# smooth.basis: Construct a functional data object by smoothing data using a... In fda: Functional Data Analysis

## Description

Discrete observations on one or more curves and for one more more variables are fit with a set of smooth curves, each defined by an expansion in terms of user-selected basis functions. The fitting criterion is weighted least squares, and smoothness can be defined in terms of a roughness penalty that is specified in a variety of ways.

Data smoothing requires at a bare minimum three elements: (1) a set of observed noisy values, (b) a set of argument values associated with these data, and (c) a specification of the basis function system used to define the curves. Typical basis functions systems are splines for nonperiodic curves, and fourier series for periodic curves.

Optionally, a set covariates may be also used to take account of various non-smooth contributions to the data. Smoothing without covariates is often called nonparametric regression, and with covariates is termed semiparametric regression.

## Usage

 ```1 2``` ```smooth.basis(argvals=1:n, y, fdParobj, wtvec=NULL, fdnames=NULL, covariates=NULL, method="chol", dfscale=1, returnMatrix=FALSE) ```

## Arguments

 `argvals` a set of argument values corresponding to the observations in array `y`. In most applications these values will be common to all curves and all variables, and therefore be defined as a vector or as a matrix with a single column. But it is possible that these argument values will vary from one curve to another, and in this case `argvals` will be input as a matrix with rows corresponding to observation points and columns corresponding to curves. Argument values can even vary from one variable to another, in which case they are input as an array with dimensions corresponding to observation points, curves and variables, respectively. Note, however, that the number of observation points per curve and per variable may NOT vary. If it does, then curves and variables must be smoothed individually rather than by a single call to this function. The default value for `argvals` are the integers 1 to `n`, where `n` is the size of the first dimension in argument `y`. `y` an set of values of curves at discrete sampling points or argument values. If the set is supplied as a matrix object, the rows must correspond to argument values and columns to replications, and it will be assumed that there is only one variable per observation. If `y` is a three-dimensional array, the first dimension corresponds to argument values, the second to replications, and the third to variables within replications. If `y` is a vector, only one replicate and variable are assumed. If the data come from a single replication but multiple vectors, such as data on coordinates for a single space curve, then be sure to coerce the data into an array object by using the `as.array` function with one as the central dimension length. `fdParobj` a functional parameter object, a functional data object or a functional basis object. In the simplest case, `fdParobj` may be a functional basis object with class "basisfd" defined previously by one of the "create" functions, and in this case, no roughness penalty is used. No roughness penalty is also the consequence of supplying a functional data object with class "fd", in which case the basis system used for smoothing is that defining this object. In these two simple cases, `smooth.basis` is essentially the same as function `Data2fd`, and this type of elementary smoothing is often called "regression smoothing." However, if the object is a functional parameter object with class "fdPar", then the linear differential operator object and the smoothing parameter in this object define the roughness penalty. For further details on how the roughness penalty is defined, see the help file for "fdPar". In general, better results can be obtained using a good roughness penalty than can be obtained by merely varying the number of basis functions in the expansion. `wtvec` typically a vector of length `n` that is the length of `argvals` containing weights for the values to be smoothed, However, it may also be a symmetric matrix of order `n`. If `wtvec` is a vector, all values must be positive, and if it is a symmetric matrix, this must be positive definite. Defaults to all weights equal to 1. `fdnames` a list of length 3 containing character vectors of names for the following: args name for each observation or point in time at which data are collected for each 'rep', unit or subject. reps name for each 'rep', unit or subject. fun name for each 'fun' or (response) variable measured repeatedly (per 'args') for each 'rep'. `covariates` The observed values in `y` are assumed to be primarily determined by the height of the curve being estimates, but from time to time certain values can also be influenced by other known variables. For example, multi-year sets of climate variables may be also determined by the presence of absence of an El Nino event, or a volcanic eruption. One or more of these covariates can be supplied as an `n` by `p` matrix, where `p` is the number of such covariates. When such covariates are available, the smoothing is called "semi-parametric." Matrices or arrays of regression coefficients are then estimated that define the impacts of each of these covariates for each curve and each variable. `method` by default the function uses the usual textbook equations for computing the coefficients of the basis function expansions. But, as in regression analysis, a price is paid in terms of rounding error for such computations since they involved cross-products of basis function values. Optionally, if `method` is set equal to the string "qr", the computation uses an algorithm based on the qr-decomposition which is more accurate, but will require substantially more computing time when `n` is large, meaning more than 500 or so. The default is "chol", referring the Choleski decomposition of a symmetric positive definite matrix. `dfscale` the generalized cross-validation or "gcv" criterion that is often used to determine the size of the smoothing parameter involves the subtraction of an measue of degrees of freedom from `n`. Chong Gu has argued that multiplying this degrees of freedom measure by a constant slightly greater than 1, such as 1.2, can produce better decisions about the level of smoothing to be used. The default value is, however, 1.0. `returnMatrix` logical: If TRUE, a two-dimensional is returned using a special class from the Matrix package.

## Details

A roughness penalty is a quantitative measure of the roughness of a curve that is designed to fit the data. For this function, this penalty consists of the product of two parts. The first is an approximate integral over the argument range of the square of a derivative of the curve. A typical choice of derivative order is 2, whose square is often called the local curvature of the function. Since a rough function has high curvature over most of the function's range, the integrated square of of the second derivature quantifies the total curvature of the function, and hence its roughness. The second factor is a positive constant called the bandwidth of smoothing parameter, and given the variable name `lambda` here.

In more sophisticated uses of `smooth.basis`, a derivative may be replaced by a linear combination of two or more order of derivative, with the coefficients of this combination themselves possibly varying over the argument range. Such a structure is called a "linear differential operator", and a clever choice of operator can result in much improved smoothing.

The rougnhness penalty is added to the weighted error sum of squares and the composite is minimized, usually in conjunction with a high dimensional basis expansion such as a spline function defined by placing a knot at every observation point. Consequently, the smoothing parameter controls the relative emphasis placed on fitting the data versus smoothness; when large, the fitted curve is more smooth, but the data fit worse, and when small, the fitted curve is more rough, but the data fit much better. Typically smoothing parameter `lambda` is manipulated on a logarithmic scale by, for example, defining it as a power of 10.

A good compromise `lambda` value can be difficult to define, and minimizing the generalized cross-validation or "gcv" criterion that is output by `smooth.basis` is a popular strategy for making this choice, although by no means foolproof. One may also explore `lambda` values for a few log units up and down from this minimizing value to see what the smoothing function and its derivatives look like. The function `plotfit.fd` was designed for this purpose.

The size of common logarithm of the minimizing value of `lambda` can vary widely, and spline functions depends critically on the typical spacing between knots. While there is typically a "natural" measurement scale for the argument, such as time in milliseconds, seconds, and so forth, it is better from a computational perspective to choose an argument scaling that gives knot spacings not too different from one.

An alternative to using `smooth.basis` is to first represent the data in a basis system with reasonably high resolution using `Data2fd`, and then smooth the resulting functional data object using function `smooth.fd`.

In warning and error messages, you may see reference to functions `smooth.basis1, smooth.basis2`, and `smooth.basis3`. These functions are defined within `smooth.basis`, and are not normally to be called by users.

The "qr" algorithm option defined by the "method" parameter will not normally be needed, but if a warning of a near singularity in the coefficient calculations appears, this choice may be a cure.

## Value

an object of class `fdSmooth`, which is a named list of length 8 with the following components:

 `fd` a functional data object containing a smooth of the data. `df` a degrees of freedom measure of the smooth `gcv` the value of the generalized cross-validation or GCV criterion. If there are multiple curves, this is a vector of values, one per curve. If the smooth is multivariate, the result is a matrix of gcv values, with columns corresponding to variables. gcv = n*SSE/((n-df)^2) `beta` the regression coefficients associated with covariate variables. These are vector, matrix or array objects dependeing on whether there is a single curve, multiple curves or multiple curves and variables, respectively. `SSE` the error sums of squares. SSE is a vector or a matrix of the same size as GCV. `penmat` the penalty matrix. `y2cMap` the matrix mapping the data to the coefficients. `argvals, y` input arguments

`lambda2df`, `lambda2gcv`, `df2lambda`, `plot.fd`, `project.basis`, `smooth.fd`, `smooth.monotone`, `smooth.pos`, `smooth.basisPar` `Data2fd`,
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610``` ```## ######## Simulated data example 1: a simple regression smooth ######## ## # Warning: In this and all simulated data examples, your results # probably won't be the same as we saw when we ran the example because # random numbers depend on the seed value in effect at the time of the # analysis. # # Set up 51 observation points equally spaced between 0 and 1 n = 51 argvals = seq(0,1,len=n) # The true curve values are sine function values with period 1/2 x = sin(4*pi*argvals) # Add independent Gaussian errors with std. dev. 0.2 to the true values sigerr = 0.2 y = x + rnorm(x)*sigerr # When we ran this code, we got these values of y (rounded to two # decimals): y = c(0.27, 0.05, 0.58, 0.91, 1.07, 0.98, 0.54, 0.94, 1.13, 0.64, 0.64, 0.60, 0.24, 0.15, -0.20, -0.63, -0.40, -1.22, -1.11, -0.76, -1.11, -0.69, -0.54, -0.50, -0.35, -0.15, 0.27, 0.35, 0.65, 0.75, 0.75, 0.91, 1.04, 1.04, 1.04, 0.46, 0.30, -0.01, -0.19, -0.42, -0.63, -0.78, -1.01, -1.08, -0.91, -0.92, -0.72, -0.84, -0.38, -0.23, 0.02) # Set up a B-spline basis system of order 4 (piecewise cubic) and with # knots at 0, 0.1, ..., 0.9 and 1.0, and plot the basis functions nbasis = 13 basisobj = create.bspline.basis(c(0,1),nbasis) plot(basisobj) # Smooth the data, outputting only the functional data object for the # fitted curve. Note that in this simple case we can supply the basis # object as the "fdParobj" parameter ys = smooth.basis(argvals=argvals, y=y, fdParobj=basisobj) Ys = smooth.basis(argvals=argvals, y=y, fdParobj=basisobj, returnMatrix=TRUE) # Ys[[7]] = Ys\$y2cMap is sparse; everything else is the same all.equal(ys[-7], Ys[-7]) xfd = ys\$fd Xfd = Ys\$fd # Plot the curve along with the data plotfit.fd(y, argvals, xfd) # Compute the root-mean-squared-error (RMSE) of the fit relative to the # truth RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) print(RMSE) # We obtained 0.069 # RMSE = 0.069 seems good relative to the standard error of 0.2. # Range through numbers of basis functions from 4 to 12 to see if we # can do better. We want the best RMSE, but we also want the smallest # number of basis functions, which in this case is the degrees of # freedom for error (df). Small df implies a stable estimate. # Note: 4 basis functions is as small as we can use without changing the # order of the spline. Also display the gcv statistic to see what it # likes. for (nbasis in 4:12) { basisobj = create.bspline.basis(c(0,1),nbasis) ys = smooth.basis(argvals, y, basisobj) xfd = ys\$fd gcv = ys\$gcv RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) # progress report: # cat(paste(nbasis,round(RMSE,3),round(gcv,3),"\n")) } # We got RMSE = 0.062 for 10 basis functions as optimal, but gcv liked # almost the same thing, namely 9 basis functions. Both RMSE and gcv # agreed emphatically that 7 or fewer basis functions was not enough. # Unlike RMSE, however, gcv does not depend on knowing the truth. # Plot the result for 10 basis functions along with "*" at the true # values nbasis = 10 basisobj = create.bspline.basis(c(0,1),10) xfd = smooth.basis(argvals, y, basisobj)\$fd plotfit.fd(y, argvals, xfd) points(argvals,x, pch="*") # Homework: # Repeat all this with various values of sigerr and various values of n ## ####### Simulated data example 2: a roughness-penalized smooth ######## ## # A roughness penalty approach is more flexible, allowing continuous # control over smoothness and degrees of freedom, can be adapted to # known features in the curve, and will generally provide better RMSE # for given degrees of freedom. # It does require a bit more effort, though. # First, we define a little display function for showing how # df, gcv and RMSE depend on the log10 smoothing parameter plotGCVRMSE.fd = function(lamlow, lamhi, lamdel, x, argvals, y, fdParobj, wtvec=NULL, fdnames=NULL, covariates=NULL) { loglamvec = seq(lamlow, lamhi, lamdel) loglamout = matrix(0,length(loglamvec),4) m = 0 for (loglambda in loglamvec) { m = m + 1 loglamout[m,1] = loglambda fdParobj\$lambda = 10^(loglambda) smoothlist = smooth.basis(argvals, y, fdParobj, wtvec=wtvec, fdnames=fdnames, covariates=covariates) xfd = smoothlist\$fd # the curve smoothing the data loglamout[m,2] = smoothlist\$df # degrees of freedom in the smoothing curve loglamout[m,3] = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) loglamout[m,4] = mean(smoothlist\$gcv) # the mean of the N gcv values } cat("log10 lambda, deg. freedom, RMSE, gcv\n") for (i in 1:m) { cat(format(round(loglamout[i,],3))) cat("\n") } par(mfrow=c(3,1)) plot(loglamvec, loglamout[,2], type="b") title("Degrees of freedom") plot(loglamvec, loglamout[,3], type="b") title("RMSE") plot(loglamvec, loglamout[,4], type="b") title("Mean gcv") return(loglamout) } # Use the data that you used in Example 1, or run the following code: n = 51 argvals = seq(0,1,len=n) x = sin(4*pi*argvals) sigerr = 0.2 err = matrix(rnorm(x),n,1)*sigerr y = x + err # We now set up a basis system with a knot at every data point. # The number of basis functions will equal the number of interior knots # plus the order, which in this case is still 4. # 49 interior knots + order 4 = 53 basis functions nbasis = n + 2 basisobj = create.bspline.basis(c(0,1),nbasis) # Note that there are more basis functions than observed values. A # basis like this is called "super-saturated." We have to use a # positive smoothing parameter for it to work. Set up an object of # class "fdPar" that penalizes the total squared second derivative, # using a smoothing parameter that is set here to 10^(-4.5). lambda = 10^(-4.5) fdParobj = fdPar(fdobj=basisobj, Lfdobj=2, lambda=lambda) # Smooth the data, outputting a list containing various quantities smoothlist = smooth.basis(argvals, y, fdParobj) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) cat(round(c(df,RMSE,gcv),3),"\n") plotfit.fd(y, argvals, xfd) points(argvals,x, pch="*") # Repeat these analyses for a range of log10(lambda) values by running # the function plotGCVRMSE that we defined above. loglamout = plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj) # When we ran this example, the optimal RMSE was 0.073, and was achieved # for log10(lambda) = -4.25 or lambda = 0.000056. At this level of # smoothing, the degrees of freedom index was 10.6, a value close to # the 10 degrees of freedom that we saw for regression smoothing. The # RMSE value is slightly higher than the regression analysis result, as # is the degrees of freedom associated with the optimal value. # Roughness penalty will, as we will see later, do better than # regression smoothing; but with slightly red faces we say, "That's # life with random data!" The gcv statistic agreed with RMSE on the # optimal smoothing level, which is great because it does not need to # know the true values. Note that gcv is emphatic about when there is # too much smoothing, but rather vague about when we have # under-smoothed the data. # Homework: # Compute average results taken across 100 sets of random data for each # level of smoothing parameter lambda, and for each number of basis # functions for regression smoothing. ## ## Simulated data example 3: ## a roughness-penalized smooth of a sample of curves ## n = 51 # number of observations per curve N = 100 # number of curves argvals = seq(0,1,len=n) # The true curve values are linear combinations of fourier function # values. # Set up the fourier basis nfourierbasis = 13 fourierbasis = create.fourier.basis(c(0,1),nfourierbasis) fourierbasismat = eval.basis(argvals,fourierbasis) # Set up some random coefficients, with declining contributions from # higher order basis functions basiswt = matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N) xcoef = matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt xfd = fd(xcoef, fourierbasis) x = eval.fd(argvals, xfd) # Add independent Gaussian noise to the data with std. dev. 0.2 sigerr = 0.2 y = x + matrix(rnorm(c(x)),n,N)*sigerr # Set up spline basis system nbasis = n + 2 basisobj = create.bspline.basis(c(0,1),nbasis) # Set up roughness penalty with smoothing parameter 10^(-5) lambda = 10^(-5) fdParobj = fdPar(fdobj=basisobj, Lfdobj=2, lambda=lambda) # Smooth the data, outputting a list containing various quantities smoothlist = smooth.basis(argvals, y, fdParobj) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) # Display the results, note that a gcv value is returned for EACH curve, # and therefore that a mean gcv result is reported cat(round(c(df,RMSE,mean(gcv)),3),"\n") # the fits are plotted interactively by plotfit.fd ... click to advance # plot plotfit.fd(y, argvals, xfd) # Repeat these results for a range of log10(lambda) values loglamout = plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj) # Our results were: # log10 lambda, deg. freedom, RMSE, GCV # -6.000 30.385 0.140 0.071 # -5.750 26.750 0.131 0.066 # -5.500 23.451 0.123 0.062 # -5.250 20.519 0.116 0.059 # -5.000 17.943 0.109 0.056 # -4.750 15.694 0.104 0.054 # -4.500 13.738 0.101 0.053 # -4.250 12.038 0.102 0.054 # -4.000 10.564 0.108 0.055 # -3.750 9.286 0.120 0.059 # -3.500 8.177 0.139 0.065 # -3.250 7.217 0.164 0.075 # -3.000 6.385 0.196 0.088 # RMSE and gcv both indicate an optimal smoothing level of # log10(lambda) = -4.5 corresponding to 13.7 df. When we repeated the # analysis using regression smoothing with 14 basis functions, we # obtained RMSE = 0.110, about 10 percent larger than the value of # 0.101 in the roughness penalty result. Smooth the data, outputting a # list containing various quantities # Homework: # Sine functions have a curvature that doesn't vary a great deal over # the range the curve. Devise some test functions with sharp local # curvature, such as Gaussian densities with standard deviations that # are small relative to the range of the observations. Compare # regression and roughness penalty smoothing in these situations. if(!CRAN()){ ## ####### Simulated data example 4: a roughness-penalized smooth ######## ## with correlated error ## # These three examples make GCV look pretty good as a basis for # selecting the smoothing parameter lambda. BUT GCV is based an # assumption of independent errors, and in reality, functional data # often have autocorrelated errors, with an autocorrelation that is # usually positive among neighboring observations. Positively # correlated random values tend to exhibit slowly varying values that # have long runs on one side or the other of their baseline, and # therefore can look like trend in the data that needs to be reflected # in the smooth curve. This code sets up the error correlation matrix # for first-order autoregressive errors, or AR(1). rho = 0.9 n = 51 argvals = seq(0,1,len=n) x = sin(4*pi*argvals) Rerr = matrix(0,n,n) for (i in 1:n) { for (j in 1:n) Rerr[i,j] = rho^abs(i-j) } # Compute the Choleski factor of the correlation matrix Lerr = chol(Rerr) # set up some data # Generate auto-correlated errors by multipling independent errors by # the transpose of the Choleski factor sigerr = 0.2 err = as.vector(crossprod(Lerr,matrix(rnorm(x),n,1))*sigerr) # See the long-run errors that are genrated plot(argvals, err) y = x + err # Our values of y were: y = c(-0.03, 0.36, 0.59, 0.71, 0.97, 1.2, 1.1, 0.96, 0.79, 0.68, 0.56, 0.25, 0.01,-0.43,-0.69, -1, -1.09,-1.29,-1.16,-1.09, -0.93, -0.9,-0.78,-0.47, -0.3,-0.01, 0.29, 0.47, 0.77, 0.85, 0.87, 0.97, 0.9, 0.72, 0.48, 0.25,-0.17,-0.39,-0.47,-0.71, -1.07,-1.28,-1.33,-1.39,-1.45, -1.3,-1.25,-1.04,-0.82,-0.55, -0.2) # Retaining the above data, now set up a basis system with a knot at # every data point: the number of basis functions will equal the # number of interior knots plus the order, which in this case is still # 4. # 19 interior knots + order 4 = 23 basis functions nbasis = n + 2 basisobj = create.bspline.basis(c(0,1),nbasis) fdParobj = fdPar(basisobj) # Smooth these results for a range of log10(lambda) values loglamout = plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj) # Our results without weighting were: # -6.000 30.385 0.261 0.004 # -5.750 26.750 0.260 0.005 # -5.500 23.451 0.259 0.005 # -5.250 20.519 0.258 0.005 # -5.000 17.943 0.256 0.005 # -4.750 15.694 0.255 0.006 # -4.500 13.738 0.252 0.006 # -4.250 12.038 0.249 0.007 # -4.000 10.564 0.246 0.010 # -3.750 9.286 0.244 0.015 # -3.500 8.177 0.248 0.028 # -3.250 7.217 0.267 0.055 # -3.000 6.385 0.310 0.102 # Now GCV still is firm on the fact that log10(lambda) over -4 is # over-smoothing, but is quite unhelpful about what constitutes # undersmoothing. In real data applications you will have to make the # final call. Now set up a weight matrix equal to the inverse of the # correlation matrix wtmat = solve(Rerr) # Smooth these results for a range of log10(lambda) values using the # weight matrix loglamout = plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj, wtvec=wtmat) # Our results with weighting were: # -6.000 46.347 0.263 0.005 # -5.750 43.656 0.262 0.005 # -5.500 40.042 0.261 0.005 # -5.250 35.667 0.259 0.005 # -5.000 30.892 0.256 0.005 # -4.750 26.126 0.251 0.006 # -4.500 21.691 0.245 0.008 # -4.250 17.776 0.237 0.012 # -4.000 14.449 0.229 0.023 # -3.750 11.703 0.231 0.045 # -3.500 9.488 0.257 0.088 # -3.250 7.731 0.316 0.161 # -3.000 6.356 0.397 0.260 # GCV is still not clear about what the right smoothing level is. # But, comparing the two results, we see an optimal RMSE without # smoothing of 0.244 at log10(lambda) = -3.75, and with smoothing 0.229 # at log10(lambda) = -4. Weighting improved the RMSE. At # log10(lambda) = -4 the improvement is 9 percent. # Smooth the data with and without the weight matrix at log10(lambda) = # -4 fdParobj = fdPar(basisobj, 2, 10^(-4)) smoothlistnowt = smooth.basis(argvals, y, fdParobj) fdobjnowt = smoothlistnowt\$fd # the curve smoothing the data df = smoothlistnowt\$df # the degrees of freedom in the smoothing curve GCV = smoothlistnowt\$gcv # the value of the GCV statistic RMSE = sqrt(mean((eval.fd(argvals, fdobjnowt) - x)^2)) cat(round(c(df,RMSE,GCV),3),"\n") smoothlistwt = smooth.basis(argvals, y, fdParobj, wtvec=wtmat) fdobjwt = smoothlistwt\$fd # the curve smoothing the data df = smoothlistwt\$df # the degrees of freedom in the smoothing curve GCV = smoothlistwt\$gcv # the value of the GCV statistic RMSE = sqrt(mean((eval.fd(argvals, fdobjwt) - x)^2)) cat(round(c(df,RMSE,GCV),3),"\n") par(mfrow=c(2,1)) plotfit.fd(y, argvals, fdobjnowt) plotfit.fd(y, argvals, fdobjwt) par(mfrow=c(1,1)) plot(fdobjnowt) lines(fdobjwt,lty=2) points(argvals, y) # Homework: # Repeat these analyses with various values of rho, perhaps using # multiple curves to get more stable indications of relative # performance. Be sure to include some negative rho's. ## ######## Simulated data example 5: derivative estimation ######## ## # Functional data analyses often involve estimating derivatives. Here # we repeat the analyses of Example 2, but this time focussing our # attention on the estimation of the first and second derivative. n = 51 argvals = seq(0,1,len=n) x = sin(4*pi*argvals) Dx = 4*pi*cos(4*pi*argvals) D2x = -(4*pi)^2*sin(4*pi*argvals) sigerr = 0.2 y = x + rnorm(x)*sigerr # We now use order 6 splines so that we can control the curvature of # the second derivative, which therefore involves penalizing the # derivative of order four. norder = 6 nbasis = n + norder - 2 basisobj = create.bspline.basis(c(0,1),nbasis,norder) # Note that there are more basis functions than observed values. A # basis like this is called "super-saturated." We have to use a # positive smoothing parameter for it to work. Set up an object of # class "fdPar" that penalizes the total squared fourth derivative. The # smoothing parameter that is set here to 10^(-10), because the squared # fourth derivative is a much larger number than the squared second # derivative. lambda = 10^(-10) fdParobj = fdPar(fdobj=basisobj, Lfdobj=4, lambda=lambda) # Smooth the data, outputting a list containing various quantities smoothlist = smooth.basis(argvals, y, fdParobj) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic Dxhat = eval.fd(argvals, xfd, Lfd=1) D2xhat = eval.fd(argvals, xfd, Lfd=2) RMSED = sqrt(mean((Dxhat - Dx )^2)) RMSED2 = sqrt(mean((D2xhat - D2x)^2)) cat(round(c(df,RMSED,RMSED2,gcv),3),"\n") # Four plots of the results row-wise: data fit, first derivative fit, # second derivative fit, second vs. first derivative fit # (phase-plane plot) par(mfrow=c(2,2)) plotfit.fd(y, argvals, xfd) plot(argvals, Dxhat, type="p", pch="o") points(argvals, Dx, pch="*") title("first derivative approximation") plot(argvals, D2xhat, type="p", pch="o") points(argvals, D2x, pch="*") title("second derivative approximation") plot(Dxhat, D2xhat, type="p", pch="o") points(Dx, D2x, pch="*") title("second against first derivative") # This illustrates an inevitable problem with spline basis functions; # because they are not periodic, they fail to capture derivative # information well at the ends of the interval. The true phase-plane # plot is an ellipse, but the phase-plane plot of the estimated # derivatives here is only a rough approximtion, and breaks down at the # left boundary. # Homework: # Repeat these results with smaller standard errors. # Repeat these results, but this time use a fourier basis with no # roughness penalty, and find the number of basis functions that gives # the best result. The right answer to this question is, of course, 3, # if we retain the constant term, even though it is here not needed. # Compare the smoothing parameter preferred by RMSE for a derivative to # that preferred by the RMSE for the function itself, and to that # preferred by gcv. ## Simulated data example 6: ## a better roughness penalty for derivative estimation ## # We want to see if we can improve the spline fit. # We know from elementary calculus as well as the code above that # (4*pi)^2 sine(2*p*x) = -D2 sine(2*p*x), so that # Lx = D2x + (4*pi)^2 x is zero for a sine or a cosine curve. # We now penalize roughness using this "smart" roughness penalty # Here we set up a linear differential operator object that defines # this penalty constbasis = create.constant.basis(c(0,1)) xcoef.fd = fd((4*pi)^2, constbasis) Dxcoef.fd = fd(0, constbasis) bwtlist = vector("list", 2) bwtlist[[1]] = xcoef.fd bwtlist[[2]] = Dxcoef.fd Lfdobj = Lfd(nderiv=2, bwtlist=bwtlist) # Now we use a much larger value of lambda to reflect our confidence # in power of calculus to solve problems! lambda = 10^(0) fdParobj = fdPar(fdobj=basisobj, Lfdobj=Lfdobj, lambda=lambda) smoothlist = smooth.basis(argvals, y, fdParobj) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic Dxhat = eval.fd(argvals, xfd, Lfd=1) D2xhat = eval.fd(argvals, xfd, Lfd=2) RMSED = sqrt(mean((Dxhat - Dx )^2)) RMSED2 = sqrt(mean((D2xhat - D2x)^2)) cat(round(c(df,RMSED,RMSED2,gcv),3),"\n") # Four plots of the results row-wise: data fit, first derivative fit, # second derivative fit, second versus first derivative fit # (phase-plane plot) par(mfrow=c(2,2)) plotfit.fd(y, argvals, xfd) plot(argvals, Dxhat, type="p", pch="o") points(argvals, Dx, pch="*") title("first derivative approximation") plot(argvals, D2xhat, type="p", pch="o") points(argvals, D2x, pch="*") title("second derivative approximation") plot(Dxhat, D2xhat, type="p", pch="o") points(Dx, D2x, pch="*") title("second against first derivative") # The results are nearly perfect in spite of the fact that we are not using # periodic basis functions. Notice, too, that we have used 2.03 # degrees of freedom, which is close to what we would use for the ideal # fourier series basis with the constant term dropped. # Homework: # These results depended on us knowing the right period, of course. # The data would certainly allow us to estimate the period 1/2 closely, # but try various other periods by replacing 1/2 by other values. # Alternatively, change x by adding a small amount of, say, linear trend. # How much trend do you have to add to seriously handicap the results? ## ######## Simulated data example 7: Using covariates ######## ## # Now we simulate data that are defined by a sine curve, but where the # the first 20 observed values are shifted upwards by 0.5, and the # second shifted downwards by -0.2. The two covariates are indicator # or dummy variables, and the estimated regression coefficients will # indicate the shifts as estimated from the data. n = 51 argvals = seq(0,1,len=n) x = sin(4*pi*argvals) sigerr = 0.2 y = x + rnorm(x)*sigerr # the n by p matrix of covariate values, p being here 2 p = 2 zmat = matrix(0,n,p) zmat[ 1:11,1] = 1 zmat[11:20,2] = 1 # The true values of the regression coefficients beta0 = matrix(c(0.5,-0.2),p,1) y = y + zmat # The same basis system and smoothing process as used in Example 2 nbasis = n + 2 basisobj = create.bspline.basis(c(0,1),nbasis) lambda = 10^(-4) fdParobj = fdPar(basisobj, 2, lambda) # Smooth the data, outputting a list containing various quantities smoothlist = smooth.basis(argvals, y, fdParobj, covariates=zmat) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic beta = smoothlist\$beta # the regression coefficients RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) cat(round(c(beta,df,RMSE,gcv),3),"\n") par(mfrow=c(1,1)) plotfit.fd(y, argvals, xfd) points(argvals,x, pch="*") print(beta) # The recovery of the smooth curve is fine, as in Example 2. The # shift of the first 10 observations was estimated to be 0.62 in our run, # and the shift of the second 20 was estimated to be -0.42. These # estimates are based on only 10 observations, and these estimates are # therefore quite reasonable. # Repeat these analyses for a range of log10(lambda) values loglamout = plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj, covariates=zmat) # Homework: # Try an example where the covariate values are themselves are # generated by a smooth known curve. ## ## Simulated data example 8: ## a roughness-penalized smooth of a sample of curves and ## variable observation points ## n = 51 # number of observations per curve N = 100 # number of curves argvals = matrix(0,n,N) for (i in 1:N) argvals[,i] = sort(runif(1:n)) # The true curve values are linear combinations of fourier function # values. # Set up the fourier basis nfourierbasis = 13 fourierbasis = create.fourier.basis(c(0,1),nfourierbasis) # Set up some random coefficients, with declining contributions from # higher order basis functions basiswt = matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N) xcoef = matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt xfd = fd(xcoef, fourierbasis) x = matrix(0,n,N) for (i in 1:N) x[,i] = eval.fd(argvals[,i], xfd[i]) # Add independent Gaussian noise to the data with std. dev. 0.2 sigerr = 0.2 y = x + matrix(rnorm(c(x)),n,N)*sigerr # Set up spline basis system nbasis = n + 2 basisobj = create.bspline.basis(c(0,1),nbasis) # Set up roughness penalty with smoothing parameter 10^(-5) lambda = 10^(-5) fdParobj = fdPar(fdobj=basisobj, Lfdobj=2, lambda=lambda) # Smooth the data, outputting a list containing various quantities smoothlist = smooth.basis(argvals, y, fdParobj) xfd = smoothlist\$fd # the curve smoothing the data df = smoothlist\$df # the degrees of freedom in the smoothing curve gcv = smoothlist\$gcv # the value of the gcv statistic #RMSE = sqrt(mean((eval.fd(argvals, xfd) - x)^2)) eval.x <- eval.fd(argvals, xfd) e.xfd <- (eval.x-x) mean.e2 <- mean(e.xfd^2) RMSE = sqrt(mean.e2) # Display the results, note that a gcv value is returned for EACH # curve, and therefore that a mean gcv result is reported cat(round(c(df,RMSE,mean(gcv)),3),"\n") # Function plotfit.fd is not equipped to handle a matrix of argvals, # but can always be called within a loop to plot each curve in turn. # Although a call to function plotGCVRMSE.fd works, the computational # overhead is substantial, and we omit this here. ## ## Real data example 9. gait ## # These data involve two variables in addition to multiple curves gaittime <- (1:20)/21 gaitrange <- c(0,1) gaitbasis <- create.fourier.basis(gaitrange,21) lambda <- 10^(-11.5) harmaccelLfd <- vec2Lfd(c(0, 0, (2*pi)^2, 0)) gaitfdPar <- fdPar(gaitbasis, harmaccelLfd, lambda) gaitSmooth <- smooth.basis(gaittime, gait, gaitfdPar) gaitfd <- gaitSmooth\$fd ## Not run: # by default creates multiple plots, asking for a click between plots plotfit.fd(gait, gaittime, gaitfd) ## End(Not run) } # end of if (!CRAN) ```