R: Projection Pursuit Regression

ppr {modreg} | R Documentation

## Projection Pursuit Regression

### Description

Fit a projection pursuit regression model.

### Usage

    ppr(formula, data = sys.parent(), weights,
        subset, na.action, contrasts = NULL,
        ww = rep(1,q), nterms, max.terms = nterms, optlevel = 2,
        sm.method = c("supsmu", "spline", "gcvspline"),
        bass = 0, span = 0, df = 5, gcvpen = 1)

    ppr(x, y, weights = rep(1,n),
        ww = rep(1,q), nterms, max.terms = nterms, optlevel = 2,
        sm.method = c("supsmu", "spline", "gcvspline"),
        bass = 0, span = 0, df = 5, gcvpen = 1)

### Arguments

`formula` | a formula specifying one or more response variables and the explanatory variables.
`x` | matrix of explanatory variables. Rows represent observations, and columns represent variables. Missing values are not accepted.
`y` | matrix of response variables. Rows represent observations, and columns represent variables. Missing values are not accepted.
`nterms` | number of terms to include in the final model.
`data` | data frame from which variables specified in `formula` are preferentially to be taken.
`weights` | a vector of weights `w_i` for each case.
`ww` | a vector of weights for each response, so the fit criterion is the sum over case `i` and responses `j` of `w_i ww_j (y_ij - fit_ij)^2` divided by the sum of `w_i`.
`subset` | an index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)
`na.action` | a function to specify the action to be taken if `NA`s are found. The default action is for the procedure to fail. An alternative is `na.omit`, which leads to rejection of cases with missing values on any required variable. (NOTE: If given, this argument must be named.)
`contrasts` | the contrasts to be used when any factor explanatory variables are coded.
`max.terms` | maximum number of terms to choose from when building the model.
`optlevel` | integer from 0 to 3 which determines the thoroughness of an optimization routine in the SMART program. See the Details section.
`sm.method` | the method used for smoothing the ridge functions. The default is to use Friedman's super smoother `supsmu`. The alternatives are to use the smoothing spline code underlying `smooth.spline`, either with a specified (equivalent) degrees of freedom for each ridge functions, or to allow the smoothness to be chosen by GCV.
`bass` | super smoother bass tone control used with automatic span selection (see `supsmu`); the range of values is 0 to 10, with larger values resulting in increased smoothing.
`span` | super smoother span control (see `supsmu`). The default, `0`, results in automatic span selection by local cross validation. `span` can also take a value in `(0, 1]`.
`df` | if `sm.method` is `"spline"` specifies the smoothness of each ridge term via the requested equivalent degrees of freedom.
`gcvpen` | if `sm.method` is `"gcvspline"` this is the penalty used in the GCV selection for each degree of freedom used.

### Details

The basic method is given by Friedman (1984), and is essentially the same code used by S-PLUS's `ppreg`. This code is extremely sensitive to the compiler used.

The algorithm first adds up to `max.terms` ridge terms one at a time; it will use less if it is unable to find a term to add that makes sufficient difference. It then removes the least ``important'' term at each step until `nterm` terms are left.

The levels of optimization (argument `optlevel`) differ in how thoroughly the models are refitted during this process. At level 0 the existing ridge terms are not refitted. At level 1 the projection directions are not refitted, but the ridge functions and the regression coefficients are. Levels 2 and 3 refit all the terms and are equivalent for one response; level 3 is more careful to re-balance the contributions from each regressor at each step and so is a little less likely to converge to a saddle point of the sum of squares criterion.

### Value

A list with the following components, many of which are for use by the method functions.

`call` | the matched call
`p` | the number of explanatory variables (after any coding)
`q` | the number of response variables
`mu` | the argument `nterms`
`ml` | the argument `max.terms`
`gof` | the overall residual (weighted) sum of squares for the selected model
`gofn` | the overall residual (weighted) sum of squares against the number of terms, up to `max.terms`. Will be invalid (and zero) for less than `nterms`.
`df` | the argument `df`
`edf` | if `sm.method` is `"spline"` or `"gcvspline"` the equivalent number of degrees of freedom for each ridge term used.
`xnames` | the names of the explanatory variables
`ynames` | the names of the response variables
`alpha` | a matrix of the projection directions, with a column for each ridge term
`beta` | a matrix of the coefficients applied for each response to the ridge terms: the rows are the responses and the columns the ridge terms
`yb` | the weighted means of each response
`ys` | the overall scale factor used: internally the responses are divided by `ys` to have unit total weighted sum of squares.
`fitted.values` | the fitted values, as a matrix if `q > 1`.
`residuals` | the residuals, as a matrix if `q > 1`.
`smod` | internal work array, which includes the ridge functions evaluated at the training set points.

### References

Friedman, J. H. and Stuetzle, W. (1981) Projection pursuit regression. Journal of the American Statistical Association, 76, 817–823.

Friedman, J. H. (1984) SMART User's Guide. Laboratory for Computational Statistics, Stanford University Technical Report No. 1.

Venables, W. N. & Ripley, B. D. (1999) Modern Applied Statistics with S-PLUS. Springer.

### See Also

`plot.ppr`, `supsmu`, `smooth.spline`

### Examples

    # Note: your numerical values may differ
    data(rock)
    attach(rock)
    area1 <- area/10000; peri1 <- peri/10000
    rock.ppr <- ppr(log(perm) ~ area1 + peri1 + shape,
                    data = rock, nterms = 2, max.terms = 5)
    rock.ppr
    # Call:
    # ppr.formula(formula = log(perm) ~ area1 + peri1 + shape, data = rock,
    #     nterms = 2, max.terms = 5)
    #
    # Goodness of fit:
    #  2 terms  3 terms  4 terms  5 terms
    # 8.737806 5.289517 4.745799 4.490378

    summary(rock.ppr)
    # .....  (same as above)
    # .....
    #
    # Projection direction vectors:
    #       term 1      term 2
    # area1  0.34357179  0.37071027
    # peri1 -0.93781471 -0.61923542
    # shape  0.04961846  0.69218595
    #
    # Coefficients of ridge terms:
    #    term 1    term 2
    # 1.6079271 0.5460971

    par(mfrow=c(3,2))# maybe: , pty="s")
    plot(rock.ppr, main="ppr(log(perm)~ ., nterms=2, max.terms=5)")
    plot(update(rock.ppr, bass=5), main = "update(..., bass = 5)")
    plot(update(rock.ppr, sm.method="gcv", gcvpen=2),
         main = "update(..., sm.method=\"gcv\", gcvpen=2)")

    detach()

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