> library(splines)
> ?ns
> x = runif(100,0,1)
> y = 5*sin(2*pi*x) + rnorm(100)
> plot(x,y)
> ns(x,4)
                  1            2          3           4
  [1,] 2.671861e-01 -0.138551081 0.49022434 -0.34804570
  [2,] 9.011021e-02  0.555378122 0.24138900  0.11312267
  [3,] 5.178343e-03  0.196740200 0.36472862  0.43335284
  [4,] 4.043862e-02 -0.123004339 0.42411239 -0.30110805
  [5,] 2.220834e-02  0.371721186 0.31468503  0.29138544
  [6,] 0.000000e+00 -0.129652843 0.44703608  0.68261676
  [7,] 5.537769e-01  0.333022253 0.12368923 -0.07981118
  [8,] 4.606186e-01  0.445180658 0.12385809 -0.06135772
  [9,] 3.271682e-03  0.152874223 0.37629311  0.46756098
 [10,] 4.417745e-01  0.463269866 0.12584357 -0.05779864
 [11,] 1.518089e-01 -0.148422992 0.51221532 -0.36365869
 [12,] 3.189043e-02 -0.116277917 0.40092005 -0.28464213
 [13,] 4.741783e-01 -0.073057396 0.36928163 -0.26217973

###  lines omitted 

 [96,] 3.069546e-04 -0.027854273 0.09604005 -0.06818577
 [97,] 6.550887e-02 -0.135784256 0.46817687 -0.33239261
 [98,] 6.051078e-01  0.038421605 0.23805525 -0.16901263
 [99,] 4.046039e-01 -0.102964369 0.41897989 -0.29746412
[100,] 3.158218e-03 -0.059382867 0.20474896 -0.14536609
attr(,"degree")
[1] 3
attr(,"knots")
      25%       50%       75% 
0.1966724 0.4646389 0.7794378 
attr(,"Boundary.knots")
[1] 0.002084014 0.996848385
attr(,"intercept")
[1] FALSE
attr(,"class")
[1] "ns"    "basis"
> bs(x,df=4)
                  1            2            3            4
  [1,] 5.867781e-01 0.2989735352 4.367694e-02 0.000000e+00
  [2,] 1.584162e-02 0.1718021744 5.752804e-01 2.370758e-01
  [3,] 9.103670e-04 0.0311895583 3.473825e-01 6.205175e-01
  [4,] 5.544673e-01 0.1047704134 6.181467e-03 0.000000e+00
  [5,] 3.904288e-03 0.0765218635 4.787427e-01 4.408311e-01
  [6,] 0.000000e+00 0.0000000000 0.000000e+00 1.000000e+00
  [7,] 1.531179e-01 0.4652404237 3.749727e-01 6.668940e-03
  [8,] 1.064774e-01 0.4182056357 4.531741e-01 2.214282e-02
  [9,] 5.751708e-04 0.0233328785 3.088990e-01 6.671929e-01
 [10,] 9.931178e-02 0.4082537307 4.661521e-01 2.628238e-02
 [11,] 6.135264e-01 0.2192362849 2.331001e-02 0.000000e+00
 [12,] 5.346601e-01 0.0910963792 4.874787e-03 0.000000e+00
 [13,] 4.709852e-01 0.4179157117 9.908703e-02 0.000000e+00
 [14,] 0.000000e+00 0.0000000000 0.000000e+00 0.000000e+00

### lines omitted

 [97,] 5.884165e-01 0.1383396717 1.001372e-02 0.000000e+00
 [98,] 3.373973e-01 0.4858477601 1.764745e-01 0.000000e+00
 [99,] 5.184430e-01 0.3800074068 7.634485e-02 0.000000e+00
[100,] 3.207814e-01 0.0218266623 4.827668e-04 0.000000e+00
attr(,"degree")
[1] 3
attr(,"knots")
      50% 
0.4646389 
attr(,"Boundary.knots")
[1] 0.002084014 0.996848385
attr(,"intercept")
[1] FALSE
attr(,"class")
[1] "bs"    "basis"


## Note that "bs" only has 1 knot, because we specified 4 df
## whereas with 4 df "ns" can have 3 knots because the function
## is linear at the endpoints

> basis = ns(x,df=4)
> newx = runif(5)
> predict(basis,newx)
              1          2         3           4
[1,] 0.02386236  0.3815322 0.3115888  0.28301659
[2,] 0.57319439  0.3022502 0.1269952 -0.08483746
[3,] 0.55184207  0.3358852 0.1234545 -0.07935517
[4,] 0.19476755 -0.1471081 0.5098843 -0.36200371
[5,] 0.28219103  0.5723714 0.1587831 -0.01786905
attr(,"degree")
[1] 3
attr(,"knots")
      25%       50%       75% 
0.1966724 0.4646389 0.7794378 
attr(,"Boundary.knots")
[1] 0.002084014 0.996848385
attr(,"intercept")
[1] FALSE
attr(,"class")
[1] "ns"    "basis"

## regression splines

> junk = lm(y~ns(x,df=4))
> plot(x,junk$fitted, pch="*")
> points(x,y)
> points(x,sin(2*pi*x),col="red")
> points(x,5*sin(2*pi*x),col="red")
> points(x,lm(y~ns(x,df=2))$fitted, col="blue")
> points(x,lm(y~ns(x,df=3))$fitted, col="blue")
> points(x,lm(y~ns(x,df=30))$fitted, col="blue")


## smoothing splines
> junk2 = gam(y~s(x))
> junk2

Family: gaussian 
Link function: identity 

Formula:
y ~ s(x)

Estimated degrees of freedom:
 5.577818   total =  6.577818 

GCV score:  1.045801 
> plot(x,junk2$fitted.value)
> points(x,y,col="blue")

# compare to regression splines

points(x,junk$fitted.value,col="red")