# Create smooth functions that fit scatterplot data. The function converts scatter plot data into one or a set of curves that do a satisfactory job of representing their corresponding abscissa and ordinate values by smooth curves. It returns a list object of class `fdSmooth` that contains curves defined by functional data objects of class `fd` as well as a variety of other results related to the data smoothing process. The function tries to do as much for the user as possible before setting up a call to function `smooth.basisPar`. This is achieved by an initial call to function `argvalsSwap` that examines arguments `argvals` that contains abscissa values and `y` that contains ordinate values. If the arguments are provided in an order that is not the one above, `Data2fd` attempts to swap argument positions to provide the correct order. A warning message is returned if this swap takes place. Any such automatic decision, though, has the possibility of being wrong, and the results should be carefully checked. Preferably, the order of the arguments should be respected: `argvals` comes first and `y` comes second. Be warned that the result that the `fdSmooth` object that is returned may not define a satisfactory smooth of the data, and consequently that it may be necessary to use the more sophisticated data smoothing function `smooth.basis` instead. Its help file provides a great deal more information than is provided here. ```r Data2fd(argvals=NULL, y=NULL, basisobj=NULL, nderiv=NULL, lambda=3e-8/diff(as.numeric(range(argvals))), fdnames=NULL, covariates=NULL, method="chol") ``` ## Arguments - `argvals`: a set of argument values. If this is a vector, the same set of argument values is used for all columns of `y`. If `argvals` is a matrix, the columns correspond to the columns of `y`, and contain the argument values for that replicate or case. Dimensions for `argvals` must match the first dimensions of `y`, though `y` can have more dimensions. For example, if dim(y) = c(9, 5, 2), `argvals` can be a vector of length 9 or a matrix of dimensions c(9, 5) or an array of dimensions c(9, 5, 2). - `y`: an array containing sampled values of curves. If `y` is a vector, only one replicate and variable are assumed. If `y` is a matrix, rows must correspond to argument values and columns to replications or cases, and it will be assumed that there is only one variable per observation. If `y` is a three-dimensional array, the first dimension (rows) corresponds to argument values, the second (columns) to replications, and the third (layers) to variables within replications. Missing values are permitted, and the number of values may vary from one replication to another. If this is the case, the number of rows must equal the maximum number of argument values, and columns of `y` having fewer values must be padded out with NA's. - `basisobj`: An object of one of the following classes: - **basisfd**: a functional basis object (class `basisfd`). - **fd**: a functional data object (class `fd`), from which its `basis` component is extracted. - **fdPar**: a functional parameter object (class `fdPar`), from which its `basis` component is extracted. - **integer**: an integer giving the order of a B-spline basis, `create.bspline.basis(argvals, norder=basisobj)` - **numeric vector**: specifying the knots for a B-spline basis, `create.bspline.basis(basisobj)` - **NULL**: Defaults to create.bspline.basis(argvals). - `nderiv`: Smoothing typically specified as an integer order for the derivative whose square is integrated and weighted by `lambda` to smooth. By default, `if basisobj[['type']] == 'bspline'`, the smoothing operator is `int2Lfd(max(0, norder-2)).` A general linear differential operator can also be supplied. - `lambda`: weight on the smoothing operator specified by `nderiv`. - `fdnames`: Either a character vector of length 3 or a named list of length 3. In either case, the three elements correspond to the following: - **argname**: name of the argument, e.g. "time" or "age". - **repname**: a description of the cases, e.g. "reps" or "weather stations". - **value**: the name of the observed function value, e.g. "temperature". If fdnames is a list, the components provide labels for the levels of the corresponding dimension of `y`. - `covariates`: the observed values in `y` are assumed to be primarily determined by the height of the curve being estimated. However, 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 timewhen `n` is large, meaning more than 500 or so. The defaultis "chol", referring the Choleski decomposition of a symmetric positive definite matrix. ## Details This function tends to be used in rather simple applications where there is no need to control the roughness of the resulting curve with any great finesse. The roughness is essentially controlled by how many basis functions are used. In more sophisticated applications, it would be better to use the function `smooth.basisPar`. It may happen that a value in argvals is outside the basis object interval due to rounding error and other causes of small violations. The code tests for this and pulls these near values into the interval if they are within 1e-7 times the interval width. ## Returns an S3 list object of class `fdSmooth` defined in file `smooth.basis1.R` containing: - **fd**: the functional data object defining smooth B-spline curves that fit the data - **df**: the degrees of freedom in the fits - **gcv**: a generalized cross-validation value - **beta**: the semi-parametric coefficient array - **SSE**: Error sum of squares - **penmat**: the symmetric matrix defining roughness penalty - **argvals**: the argument values - **y**: the array of ordinate values in the scatter-plot ## References Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), **Functional data analysis with R and Matlab**, Springer, New York. Ramsay, James O., and Silverman, Bernard W. (2005), **Functional Data Analysis, 2nd ed.**, Springer, New York. Ramsay, James O., and Silverman, Bernard W. (2002), **Applied Functional Data Analysis**, Springer, New York. ## See Also `smooth.basisPar`, `smooth.basis`, `smooth.basis1`, `project.basis`, `smooth.fd`, `smooth.monotone`, `smooth.pos`, `day.5` ## Examples ```r ## ## Simplest possible example: constant function ## # 1 basis, order 1 = degree 0 = constant function b1.1 <- create.bspline.basis(nbasis=1, norder=1) # data values: 1 and 2, with a mean of 1.5 y12 <- 1:2 # smooth data, giving a constant function with value 1.5 fd1.1 <- Data2fd(y12, basisobj=b1.1) oldpar <- par(no.readonly=TRUE) plot(fd1.1) # now repeat the analysis with some smoothing, which moves the # toward 0. fd1.5 <- Data2fd(y12, basisobj=b1.1, lambda=0.5) # values of the smooth: # fd1.1 = sum(y12)/(n+lambda*integral(over arg=0 to 1 of 1)) # = 3 / (2+0.5) = 1.2 #. JR ... Data2fd returns an fdsmooth object and a member of the # is an fd smooth object. Calls to functions expecting # an fd object require attaching $fd to the fdsmooth object # this is required in lines 268, 311 and 337 eval.fd(seq(0, 1, .2), fd1.5) ## ## step function smoothing ## # 2 step basis functions: order 1 = degree 0 = step functions b1.2 <- create.bspline.basis(nbasis=2, norder=1) # fit the data without smoothing fd1.2 <- Data2fd(1:2, basisobj=b1.2) # plot the result: A step function: 1 to 0.5, then 2 op <- par(mfrow=c(2,1)) plot(b1.2, main='bases') plot(fd1.2, main='fit') par(op) ## ## Simple oversmoothing ## # 3 step basis functions: order 1 = degree 0 = step functions b1.3 <- create.bspline.basis(nbasis=3, norder=1) # smooth the data with smoothing fd1.3 <- Data2fd(y12, basisobj=b1.3, lambda=0.5) # plot the fit along with the points plot(0:1, c(0, 2), type='n') points(0:1, y12) lines(fd1.3) # Fit = penalized least squares with penalty = # = lambda * integral(0:1 of basis^2), # which shrinks the points towards 0. # X1.3 = matrix(c(1,0, 0,0, 0,1), 2) # XtX = crossprod(X1.3) = diag(c(1, 0, 1)) # penmat = diag(3)/3 # = 3x3 matrix of integral(over arg=0:1 of basis[i]*basis[j]) # Xt.y = crossprod(X1.3, y12) = c(1, 0, 2) # XtX + lambda*penmat = diag(c(7, 1, 7)/6 # so coef(fd1.3.5) = solve(XtX + lambda*penmat, Xt.y) # = c(6/7, 0, 12/7) ## ## linear spline fit ## # 3 bases, order 2 = degree 1 b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1)) # interpolate the values 0, 2, 1 fd2.3 <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=0) # display the coefficients round(fd2.3$coefs, 4) # plot the results op <- par(mfrow=c(2,1)) plot(b2.3, main='bases') plot(fd2.3, main='fit') par(op) # apply some smoothing fd2.3. <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=1) op <- par(mfrow=c(2,1)) plot(b2.3, main='bases') plot(fd2.3., main='fit', ylim=c(0,2)) par(op) all.equal( unclass(fd2.3)[-1], unclass(fd2.3.)[-1]) ##** CONCLUSION: ##** The only differences between fd2.3 and fd2.3. ##** are the coefficients, as we would expect. ## ## quadratic spline fit ## # 4 bases, order 3 = degree 2 = continuous, bounded, locally quadratic b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1)) # fit values c(0,4,2,3) without interpolation fd3.4 <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=0) round(fd3.4$coefs, 4) op <- par(mfrow=c(2,1)) plot(b3.4) plot(fd3.4) points(c(0,1/3,2/3,1), c(0,4,2,3)) par(op) # try smoothing fd3.4. <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=1) round(fd3.4.$coef, 4) op <- par(mfrow=c(2,1)) plot(b3.4) plot(fd3.4., ylim=c(0,4)) points(seq(0,1,len=4), c(0,4,2,3)) par(op) ## ## Two simple Fourier examples ## gaitbasis3 <- create.fourier.basis(nbasis=5) gaitfd3 <- Data2fd(seq(0,1,len=20), gait, basisobj=gaitbasis3) # plotfit.fd(gait, seq(0,1,len=20), gaitfd3) # set up the fourier basis daybasis <- create.fourier.basis(c(0, 365), nbasis=65) # Make temperature fd object # Temperature data are in 12 by 365 matrix tempav # See analyses of weather data. tempfd <- Data2fd(CanadianWeather$dailyAv[,,"Temperature.C"], day.5, daybasis) # plot the temperature curves par(mfrow=c(1,1)) plot(tempfd) ## ## argvals of class Date and POSIXct ## # These classes of time can generate very large numbers when converted to # numeric vectors. For basis systems such as polynomials or splines, # severe rounding error issues can arise if the time interval for the # data is very large. To offset this, it is best to normalize the # numeric version of the data before analyzing them. # Date class time unit is one day, divide by 365.25. invasion1 <- as.Date('1775-09-04') invasion2 <- as.Date('1812-07-12') earlyUS.Canada <- as.numeric(c(invasion1, invasion2))/365.25 BspInvasion <- create.bspline.basis(earlyUS.Canada) earlyYears <- seq(invasion1, invasion2, length.out=7) earlyQuad <- (as.numeric(earlyYears-invasion1)/365.25)^2 earlyYears <- as.numeric(earlyYears)/365.25 fitQuad <- Data2fd(earlyYears, earlyQuad, BspInvasion) # POSIXct: time unit is one second, divide by 365.25*24*60*60 rescale <- 365.25*24*60*60 AmRev.ct <- as.POSIXct1970(c('1776-07-04', '1789-04-30')) BspRev.ct <- create.bspline.basis(as.numeric(AmRev.ct)/rescale) AmRevYrs.ct <- seq(AmRev.ct[1], AmRev.ct[2], length.out=14) AmRevLin.ct <- as.numeric(AmRevYrs.ct-AmRev.ct[1]) AmRevYrs.ct <- as.numeric(AmRevYrs.ct)/rescale AmRevLin.ct <- as.numeric(AmRevLin.ct)/rescale fitLin.ct <- Data2fd(AmRevYrs.ct, AmRevLin.ct, BspRev.ct) par(oldpar) ```

Data2fd function