# Create a Kernel Matrix Create a kernel matrix to be used in a 2-D convolution smooth (e.g., using kernel2dsmooth). ```r kernel2dmeitsjer(type = "gauss", ...) ``` ## Arguments - `type`: character name of the kernel to be created. - ``...``: Other arguments to the specific kernel type. See Details section below. ## Details The specific types of kernels that can be made are as follows. In each case, h=||x-x_c|| is the distance from the center of the kernel. Every kernel that requires single numerics nx and ny be specified returns an nx by ny metrix. Distances h are found by setting up a grid based on 1 to nx and 1 to ny denoting the points as (xgrid, ygrid), finding the center of the grid as (x.center, y.center)=(nx/2,ny/2), and then h = sqrt( (xgrid - x.center)^2 + (ygrid - y.center)^2). For kernels that better reflect distance (e.g., using great-circle distance, anisotropic distances, etc.), the matrix h can be passed instead of nx and ny, but only for those kernels that take h as an argument. In each case with sigma as an argument, sigma is the smoothing parameter. There are many kernel functions allowed here, and not all of them make sense for every purpose. average gives a kernel that will give an average of the nearest neighbors in each direction (can take an average grid points further in the x- direction than the y-direction, and vice versa). Requires that nx and ny be specified, and the resulting kernel is defined by an nx by ny matrix with each element equal to 1/(nx*ny). If nx = ny, then the result is the same as the boxcar kernel below. boxcar the boxcar kernel is an n by n matrix of 1/n^2. This results in a neighborhood smoothing when used with kernel2dsmooth giving the type of smoothed fields utilized, e.g., in Roberts and Lean (2008) and Ebert (2008). Requires that n be specified. Note that usually the boxcar is a square matrix of ones, which gives the sum of the nearest n^2 grid points. This gives the average. cauchy The Cauchy kernel is given by K(sigma)=1/(1+h^2/sigma). Requires the arguments nx, ny and sigma. See Souza (2010) for more details. disk gives a circular averaging (pill-box) kernel (aka, disk kernel). Similar to average or boxcar , but this kernel accounts for a specific distance in all directions from the center (i.e., an average of grid squares within a circular radius of the central point). This results in the convolution radius smoothing applied in Davis et al. (2006a,2006b). Requires that r (the desired radius) be supplied, and a square matrix of appropriate dimension is returned. epanechnikov The Epanechnikov kernel is defined by max(0, 3/4*(1-h/(sigma^2))). See, e.g., Hastie and Tibshirani (1990). Requires arguments nx, ny, and sigma. exponential The exponential kernel is given by K(sigma) = a*exp(-h/(2*sigma)). Requires the arguments nx, ny and sigma, and optionally takes the argument a (default is a=1). An nx by ny matrix is returned. See Souza (2010) for more details. gauss The Gaussian kernel defined by K(sigma) = 1/(2*pi*sigma^2)*exp(-h/(2*sigma)). Requires the arguments nx, ny and sigma be specified. The convolution with this kernel results in a Gaussian smoothed field as used in the practically perfect hindcast method of Brooks et al. (1998) (see also Ebert 2008) and studied by Sobash et al (2011) for spatial forecast verification purposes. Returns an nx by ny matrix. laplacian Laplacian Operator kernel, which gives the sum of second partial derivatives for each direction. It is often used for edge detection because it identifies areas of rapid intensity change. Typically, it is first applied to a field that has been smoothed first by a Gaussian kernel smoother (or an approximation thereof; cf. type LoG below). This method optionally the parameter alpha, which controls the shape of the Laplacian kernel, which must be between 0 and 1 (inclusive), or else it will be set to 0 (if < 0) or 1 (if > 1). Returns a 3 by 3 kernel matrix. LoG Laplacian of Gaussian kernel. This combines the Laplacian Operator kernel with that of a Gaussian kernel. The form is given by K(sigma) = -1/(pi*sigma^4)*exp(-h/(2*sigma^2))*(1-h/(2*sigma^2)). Requires the arguments nx, ny and sigma be specified. Returns an nx by ny matrix. minvar A minimum variance kernel, which is given by 3/8*(3 - 5*h/sigma^2) if h <= 1, and zero otherwise (see, e.g., Hastie and Tibshirani, 1990). Requires the arguments nx, ny, and sigma be specified. Returns an nx by ny matrix. multiquad The multiquadratic kernel is similar to the rational quadratic kernel, and is given by K(a) = sqrt(h + a^2). The inverse is given by 1/K(a). Requires the arguments nx, ny and a be specified. Optionally takes a logical named inverse determining whether to return the inverse multiquadratic kernel or not. prewitt Prewitt filter kernel, which emphasizes horizontal (vertical) edges through approximation of a vertical (horizontal) gradient. Optionally takes a logical argument named transpose, which if FALSE (default) emphasis is on horizontal, and if TRUE emphasis is on vertical. Returns a 3 by 3 matrix whose first row is all ones, second row is all zeros, and third row is all negative ones for the transpose=FALSE case, and the transpose of this matrix in the transpose=TRUE case. power The power kernel is defined by K(p) = -h^p. The log power kernel is similarly defined as K(p) = -log(h^p+1). Requires specification of the arguments nx, ny and p. Alternatively takes the logical do.log to determine whether the log power kernel should be returned (TRUE) or not (FALSE). Default if not passed is to do the power kernel. Returns an nx by ny matrix. See Souza (2010) for more details. radial The radial kernel is returns a*|h|^(2*m-d)*log(|h|) if d is even and a*|h|^(2*m-d) otherwise. Requires arguments a, m, d nx and ny. Replaces any missing values with zero. ratquad The rational quadratic kernel is used as an alternative to the Gaussian, and is given by K(a) = 1 - h/(h+a). Requires the arguments nx, ny and a, and returns an nx by ny matrix. See Souza (2010) for more details. sobel Same as prewitt, except that the elements 1,2 and 3,2 are replaced by two and neative two, resp. student The generalized Student's t kernel is defined by K(p)=1/(1+h^p). Requires the arguments nx, ny and p be specified. Returns an nx by ny matrix. See Souza (2010) for more details. unsharp Unsharp contrast enhancement filter. This is simply given by a 3 by 3 matrix of al zeros, except for a one in the center subtracted by a laplacian operator kernel matrix. Requires the same arguments as for laplacian . Returns a 3 by 3 matrix. wave The wave kernel is defined by K(phi) = phi/h * sin( h/phi). Requires arguments nx, ny and phi be specified. Returns an nx by ny matrix. oval The oval kernel is like it sounds, it yields an oval-shaped kernel. Allows arguments a (scale in x-direction), b (scale in y-direction), n (size of kernel in x-direction) and m (size of kernel in y-direction). Default for a and b is 1, and default for n is the maximum of a and b plus two. The default for m is to be identical to n. ## Returns matrix of dimension determined by the specific type of kernel, and possibly user passed arguments giving the kernel to be used by `kernel2dsmooth`. ## References Brooks, H. E., Kay, M., and Hart, J. A. (1998) Objective limits on forecasting skill of rare events. **19th Conf. Severe Local Storms**. Amer. Met. Soc., 552--555. Davis, C. A., Brown, B. G. and Bullock, R. G. (2006a) Object-based verification of precipitation forecasts, Part I: Methodology and application to mesoscale rain areas. **Mon. Wea. Rev.**, 134 , 1772--1784. Davis, C. A., Brown, B. G. and Bullock, R. G. (2006b) Object-based verification of precipitation forecasts, Part II: Application to convective rain systems. **Mon. Wea. Rev.**, 134 , 1785--1795. Ebert, E. E. (2008) Fuzzy verification of high resolution gridded forecasts: A review and proposed framework. **Meteorol. Appl.**, 15 , 51-64. tools:::Rd_expr_doi("10.1002/met.25") . Hastie, T. J. and Tibshirani, R. J. (1990) **Generalized Additive Models**. Chapman and Hall/CRC Monographs on Statistics and Applied Probability 43, 335pp. Roberts, N. M. and Lean, H. W. (2008) Scale-selective verification of rainfall accumulations from high-resolution forecasts of convective events. **Mon. Wea. Rev.**, 136 , 78--97. tools:::Rd_expr_doi("10.1175/2007MWR2123.1") . Sobash, R. A., Kain, J. S. Bright, D. R. Dean, A. R. Coniglio, M. C. and Weiss, S. J. (2011) Probabilistic forecast guidance for severe thunderstorms based on the identification of extreme phenomena in convection-allowing model forecasts. **Wea. Forecasting**, 26 , 714--728. Souza, C. R. (2010) **Kernel Functions for Machine Learning Applications**. 17 Mar 2010. Web. [http://crsouza.blogspot.com/2010/03/kernel-functions-for-machine-learning.html](http://crsouza.blogspot.com/2010/03/kernel-functions-for-machine-learning.html). ## Author(s) Eric Gilleland ## See Also `fft`, `kernel2dsmooth` ## Examples ```r x <- matrix( 0, 10, 12) x[4,5] <- 1 kmat <- kernel2dmeitsjer( "average", nx=7, ny=5) kernel2dsmooth( x, K=kmat) ## ## Can also call 'kernel2dsmooth' directly. ## kernel2dsmooth( x, kernel.type="boxcar", n=5) kernel2dsmooth( x, kernel.type="cauchy", sigma=20, nx=10, ny=12) kernel2dsmooth( x, kernel.type="disk", r=3) kernel2dsmooth( x, kernel.type="epanechnikov", nx=10, ny=12, sigma=4) kernel2dsmooth( x, kernel.type="exponential", a=0.1, sigma=4, nx=10, ny=12) kernel2dsmooth( x, kernel.type="gauss", nx=10, ny=12, sigma=4) kernel2dsmooth( x, kernel.type="laplacian", alpha=0) kernel2dsmooth( x, kernel.type="LoG", nx=10, ny=12, sigma=1) kernel2dsmooth( x, kernel.type="minvar", nx=10, ny=12, sigma=4) kernel2dsmooth( x, kernel.type="multiquad", a=0.1, nx=10, ny=12) kernel2dsmooth( x, kernel.type="power", p=0.5, nx=10, ny=12) kernel2dsmooth( x, kernel.type="prewitt") kernel2dsmooth( x, kernel.type="prewitt", transpose=TRUE) kernel2dsmooth( x, kernel.type="radial", a=1, m=2, d=1, nx=10, ny=12) kernel2dsmooth( x, kernel.type="ratquad", a=0.1, nx=10, ny=12) kernel2dsmooth( x, kernel.type="sobel") kernel2dsmooth( x, kernel.type="sobel", transpose=TRUE) kernel2dsmooth( x, kernel.type="student", p=1.5, nx=10, ny=12) kernel2dsmooth( x, kernel.type="unsharp", alpha=0) kernel2dsmooth( x, kernel.type="wave", phi=45, nx=10, ny=12) ## Not run: ## the lennon image is in package 'fields'. data(lennon) kmat <- kernel2dmeitsjer( "average", nx=7, ny=5) lennon.smAvg <- kernel2dsmooth( lennon, K=kmat) ## Can also just make a call to kernel2dsmooth, which ## will call this function. lennon.smBox <- kernel2dsmooth( lennon, kernel.type="boxcar", n=7) lennon.smDsk <- kernel2dsmooth( lennon, kernel.type="disk", r=5) par( mfrow=c(2,2), mar=rep(0.1,4)) image.plot( lennon, col=tim.colors(256), axes=FALSE) image.plot( lennon.smAvg, col=tim.colors(256), axes=FALSE) image.plot( lennon.smBox, col=tim.colors(256), axes=FALSE) image.plot( lennon.smDsk, col=tim.colors(256), axes=FALSE) lennon.smEpa <- kernel2dsmooth( lennon, kernel.type="epanechnikov", nx=10, ny=10, sigma=20) lennon.smGau <- kernel2dsmooth( lennon, kernel.type="gauss", nx=10, ny=10, sigma=20) lennon.smMvr <- kernel2dsmooth( lennon, kernel.type="minvar", nx=10, ny=10, sigma=20) par( mfrow=c(2,2), mar=rep(0.1,4)) image.plot( lennon, col=tim.colors(256), axes=FALSE) image.plot( lennon.smEpa, col=tim.colors(256), axes=FALSE) image.plot( lennon.smGau, col=tim.colors(256), axes=FALSE) image.plot( lennon.smMvr, col=tim.colors(256), axes=FALSE) lennon.smLa0 <- kernel2dsmooth( lennon, kernel.type="laplacian", alpha=0) lennon.smLap <- kernel2dsmooth( lennon, kernel.type="laplacian", alpha=0.999) lennon.smLoG <- kernel2dsmooth( lennon, kernel.type="LoG", nx=10, ny=10, sigma=20) par( mfrow=c(2,2), mar=rep(0.1,4)) image.plot( lennon, col=tim.colors(256), axes=FALSE) image.plot( lennon.smLa0, col=tim.colors(256), axes=FALSE) image.plot( lennon.smLap, col=tim.colors(256), axes=FALSE) image.plot( lennon.smLoG, col=tim.colors(256), axes=FALSE) lennon.smPrH <- kernel2dsmooth( lennon, kernel.type="prewitt") lennon.smPrV <- kernel2dsmooth( lennon, kernel.type="prewitt", transpose=TRUE) lennon.smSoH <- kernel2dsmooth( lennon, kernel.type="sobel") lennon.smSoV <- kernel2dsmooth( lennon, kernel.type="sobel", transpose=TRUE) par( mfrow=c(2,2), mar=rep(0.1,4)) image.plot( lennon.smPrH, col=tim.colors(256), axes=FALSE) image.plot( lennon.smPrV, col=tim.colors(256), axes=FALSE) image.plot( lennon.smSoH, col=tim.colors(256), axes=FALSE) image.plot( lennon.smSoV, col=tim.colors(256), axes=FALSE) lennon.smUsh <- kernel2dsmooth( lennon, kernel.type="unsharp", alpha=0.999) par( mfrow=c(2,1), mar=rep(0.1,4)) image.plot( lennon, col=tim.colors(256), axes=FALSE) image.plot( lennon.smUsh, col=tim.colors(256), axes=FALSE) lennon.smRad1 <- kernel2dsmooth( lennon, kernel.type="radial", a=2, m=2, d=1, nx=10, ny=10) lennon.smRad2 <- kernel2dsmooth( lennon, kernel.type="radial", a=2, m=2, d=2, nx=10, ny=10) par( mfrow=c(2,1), mar=rep(0.1,4)) image.plot( lennon.smRad1, col=tim.colors(256), axes=FALSE) image.plot( lennon.smRad2, col=tim.colors(256), axes=FALSE) lennon.smRQd <- kernel2dsmooth( lennon, kernel.type="ratquad", a=0.5, nx=10, ny=10) lennon.smExp <- kernel2dsmooth( lennon, kernel.type="exponential", a=0.5, sigma=20, nx=10, ny=10) lennon.smMQd <- kernel2dsmooth( lennon, kernel.type="multiquad", a=0.5, nx=10, ny=10) par( mfrow=c(2,2), mar=rep(0.1,4)) image.plot( lennon.smGau, col=tim.colors(256), axes=FALSE) image.plot( lennon.smRQd, col=tim.colors(256), axes=FALSE) image.plot( lennon.smExp, col=tim.colors(256), axes=FALSE) image.plot( lennon.smMQd, col=tim.colors(256), axes=FALSE) lennon.smIMQ <- kernel2dsmooth( lennon, kernel.type="multiquad", a=0.5, nx=10, ny=10, inverse=TRUE) par( mfrow=c(2,1), mar=rep(0.1,4)) image.plot( lennon.smMQd, col=tim.colors(256), axes=FALSE) image.plot( lennon.smIMQ, col=tim.colors(256), axes=FALSE) lennon.smWav <- kernel2dsmooth( lennon, kernel.type="wave", phi=45, nx=10, ny=10) par( mfrow=c(1,1), mar=rep(0.1,4)) image.plot( lennon.smWav, col=tim.colors(256), axes=FALSE) lennon.smPow <- kernel2dsmooth( lennon, kernel.type="power", p=0.5, nx=10, ny=10) lennon.smLpw <- kernel2dsmooth( lennon, kernel.type="power", p=0.5, nx=10, ny=10, do.log=TRUE) par( mfrow=c(2,1), mar=rep(0.1,4)) image.plot( lennon.smPow, col=tim.colors(256), axes=FALSE) image.plot( lennon.smLpw, col=tim.colors(256), axes=FALSE) lennon.smCau <- kernel2dsmooth( lennon, kernel.type="cauchy", sigma=20, nx=10, ny=10) lennon.smStd <- kernel2dsmooth( lennon, kernel.type="student", p=1.5, nx=10, ny=10) par( mfrow=c(2,1), mar=rep(0.1,4)) image.plot( lennon.smCau, col=tim.colors(256), axes=FALSE) image.plot( lennon.smStd, col=tim.colors(256), axes=FALSE) image.plot( lennon, kernel.type = "oval", n = 10, m = 15, a = 6, b = 3 ) ## End(Not run) ```

kernel2dmeitsjer function