Fit a bridge coefficient path

Fit a bridge-penalized least-squares path using local quadratic approximation.

Usage

bridge(
  X,
  y,
  q = 1,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  lambda,
  eta = 1e-07,
  converge = 10^10
)

Arguments

X: Numeric design matrix with n observation rows and p predictor columns. Supply finite values and predictors with nonzero variation; omit an intercept column. A data frame is converted using model.matrix(~0+., X); restrictions and future predictions must use the resulting predictor order.
y: Numeric response vector of length n, containing finite values.
q: Bridge-penalty exponent, a finite number in (0, 2]. Values 1 and 2 specify the L1 and L2 penalty forms. Estimation uses local quadratic approximation (LQA).
lambda.min: Positive absolute endpoint of the automatically generated penalty grid. The default is 0.001 when n > p, and 0.05 otherwise. This argument is a penalty value, not a ratio. In CV results, the field also named lambda.min instead denotes the selected penalty.
nlambda: Number of automatically generated penalty values, an integer of at least 2. The default is 100; ignored when lambda is supplied.
lambda: Optional nonempty numeric vector of finite, nonnegative penalty values. Its order is retained. When supplied, it determines the fitted path and overrides nlambda and the grid argument lambda.min.
eta: Positive numerical tolerance, with default 1e-7. Used for coefficient elimination and stopping on squared coefficient changes.
converge: Nonnegative initial convergence-distance value, retained for compatibility. The default is 1e10. Iteration stops when the updated squared coefficient change is at most eta, or at the historical cap.

Value

A list of class bridge with components:

betas: Numeric coefficient matrix with p + 1 rows and length(lambda) columns, one column per fitted penalty. The first row is (Intercept); the remaining rows are original-scale slopes in predictor order. Column names identify the fitted penalties.
lambda: Numeric vector of penalties, in fitted column order.

Details

Predictors are centered and divided by their centered root-mean-square (RMS) deviations, using the observation count n as the divisor. The response is centered. The intercept is reconstructed after fitting, and returned coefficients are expressed on the original predictor scale. The generated penalties are logarithmically spaced between an internally computed starting value and the absolute endpoint lambda.min.

Let $Z$ be the $n$ by $p$ centered, RMS-standardized design, $y_c$ the centered response vector of length $n$ , and $b$ the $p$ -dimensional standardized slope vector. For exponent $q \in (0,2]$ and penalty weight $\lambda \ge 0$ , Equation (1) defines the penalized least-squares target: $\min_{b} \left\{\|y_c-Zb\|_2^2+\lambda\sum_{j=1}^{p}|b_j|^q\right\}. \qquad (1)$ In Equation (1), $\|\cdot\|_2^2$ is the squared Euclidean norm, so its first term is the unnormalized residual sum of squares. The index $j$ runs over the predictors; the second term is the bridge penalty, with $\lambda$ controlling its weight relative to residual error.

The fitting routine applies the article's LQA updates, coefficient elimination and stopping rules. The criterion is nonconvex when $q < 1$ , so the iterative result is not a guarantee of a global minimum. The getting-started vignette explains the penalty geometry and the fixed-LQA restriction correction.

References

Yuzbasi, B., Arashi, M. and Akdeniz, F. (2021). Penalized regression via the restricted bridge estimator. Soft Computing, 25, 8401–8416. doi:10.1007/s00500-021-05763-9 .

Author

Bahadir Yuzbasi
Maintainer: Bahadir Yuzbasi b.yzb@hotmail.com

Examples

# Reproducible synthetic demonstration: 100 observations, eight predictors.
set.seed(2019)
beta <- c(3, 1.5, 0, 0, 2, 0, 0, 0)
p <- length(beta)
beta <- matrix(beta, nrow = p, ncol = 1)
n <- 100
X <- matrix(rnorm(n * p), nrow = n, ncol = p)
y <- as.numeric(X %*% beta + rnorm(n))

# Five penalties keep this help example fast; the default is 100.
# L1 and L2 penalty forms on the same observations.
model1 <- bridge(X, y, q = 1, nlambda = 5)
model2 <- bridge(X, y, q = 2, nlambda = 5)
dim(model1$betas)
#> [1] 9 5
# Extract one fitted column and predict five observations.
selected <- model1$lambda[2]
coef(model1, s = selected)
#>             26.8035044402758
#> (Intercept)    -3.121540e-02
#> V1              2.848999e+00
#> V2              1.433841e+00
#> V3              1.282147e-04
#> V4             -1.150407e-05
#> V5              1.879705e+00
#> V6             -1.064341e-04
#> V7              8.497907e-08
#> V8             -1.609602e-05
predict(model1, newx = X[1:5, , drop = FALSE], s = selected)
#> [1] -1.5002196  0.2180873 -1.5417858  2.4731030 -9.8607011