there is a series of x and y values I have (but not the function itself). I would like to get derivative of the unknown function by spline interpolation of the x and y values (getting the derivat...). My example EDITED
x<-c(1,2,3,4,5,6,7,8,9,10)
y<-c(0.1,0.3,0.8,0.9,0.91,0.93,0.95,0.98,0.99,0.999)
is it possible in R to interpolate and to get the functional form of the derivative? My problem is that I have only x and y values of a cdf function but would need to obtain the probability denisty function..so I want to get the derivative by spline interpolation...
The reason for the question is that I would need to obtain the pdf of that cdf so I am trying to spline interpolate the xy values of the cdf - please note that this is a simple example here and not a real cdf
I haven't found the functional form of restricted cubic splines to be particularly difficult to grasp after reading the explanation by Frank Harrell in his book: "Regression Modeling Strategies".
require(rms)
df <- data.frame( x = c(1,2,3,4,5,6,7,8,9,10),
y =c(12,2,-3,5,6,9,8,10,11,10.5))
ols( y ~ rcs(x, 3), df)
#--------------
Linear Regression Model
ols(formula = y ~ rcs(x, 3), data = df)
Model Likelihood Discrimination
Ratio Test Indexes
Obs 10 LR chi2 3.61 R2 0.303
sigma 4.4318 d.f. 2 R2 adj 0.104
d.f. 7 Pr(> chi2) 0.1646 g 2.811
Residuals
Min 1Q Median 3Q Max
-8.1333 -1.1625 0.5333 0.9833 6.9000
Coef S.E. t Pr(>|t|)
Intercept 5.0833 4.2431 1.20 0.2699
x 0.0167 1.1046 0.02 0.9884
x' 1.0000 1.3213 0.76 0.4738
#----------
The rms
package has an odd system for storing summary information that needs to be done for some of its special
dd <- datadist(df)
options(datadist="dd")
mymod <- ols( y ~ rcs(x, 3), df)
# cannot imagine that more than 3 knots would make sense in such a small example
Function(mymod)
# --- reformatted to allow inspection of separate terms
function(x = 5.5) {5.0833333+0.016666667* x +
1*pmax(x-5, 0)^3 -
2*pmax(x-5.5, 0)^3 +
1*pmax(x-6, 0)^3 }
<environment: 0x1304ad940>
The zeros in the pmax
functions basically suppress any contribution to the total from the term when the x value is less than the knots ( 5, 5.5 and 6 in this case)
Compare three versus four knots (and if you wanted smooth curves then include a finer grained ...-
data argument to Predict
):
png()
plot(df$x,df$y )
mymod <- ols( y ~ rcs(x, 3), df)
lines(df$x, predict(mymod) ,col="blue")
mymod <- ols( y ~ rcs(x, 4), df)
lines(df$x, predict(mymod) ,col="red")
dev.off()
Take a look at monotone cubic splines, which are nondecreasing by construction. A web search for "monotone cubic spline R" turns up some hits. I haven't used any of the packages mentioned.
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