894aebd6dfa90dd49e474e0177baed7e9141bb17
[morpheus.git] / pkg / R / optimParams.R
1 #' Wrapper function for OptimParams class
2 #'
3 #' @param K Number of populations.
4 #' @param link The link type, 'logit' or 'probit'.
5 #' @param X Data matrix of covariables
6 #' @param Y Output as a binary vector
7 #'
8 #' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
9 #' outputs the list of optimized parameters
10 #' \itemize{
11 #' \item p: proportions, size K
12 #' \item β: regression matrix, size dxK
13 #' \item b: intercepts, size K
14 #' }
15 #' θ0 is a vector containing respectively the K-1 first elements of p, then β by
16 #' columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
17 #'
18 #' @seealso \code{multiRun} to estimate statistics based on β, and
19 #' \code{generateSampleIO} for I/O random generation.
20 #'
21 #' @examples
22 #' # Optimize parameters from estimated μ
23 #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
24 #' μ = computeMu(io$X, io$Y, list(K=2))
25 #' o <- optimParams(io$X, io$Y, 2, "logit")
26 #' θ0 <- list(p=1/2, β=μ, b=c(0,0))
27 #' par0 <- o$run(θ0)
28 #' # Compare with another starting point
29 #' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
30 #' par1 <- o$run(θ1)
31 #' o$f( o$linArgs(par0) )
32 #' o$f( o$linArgs(par1) )
33 #' @export
34 optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL)
35 {
36 # Check arguments
37 if (!is.matrix(X) || any(is.na(X)))
38 stop("X: numeric matrix, no NAs")
39 if (!is.numeric(Y) || any(is.na(Y)) || any(Y!=0 & Y!=1))
40 stop("Y: binary vector with 0 and 1 only")
41 link <- match.arg(link)
42 if (!is.numeric(K) || K!=floor(K) || K < 2)
43 stop("K: integer >= 2")
44
45 if (is.null(M))
46 {
47 # Precompute empirical moments
48 Mtmp <- computeMoments(X, Y)
49 M1 <- as.double(Mtmp[[1]])
50 M2 <- as.double(Mtmp[[2]])
51 M3 <- as.double(Mtmp[[3]])
52 M <- c(M1, M2, M3)
53 }
54 else
55 M <- c(M[[1]], M[[2]], M[[3]])
56
57 # Build and return optimization algorithm object
58 methods::new("OptimParams", "li"=link, "X"=X,
59 "Y"=as.integer(Y), "K"=as.integer(K), "Mhat"=as.double(M))
60 }
61
62 #' Encapsulated optimization for p (proportions), β and b (regression parameters)
63 #'
64 #' Optimize the parameters of a mixture of logistic regressions model, possibly using
65 #' \code{mu <- computeMu(...)} as a partial starting point.
66 #'
67 #' @field li Link function, 'logit' or 'probit'
68 #' @field X Data matrix of covariables
69 #' @field Y Output as a binary vector
70 #' @field K Number of populations
71 #' @field d Number of dimensions
72 #' @field W Weights matrix (iteratively refined)
73 #'
74 setRefClass(
75 Class = "OptimParams",
76
77 fields = list(
78 # Inputs
79 li = "character", #link function
80 X = "matrix",
81 Y = "numeric",
82 Mhat = "numeric", #vector of empirical moments
83 # Dimensions
84 K = "integer",
85 n = "integer",
86 d = "integer",
87 # Weights matrix (generalized least square)
88 W = "matrix"
89 ),
90
91 methods = list(
92 initialize = function(...)
93 {
94 "Check args and initialize K, d, W"
95
96 callSuper(...)
97 if (!hasArg("X") || !hasArg("Y") || !hasArg("K")
98 || !hasArg("li") || !hasArg("Mhat"))
99 {
100 stop("Missing arguments")
101 }
102
103 n <<- nrow(X)
104 d <<- ncol(X)
105 W <<- diag(d+d^2+d^3) #initialize at W = Identity
106 },
107
108 expArgs = function(v)
109 {
110 "Expand individual arguments from vector v into a list"
111
112 list(
113 # p: dimension K-1, need to be completed
114 "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
115 "β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
116 "b" = v[(K+d*K):(K+(d+1)*K-1)])
117 },
118
119 linArgs = function(L)
120 {
121 "Linearize vectors+matrices from list L into a vector"
122
123 # β linearized row by row, to match derivatives order
124 c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
125 },
126
127 computeW = function(θ)
128 {
129 require(MASS)
130 dd <- d + d^2 + d^3
131 M <- Moments(θ)
132 Omega <- matrix( .C("Compute_Omega",
133 X=as.double(X), Y=as.integer(Y), M=as.double(M),
134 pn=as.integer(n), pd=as.integer(d),
135 W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
136 MASS::ginv(Omega)
137 },
138
139 Moments = function(θ)
140 {
141 "Vector of moments, of size d+d^2+d^3"
142
143 p <- θ$p
144 β <- θ$β
145 λ <- sqrt(colSums(β^2))
146 b <- θ$b
147
148 # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
149 β2 <- apply(β, 2, function(col) col %o% col)
150 β3 <- apply(β, 2, function(col) col %o% col %o% col)
151
152 c(
153 β %*% (p * .G(li,1,λ,b)),
154 β2 %*% (p * .G(li,2,λ,b)),
155 β3 %*% (p * .G(li,3,λ,b)))
156 },
157
158 f = function(θ)
159 {
160 "Product t(hat_Mi - Mi) W (hat_Mi - Mi) with Mi(theta)"
161
162 L <- expArgs(θ)
163 A <- as.matrix(Mhat - Moments(L))
164 t(A) %*% W %*% A
165 },
166
167 grad_f = function(θ)
168 {
169 "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
170
171 L <- expArgs(θ)
172 -2 * t(grad_M(L)) %*% W %*% as.matrix(Mhat - Moments(L))
173 },
174
175 grad_M = function(θ)
176 {
177 "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
178
179 p <- θ$p
180 β <- θ$β
181 λ <- sqrt(colSums(β^2))
182 μ <- sweep(β, 2, λ, '/')
183 b <- θ$b
184
185 res <- matrix(nrow=nrow(W), ncol=0)
186
187 # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
188 β2 <- apply(β, 2, function(col) col %o% col)
189 β3 <- apply(β, 2, function(col) col %o% col %o% col)
190
191 # Some precomputations
192 G1 = .G(li,1,λ,b)
193 G2 = .G(li,2,λ,b)
194 G3 = .G(li,3,λ,b)
195 G4 = .G(li,4,λ,b)
196 G5 = .G(li,5,λ,b)
197
198 # Gradient on p: K-1 columns, dim rows
199 km1 = 1:(K-1)
200 res <- cbind(res, rbind(
201 sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
202 sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
203 sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ))
204
205 for (i in 1:d)
206 {
207 # i determines the derivated matrix dβ[2,3]
208
209 dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
210 dβ_right <- matrix(0, nrow=d, ncol=K)
211 block <- i
212 dβ_right[block,] <- dβ_right[block,] + 1
213 dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
214
215 dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
216 dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
217 sweep(dβ_right, 2, β[j,], '*')
218 }) )
219 block <- ((i-1)*d+1):(i*d)
220 dβ2_right[block,] <- dβ2_right[block,] + β
221 dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
222
223 dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
224 dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
225 sweep(dβ2_right, 2, β[j,], '*')
226 }) )
227 block <- ((i-1)*d*d+1):(i*d*d)
228 dβ3_right[block,] <- dβ3_right[block,] + β2
229 dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
230
231 res <- cbind(res, rbind(dβ, dβ2, dβ3))
232 }
233
234 # Gradient on b
235 res <- cbind(res, rbind(
236 sweep(β, 2, p * G2, '*'),
237 sweep(β2, 2, p * G3, '*'),
238 sweep(β3, 2, p * G4, '*') ))
239
240 res
241 },
242
243 run = function(θ0)
244 {
245 "Run optimization from θ0 with solver..."
246
247 if (!is.list(θ0))
248 stop("θ0: list")
249 if (is.null(θ0$β))
250 stop("At least θ0$β must be provided")
251 if (!is.matrix(θ0$β) || any(is.na(θ0$β))
252 || nrow(θ0$β) != d || ncol(θ0$β) != K)
253 {
254 stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
255 }
256 if (is.null(θ0$p))
257 θ0$p = rep(1/K, K-1)
258 else if (!is.numeric(θ0$p) || length(θ0$p) != K-1
259 || any(is.na(θ0$p)) || sum(θ0$p) > 1)
260 {
261 stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
262 }
263 if (is.null(θ0$b))
264 θ0$b = rep(0, K)
265 else if (!is.numeric(θ0$b) || length(θ0$b) != K || any(is.na(θ0$b)))
266 stop("θ0$b: length K, no NA")
267 # TODO: stopping condition? N iterations? Delta <= epsilon ?
268 loopMax <- 2
269 for (loop in 1:loopMax)
270 {
271 op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
272 ui=cbind(
273 rbind( rep(-1,K-1), diag(K-1) ),
274 matrix(0, nrow=K, ncol=(d+1)*K) ),
275 ci=c(-1,rep(0,K-1)) )
276 if (loop < loopMax) #avoid computing an extra W
277 W <<- computeW(expArgs(op_res$par))
278 #print(op_res$value) #debug
279 #print(expArgs(op_res$par)) #debug
280 }
281
282 expArgs(op_res$par)
283 }
284 )
285 )
286
287 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
288 # = E[g^{(order)}(z)] with z~N(b,diag(λ))
289 # by numerically evaluating the integral.
290 #
291 # @param link Link, 'logit' or 'probit'
292 # @param order Order of derivative
293 # @param λ Norm of columns of β
294 # @param b Intercept
295 #
296 .G <- function(link, order, λ, b)
297 {
298 # NOTE: weird "integral divergent" error on inputs:
299 # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
300 # Switch to pracma package for that (but it seems slow...)
301 sapply( seq_along(λ), function(k) {
302 res <- NULL
303 tryCatch({
304 # Fast code, may fail:
305 res <- stats::integrate(
306 function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
307 lower=-Inf, upper=Inf )$value
308 }, error = function(e) {
309 # Robust slow code, no fails observed:
310 sink("/dev/null") #pracma package has some useless printed outputs...
311 res <- pracma::integral(
312 function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
313 xmin=-Inf, xmax=Inf, method="Kronrod")
314 sink()
315 })
316 res
317 })
318 }
319
320 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
321 #
322 .deriv <- list(
323 "probit"=list(
324 # 'probit' derivatives list;
325 # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
326 function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
327 function(x) exp(-x^2/2)/(sqrt(2*pi)) * -x, #g''
328 function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1), #g^(3)
329 function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x), #g^(4)
330 function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
331 ),
332 "logit"=list(
333 # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
334 # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
335 function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
336 function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
337 function(x) {e=exp(x); .zin(e*( e^2 - 4*e + 1) /(e+1)^4)}, #g^(3)
338 function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
339 function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)} #g^(5)
340 )
341 )
342
343 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
344 #
345 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
346 #
347 .zin <- function(x)
348 {
349 x[is.nan(x)] <- 0.
350 x
351 }