pbs: Periodic B-Spline Basis for Polynomial Splines
In pbs: Periodic B Splines
Description
Usage
Arguments
Value
References
See Also
Examples
Description
Generate the periodic B-spline basis matrix for a polynomial spline.
Usage
1
2
Arguments
x
the predictor variable. Missing values are allowed.
df
degrees of freedom; one can specify 'df' rather than 'knots';
'pbs()' then chooses 'df - 1' knots at suitable quantiles
of 'x' (which will ignore missing values).
knots
the _internal_ breakpoints that define the spline.
The number of internal knots
must be greater than or equal to degree
polynomial regression. See also
'Boundary.knots'.
degree
degree of the piecewise polynomial-default is 3 for cubic
splines.
intercept
if 'TRUE', an intercept is included in the basis; default is
'FALSE'.
Boundary.knots
boundary points at which to set the period of the perodic
B-spline basis(default the range of the data). If both 'knots'
and 'Boundary.knots' are supplied, the basis parameters do not
depend on 'x'. Data CAN NOT be extended beyond 'Boundary.knots'.
Typical Bourday knots are start and end values of period.
periodic
if "TRUE", periodic basis is generated. Default is TRUE.
This is disabled. DO NOT USE.
Value
A matrix of dimension 'length(x) * (df)', where either 'df' was
supplied or if 'knots' were supplied, 'df = length(knots) +
intercept'. Attributes are returned that correspond to the
arguments to 'pbs', and explicitly give the 'knots',
'Boundary.knots' etc for use by 'predict.pbs()'.
'pbs()' is based on the function 'spline.des()' in package splines.
It generates a basis matrix for representing the family of piecewise
polynomials with the specified interior knots and degree, evaluated
at the values of 'x'. A primary use is in modeling formulas to
directly specify a piecewise polynomial term in a model.
References
Package splines in Base R. pbs is actually modified from bs function
See Also
'bs', 'ns', 'predict.pbs'
Examples
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require(stats); require(graphics); require(splines)
x = seq(1,628)/100
z = rep(seq(1, 314)/100, 2)
pbs(x, df = 5, Boundary.knots = c(0, 2*pi))
pbs(x, knots=c(pi/2, pi, pi*3/2), Boundary.knots = c(0, 2*pi))
#### example of one periodic functions
y= sin(x) + cos(2*x) +
rnorm(628, 0, 0.1) ## x has a period of 2*pi
## df method, need to use large enough df to get a better fit.
## May use max loglik to choose optimal df
summary( fm1 <- lm(y ~ pbs(x, df = 10, Boundary.knots = c(0, 2*pi))) )
plot(x, y, xlab = "x", ylab = "sin(x)", pch="x", cex=.5)
lines(x, predict(fm1, data.frame(x=x, z=z)), col='blue')
lines(x, sin(x) + cos(2*x), col='red')
## knots methods, usually selected at turning points
summary( fm2 <- lm(y ~ pbs(x, knots=c(pi/2, pi, pi*3/2),
Boundary.knots = c(0, 2*pi)))
)
plot(x, y, xlab = "x", ylab = "sin(x)", pch="x", cex=.5)
lines(x, predict(fm2, data.frame(x=x, z=z)), col='blue')
lines(x, sin(x) + cos(2*x), col='red')
#### example of two periodic functions
x0 = seq(1,628, by=4)/100
z0 = seq(1, 314, by=3)/100
x = rep(x0, each=length(z0))
z = rep(z0, length(x0))
y = sin(x) + cos(2*z) +
rnorm(length(x), 0, 0.1) ## x has a period of 2*pi and z of pi
summary( fm3 <- lm(y ~ pbs(x, df = 5, Boundary.knots = c(0, 2*pi))+
pbs(z, df = 5, Boundary.knots = c(0, pi)))
)
plot(sin(x) + cos(2*3), predict(fm3, data.frame(x=x, z=3)))
summary(sin(x) + cos(2*3)- predict(fm3, data.frame(x=x, z=3)))
## End(Not run)
Example output
Loading required package: splines
1 2 3 4 5
[1,] 0.000000e+00 0.000000e+00 1.633648e-01 6.659058e-01 1.707292e-01
[2,] 0.000000e+00 0.000000e+00 1.587197e-01 6.656678e-01 1.756113e-01
[3,] 0.000000e+00 0.000000e+00 1.541634e-01 6.652538e-01 1.805789e-01
[4,] 0.000000e+00 0.000000e+00 1.496952e-01 6.646662e-01 1.856294e-01
[5,] 0.000000e+00 0.000000e+00 1.453141e-01 6.639078e-01 1.907601e-01
[6,] 0.000000e+00 0.000000e+00 1.410194e-01 6.629810e-01 1.959686e-01
[7,] 0.000000e+00 0.000000e+00 1.368102e-01 6.618884e-01 2.012522e-01
[8,] 0.000000e+00 0.000000e+00 1.326855e-01 6.606326e-01 2.066084e-01
[9,] 0.000000e+00 0.000000e+00 1.286446e-01 6.592162e-01 2.120345e-01
[10,] 0.000000e+00 0.000000e+00 1.246866e-01 6.576418e-01 2.175281e-01
[11,] 0.000000e+00 0.000000e+00 1.208106e-01 6.559119e-01 2.230864e-01
[12,] 0.000000e+00 0.000000e+00 1.170158e-01 6.540292e-01 2.287070e-01
[13,] 0.000000e+00 0.000000e+00 1.133013e-01 6.519961e-01 2.343873e-01
[14,] 0.000000e+00 0.000000e+00 1.096663e-01 6.498154e-01 2.401246e-01
[15,] 0.000000e+00 0.000000e+00 1.061098e-01 6.474895e-01 2.459164e-01
[16,] 0.000000e+00 0.000000e+00 1.026311e-01 6.450210e-01 2.517601e-01
[17,] 0.000000e+00 0.000000e+00 9.922921e-02 6.424125e-01 2.576532e-01
[18,] 0.000000e+00 0.000000e+00 9.590337e-02 6.396667e-01 2.635930e-01
[19,] 0.000000e+00 0.000000e+00 9.265269e-02 6.367860e-01 2.695769e-01
[20,] 0.000000e+00 0.000000e+00 8.947631e-02 6.337730e-01 2.756025e-01
[21,] 0.000000e+00 0.000000e+00 8.637337e-02 6.306304e-01 2.816671e-01
[22,] 0.000000e+00 0.000000e+00 8.334300e-02 6.273607e-01 2.877681e-01
[23,] 0.000000e+00 0.000000e+00 8.038436e-02 6.239665e-01 2.939030e-01
[24,] 0.000000e+00 0.000000e+00 7.749658e-02 6.204503e-01 3.000691e-01
[25,] 0.000000e+00 0.000000e+00 7.467880e-02 6.168148e-01 3.062640e-01
[26,] 0.000000e+00 0.000000e+00 7.193017e-02 6.130624e-01 3.124849e-01
[27,] 0.000000e+00 0.000000e+00 6.924983e-02 6.091959e-01 3.187294e-01
[28,] 0.000000e+00 0.000000e+00 6.663691e-02 6.052178e-01 3.249948e-01
[29,] 0.000000e+00 0.000000e+00 6.409055e-02 6.011306e-01 3.312786e-01
[30,] 0.000000e+00 0.000000e+00 6.160991e-02 5.969369e-01 3.375782e-01
[31,] 0.000000e+00 0.000000e+00 5.919411e-02 5.926394e-01 3.438910e-01
[32,] 0.000000e+00 0.000000e+00 5.684231e-02 5.882405e-01 3.502144e-01
[33,] 0.000000e+00 0.000000e+00 5.455364e-02 5.837429e-01 3.565459e-01
[34,] 0.000000e+00 0.000000e+00 5.232724e-02 5.791491e-01 3.628828e-01
[35,] 0.000000e+00 0.000000e+00 5.016226e-02 5.744618e-01 3.692226e-01
[36,] 0.000000e+00 0.000000e+00 4.805783e-02 5.696834e-01 3.755627e-01
[37,] 0.000000e+00 0.000000e+00 4.601309e-02 5.648167e-01 3.819006e-01
[38,] 0.000000e+00 0.000000e+00 4.402720e-02 5.598640e-01 3.882336e-01
[39,] 0.000000e+00 0.000000e+00 4.209928e-02 5.548282e-01 3.945592e-01
[40,] 0.000000e+00 0.000000e+00 4.022848e-02 5.497116e-01 4.008747e-01
[41,] 0.000000e+00 0.000000e+00 3.841394e-02 5.445169e-01 4.071777e-01
[42,] 0.000000e+00 0.000000e+00 3.665480e-02 5.392467e-01 4.134655e-01
[43,] 0.000000e+00 0.000000e+00 3.495021e-02 5.339035e-01 4.197355e-01
[44,] 0.000000e+00 0.000000e+00 3.329930e-02 5.284900e-01 4.259852e-01
[45,] 0.000000e+00 0.000000e+00 3.170121e-02 5.230086e-01 4.322120e-01
[46,] 0.000000e+00 0.000000e+00 3.015509e-02 5.174621e-01 4.384133e-01
[47,] 0.000000e+00 0.000000e+00 2.866007e-02 5.118529e-01 4.445866e-01
[48,] 0.000000e+00 0.000000e+00 2.721531e-02 5.061836e-01 4.507291e-01
[49,] 0.000000e+00 0.000000e+00 2.581993e-02 5.004568e-01 4.568384e-01
[50,] 0.000000e+00 0.000000e+00 2.447308e-02 4.946752e-01 4.629120e-01
[51,] 0.000000e+00 0.000000e+00 2.317390e-02 4.888412e-01 4.689471e-01
[52,] 0.000000e+00 0.000000e+00 2.192154e-02 4.829574e-01 4.749412e-01
[53,] 0.000000e+00 0.000000e+00 2.071513e-02 4.770265e-01 4.808918e-01
[54,] 0.000000e+00 0.000000e+00 1.955381e-02 4.710510e-01 4.867962e-01
[55,] 0.000000e+00 0.000000e+00 1.843673e-02 4.650334e-01 4.926520e-01
[56,] 0.000000e+00 0.000000e+00 1.736302e-02 4.589765e-01 4.984564e-01
[57,] 0.000000e+00 0.000000e+00 1.633183e-02 4.528826e-01 5.042069e-01
[58,] 0.000000e+00 0.000000e+00 1.534230e-02 4.467545e-01 5.099010e-01
[59,] 0.000000e+00 0.000000e+00 1.439357e-02 4.405947e-01 5.155361e-01
[60,] 0.000000e+00 0.000000e+00 1.348478e-02 4.344057e-01 5.211095e-01
[61,] 0.000000e+00 0.000000e+00 1.261507e-02 4.281902e-01 5.266187e-01
[62,] 0.000000e+00 0.000000e+00 1.178359e-02 4.219508e-01 5.320612e-01
[63,] 0.000000e+00 0.000000e+00 1.098946e-02 4.156899e-01 5.374343e-01
[64,] 0.000000e+00 0.000000e+00 1.023185e-02 4.094102e-01 5.427354e-01
[65,] 0.000000e+00 0.000000e+00 9.509874e-03 4.031143e-01 5.479621e-01
[66,] 0.000000e+00 0.000000e+00 8.822690e-03 3.968047e-01 5.531116e-01
[67,] 0.000000e+00 0.000000e+00 8.169433e-03 3.904841e-01 5.581815e-01
[68,] 0.000000e+00 0.000000e+00 7.549245e-03 3.841549e-01 5.631691e-01
[69,] 0.000000e+00 0.000000e+00 6.961267e-03 3.778198e-01 5.680718e-01
[70,] 0.000000e+00 0.000000e+00 6.404641e-03 3.714814e-01 5.728871e-01
[71,] 0.000000e+00 0.000000e+00 5.878506e-03 3.651422e-01 5.776125e-01
[72,] 0.000000e+00 0.000000e+00 5.382005e-03 3.588048e-01 5.822452e-01
[73,] 0.000000e+00 0.000000e+00 4.914279e-03 3.524718e-01 5.867828e-01
[74,] 0.000000e+00 0.000000e+00 4.474468e-03 3.461458e-01 5.912226e-01
[75,] 0.000000e+00 0.000000e+00 4.061713e-03 3.398293e-01 5.955621e-01
[76,] 0.000000e+00 0.000000e+00 3.675156e-03 3.335250e-01 5.997987e-01
[77,] 0.000000e+00 0.000000e+00 3.313937e-03 3.272353e-01 6.039299e-01
[78,] 0.000000e+00 0.000000e+00 2.977199e-03 3.209629e-01 6.079529e-01
[79,] 0.000000e+00 0.000000e+00 2.664080e-03 3.147104e-01 6.118653e-01
[80,] 0.000000e+00 0.000000e+00 2.373724e-03 3.084803e-01 6.156645e-01
[81,] 0.000000e+00 0.000000e+00 2.105271e-03 3.022753e-01 6.193479e-01
[82,] 0.000000e+00 0.000000e+00 1.857861e-03 2.960978e-01 6.229129e-01
[83,] 0.000000e+00 0.000000e+00 1.630637e-03 2.899505e-01 6.263570e-01
[84,] 0.000000e+00 0.000000e+00 1.422739e-03 2.838359e-01 6.296774e-01
[85,] 0.000000e+00 0.000000e+00 1.233307e-03 2.777566e-01 6.328718e-01
[86,] 0.000000e+00 0.000000e+00 1.061484e-03 2.717153e-01 6.359375e-01
[87,] 0.000000e+00 0.000000e+00 9.064104e-04 2.657145e-01 6.388718e-01
[88,] 0.000000e+00 0.000000e+00 7.672270e-04 2.597567e-01 6.416723e-01
[89,] 0.000000e+00 0.000000e+00 6.430749e-04 2.538445e-01 6.443364e-01
[90,] 0.000000e+00 0.000000e+00 5.330952e-04 2.479806e-01 6.468614e-01
[91,] 0.000000e+00 0.000000e+00 4.364291e-04 2.421674e-01 6.492448e-01
[92,] 0.000000e+00 0.000000e+00 3.522176e-04 2.364077e-01 6.514841e-01
[93,] 0.000000e+00 0.000000e+00 2.796017e-04 2.307038e-01 6.535765e-01
[94,] 0.000000e+00 0.000000e+00 2.177225e-04 2.250585e-01 6.555196e-01
[95,] 0.000000e+00 0.000000e+00 1.657210e-04 2.194744e-01 6.573108e-01
[96,] 0.000000e+00 0.000000e+00 1.227384e-04 2.139539e-01 6.589475e-01
[97,] 0.000000e+00 0.000000e+00 8.791571e-05 2.084996e-01 6.604271e-01
[98,] 0.000000e+00 0.000000e+00 6.039396e-05 2.031142e-01 6.617470e-01
[99,] 0.000000e+00 0.000000e+00 3.931423e-05 1.978002e-01 6.629047e-01
[100,] 0.000000e+00 0.000000e+00 2.381759e-05 1.925603e-01 6.638975e-01
[101,] 0.000000e+00 0.000000e+00 1.304510e-05 1.873969e-01 6.647229e-01
[102,] 0.000000e+00 0.000000e+00 6.137816e-06 1.823126e-01 6.653784e-01
[103,] 0.000000e+00 0.000000e+00 2.236813e-06 1.773101e-01 6.658612e-01
[104,] 0.000000e+00 0.000000e+00 4.831517e-07 1.723919e-01 6.661689e-01
[105,] 0.000000e+00 0.000000e+00 1.789451e-08 1.675606e-01 6.662989e-01
[106,] 1.825618e-08 0.000000e+00 0.000000e+00 1.628186e-01 6.662487e-01
[107,] 4.929168e-07 0.000000e+00 0.000000e+00 1.581670e-01 6.660183e-01
[108,] 2.282022e-06 0.000000e+00 0.000000e+00 1.536047e-01 6.656102e-01
[109,] 6.261869e-06 0.000000e+00 0.000000e+00 1.491311e-01 6.650270e-01
[110,] 1.330875e-05 0.000000e+00 0.000000e+00 1.447452e-01 6.642714e-01
[111,] 2.429897e-05 0.000000e+00 0.000000e+00 1.404461e-01 6.633459e-01
[112,] 4.010883e-05 0.000000e+00 0.000000e+00 1.362330e-01 6.622532e-01
[113,] 6.161460e-05 0.000000e+00 0.000000e+00 1.321050e-01 6.609958e-01
[114,] 8.969261e-05 0.000000e+00 0.000000e+00 1.280613e-01 6.595764e-01
[115,] 1.252191e-04 0.000000e+00 0.000000e+00 1.241009e-01 6.579976e-01
[116,] 1.690705e-04 0.000000e+00 0.000000e+00 1.202230e-01 6.562621e-01
[117,] 2.221229e-04 0.000000e+00 0.000000e+00 1.164268e-01 6.543723e-01
[118,] 2.852528e-04 0.000000e+00 0.000000e+00 1.127113e-01 6.523310e-01
[119,] 3.593364e-04 0.000000e+00 0.000000e+00 1.090758e-01 6.501407e-01
[120,] 4.452500e-04 0.000000e+00 0.000000e+00 1.055192e-01 6.478040e-01
[121,] 5.438698e-04 0.000000e+00 0.000000e+00 1.020409e-01 6.453237e-01
[122,] 6.560723e-04 0.000000e+00 0.000000e+00 9.863979e-02 6.427022e-01
[123,] 7.827337e-04 0.000000e+00 0.000000e+00 9.531514e-02 6.399421e-01
[124,] 9.247302e-04 0.000000e+00 0.000000e+00 9.206604e-02 6.370462e-01
[125,] 1.082938e-03 0.000000e+00 0.000000e+00 8.889164e-02 6.340170e-01
[126,] 1.258234e-03 0.000000e+00 0.000000e+00 8.579105e-02 6.308571e-01
[127,] 1.451494e-03 0.000000e+00 0.000000e+00 8.276342e-02 6.275691e-01
[128,] 1.663594e-03 0.000000e+00 0.000000e+00 7.980786e-02 6.241557e-01
[129,] 1.895411e-03 0.000000e+00 0.000000e+00 7.692353e-02 6.206194e-01
[130,] 2.147821e-03 0.000000e+00 0.000000e+00 7.410953e-02 6.169629e-01
[131,] 2.421700e-03 0.000000e+00 0.000000e+00 7.136502e-02 6.131888e-01
[132,] 2.717925e-03 0.000000e+00 0.000000e+00 6.868912e-02 6.092996e-01
[133,] 3.037372e-03 0.000000e+00 0.000000e+00 6.608095e-02 6.052981e-01
[134,] 3.380917e-03 0.000000e+00 0.000000e+00 6.353966e-02 6.011867e-01
[135,] 3.749436e-03 0.000000e+00 0.000000e+00 6.106438e-02 5.969682e-01
[136,] 4.143806e-03 0.000000e+00 0.000000e+00 5.865422e-02 5.926451e-01
[137,] 4.564903e-03 0.000000e+00 0.000000e+00 5.630834e-02 5.882201e-01
[138,] 5.013603e-03 0.000000e+00 0.000000e+00 5.402585e-02 5.836957e-01
[139,] 5.490783e-03 0.000000e+00 0.000000e+00 5.180589e-02 5.790746e-01
[140,] 5.997319e-03 0.000000e+00 0.000000e+00 4.964760e-02 5.743593e-01
[141,] 6.534087e-03 0.000000e+00 0.000000e+00 4.755010e-02 5.695526e-01
[142,] 7.101964e-03 0.000000e+00 0.000000e+00 4.551252e-02 5.646569e-01
[143,] 7.701826e-03 0.000000e+00 0.000000e+00 4.353400e-02 5.596750e-01
[144,] 8.334548e-03 0.000000e+00 0.000000e+00 4.161366e-02 5.546093e-01
[145,] 9.001008e-03 0.000000e+00 0.000000e+00 3.975065e-02 5.494626e-01
[146,] 9.702082e-03 0.000000e+00 0.000000e+00 3.794408e-02 5.442375e-01
[147,] 1.043865e-02 0.000000e+00 0.000000e+00 3.619310e-02 5.389365e-01
[148,] 1.121158e-02 0.000000e+00 0.000000e+00 3.449683e-02 5.335623e-01
[149,] 1.202175e-02 0.000000e+00 0.000000e+00 3.285441e-02 5.281174e-01
[150,] 1.287004e-02 0.000000e+00 0.000000e+00 3.126496e-02 5.226046e-01
[151,] 1.375733e-02 0.000000e+00 0.000000e+00 2.972762e-02 5.170263e-01
[152,] 1.468449e-02 0.000000e+00 0.000000e+00 2.824152e-02 5.113853e-01
[153,] 1.565239e-02 0.000000e+00 0.000000e+00 2.680580e-02 5.056841e-01
[154,] 1.666192e-02 0.000000e+00 0.000000e+00 2.541957e-02 4.999253e-01
[155,] 1.771395e-02 0.000000e+00 0.000000e+00 2.408198e-02 4.941116e-01
[156,] 1.880936e-02 0.000000e+00 0.000000e+00 2.279216e-02 4.882455e-01
[157,] 1.994902e-02 0.000000e+00 0.000000e+00 2.154923e-02 4.823297e-01
[158,] 2.113381e-02 0.000000e+00 0.000000e+00 2.035233e-02 4.763668e-01
[159,] 2.236460e-02 0.000000e+00 0.000000e+00 1.920059e-02 4.703593e-01
[160,] 2.364228e-02 0.000000e+00 0.000000e+00 1.809315e-02 4.643100e-01
[161,] 2.496772e-02 0.000000e+00 0.000000e+00 1.702913e-02 4.582214e-01
[162,] 2.634179e-02 0.000000e+00 0.000000e+00 1.600766e-02 4.520961e-01
[163,] 2.776537e-02 0.000000e+00 0.000000e+00 1.502788e-02 4.459368e-01
[164,] 2.923933e-02 0.000000e+00 0.000000e+00 1.408892e-02 4.397460e-01
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attr(,"degree")
[1] 3
attr(,"knots")
16.66667% 33.33333% 50% 66.66667% 83.33333%
1.055 2.100 3.145 4.190 5.235
attr(,"Boundary.knots")
[1] 0.000000 6.283185
attr(,"intercept")
[1] FALSE
attr(,"periodic")
[1] TRUE
attr(,"class")
[1] "pbs" "basis"
1 2 3
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[450,] 6.496206e-01 1.077902e-01 4.119885e-04
[451,] 6.511352e-01 1.101883e-01 3.564919e-04
[452,] 6.525843e-01 1.126216e-01 3.062173e-04
[453,] 6.539673e-01 1.150906e-01 2.609065e-04
[454,] 6.552833e-01 1.175954e-01 2.203016e-04
[455,] 6.565317e-01 1.201362e-01 1.841445e-04
[456,] 6.577115e-01 1.227134e-01 1.521773e-04
[457,] 6.588221e-01 1.253272e-01 1.241419e-04
[458,] 6.598626e-01 1.279778e-01 9.978026e-05
[459,] 6.608324e-01 1.306656e-01 7.883445e-05
[460,] 6.617305e-01 1.333907e-01 6.104644e-05
[461,] 6.625563e-01 1.361535e-01 4.615819e-05
[462,] 6.633090e-01 1.389541e-01 3.391171e-05
[463,] 6.639878e-01 1.417929e-01 2.404897e-05
[464,] 6.645918e-01 1.446701e-01 1.631197e-05
[465,] 6.651205e-01 1.475860e-01 1.044270e-05
[466,] 6.655729e-01 1.505407e-01 6.183138e-06
[467,] 6.659483e-01 1.535347e-01 3.275277e-06
[468,] 6.662459e-01 1.565681e-01 1.461103e-06
[469,] 6.664650e-01 1.596411e-01 4.826053e-07
[470,] 6.666047e-01 1.627542e-01 8.177034e-08
[471,] 6.666644e-01 1.659074e-01 5.863094e-10
[472,] 6.666432e-01 1.691010e-01 1.895910e-08
[473,] 6.665417e-01 1.723346e-01 2.348782e-07
[474,] 6.663604e-01 1.756073e-01 9.051831e-07
[475,] 6.661002e-01 1.789184e-01 2.287886e-06
[476,] 6.657619e-01 1.822672e-01 4.641000e-06
[477,] 6.653462e-01 1.856527e-01 8.222536e-06
[478,] 6.648539e-01 1.890744e-01 1.329051e-05
[479,] 6.642858e-01 1.925313e-01 2.010293e-05
[480,] 6.636426e-01 1.960228e-01 2.891780e-05
[481,] 6.629251e-01 1.995480e-01 3.999316e-05
[482,] 6.621342e-01 2.031062e-01 5.358699e-05
[483,] 6.612705e-01 2.066966e-01 6.995732e-05
[484,] 6.603349e-01 2.103184e-01 8.936216e-05
[485,] 6.593281e-01 2.139708e-01 1.120595e-04
[486,] 6.582508e-01 2.176532e-01 1.383074e-04
[487,] 6.571040e-01 2.213646e-01 1.683639e-04
[488,] 6.558883e-01 2.251044e-01 2.024868e-04
[489,] 6.546045e-01 2.288717e-01 2.409344e-04
[490,] 6.532534e-01 2.326658e-01 2.839646e-04
[491,] 6.518358e-01 2.364859e-01 3.318353e-04
[492,] 6.503524e-01 2.403313e-01 3.848047e-04
[493,] 6.488040e-01 2.442011e-01 4.431306e-04
[494,] 6.471914e-01 2.480945e-01 5.070712e-04
[495,] 6.455154e-01 2.520109e-01 5.768845e-04
[496,] 6.437767e-01 2.559495e-01 6.528284e-04
[497,] 6.419761e-01 2.599094e-01 7.351609e-04
[498,] 6.401144e-01 2.638898e-01 8.241402e-04
[499,] 6.381923e-01 2.678901e-01 9.200241e-04
[500,] 6.362107e-01 2.719095e-01 1.023071e-03
[501,] 6.341703e-01 2.759471e-01 1.133538e-03
[502,] 6.320718e-01 2.800022e-01 1.251684e-03
[503,] 6.299162e-01 2.840740e-01 1.377767e-03
[504,] 6.277040e-01 2.881618e-01 1.512045e-03
[505,] 6.254362e-01 2.922647e-01 1.654775e-03
[506,] 6.231134e-01 2.963820e-01 1.806216e-03
[507,] 6.207364e-01 3.005130e-01 1.966626e-03
[508,] 6.183061e-01 3.046568e-01 2.136263e-03
[509,] 6.158232e-01 3.088126e-01 2.315384e-03
[510,] 6.132885e-01 3.129798e-01 2.504249e-03
[511,] 6.107027e-01 3.171575e-01 2.703114e-03
[512,] 6.080666e-01 3.213449e-01 2.912238e-03
[513,] 6.053811e-01 3.255414e-01 3.131879e-03
[514,] 6.026468e-01 3.297460e-01 3.362295e-03
[515,] 5.998645e-01 3.339581e-01 3.603743e-03
[516,] 5.970350e-01 3.381768e-01 3.856483e-03
[517,] 5.941592e-01 3.424014e-01 4.120772e-03
[518,] 5.912377e-01 3.466311e-01 4.396867e-03
[519,] 5.882713e-01 3.508651e-01 4.685027e-03
[520,] 5.852609e-01 3.551027e-01 4.985511e-03
[521,] 5.822071e-01 3.593431e-01 5.298575e-03
[522,] 5.791108e-01 3.635855e-01 5.624478e-03
[523,] 5.759727e-01 3.678291e-01 5.963479e-03
[524,] 5.727937e-01 3.720732e-01 6.315834e-03
[525,] 5.695744e-01 3.763170e-01 6.681802e-03
[526,] 5.663157e-01 3.805597e-01 7.061641e-03
[527,] 5.630183e-01 3.848005e-01 7.455610e-03
[528,] 5.596830e-01 3.890387e-01 7.863965e-03
[529,] 5.563106e-01 3.932736e-01 8.286965e-03
[530,] 5.529018e-01 3.975042e-01 8.724869e-03
[531,] 5.494575e-01 4.017298e-01 9.177934e-03
[532,] 5.459784e-01 4.059498e-01 9.646417e-03
[533,] 5.424652e-01 4.101632e-01 1.013058e-02
[534,] 5.389188e-01 4.143694e-01 1.063067e-02
[535,] 5.353399e-01 4.185675e-01 1.114696e-02
[536,] 5.317293e-01 4.227568e-01 1.167970e-02
[537,] 5.280878e-01 4.269364e-01 1.222915e-02
[538,] 5.244162e-01 4.311058e-01 1.279557e-02
[539,] 5.207151e-01 4.352639e-01 1.337921e-02
[540,] 5.169854e-01 4.394102e-01 1.398034e-02
[541,] 5.132279e-01 4.435437e-01 1.459920e-02
[542,] 5.094434e-01 4.476638e-01 1.523607e-02
[543,] 5.056325e-01 4.517696e-01 1.589119e-02
[544,] 5.017962e-01 4.558604e-01 1.656483e-02
[545,] 4.979351e-01 4.599355e-01 1.725724e-02
[546,] 4.940501e-01 4.639939e-01 1.796868e-02
[547,] 4.901418e-01 4.680350e-01 1.869941e-02
[548,] 4.862112e-01 4.720581e-01 1.944969e-02
[549,] 4.822589e-01 4.760622e-01 2.021978e-02
[550,] 4.782857e-01 4.800466e-01 2.100992e-02
[551,] 4.742924e-01 4.840107e-01 2.182039e-02
[552,] 4.702799e-01 4.879535e-01 2.265144e-02
[553,] 4.662488e-01 4.918743e-01 2.350333e-02
[554,] 4.621999e-01 4.957724e-01 2.437631e-02
[555,] 4.581340e-01 4.996469e-01 2.527064e-02
[556,] 4.540519e-01 5.034972e-01 2.618659e-02
[557,] 4.499543e-01 5.073223e-01 2.712440e-02
[558,] 4.458421e-01 5.111216e-01 2.808434e-02
[559,] 4.417159e-01 5.148943e-01 2.906667e-02
[560,] 4.375767e-01 5.186396e-01 3.007164e-02
[561,] 4.334251e-01 5.223567e-01 3.109952e-02
[562,] 4.292619e-01 5.260448e-01 3.215055e-02
[563,] 4.250879e-01 5.297032e-01 3.322500e-02
[564,] 4.209038e-01 5.333311e-01 3.432312e-02
[565,] 4.167106e-01 5.369278e-01 3.544518e-02
[566,] 4.125088e-01 5.404924e-01 3.659143e-02
[567,] 4.082993e-01 5.440242e-01 3.776213e-02
[568,] 4.040829e-01 5.475224e-01 3.895754e-02
[569,] 3.998603e-01 5.509862e-01 4.017791e-02
[570,] 3.956324e-01 5.544149e-01 4.142351e-02
[571,] 3.913998e-01 5.578077e-01 4.269459e-02
[572,] 3.871634e-01 5.611637e-01 4.399141e-02
[573,] 3.829239e-01 5.644823e-01 4.531422e-02
[574,] 3.786822e-01 5.677627e-01 4.666329e-02
[575,] 3.744389e-01 5.710041e-01 4.803888e-02
[576,] 3.701949e-01 5.742057e-01 4.944123e-02
[577,] 3.659509e-01 5.773667e-01 5.087062e-02
[578,] 3.617077e-01 5.804863e-01 5.232729e-02
[579,] 3.574661e-01 5.835639e-01 5.381151e-02
[580,] 3.532269e-01 5.865986e-01 5.532353e-02
[581,] 3.489908e-01 5.895896e-01 5.686362e-02
[582,] 3.447585e-01 5.925362e-01 5.843202e-02
[583,] 3.405310e-01 5.954376e-01 6.002900e-02
[584,] 3.363089e-01 5.982930e-01 6.165482e-02
[585,] 3.320931e-01 6.011017e-01 6.330973e-02
[586,] 3.278842e-01 6.038628e-01 6.499400e-02
[587,] 3.236831e-01 6.065757e-01 6.670787e-02
[588,] 3.194905e-01 6.092394e-01 6.845161e-02
[589,] 3.153073e-01 6.118534e-01 7.022548e-02
[590,] 3.111342e-01 6.144166e-01 7.202973e-02
[591,] 3.069719e-01 6.169285e-01 7.386462e-02
[592,] 3.028213e-01 6.193883e-01 7.573041e-02
[593,] 2.986831e-01 6.217950e-01 7.762736e-02
[594,] 2.945581e-01 6.241481e-01 7.955573e-02
[595,] 2.904470e-01 6.264467e-01 8.151577e-02
[596,] 2.863507e-01 6.286900e-01 8.350774e-02
[597,] 2.822699e-01 6.308772e-01 8.553190e-02
[598,] 2.782054e-01 6.330077e-01 8.758851e-02
[599,] 2.741580e-01 6.350805e-01 8.967782e-02
[600,] 2.701283e-01 6.370950e-01 9.180010e-02
[601,] 2.661173e-01 6.390503e-01 9.395561e-02
[602,] 2.621257e-01 6.409458e-01 9.614459e-02
[603,] 2.581542e-01 6.427805e-01 9.836731e-02
[604,] 2.542037e-01 6.445538e-01 1.006240e-01
[605,] 2.502749e-01 6.462649e-01 1.029150e-01
[606,] 2.463685e-01 6.479129e-01 1.052405e-01
[607,] 2.424854e-01 6.494972e-01 1.076007e-01
[608,] 2.386264e-01 6.510169e-01 1.099960e-01
[609,] 2.347921e-01 6.524713e-01 1.124266e-01
[610,] 2.309834e-01 6.538596e-01 1.148927e-01
[611,] 2.272010e-01 6.551810e-01 1.173946e-01
[612,] 2.234458e-01 6.564348e-01 1.199326e-01
[613,] 2.197185e-01 6.576201e-01 1.225068e-01
[614,] 2.160198e-01 6.587362e-01 1.251177e-01
[615,] 2.123506e-01 6.597824e-01 1.277654e-01
[616,] 2.087116e-01 6.607578e-01 1.304502e-01
[617,] 2.051036e-01 6.616617e-01 1.331723e-01
[618,] 2.015274e-01 6.624932e-01 1.359321e-01
[619,] 1.979837e-01 6.632518e-01 1.387297e-01
[620,] 1.944734e-01 6.639364e-01 1.415655e-01
[621,] 1.909971e-01 6.645465e-01 1.444396e-01
[622,] 1.875557e-01 6.650812e-01 1.473523e-01
[623,] 1.841499e-01 6.655397e-01 1.503040e-01
[624,] 1.807805e-01 6.659212e-01 1.532948e-01
[625,] 1.774483e-01 6.662251e-01 1.563250e-01
[626,] 1.741541e-01 6.664504e-01 1.593950e-01
[627,] 1.708986e-01 6.665965e-01 1.625048e-01
[628,] 1.676826e-01 6.666626e-01 1.656548e-01
attr(,"degree")
[1] 3
attr(,"knots")
[1] 1.570796 3.141593 4.712389
attr(,"Boundary.knots")
[1] 0.000000 6.283185
attr(,"intercept")
[1] FALSE
attr(,"periodic")
[1] TRUE
attr(,"class")
[1] "pbs" "basis"
Call:
lm(formula = y ~ pbs(x, df = 10, Boundary.knots = c(0, 2 * pi)))
Residuals:
Min 1Q Median 3Q Max
-0.38317 -0.06361 0.00231 0.06834 0.30621
Coefficients:
Estimate Std. Error t value
(Intercept) 0.15559 0.02989 5.204
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))1 -0.28780 0.05388 -5.342
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))2 0.48547 0.03441 14.108
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))3 1.15031 0.04594 25.039
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))4 0.63945 0.04053 15.779
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))5 -1.16418 0.04286 -27.163
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))6 -2.37922 0.04286 -55.510
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))7 -1.94186 0.04047 -47.977
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))8 -0.18792 0.04591 -4.093
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))9 1.11324 0.03442 32.342
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))10 0.91646 0.05391 17.001
Pr(>|t|)
(Intercept) 2.65e-07 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))1 1.30e-07 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))2 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))3 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))4 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))5 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))6 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))7 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))8 4.82e-05 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))9 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))10 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1017 on 617 degrees of freedom
Multiple R-squared: 0.9899, Adjusted R-squared: 0.9898
F-statistic: 6075 on 10 and 617 DF, p-value: < 2.2e-16
Call:
lm(formula = y ~ pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0,
2 * pi)))
Residuals:
Min 1Q Median 3Q Max
-0.39626 -0.06549 0.00058 0.06692 0.31204
Coefficients:
Estimate
(Intercept) 3.04079
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 -7.60160
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.01498
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 -4.55223
Std. Error
(Intercept) 0.01999
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 0.03713
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.01751
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 0.03713
t value
(Intercept) 152.151
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 -204.731
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.855
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 -122.604
Pr(>|t|)
(Intercept) <2e-16
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 <2e-16
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.393
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 <2e-16
(Intercept) ***
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 ***
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1019 on 624 degrees of freedom
Multiple R-squared: 0.9898, Adjusted R-squared: 0.9898
F-statistic: 2.019e+04 on 3 and 624 DF, p-value: < 2.2e-16
Call:
lm(formula = y ~ pbs(x, df = 5, Boundary.knots = c(0, 2 * pi)) +
pbs(z, df = 5, Boundary.knots = c(0, pi)))
Residuals:
Min 1Q Median 3Q Max
-0.35669 -0.06836 0.00019 0.06792 0.39867
Coefficients:
Estimate Std. Error t value
(Intercept) 0.440686 0.006205 71.024
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))1 -1.027707 0.008034 -127.925
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))2 -2.069734 0.004718 -438.710
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))3 -2.097653 0.007288 -287.819
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))4 -1.042592 0.004728 -220.530
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))5 -0.004846 0.008023 -0.604
pbs(z, df = 5, Boundary.knots = c(0, pi))1 -0.598482 0.008035 -74.489
pbs(z, df = 5, Boundary.knots = c(0, pi))2 -0.002906 0.004734 -0.614
pbs(z, df = 5, Boundary.knots = c(0, pi))3 1.200186 0.007297 164.474
pbs(z, df = 5, Boundary.knots = c(0, pi))4 1.803385 0.004705 383.305
pbs(z, df = 5, Boundary.knots = c(0, pi))5 1.199290 0.008052 148.939
Pr(>|t|)
(Intercept) <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))1 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))2 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))3 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))4 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))5 0.546
pbs(z, df = 5, Boundary.knots = c(0, pi))1 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))2 0.539
pbs(z, df = 5, Boundary.knots = c(0, pi))3 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))4 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))5 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.101 on 16474 degrees of freedom
Multiple R-squared: 0.9899, Adjusted R-squared: 0.9899
F-statistic: 1.616e+05 on 10 and 16474 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.0053448 -0.0012041 -0.0004680 -0.0005027 0.0008607 0.0031570
pbs documentation built on May 1, 2019, 9:17 p.m.
R Package Documentation
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Description Usage Arguments Value References See Also Examples
Description
Generate the periodic B-spline basis matrix for a polynomial spline.
Usage
1 2 |
Arguments
x |
the predictor variable. Missing values are allowed. |
df |
degrees of freedom; one can specify 'df' rather than 'knots'; 'pbs()' then chooses 'df - 1' knots at suitable quantiles of 'x' (which will ignore missing values). |
knots |
the _internal_ breakpoints that define the spline. The number of internal knots must be greater than or equal to degree polynomial regression. See also 'Boundary.knots'. |
degree |
degree of the piecewise polynomial-default is 3 for cubic splines. |
intercept |
if 'TRUE', an intercept is included in the basis; default is 'FALSE'. |
Boundary.knots |
boundary points at which to set the period of the perodic B-spline basis(default the range of the data). If both 'knots' and 'Boundary.knots' are supplied, the basis parameters do not depend on 'x'. Data CAN NOT be extended beyond 'Boundary.knots'. Typical Bourday knots are start and end values of period. |
periodic |
if "TRUE", periodic basis is generated. Default is TRUE. This is disabled. DO NOT USE. |
Value
A matrix of dimension 'length(x) * (df)', where either 'df' was supplied or if 'knots' were supplied, 'df = length(knots) + intercept'. Attributes are returned that correspond to the arguments to 'pbs', and explicitly give the 'knots', 'Boundary.knots' etc for use by 'predict.pbs()'.
'pbs()' is based on the function 'spline.des()' in package splines. It generates a basis matrix for representing the family of piecewise polynomials with the specified interior knots and degree, evaluated at the values of 'x'. A primary use is in modeling formulas to directly specify a piecewise polynomial term in a model.
References
Package splines in Base R. pbs is actually modified from bs function
See Also
'bs', 'ns', 'predict.pbs'
Examples
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 | require(stats); require(graphics); require(splines)
x = seq(1,628)/100
z = rep(seq(1, 314)/100, 2)
pbs(x, df = 5, Boundary.knots = c(0, 2*pi))
pbs(x, knots=c(pi/2, pi, pi*3/2), Boundary.knots = c(0, 2*pi))
#### example of one periodic functions
y= sin(x) + cos(2*x) +
rnorm(628, 0, 0.1) ## x has a period of 2*pi
## df method, need to use large enough df to get a better fit.
## May use max loglik to choose optimal df
summary( fm1 <- lm(y ~ pbs(x, df = 10, Boundary.knots = c(0, 2*pi))) )
plot(x, y, xlab = "x", ylab = "sin(x)", pch="x", cex=.5)
lines(x, predict(fm1, data.frame(x=x, z=z)), col='blue')
lines(x, sin(x) + cos(2*x), col='red')
## knots methods, usually selected at turning points
summary( fm2 <- lm(y ~ pbs(x, knots=c(pi/2, pi, pi*3/2),
Boundary.knots = c(0, 2*pi)))
)
plot(x, y, xlab = "x", ylab = "sin(x)", pch="x", cex=.5)
lines(x, predict(fm2, data.frame(x=x, z=z)), col='blue')
lines(x, sin(x) + cos(2*x), col='red')
#### example of two periodic functions
x0 = seq(1,628, by=4)/100
z0 = seq(1, 314, by=3)/100
x = rep(x0, each=length(z0))
z = rep(z0, length(x0))
y = sin(x) + cos(2*z) +
rnorm(length(x), 0, 0.1) ## x has a period of 2*pi and z of pi
summary( fm3 <- lm(y ~ pbs(x, df = 5, Boundary.knots = c(0, 2*pi))+
pbs(z, df = 5, Boundary.knots = c(0, pi)))
)
plot(sin(x) + cos(2*3), predict(fm3, data.frame(x=x, z=3)))
summary(sin(x) + cos(2*3)- predict(fm3, data.frame(x=x, z=3)))
## End(Not run)
|
Example output
Loading required package: splines
1 2 3 4 5
[1,] 0.000000e+00 0.000000e+00 1.633648e-01 6.659058e-01 1.707292e-01
[2,] 0.000000e+00 0.000000e+00 1.587197e-01 6.656678e-01 1.756113e-01
[3,] 0.000000e+00 0.000000e+00 1.541634e-01 6.652538e-01 1.805789e-01
[4,] 0.000000e+00 0.000000e+00 1.496952e-01 6.646662e-01 1.856294e-01
[5,] 0.000000e+00 0.000000e+00 1.453141e-01 6.639078e-01 1.907601e-01
[6,] 0.000000e+00 0.000000e+00 1.410194e-01 6.629810e-01 1.959686e-01
[7,] 0.000000e+00 0.000000e+00 1.368102e-01 6.618884e-01 2.012522e-01
[8,] 0.000000e+00 0.000000e+00 1.326855e-01 6.606326e-01 2.066084e-01
[9,] 0.000000e+00 0.000000e+00 1.286446e-01 6.592162e-01 2.120345e-01
[10,] 0.000000e+00 0.000000e+00 1.246866e-01 6.576418e-01 2.175281e-01
[11,] 0.000000e+00 0.000000e+00 1.208106e-01 6.559119e-01 2.230864e-01
[12,] 0.000000e+00 0.000000e+00 1.170158e-01 6.540292e-01 2.287070e-01
[13,] 0.000000e+00 0.000000e+00 1.133013e-01 6.519961e-01 2.343873e-01
[14,] 0.000000e+00 0.000000e+00 1.096663e-01 6.498154e-01 2.401246e-01
[15,] 0.000000e+00 0.000000e+00 1.061098e-01 6.474895e-01 2.459164e-01
[16,] 0.000000e+00 0.000000e+00 1.026311e-01 6.450210e-01 2.517601e-01
[17,] 0.000000e+00 0.000000e+00 9.922921e-02 6.424125e-01 2.576532e-01
[18,] 0.000000e+00 0.000000e+00 9.590337e-02 6.396667e-01 2.635930e-01
[19,] 0.000000e+00 0.000000e+00 9.265269e-02 6.367860e-01 2.695769e-01
[20,] 0.000000e+00 0.000000e+00 8.947631e-02 6.337730e-01 2.756025e-01
[21,] 0.000000e+00 0.000000e+00 8.637337e-02 6.306304e-01 2.816671e-01
[22,] 0.000000e+00 0.000000e+00 8.334300e-02 6.273607e-01 2.877681e-01
[23,] 0.000000e+00 0.000000e+00 8.038436e-02 6.239665e-01 2.939030e-01
[24,] 0.000000e+00 0.000000e+00 7.749658e-02 6.204503e-01 3.000691e-01
[25,] 0.000000e+00 0.000000e+00 7.467880e-02 6.168148e-01 3.062640e-01
[26,] 0.000000e+00 0.000000e+00 7.193017e-02 6.130624e-01 3.124849e-01
[27,] 0.000000e+00 0.000000e+00 6.924983e-02 6.091959e-01 3.187294e-01
[28,] 0.000000e+00 0.000000e+00 6.663691e-02 6.052178e-01 3.249948e-01
[29,] 0.000000e+00 0.000000e+00 6.409055e-02 6.011306e-01 3.312786e-01
[30,] 0.000000e+00 0.000000e+00 6.160991e-02 5.969369e-01 3.375782e-01
[31,] 0.000000e+00 0.000000e+00 5.919411e-02 5.926394e-01 3.438910e-01
[32,] 0.000000e+00 0.000000e+00 5.684231e-02 5.882405e-01 3.502144e-01
[33,] 0.000000e+00 0.000000e+00 5.455364e-02 5.837429e-01 3.565459e-01
[34,] 0.000000e+00 0.000000e+00 5.232724e-02 5.791491e-01 3.628828e-01
[35,] 0.000000e+00 0.000000e+00 5.016226e-02 5.744618e-01 3.692226e-01
[36,] 0.000000e+00 0.000000e+00 4.805783e-02 5.696834e-01 3.755627e-01
[37,] 0.000000e+00 0.000000e+00 4.601309e-02 5.648167e-01 3.819006e-01
[38,] 0.000000e+00 0.000000e+00 4.402720e-02 5.598640e-01 3.882336e-01
[39,] 0.000000e+00 0.000000e+00 4.209928e-02 5.548282e-01 3.945592e-01
[40,] 0.000000e+00 0.000000e+00 4.022848e-02 5.497116e-01 4.008747e-01
[41,] 0.000000e+00 0.000000e+00 3.841394e-02 5.445169e-01 4.071777e-01
[42,] 0.000000e+00 0.000000e+00 3.665480e-02 5.392467e-01 4.134655e-01
[43,] 0.000000e+00 0.000000e+00 3.495021e-02 5.339035e-01 4.197355e-01
[44,] 0.000000e+00 0.000000e+00 3.329930e-02 5.284900e-01 4.259852e-01
[45,] 0.000000e+00 0.000000e+00 3.170121e-02 5.230086e-01 4.322120e-01
[46,] 0.000000e+00 0.000000e+00 3.015509e-02 5.174621e-01 4.384133e-01
[47,] 0.000000e+00 0.000000e+00 2.866007e-02 5.118529e-01 4.445866e-01
[48,] 0.000000e+00 0.000000e+00 2.721531e-02 5.061836e-01 4.507291e-01
[49,] 0.000000e+00 0.000000e+00 2.581993e-02 5.004568e-01 4.568384e-01
[50,] 0.000000e+00 0.000000e+00 2.447308e-02 4.946752e-01 4.629120e-01
[51,] 0.000000e+00 0.000000e+00 2.317390e-02 4.888412e-01 4.689471e-01
[52,] 0.000000e+00 0.000000e+00 2.192154e-02 4.829574e-01 4.749412e-01
[53,] 0.000000e+00 0.000000e+00 2.071513e-02 4.770265e-01 4.808918e-01
[54,] 0.000000e+00 0.000000e+00 1.955381e-02 4.710510e-01 4.867962e-01
[55,] 0.000000e+00 0.000000e+00 1.843673e-02 4.650334e-01 4.926520e-01
[56,] 0.000000e+00 0.000000e+00 1.736302e-02 4.589765e-01 4.984564e-01
[57,] 0.000000e+00 0.000000e+00 1.633183e-02 4.528826e-01 5.042069e-01
[58,] 0.000000e+00 0.000000e+00 1.534230e-02 4.467545e-01 5.099010e-01
[59,] 0.000000e+00 0.000000e+00 1.439357e-02 4.405947e-01 5.155361e-01
[60,] 0.000000e+00 0.000000e+00 1.348478e-02 4.344057e-01 5.211095e-01
[61,] 0.000000e+00 0.000000e+00 1.261507e-02 4.281902e-01 5.266187e-01
[62,] 0.000000e+00 0.000000e+00 1.178359e-02 4.219508e-01 5.320612e-01
[63,] 0.000000e+00 0.000000e+00 1.098946e-02 4.156899e-01 5.374343e-01
[64,] 0.000000e+00 0.000000e+00 1.023185e-02 4.094102e-01 5.427354e-01
[65,] 0.000000e+00 0.000000e+00 9.509874e-03 4.031143e-01 5.479621e-01
[66,] 0.000000e+00 0.000000e+00 8.822690e-03 3.968047e-01 5.531116e-01
[67,] 0.000000e+00 0.000000e+00 8.169433e-03 3.904841e-01 5.581815e-01
[68,] 0.000000e+00 0.000000e+00 7.549245e-03 3.841549e-01 5.631691e-01
[69,] 0.000000e+00 0.000000e+00 6.961267e-03 3.778198e-01 5.680718e-01
[70,] 0.000000e+00 0.000000e+00 6.404641e-03 3.714814e-01 5.728871e-01
[71,] 0.000000e+00 0.000000e+00 5.878506e-03 3.651422e-01 5.776125e-01
[72,] 0.000000e+00 0.000000e+00 5.382005e-03 3.588048e-01 5.822452e-01
[73,] 0.000000e+00 0.000000e+00 4.914279e-03 3.524718e-01 5.867828e-01
[74,] 0.000000e+00 0.000000e+00 4.474468e-03 3.461458e-01 5.912226e-01
[75,] 0.000000e+00 0.000000e+00 4.061713e-03 3.398293e-01 5.955621e-01
[76,] 0.000000e+00 0.000000e+00 3.675156e-03 3.335250e-01 5.997987e-01
[77,] 0.000000e+00 0.000000e+00 3.313937e-03 3.272353e-01 6.039299e-01
[78,] 0.000000e+00 0.000000e+00 2.977199e-03 3.209629e-01 6.079529e-01
[79,] 0.000000e+00 0.000000e+00 2.664080e-03 3.147104e-01 6.118653e-01
[80,] 0.000000e+00 0.000000e+00 2.373724e-03 3.084803e-01 6.156645e-01
[81,] 0.000000e+00 0.000000e+00 2.105271e-03 3.022753e-01 6.193479e-01
[82,] 0.000000e+00 0.000000e+00 1.857861e-03 2.960978e-01 6.229129e-01
[83,] 0.000000e+00 0.000000e+00 1.630637e-03 2.899505e-01 6.263570e-01
[84,] 0.000000e+00 0.000000e+00 1.422739e-03 2.838359e-01 6.296774e-01
[85,] 0.000000e+00 0.000000e+00 1.233307e-03 2.777566e-01 6.328718e-01
[86,] 0.000000e+00 0.000000e+00 1.061484e-03 2.717153e-01 6.359375e-01
[87,] 0.000000e+00 0.000000e+00 9.064104e-04 2.657145e-01 6.388718e-01
[88,] 0.000000e+00 0.000000e+00 7.672270e-04 2.597567e-01 6.416723e-01
[89,] 0.000000e+00 0.000000e+00 6.430749e-04 2.538445e-01 6.443364e-01
[90,] 0.000000e+00 0.000000e+00 5.330952e-04 2.479806e-01 6.468614e-01
[91,] 0.000000e+00 0.000000e+00 4.364291e-04 2.421674e-01 6.492448e-01
[92,] 0.000000e+00 0.000000e+00 3.522176e-04 2.364077e-01 6.514841e-01
[93,] 0.000000e+00 0.000000e+00 2.796017e-04 2.307038e-01 6.535765e-01
[94,] 0.000000e+00 0.000000e+00 2.177225e-04 2.250585e-01 6.555196e-01
[95,] 0.000000e+00 0.000000e+00 1.657210e-04 2.194744e-01 6.573108e-01
[96,] 0.000000e+00 0.000000e+00 1.227384e-04 2.139539e-01 6.589475e-01
[97,] 0.000000e+00 0.000000e+00 8.791571e-05 2.084996e-01 6.604271e-01
[98,] 0.000000e+00 0.000000e+00 6.039396e-05 2.031142e-01 6.617470e-01
[99,] 0.000000e+00 0.000000e+00 3.931423e-05 1.978002e-01 6.629047e-01
[100,] 0.000000e+00 0.000000e+00 2.381759e-05 1.925603e-01 6.638975e-01
[101,] 0.000000e+00 0.000000e+00 1.304510e-05 1.873969e-01 6.647229e-01
[102,] 0.000000e+00 0.000000e+00 6.137816e-06 1.823126e-01 6.653784e-01
[103,] 0.000000e+00 0.000000e+00 2.236813e-06 1.773101e-01 6.658612e-01
[104,] 0.000000e+00 0.000000e+00 4.831517e-07 1.723919e-01 6.661689e-01
[105,] 0.000000e+00 0.000000e+00 1.789451e-08 1.675606e-01 6.662989e-01
[106,] 1.825618e-08 0.000000e+00 0.000000e+00 1.628186e-01 6.662487e-01
[107,] 4.929168e-07 0.000000e+00 0.000000e+00 1.581670e-01 6.660183e-01
[108,] 2.282022e-06 0.000000e+00 0.000000e+00 1.536047e-01 6.656102e-01
[109,] 6.261869e-06 0.000000e+00 0.000000e+00 1.491311e-01 6.650270e-01
[110,] 1.330875e-05 0.000000e+00 0.000000e+00 1.447452e-01 6.642714e-01
[111,] 2.429897e-05 0.000000e+00 0.000000e+00 1.404461e-01 6.633459e-01
[112,] 4.010883e-05 0.000000e+00 0.000000e+00 1.362330e-01 6.622532e-01
[113,] 6.161460e-05 0.000000e+00 0.000000e+00 1.321050e-01 6.609958e-01
[114,] 8.969261e-05 0.000000e+00 0.000000e+00 1.280613e-01 6.595764e-01
[115,] 1.252191e-04 0.000000e+00 0.000000e+00 1.241009e-01 6.579976e-01
[116,] 1.690705e-04 0.000000e+00 0.000000e+00 1.202230e-01 6.562621e-01
[117,] 2.221229e-04 0.000000e+00 0.000000e+00 1.164268e-01 6.543723e-01
[118,] 2.852528e-04 0.000000e+00 0.000000e+00 1.127113e-01 6.523310e-01
[119,] 3.593364e-04 0.000000e+00 0.000000e+00 1.090758e-01 6.501407e-01
[120,] 4.452500e-04 0.000000e+00 0.000000e+00 1.055192e-01 6.478040e-01
[121,] 5.438698e-04 0.000000e+00 0.000000e+00 1.020409e-01 6.453237e-01
[122,] 6.560723e-04 0.000000e+00 0.000000e+00 9.863979e-02 6.427022e-01
[123,] 7.827337e-04 0.000000e+00 0.000000e+00 9.531514e-02 6.399421e-01
[124,] 9.247302e-04 0.000000e+00 0.000000e+00 9.206604e-02 6.370462e-01
[125,] 1.082938e-03 0.000000e+00 0.000000e+00 8.889164e-02 6.340170e-01
[126,] 1.258234e-03 0.000000e+00 0.000000e+00 8.579105e-02 6.308571e-01
[127,] 1.451494e-03 0.000000e+00 0.000000e+00 8.276342e-02 6.275691e-01
[128,] 1.663594e-03 0.000000e+00 0.000000e+00 7.980786e-02 6.241557e-01
[129,] 1.895411e-03 0.000000e+00 0.000000e+00 7.692353e-02 6.206194e-01
[130,] 2.147821e-03 0.000000e+00 0.000000e+00 7.410953e-02 6.169629e-01
[131,] 2.421700e-03 0.000000e+00 0.000000e+00 7.136502e-02 6.131888e-01
[132,] 2.717925e-03 0.000000e+00 0.000000e+00 6.868912e-02 6.092996e-01
[133,] 3.037372e-03 0.000000e+00 0.000000e+00 6.608095e-02 6.052981e-01
[134,] 3.380917e-03 0.000000e+00 0.000000e+00 6.353966e-02 6.011867e-01
[135,] 3.749436e-03 0.000000e+00 0.000000e+00 6.106438e-02 5.969682e-01
[136,] 4.143806e-03 0.000000e+00 0.000000e+00 5.865422e-02 5.926451e-01
[137,] 4.564903e-03 0.000000e+00 0.000000e+00 5.630834e-02 5.882201e-01
[138,] 5.013603e-03 0.000000e+00 0.000000e+00 5.402585e-02 5.836957e-01
[139,] 5.490783e-03 0.000000e+00 0.000000e+00 5.180589e-02 5.790746e-01
[140,] 5.997319e-03 0.000000e+00 0.000000e+00 4.964760e-02 5.743593e-01
[141,] 6.534087e-03 0.000000e+00 0.000000e+00 4.755010e-02 5.695526e-01
[142,] 7.101964e-03 0.000000e+00 0.000000e+00 4.551252e-02 5.646569e-01
[143,] 7.701826e-03 0.000000e+00 0.000000e+00 4.353400e-02 5.596750e-01
[144,] 8.334548e-03 0.000000e+00 0.000000e+00 4.161366e-02 5.546093e-01
[145,] 9.001008e-03 0.000000e+00 0.000000e+00 3.975065e-02 5.494626e-01
[146,] 9.702082e-03 0.000000e+00 0.000000e+00 3.794408e-02 5.442375e-01
[147,] 1.043865e-02 0.000000e+00 0.000000e+00 3.619310e-02 5.389365e-01
[148,] 1.121158e-02 0.000000e+00 0.000000e+00 3.449683e-02 5.335623e-01
[149,] 1.202175e-02 0.000000e+00 0.000000e+00 3.285441e-02 5.281174e-01
[150,] 1.287004e-02 0.000000e+00 0.000000e+00 3.126496e-02 5.226046e-01
[151,] 1.375733e-02 0.000000e+00 0.000000e+00 2.972762e-02 5.170263e-01
[152,] 1.468449e-02 0.000000e+00 0.000000e+00 2.824152e-02 5.113853e-01
[153,] 1.565239e-02 0.000000e+00 0.000000e+00 2.680580e-02 5.056841e-01
[154,] 1.666192e-02 0.000000e+00 0.000000e+00 2.541957e-02 4.999253e-01
[155,] 1.771395e-02 0.000000e+00 0.000000e+00 2.408198e-02 4.941116e-01
[156,] 1.880936e-02 0.000000e+00 0.000000e+00 2.279216e-02 4.882455e-01
[157,] 1.994902e-02 0.000000e+00 0.000000e+00 2.154923e-02 4.823297e-01
[158,] 2.113381e-02 0.000000e+00 0.000000e+00 2.035233e-02 4.763668e-01
[159,] 2.236460e-02 0.000000e+00 0.000000e+00 1.920059e-02 4.703593e-01
[160,] 2.364228e-02 0.000000e+00 0.000000e+00 1.809315e-02 4.643100e-01
[161,] 2.496772e-02 0.000000e+00 0.000000e+00 1.702913e-02 4.582214e-01
[162,] 2.634179e-02 0.000000e+00 0.000000e+00 1.600766e-02 4.520961e-01
[163,] 2.776537e-02 0.000000e+00 0.000000e+00 1.502788e-02 4.459368e-01
[164,] 2.923933e-02 0.000000e+00 0.000000e+00 1.408892e-02 4.397460e-01
[165,] 3.076456e-02 0.000000e+00 0.000000e+00 1.318991e-02 4.335264e-01
[166,] 3.234194e-02 0.000000e+00 0.000000e+00 1.232997e-02 4.272805e-01
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attr(,"degree")
[1] 3
attr(,"knots")
16.66667% 33.33333% 50% 66.66667% 83.33333%
1.055 2.100 3.145 4.190 5.235
attr(,"Boundary.knots")
[1] 0.000000 6.283185
attr(,"intercept")
[1] FALSE
attr(,"periodic")
[1] TRUE
attr(,"class")
[1] "pbs" "basis"
1 2 3
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[451,] 6.511352e-01 1.101883e-01 3.564919e-04
[452,] 6.525843e-01 1.126216e-01 3.062173e-04
[453,] 6.539673e-01 1.150906e-01 2.609065e-04
[454,] 6.552833e-01 1.175954e-01 2.203016e-04
[455,] 6.565317e-01 1.201362e-01 1.841445e-04
[456,] 6.577115e-01 1.227134e-01 1.521773e-04
[457,] 6.588221e-01 1.253272e-01 1.241419e-04
[458,] 6.598626e-01 1.279778e-01 9.978026e-05
[459,] 6.608324e-01 1.306656e-01 7.883445e-05
[460,] 6.617305e-01 1.333907e-01 6.104644e-05
[461,] 6.625563e-01 1.361535e-01 4.615819e-05
[462,] 6.633090e-01 1.389541e-01 3.391171e-05
[463,] 6.639878e-01 1.417929e-01 2.404897e-05
[464,] 6.645918e-01 1.446701e-01 1.631197e-05
[465,] 6.651205e-01 1.475860e-01 1.044270e-05
[466,] 6.655729e-01 1.505407e-01 6.183138e-06
[467,] 6.659483e-01 1.535347e-01 3.275277e-06
[468,] 6.662459e-01 1.565681e-01 1.461103e-06
[469,] 6.664650e-01 1.596411e-01 4.826053e-07
[470,] 6.666047e-01 1.627542e-01 8.177034e-08
[471,] 6.666644e-01 1.659074e-01 5.863094e-10
[472,] 6.666432e-01 1.691010e-01 1.895910e-08
[473,] 6.665417e-01 1.723346e-01 2.348782e-07
[474,] 6.663604e-01 1.756073e-01 9.051831e-07
[475,] 6.661002e-01 1.789184e-01 2.287886e-06
[476,] 6.657619e-01 1.822672e-01 4.641000e-06
[477,] 6.653462e-01 1.856527e-01 8.222536e-06
[478,] 6.648539e-01 1.890744e-01 1.329051e-05
[479,] 6.642858e-01 1.925313e-01 2.010293e-05
[480,] 6.636426e-01 1.960228e-01 2.891780e-05
[481,] 6.629251e-01 1.995480e-01 3.999316e-05
[482,] 6.621342e-01 2.031062e-01 5.358699e-05
[483,] 6.612705e-01 2.066966e-01 6.995732e-05
[484,] 6.603349e-01 2.103184e-01 8.936216e-05
[485,] 6.593281e-01 2.139708e-01 1.120595e-04
[486,] 6.582508e-01 2.176532e-01 1.383074e-04
[487,] 6.571040e-01 2.213646e-01 1.683639e-04
[488,] 6.558883e-01 2.251044e-01 2.024868e-04
[489,] 6.546045e-01 2.288717e-01 2.409344e-04
[490,] 6.532534e-01 2.326658e-01 2.839646e-04
[491,] 6.518358e-01 2.364859e-01 3.318353e-04
[492,] 6.503524e-01 2.403313e-01 3.848047e-04
[493,] 6.488040e-01 2.442011e-01 4.431306e-04
[494,] 6.471914e-01 2.480945e-01 5.070712e-04
[495,] 6.455154e-01 2.520109e-01 5.768845e-04
[496,] 6.437767e-01 2.559495e-01 6.528284e-04
[497,] 6.419761e-01 2.599094e-01 7.351609e-04
[498,] 6.401144e-01 2.638898e-01 8.241402e-04
[499,] 6.381923e-01 2.678901e-01 9.200241e-04
[500,] 6.362107e-01 2.719095e-01 1.023071e-03
[501,] 6.341703e-01 2.759471e-01 1.133538e-03
[502,] 6.320718e-01 2.800022e-01 1.251684e-03
[503,] 6.299162e-01 2.840740e-01 1.377767e-03
[504,] 6.277040e-01 2.881618e-01 1.512045e-03
[505,] 6.254362e-01 2.922647e-01 1.654775e-03
[506,] 6.231134e-01 2.963820e-01 1.806216e-03
[507,] 6.207364e-01 3.005130e-01 1.966626e-03
[508,] 6.183061e-01 3.046568e-01 2.136263e-03
[509,] 6.158232e-01 3.088126e-01 2.315384e-03
[510,] 6.132885e-01 3.129798e-01 2.504249e-03
[511,] 6.107027e-01 3.171575e-01 2.703114e-03
[512,] 6.080666e-01 3.213449e-01 2.912238e-03
[513,] 6.053811e-01 3.255414e-01 3.131879e-03
[514,] 6.026468e-01 3.297460e-01 3.362295e-03
[515,] 5.998645e-01 3.339581e-01 3.603743e-03
[516,] 5.970350e-01 3.381768e-01 3.856483e-03
[517,] 5.941592e-01 3.424014e-01 4.120772e-03
[518,] 5.912377e-01 3.466311e-01 4.396867e-03
[519,] 5.882713e-01 3.508651e-01 4.685027e-03
[520,] 5.852609e-01 3.551027e-01 4.985511e-03
[521,] 5.822071e-01 3.593431e-01 5.298575e-03
[522,] 5.791108e-01 3.635855e-01 5.624478e-03
[523,] 5.759727e-01 3.678291e-01 5.963479e-03
[524,] 5.727937e-01 3.720732e-01 6.315834e-03
[525,] 5.695744e-01 3.763170e-01 6.681802e-03
[526,] 5.663157e-01 3.805597e-01 7.061641e-03
[527,] 5.630183e-01 3.848005e-01 7.455610e-03
[528,] 5.596830e-01 3.890387e-01 7.863965e-03
[529,] 5.563106e-01 3.932736e-01 8.286965e-03
[530,] 5.529018e-01 3.975042e-01 8.724869e-03
[531,] 5.494575e-01 4.017298e-01 9.177934e-03
[532,] 5.459784e-01 4.059498e-01 9.646417e-03
[533,] 5.424652e-01 4.101632e-01 1.013058e-02
[534,] 5.389188e-01 4.143694e-01 1.063067e-02
[535,] 5.353399e-01 4.185675e-01 1.114696e-02
[536,] 5.317293e-01 4.227568e-01 1.167970e-02
[537,] 5.280878e-01 4.269364e-01 1.222915e-02
[538,] 5.244162e-01 4.311058e-01 1.279557e-02
[539,] 5.207151e-01 4.352639e-01 1.337921e-02
[540,] 5.169854e-01 4.394102e-01 1.398034e-02
[541,] 5.132279e-01 4.435437e-01 1.459920e-02
[542,] 5.094434e-01 4.476638e-01 1.523607e-02
[543,] 5.056325e-01 4.517696e-01 1.589119e-02
[544,] 5.017962e-01 4.558604e-01 1.656483e-02
[545,] 4.979351e-01 4.599355e-01 1.725724e-02
[546,] 4.940501e-01 4.639939e-01 1.796868e-02
[547,] 4.901418e-01 4.680350e-01 1.869941e-02
[548,] 4.862112e-01 4.720581e-01 1.944969e-02
[549,] 4.822589e-01 4.760622e-01 2.021978e-02
[550,] 4.782857e-01 4.800466e-01 2.100992e-02
[551,] 4.742924e-01 4.840107e-01 2.182039e-02
[552,] 4.702799e-01 4.879535e-01 2.265144e-02
[553,] 4.662488e-01 4.918743e-01 2.350333e-02
[554,] 4.621999e-01 4.957724e-01 2.437631e-02
[555,] 4.581340e-01 4.996469e-01 2.527064e-02
[556,] 4.540519e-01 5.034972e-01 2.618659e-02
[557,] 4.499543e-01 5.073223e-01 2.712440e-02
[558,] 4.458421e-01 5.111216e-01 2.808434e-02
[559,] 4.417159e-01 5.148943e-01 2.906667e-02
[560,] 4.375767e-01 5.186396e-01 3.007164e-02
[561,] 4.334251e-01 5.223567e-01 3.109952e-02
[562,] 4.292619e-01 5.260448e-01 3.215055e-02
[563,] 4.250879e-01 5.297032e-01 3.322500e-02
[564,] 4.209038e-01 5.333311e-01 3.432312e-02
[565,] 4.167106e-01 5.369278e-01 3.544518e-02
[566,] 4.125088e-01 5.404924e-01 3.659143e-02
[567,] 4.082993e-01 5.440242e-01 3.776213e-02
[568,] 4.040829e-01 5.475224e-01 3.895754e-02
[569,] 3.998603e-01 5.509862e-01 4.017791e-02
[570,] 3.956324e-01 5.544149e-01 4.142351e-02
[571,] 3.913998e-01 5.578077e-01 4.269459e-02
[572,] 3.871634e-01 5.611637e-01 4.399141e-02
[573,] 3.829239e-01 5.644823e-01 4.531422e-02
[574,] 3.786822e-01 5.677627e-01 4.666329e-02
[575,] 3.744389e-01 5.710041e-01 4.803888e-02
[576,] 3.701949e-01 5.742057e-01 4.944123e-02
[577,] 3.659509e-01 5.773667e-01 5.087062e-02
[578,] 3.617077e-01 5.804863e-01 5.232729e-02
[579,] 3.574661e-01 5.835639e-01 5.381151e-02
[580,] 3.532269e-01 5.865986e-01 5.532353e-02
[581,] 3.489908e-01 5.895896e-01 5.686362e-02
[582,] 3.447585e-01 5.925362e-01 5.843202e-02
[583,] 3.405310e-01 5.954376e-01 6.002900e-02
[584,] 3.363089e-01 5.982930e-01 6.165482e-02
[585,] 3.320931e-01 6.011017e-01 6.330973e-02
[586,] 3.278842e-01 6.038628e-01 6.499400e-02
[587,] 3.236831e-01 6.065757e-01 6.670787e-02
[588,] 3.194905e-01 6.092394e-01 6.845161e-02
[589,] 3.153073e-01 6.118534e-01 7.022548e-02
[590,] 3.111342e-01 6.144166e-01 7.202973e-02
[591,] 3.069719e-01 6.169285e-01 7.386462e-02
[592,] 3.028213e-01 6.193883e-01 7.573041e-02
[593,] 2.986831e-01 6.217950e-01 7.762736e-02
[594,] 2.945581e-01 6.241481e-01 7.955573e-02
[595,] 2.904470e-01 6.264467e-01 8.151577e-02
[596,] 2.863507e-01 6.286900e-01 8.350774e-02
[597,] 2.822699e-01 6.308772e-01 8.553190e-02
[598,] 2.782054e-01 6.330077e-01 8.758851e-02
[599,] 2.741580e-01 6.350805e-01 8.967782e-02
[600,] 2.701283e-01 6.370950e-01 9.180010e-02
[601,] 2.661173e-01 6.390503e-01 9.395561e-02
[602,] 2.621257e-01 6.409458e-01 9.614459e-02
[603,] 2.581542e-01 6.427805e-01 9.836731e-02
[604,] 2.542037e-01 6.445538e-01 1.006240e-01
[605,] 2.502749e-01 6.462649e-01 1.029150e-01
[606,] 2.463685e-01 6.479129e-01 1.052405e-01
[607,] 2.424854e-01 6.494972e-01 1.076007e-01
[608,] 2.386264e-01 6.510169e-01 1.099960e-01
[609,] 2.347921e-01 6.524713e-01 1.124266e-01
[610,] 2.309834e-01 6.538596e-01 1.148927e-01
[611,] 2.272010e-01 6.551810e-01 1.173946e-01
[612,] 2.234458e-01 6.564348e-01 1.199326e-01
[613,] 2.197185e-01 6.576201e-01 1.225068e-01
[614,] 2.160198e-01 6.587362e-01 1.251177e-01
[615,] 2.123506e-01 6.597824e-01 1.277654e-01
[616,] 2.087116e-01 6.607578e-01 1.304502e-01
[617,] 2.051036e-01 6.616617e-01 1.331723e-01
[618,] 2.015274e-01 6.624932e-01 1.359321e-01
[619,] 1.979837e-01 6.632518e-01 1.387297e-01
[620,] 1.944734e-01 6.639364e-01 1.415655e-01
[621,] 1.909971e-01 6.645465e-01 1.444396e-01
[622,] 1.875557e-01 6.650812e-01 1.473523e-01
[623,] 1.841499e-01 6.655397e-01 1.503040e-01
[624,] 1.807805e-01 6.659212e-01 1.532948e-01
[625,] 1.774483e-01 6.662251e-01 1.563250e-01
[626,] 1.741541e-01 6.664504e-01 1.593950e-01
[627,] 1.708986e-01 6.665965e-01 1.625048e-01
[628,] 1.676826e-01 6.666626e-01 1.656548e-01
attr(,"degree")
[1] 3
attr(,"knots")
[1] 1.570796 3.141593 4.712389
attr(,"Boundary.knots")
[1] 0.000000 6.283185
attr(,"intercept")
[1] FALSE
attr(,"periodic")
[1] TRUE
attr(,"class")
[1] "pbs" "basis"
Call:
lm(formula = y ~ pbs(x, df = 10, Boundary.knots = c(0, 2 * pi)))
Residuals:
Min 1Q Median 3Q Max
-0.38317 -0.06361 0.00231 0.06834 0.30621
Coefficients:
Estimate Std. Error t value
(Intercept) 0.15559 0.02989 5.204
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))1 -0.28780 0.05388 -5.342
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))2 0.48547 0.03441 14.108
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))3 1.15031 0.04594 25.039
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))4 0.63945 0.04053 15.779
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))5 -1.16418 0.04286 -27.163
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))6 -2.37922 0.04286 -55.510
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))7 -1.94186 0.04047 -47.977
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))8 -0.18792 0.04591 -4.093
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))9 1.11324 0.03442 32.342
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))10 0.91646 0.05391 17.001
Pr(>|t|)
(Intercept) 2.65e-07 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))1 1.30e-07 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))2 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))3 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))4 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))5 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))6 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))7 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))8 4.82e-05 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))9 < 2e-16 ***
pbs(x, df = 10, Boundary.knots = c(0, 2 * pi))10 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1017 on 617 degrees of freedom
Multiple R-squared: 0.9899, Adjusted R-squared: 0.9898
F-statistic: 6075 on 10 and 617 DF, p-value: < 2.2e-16
Call:
lm(formula = y ~ pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0,
2 * pi)))
Residuals:
Min 1Q Median 3Q Max
-0.39626 -0.06549 0.00058 0.06692 0.31204
Coefficients:
Estimate
(Intercept) 3.04079
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 -7.60160
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.01498
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 -4.55223
Std. Error
(Intercept) 0.01999
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 0.03713
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.01751
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 0.03713
t value
(Intercept) 152.151
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 -204.731
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.855
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 -122.604
Pr(>|t|)
(Intercept) <2e-16
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 <2e-16
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2 0.393
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 <2e-16
(Intercept) ***
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))1 ***
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))2
pbs(x, knots = c(pi/2, pi, pi * 3/2), Boundary.knots = c(0, 2 * pi))3 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1019 on 624 degrees of freedom
Multiple R-squared: 0.9898, Adjusted R-squared: 0.9898
F-statistic: 2.019e+04 on 3 and 624 DF, p-value: < 2.2e-16
Call:
lm(formula = y ~ pbs(x, df = 5, Boundary.knots = c(0, 2 * pi)) +
pbs(z, df = 5, Boundary.knots = c(0, pi)))
Residuals:
Min 1Q Median 3Q Max
-0.35669 -0.06836 0.00019 0.06792 0.39867
Coefficients:
Estimate Std. Error t value
(Intercept) 0.440686 0.006205 71.024
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))1 -1.027707 0.008034 -127.925
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))2 -2.069734 0.004718 -438.710
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))3 -2.097653 0.007288 -287.819
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))4 -1.042592 0.004728 -220.530
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))5 -0.004846 0.008023 -0.604
pbs(z, df = 5, Boundary.knots = c(0, pi))1 -0.598482 0.008035 -74.489
pbs(z, df = 5, Boundary.knots = c(0, pi))2 -0.002906 0.004734 -0.614
pbs(z, df = 5, Boundary.knots = c(0, pi))3 1.200186 0.007297 164.474
pbs(z, df = 5, Boundary.knots = c(0, pi))4 1.803385 0.004705 383.305
pbs(z, df = 5, Boundary.knots = c(0, pi))5 1.199290 0.008052 148.939
Pr(>|t|)
(Intercept) <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))1 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))2 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))3 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))4 <2e-16 ***
pbs(x, df = 5, Boundary.knots = c(0, 2 * pi))5 0.546
pbs(z, df = 5, Boundary.knots = c(0, pi))1 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))2 0.539
pbs(z, df = 5, Boundary.knots = c(0, pi))3 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))4 <2e-16 ***
pbs(z, df = 5, Boundary.knots = c(0, pi))5 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.101 on 16474 degrees of freedom
Multiple R-squared: 0.9899, Adjusted R-squared: 0.9899
F-statistic: 1.616e+05 on 10 and 16474 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.0053448 -0.0012041 -0.0004680 -0.0005027 0.0008607 0.0031570
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