# Curve Fitting Using Piecewise Polynomials with Unknown Number and Location of Knots Fit a variety of curves by a sequence of piecewise polynomials. The number and location of knots are determined by the Reversible Jump MCMC method. ```r curve.polynomial.rjmcmc(y, x, lambda, l, l0, c=0.4, gamma.shape=1e-3, gamma.rate=1e-3, maxit=10000, err=1e-8, verbose=TRUE) ``` ## Arguments - `y`: a vector of values of a response variable. - `x`: a vector of values of the corresponding explanatory variable. - `lambda`: the parameter of the Poisson prior for the number of knots. - `l`: the order of polynomials. - `l0`: the degree of continuity at the knots. - `c`: the parameter controlling the rate of changing dimension. The default value is 0.4. - `gamma.shape`: the parameter shape of the gamma prior for the error precision. The default value is 1e-3. - `gamma.rate`: the parameter rate of the gamma prior for the error precision. The default value is 1e-3. - `maxit`: the maximum number of iterations. The default value is 10000. - `err`: the iteration stops when consecutive difference in percentage of mean-squared error reaches this bound. The default value is 1e-8. - `verbose`: logical. If `TRUE`, then indicate the level of output after every 1000 iterations. The default is `TRUE`. ## Returns - **knots.save**: a list of location of knots. One component per iteration. - **betas.save**: a list of estimates of regression parameters $\beta$s. One component per iteration. - **fitted.save**: a matrix of fitted values. One column per iteration. - **sigma2.save**: a vector of estimate of error variance. One component per iteration. ## Details The method is based on Denison et al. (1998). It can be used to fit both smooth and unsmooth curves. When both `l0` and `l` are set to 3, it fits curves using cubic spline. ## Note The factor $\frac{1}{\sqrt{n}}$ was added in the likelihood ratio to penalize the ratio for dimensionality as suggested in Dimatteo et al. (2001). ## References Denison, D. G. T., Mallick, B. K., Smith, A. F. M.(1998) Automatic Bayesian Curve Fitting **JRSSB** vol. 60, no. 2 333-350 Dimatteo, I., Genovese, C. R., Kass, R. E.(2001) Bayesian Curve-fitting with Free-knot Splines **Biometrika** vol. 88, no. 4 1055-1071 ## See Also `sm.spline` ## Examples ```r ## Not run: #Example 1: smooth curve #example 3.1. (b) in Denison et al.(1998) x <- runif(200) xx <- -2 + 4*x y.truth <- sin(2*xx) + 2*exp(-16*xx^2) y <- y.truth + rnorm(200, mean=0, sd=0.3) results <- curve.polynomial.rjmcmc(y, x, lambda=1, l=2, l0=1) plot(sort(x), y.truth[order(x)], type="l") lines(sort(x), rowMeans(results$fitted.save), col="red") #Example 2: unsmooth curve #blocks in Denison et al.(1998) tj <- c(0.1, 0.13, 0.15, 0.23, 0.25, 0.4, 0.44, 0.65, 0.76, 0.78, 0.81) hj <- c(4, -5, 3, -4, 5, -4.2, 2.1, 4.3, -3.1, 2.1, -4.2) t <- seq(0, 1, len=2048) Ktmtj <- outer(t, tj, function(a,b) ifelse(a-b > 0, 1, -1)) ft <- rowSums(Ktmtj %*% diag(hj)) x <- t y <- ft + rnorm(2048, 0, 1) results <- curve.polynomial.rjmcmc(y, x, lambda=5, l=2, l0=1) plot(x, ft, type="s") lines(x, rowMeans(results$fitted.save), col="red") #Example 3: unsmooth curve #bumps in Denison et al.(1998) tj <- c(0.1, 0.13, 0.15, 0.23, 0.25, 0.4, 0.44, 0.65, 0.76, 0.78, 0.81) hj <- c(4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2)*10 wj <- c(0.005, 0.005, 0.006, 0.01, 0.01, 0.03, 0.01, 0.01, 0.005, 0.008, 0.005) t <- seq(0, 1, len=2048) ft <- rowSums((1 + abs(outer(t, tj ,"-") %*% diag(1/wj)))^(-4) %*% diag(hj)) y <- ft + rnorm(2048, 0, 1) results <- curve.polynomial.rjmcmc(y, t, lambda=5, l=2, l0=1) plot(t, ft, type="s") lines(t, rowMeans(results$fitted.save), col="red") ## End(Not run) ```

curve.polynomial.rjmcmc function