you can get the y values
from the s values. I'm relying on your statement that "s is sum of the
current y and all previous y (s3 = y1 + y2 + y3)." E.g.,
y<- c(1, 4, 6, 9, 3, 7)
s1 = 1
s2 = 4 + s1 = 5
s3 = 6 + s2 = 11
more generally
s<- cumsum(y)
Then if we only see s, we can get back the y vector by doing
c(s[1], diff(s))
which is identical to y.
So for your data, the underlying y must have been c(109, 1091, 4125,
2891) right?
Or have I completely misunderstood your problem?
Michael
On Tue, May 22, 2012 at 12:25 PM, Robbie Edwards
<robbie.edwa...@gmail.com> wrote:
Actually, I can't. I don't know the y values. Only the s and only for a
subset of the data.
Like this.
d<- data.frame(x=c(1, 4, 9, 12), s=c(109, 1200, 5325, 8216))
On Tue, May 22, 2012 at 11:57 AM, R. Michael Weylandt
<michael.weyla...@gmail.com> wrote:
You can reconstruct the y values by taking first-differences of the s
vector, no? Then it sounds like you're good to go
Best, Michael
On Tue, May 22, 2012 at 11:40 AM, Robbie Edwards
<robbie.edwa...@gmail.com> wrote:
Hi all,
Thanks for the replies, but I realize I've done a bad job explaining
my
problem. To help, I've created some sample data to explain the
problem.
df<- data.frame(x=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), y=c(109,
232,
363, 496, 625, 744, 847, 928, 981, 1000, 979, 912), s=c(109, 341, 704,
1200, 1825, 2569, 3416, 4344, 5325, 6325, 7304, 8216))
In this data frame, y results from y = x * b1 + x^2 * b2 + x^3 * b3
and
s
is sum of the current y and all previous y (s3 = y1 + y2 + y3).
I know I can find b1, b2 and b3 using:
lm(y ~ 0 + x + I(x^2) + I(x^3), data=df)
yielding...
Coefficients:
x I(x^2) I(x^3)
100 10 -1
However, I need to find b1, b2 and b3 using the s column. The reason
being, I don't actually know the values of y in the actual data set.
And
in the actual data, I only have a few of the values. Imagine this
data
is
being used a reward schedule for like a loyalty points program. y
represents the number of points needed for each level while s is the
total
number of points to reach that level. In the real problem, my data
looks
more like this:
d<- data.frame(x=c(1, 4, 9, 12), s=c(109, 1200, 5325, 8216))
Where I need to use a few sample points to help define the parameters
of
the curve.
thanks again and hopefully this makes the problem a bit clearer.
robbie
On Fri, May 18, 2012 at 7:40 PM, David Winsemius
<dwinsem...@comcast.net>wrote:
On May 18, 2012, at 1:44 PM, Robbie Edwards wrote:
Hi all,
I'm trying to model some data where the y is defined by
y = summation[1 to 50] B1 * x + B2 * x^2 + B3 * x^3
Hopefully that reads clearly for email.
cumsum( rowSums( cbind(B1 * x, B2 * x^2, B3 * x^3)))
Anyway, if it wasn't for the summation, I know I would do it like
this
lm(y ~ x + x2 + x3)
Where x2 and x3 are x^2 and x^3.
However, since each value of x is related to the previous values of
x,
I
don't know how to do this. Any help is greatly appreciated.
David Winsemius, MD
West Hartford, CT
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