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R/ispline3.R
In SCCS: The Self-Controlled Case Series Method
Defines functions
ispline3
Documented in
ispline3
########### Third integral of an I-spline ########
ispline3 <- function(x, knots1, m) { # ispline1 is a function that evaluates integral of an I-spline created by inegrating
# x is a vector of values to compute the I-spline at
# i ith I-spline, k= knots vector, m= order
# m <- order 4
# k <-knots (the knots used in computing M-splines of order 5 should be used
# in computing the I-splines)
dk <- knots1[2]-knots1[1]
k <- c(knots1[1]-dk*(4:1), knots1, knots1[length(knots1)]+dk*(1:4))
k6 <- c(knots1[1]-dk*(5:1), knots1, knots1[length(knots1)]+dk*(1:5))
k7 <- c(knots1[1]-dk*(6:1), knots1, knots1[length(knots1)]+dk*(1:6))
k8 <- c(knots1[1]-dk*(7:1), knots1, knots1[length(knots1)]+dk*(1:7))
msplinedesign8 <- dmsplinedesign(x, knots1, 8, deriv=0) # A matrix of order 8 to be used in computing the I-splines (integral of the M-splines of order 7),
# k8 = knots8 is the set of knots used to compute M-splines of order 8
d <- length(k)-m-1 # number of columuns for the I-spline matrices
resu <- matrix(0, length(x), length(k)-m-1)
for (j in 1:length(x))
for (i in 1:(length(k)-m-1))
if (x[j]>k[i+m+1]){
# resu[j,i] <- 1
resu[j,i] <- ((((x[j])^3)/(2*3)) - ((k[i+m+1]*x[j]^2))/2 + ((x[j]*k[i+m+1]^2)/2) - ((k[i+m+1]^3)/6))
+ ((k[i+m+1+1]-k[i+1])/(m+1))*( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
+ ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*( ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*(((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+3+m+1+1]-k[i+3+1])/(m+1))*(((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
} else if (x[j] < k[i+1]) {
# resu[j,i]<- 0
resu[j,i]<- 0
} else if (x[j]> k[i+1] && x[j]<=k[i+1+1]) {
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1)
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*((k6[i+m+2+1]-k6[i+1])/(m+2))*((k7[i+m+3+1]-k7[i+1])/(m+3))*((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])
} else if (x[j]>k[i+1+1] && x[j]<=k[i+2+1]) {
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1)
# resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*((((k6[i+m+2+1]-k6[i+1])/(m+2))*((((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))) + ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])))) + ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*((((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))))))
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*((((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*((((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))))))
} else if (x[j]>k[i+2+1] && x[j] <= k[i+3+1]){
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1) + (k[i+2+m+1]-k[i+2])*(msplinedesign5[j,i+2])/(m+1)
# resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])+ ((k[i+2+m+1]-k[i+2])/(m+1))*((k6[i+2+m+2]-k6[i+2])/(m+2))*((k7[i+2+m+3]-k7[i+2])/(m+3))*((k8[i+2+m+4]-k8[i+2])/(m+4))*(msplinedesign8[j,i+2])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*(( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*((((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*((((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
} else
#if (x>=k[i+3] && x < k[i+4])
{
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1) + (k[i+2+m+1]-k[i+2])*(msplinedesign5[j,i+2])/(m+1) + (k[i+3+m+1]-k[i+3])*(msplinedesign5[j,i+3])/(m+1)
#resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])+ ((k[i+2+m+1]-k[i+2])/(m+1))*((k6[i+2+m+2]-k6[i+2])/(m+2))*((k7[i+2+m+3]-k7[i+2])/(m+3))*((k8[i+2+m+4]-k8[i+2])/(m+4))*(msplinedesign8[j,i+2]) + ((k[i+3+m+1]-k[i+3])/(m+1))*((k6[i+3+m+2]-k6[i+3])/(m+2))*((k7[i+3+m+3]-k7[i+3])/(m+3))*((k8[i+3+m+4]-k8[i+3])/(m+4))*(msplinedesign8[j,i+3])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*( ( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*( (((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*( (((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*( ( ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+3+m+1+1]-k[i+3+1])/(m+1))*( ( ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))))))
}
return(resu[,1:(d-1)])
}
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Defines functions ispline3
Documented in ispline3
########### Third integral of an I-spline ########
ispline3 <- function(x, knots1, m) { # ispline1 is a function that evaluates integral of an I-spline created by inegrating
# x is a vector of values to compute the I-spline at
# i ith I-spline, k= knots vector, m= order
# m <- order 4
# k <-knots (the knots used in computing M-splines of order 5 should be used
# in computing the I-splines)
dk <- knots1[2]-knots1[1]
k <- c(knots1[1]-dk*(4:1), knots1, knots1[length(knots1)]+dk*(1:4))
k6 <- c(knots1[1]-dk*(5:1), knots1, knots1[length(knots1)]+dk*(1:5))
k7 <- c(knots1[1]-dk*(6:1), knots1, knots1[length(knots1)]+dk*(1:6))
k8 <- c(knots1[1]-dk*(7:1), knots1, knots1[length(knots1)]+dk*(1:7))
msplinedesign8 <- dmsplinedesign(x, knots1, 8, deriv=0) # A matrix of order 8 to be used in computing the I-splines (integral of the M-splines of order 7),
# k8 = knots8 is the set of knots used to compute M-splines of order 8
d <- length(k)-m-1 # number of columuns for the I-spline matrices
resu <- matrix(0, length(x), length(k)-m-1)
for (j in 1:length(x))
for (i in 1:(length(k)-m-1))
if (x[j]>k[i+m+1]){
# resu[j,i] <- 1
resu[j,i] <- ((((x[j])^3)/(2*3)) - ((k[i+m+1]*x[j]^2))/2 + ((x[j]*k[i+m+1]^2)/2) - ((k[i+m+1]^3)/6))
+ ((k[i+m+1+1]-k[i+1])/(m+1))*( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
+ ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*( ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*(((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*(((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*(((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
)
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3])))
) ))
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
+ ((k[i+3+m+1+1]-k[i+3+1])/(m+1))*(((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*(((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*((dmsplinedesign(k[i+m+1], knots1, 8, deriv=0))[1,i+3+1+3]))))))
)
} else if (x[j] < k[i+1]) {
# resu[j,i]<- 0
resu[j,i]<- 0
} else if (x[j]> k[i+1] && x[j]<=k[i+1+1]) {
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1)
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*((k6[i+m+2+1]-k6[i+1])/(m+2))*((k7[i+m+3+1]-k7[i+1])/(m+3))*((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])
} else if (x[j]>k[i+1+1] && x[j]<=k[i+2+1]) {
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1)
# resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*((((k6[i+m+2+1]-k6[i+1])/(m+2))*((((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))) + ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])))) + ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*((((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))))))
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*((((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*((((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))* (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3]))))))
} else if (x[j]>k[i+2+1] && x[j] <= k[i+3+1]){
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1) + (k[i+2+m+1]-k[i+2])*(msplinedesign5[j,i+2])/(m+1)
# resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])+ ((k[i+2+m+1]-k[i+2])/(m+1))*((k6[i+2+m+2]-k6[i+2])/(m+2))*((k7[i+2+m+3]-k7[i+2])/(m+3))*((k8[i+2+m+4]-k8[i+2])/(m+4))*(msplinedesign8[j,i+2])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*(( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*((((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*((((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*( (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) )
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) ))
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*((((k6[i+2+m+2+1]-k6[i+2+1])/(m+2)))*((((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) ) )))
} else
#if (x>=k[i+3] && x < k[i+4])
{
# resu[j,i] <- (k[i+m+1]-k[i])*(msplinedesign5[j,i])/(m+1) + (k[i+1+m+1]-k[i+1])*(msplinedesign5[j,i+1])/(m+1) + (k[i+2+m+1]-k[i+2])*(msplinedesign5[j,i+2])/(m+1) + (k[i+3+m+1]-k[i+3])*(msplinedesign5[j,i+3])/(m+1)
#resu[j,i] <- ((k[i+m+1]-k[i])/(m+1))*((k6[i+m+2]-k6[i])/(m+2))*((k7[i+m+3]-k7[i])/(m+3))*((k8[i+m+4]-k8[i])/(m+4))*(msplinedesign8[j,i]) + ((k[i+1+m+1]-k[i+1])/(m+1))*((k6[i+1+m+2]-k6[i+1])/(m+2))*((k7[i+1+m+3]-k7[i+1])/(m+3))*((k8[i+1+m+4]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1])+ ((k[i+2+m+1]-k[i+2])/(m+1))*((k6[i+2+m+2]-k6[i+2])/(m+2))*((k7[i+2+m+3]-k7[i+2])/(m+3))*((k8[i+2+m+4]-k8[i+2])/(m+4))*(msplinedesign8[j,i+2]) + ((k[i+3+m+1]-k[i+3])/(m+1))*((k6[i+3+m+2]-k6[i+3])/(m+2))*((k7[i+3+m+3]-k7[i+3])/(m+3))*((k8[i+3+m+4]-k8[i+3])/(m+4))*(msplinedesign8[j,i+3])
resu[j,i] <- ((k[i+m+1+1]-k[i+1])/(m+1))*( ( ((k6[i+m+2+1]-k6[i+1])/(m+2))*( ( ((k7[i+m+3+1]-k7[i+1])/(m+3))*( (((k8[i+m+4+1]-k8[i+1])/(m+4))*(msplinedesign8[j,i+1+3])) + (((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+1+m+1+1]-k[i+1+1])/(m+1))*( (((k6[i+1+m+2+1]-k6[i+1+1])/(m+2))*( ( ((k7[i+1+m+3+1]-k7[i+1+1])/(m+3))*((((k8[i+1+m+4+1]-k8[i+1+1])/(m+4))*(msplinedesign8[j,i+1+1+3])) + (((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+2+m+1+1]-k[i+2+1])/(m+1))*( ( ((k6[i+2+m+2+1]-k6[i+2+1])/(m+2))*( ( ((k7[i+2+m+3+1]-k7[i+2+1])/(m+3))*((((k8[i+2+m+4+1]-k8[i+2+1])/(m+4))*(msplinedesign8[j,i+2+1+3])) + (((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))
+ ((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))) )
)
+ ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3]))))))
)
+ ((k[i+3+m+1+1]-k[i+3+1])/(m+1))*( ( ((k6[i+3+m+2+1]-k6[i+3+1])/(m+2))*( (((k7[i+3+m+3+1]-k7[i+3+1])/(m+3))*((((k8[i+3+m+4+1]-k8[i+3+1])/(m+4))*(msplinedesign8[j,i+3+1+3])))))))
}
return(resu[,1:(d-1)])
}
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