# Active Learning Cohn for Sequential Design Acts on a `gp`, `dgp2`, or `dgp3` object. Current version requires squared exponential covariance (`cov = "exp2"`). Calculates ALC over the input locations `x_new` using specified reference grid. If no reference grid is specified, `x_new` is used as the reference. Optionally utilizes SNOW parallelization. User should select the point with the highest ALC to add to the design. ```r ALC(object, x_new, ref, cores) ## S3 method for class 'gp' ALC(object, x_new = NULL, ref = NULL, cores = 1) ## S3 method for class 'dgp2' ALC(object, x_new = NULL, ref = NULL, cores = 1) ## S3 method for class 'dgp3' ALC(object, x_new = NULL, ref = NULL, cores = 1) ``` ## Arguments - `object`: object of class `gp`, `dgp2`, or `dgp3` - `x_new`: matrix of possible input locations, if object has been run through `predict` the previously stored `x_new` is used - `ref`: optional reference grid for ALC approximation, if `ref = NULL` then `x_new` is used - `cores`: number of cores to utilize for SNOW parallelization ## Returns list with elements: * `value`: vector of ALC values, indices correspond to `x_new` * `time`: computation time in seconds ## Details Not yet implemented for Vecchia-approximated fits or Matern kernels. All iterations in the object are used in the calculation, so samples should be burned-in. Thinning the samples using `trim` will speed up computation. This function may be used in two ways: * Option 1: called on an object with only MCMC iterations, in which case `x_new` must be specified * Option 2: called on an object that has been predicted over, in which case the `x_new` from `predict` is used In Option 2, it is recommended to set `store_latent = TRUE` for `dgp2` and `dgp3` objects so latent mappings do not have to be re-calculated. Through `predict`, the user may specify a mean mapping (`mean_map = TRUE`) or a full sample from the MVN distribution over `w_new` (`mean_map = FALSE`). When the object has not yet been predicted over (Option 1), the mean mapping is used. SNOW parallelization reduces computation time but requires more memory storage. C code derived from the "laGP" package (Robert B Gramacy and Furong Sun). ## Examples ```r # Additional examples including real-world computer experiments are available at: # https://bitbucket.org/gramacylab/deepgp-ex/ # -------------------------------------------------------- # Example 1: toy step function, runs in less than 5 seconds # -------------------------------------------------------- f <- function(x) { if (x <= 0.4) return(-1) if (x >= 0.6) return(1) if (x > 0.4 & x < 0.6) return(10*(x-0.5)) } x <- seq(0.05, 0.95, length = 7) y <- sapply(x, f) x_new <- seq(0, 1, length = 100) # Fit model and calculate ALC fit <- fit_two_layer(x, y, nmcmc = 100, cov = "exp2") fit <- trim(fit, 50) fit <- predict(fit, x_new, cores = 1, store_latent = TRUE) alc <- ALC(fit) # -------------------------------------------------------- # Example 2: damped sine wave # -------------------------------------------------------- f <- function(x) { exp(-10*x) * (cos(10*pi*x - 1) + sin(10*pi*x - 1)) * 5 - 0.2 } # Training data x <- seq(0, 1, length = 30) y <- f(x) + rnorm(30, 0, 0.05) # Testing data xx <- seq(0, 1, length = 100) yy <- f(xx) plot(xx, yy, type = "l") points(x, y, col = 2) # Conduct MCMC (can replace fit_two_layer with fit_one_layer/fit_three_layer) # nugget estimated fit <- fit_two_layer(x, y, D = 1, nmcmc = 2000, cov = "exp2") plot(fit) fit <- trim(fit, 1000, 2) # Option 1 - calculate ALC from MCMC iterations alc <- ALC(fit, xx) # Option 2 - calculate ALC after predictions fit <- predict(fit, xx, cores = 1, store_latent = TRUE) alc <- ALC(fit) # Visualize fit plot(fit) par(new = TRUE) # overlay ALC plot(xx, alc$value, type = 'l', lty = 2, axes = FALSE, xlab = '', ylab = '') # Select next design point x_new <- xx[which.max(alc$value)] ``` ## References Sauer, A., Gramacy, R.B., & Higdon, D. (2023). Active learning for deep Gaussian process surrogates. *Technometrics, 65,* 4-18. arXiv:2012.08015 Seo, S, M Wallat, T Graepel, and K Obermayer. 2000. Gaussian Process Regression: Active Data Selection and Test Point Rejection. In Mustererkennung 2000, 2734. New York, NY: SpringerVerlag. Gramacy, RB and F Sun. (2016). laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes in R. **Journal of Statistical Software 72** (1), 1-46. doi:10.18637/jss.v072.i01

ALC function