Kernel Functions
kernels.RdThere are an assortment of (vectorized) kernel functions located in the package.
Truncated Kernel: Kernel where \(1, |x|\leq 1\) and \(0\) otherwise. If \(x=0/0\) then the value \(1\) is given.
Bartlett Kernel: Kernel where \(max(0,1-|x|), h\neq 0\). If \(x=0/0\) then the value \(1\) is given.
Parzen Kernel: Kernel where \(1 - 6 * x^2 + 6 * |x|^3, |x|<=0.5\), \( 2 * (1 - |x|)^3, 0.5<|x|<1\), and \(0, |x|>1\). If \(x=0/0\) then the value \(1\) is given.
Tukey-Hanning Kernel: Kernel where \((1 + cos(\pi x) )/2, |x|<=1\) and \(0, |x|>1\). If \(x=0/0\) then the value \(1\) is given.
Quadratic Spectral Kernel: Kernel where \(\frac{25}{12\pi^2x^2} \left(\frac{sin(6\pi x/5)}{6\pi x/5} - cos(6\pi x/5) \right)\). If \(x=0/0\) then the value \(1\) is given.
Daniell Kernel: Kernel where \(sin(pi * x) / (pi * x)*(1 + cos(pi*x)), abs(x)<=1\). If \(x=0/0\) then the value \(1\) is given.
Flat-Top Kernel: Kernel where \(min(1, max(1.1-|x|,0)),|x|\leq 1\). If \(x=0/0\) then the value \(1\) is given.
Usage
truncated_kernel(x)
bartlett_kernel(x)
parzen_kernel(x)
tukey_hanning_kernel(x)
quadratic_spectral_kernel(x)
daniell_kernel(x)
flat_top_kernel(x)References
Horvath, L., & Rice, G. (2024). Change point analysis for time series (1st ed. 2024.). Springer Nature Switzerland.
L. Horvath, P. Kokoszka, G. Rice (2014) "Testing stationarity of functional time series", Journal of Econometrics, 179(1), 66-82.
Politis, D. N. (2003). Adaptive bandwidth choice. Journal of Nonparametric Statistics, 15(4-5), 517-533.
Politis, D. N. (2011). Higher-order accurate, positive semidefinite estimation of large-sample covariance and spectral density matrices. Econometric Theory, 27(4), 703-744.
Examples
truncated_kernel(-20:20/15)
#> [1] 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
#> [39] 0 0 0
bartlett_kernel(-20:20/15)
#> [1] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
#> [7] 0.06666667 0.13333333 0.20000000 0.26666667 0.33333333 0.40000000
#> [13] 0.46666667 0.53333333 0.60000000 0.66666667 0.73333333 0.80000000
#> [19] 0.86666667 0.93333333 1.00000000 0.93333333 0.86666667 0.80000000
#> [25] 0.73333333 0.66666667 0.60000000 0.53333333 0.46666667 0.40000000
#> [31] 0.33333333 0.26666667 0.20000000 0.13333333 0.06666667 0.00000000
#> [37] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
parzen_kernel(-20:20/15)
#> [1] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000
#> [6] 0.0000000000 0.0005925926 0.0047407407 0.0160000000 0.0379259259
#> [11] 0.0740740741 0.1280000000 0.2032592593 0.3031111111 0.4240000000
#> [16] 0.5555555556 0.6871111111 0.8080000000 0.9075555556 0.9751111111
#> [21] 1.0000000000 0.9751111111 0.9075555556 0.8080000000 0.6871111111
#> [26] 0.5555555556 0.4240000000 0.3031111111 0.2032592593 0.1280000000
#> [31] 0.0740740741 0.0379259259 0.0160000000 0.0047407407 0.0005925926
#> [36] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000
#> [41] 0.0000000000
tukey_hanning_kernel(-20:20/15)
#> [1] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
#> [7] 0.01092620 0.04322727 0.09549150 0.16543470 0.25000000 0.34549150
#> [13] 0.44773577 0.55226423 0.65450850 0.75000000 0.83456530 0.90450850
#> [19] 0.95677273 0.98907380 1.00000000 0.98907380 0.95677273 0.90450850
#> [25] 0.83456530 0.75000000 0.65450850 0.55226423 0.44773577 0.34549150
#> [31] 0.25000000 0.16543470 0.09549150 0.04322727 0.01092620 0.00000000
#> [37] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
quadratic_spectral_kernel(-20:20/15)
#> [1] -0.059156968 -0.035757776 -0.004361244 0.035169330 0.082709303
#> [6] 0.137860582 0.199949461 0.268034899 0.340927244 0.417217036
#> [11] 0.495313030 0.573488238 0.649932362 0.722808762 0.790313821
#> [16] 0.850736481 0.902515643 0.944293220 0.974960742 0.993697686
#> [21] 1.000000000 0.993697686 0.974960742 0.944293220 0.902515643
#> [26] 0.850736481 0.790313821 0.722808762 0.649932362 0.573488238
#> [31] 0.495313030 0.417217036 0.340927244 0.268034899 0.199949461
#> [36] 0.137860582 0.082709303 0.035169330 -0.004361244 -0.035757776
#> [41] -0.059156968
daniell_kernel(-20:20/15)
#> [1] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
#> [7] 0.001549499 0.012915141 0.044665639 0.106728094 0.206748336 0.348636272
#> [13] 0.531517459 0.749263553 0.990699050 1.240490015 1.480625327 1.692316012
#> [19] 1.858076002 1.963717409 1.000000000 1.963717409 1.858076002 1.692316012
#> [25] 1.480625327 1.240490015 0.990699050 0.749263553 0.531517459 0.348636272
#> [31] 0.206748336 0.106728094 0.044665639 0.012915141 0.001549499 0.000000000
#> [37] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
flat_top_kernel(-20:20/15)
#> [1] 0.00000000 0.00000000 0.00000000 0.00000000 0.03333333 0.10000000
#> [7] 0.16666667 0.23333333 0.30000000 0.36666667 0.43333333 0.50000000
#> [13] 0.56666667 0.63333333 0.70000000 0.76666667 0.83333333 0.90000000
#> [19] 0.96666667 1.00000000 1.00000000 1.00000000 0.96666667 0.90000000
#> [25] 0.83333333 0.76666667 0.70000000 0.63333333 0.56666667 0.50000000
#> [31] 0.43333333 0.36666667 0.30000000 0.23333333 0.16666667 0.10000000
#> [37] 0.03333333 0.00000000 0.00000000 0.00000000 0.00000000