# vrcp: Estimation for Varying Regression Segments and Change Point... In vrcp: Change Point Estimation for Regression with Varying Segments and Heteroscedastic Variances

## Description

Estimation of two segments and a change point in 2-segment regression models with varying variances and varying types of regression segments, with or without a smoothness constraint at the change point.

## Usage

 ```1 2 3``` ```vrcp(dataset, lo, hi, smooth = c("c0", "c1", "u"), segment1 = c("L", "Q", "Log", "Exp", "NLExp"), segment2 = c("L", "Q", "Log", "Exp", "NLExp"), variance = c("Common", "Diff"), spline = c("FALSE", "TRUE"), start) ```

## Arguments

 `dataset` either a data frame, or a matrix, containing 2 columns, where the first column contains covariate values and the second column contains response values. `lo, hi` lower and upper bounds for the x value of the change point, to be set by a user. `smooth` smoothness constraint of the regression function at the change point. Default constraint is "c0," continuous at the change point. "c1" indicates that the first derivative at the change point is continuous, while "u" indicates that there is constraint at the change point. `segment1, segment2` regression model used to compute parameters in segment1(or segment2) with additive Gaussian errors. Currently allowable models are: L: Linear model. y ~ a0 + a1 * x Q: Quadratic model. y ~ a0 + a1 * x + a2 * x^2 Exp: Linearizable Exponential model. y ~ a0 + a1 * exp(x) Log: Linearizable Logarithm model. y ~ a0 + a1 * log(x) NLExp: Nonlinearizable Exponential model. y ~ a0 + a1 * exp(a2 * (x - k)), where k is the change point in x. These 5 models lead to the following 15 allowable combinations of varying types of segments, reflecting reasonable models we have seen from Degredation Science: "L"-"L", "L"-"Q", "L"-"Exp", "L"-"Log", "L"-"NLExp", "Q"-"L", "Q"-"Q", "Q"-"Exp", "Q"-"Log", and "Q"-"NLExp", "Exp"-"L", "Exp"-"Exp", "Log"-"L", "Log"-"Log", and "NLExp"-"L". `variance` variance type of the data set, either "Common" that requires the variances at two segments to be the same, or "Diff" that does NOT require them to be the same. `spline` a "TRUE" or "FALSE" logical argument, where "TRUE" shows a B-spline fit with the knot at the change point, as an extra option only available for segment combinations of "L"-"L" and "Q"-"Q". The default is "FALSE". `start` a named numeric vector or a logic status "FALSE". Default is "start=FALSE", which will compute initial values automatically based on data. "start=a vector" specifies initial values of vector parameters, typically in the case with a nonlinearizable segment as specified below, where a0, a1, a2 are regression parameters for segment1, and b0, b1, b2 are regression parameters for segment2, etc: "L"-"NLExp": 3 initial values for (a0, a1; b2) may be specified. "Q"-"NLExp": 4 initial values for (a0, a1, a2; b2) may be specified. "NLExp"-"L": 3 initial values for (b0, b1; a2) may be specified.

## Value

maxloglik: maximum log-likelihood value.

sigma2: estimated variance(s) for two segments

coe: coefficients for two regression segments, beta = (a0,a1,a2,b0,b1,b2). No a2, b2 output for linear/linearizable segment.

changepoint: change point in x value.

## References

Stephen J. Ganocy and Jiayang Sun (2015), "Heteroscedastic Change Point Analysis and Application to Footprint Data", J of Data Science, v.13.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534``` ```library(ggplot2) # Test the vrcp() using simulated data sets # Example 1: L-L model with "c0", continuity at change point and common variance # Simulate the data x1<-seq(0,2,by=0.05) x2<-seq(2.05,5,by=0.05) # The true regression functions yt1 <- 2+0.5*x1 yt2 <- -1+2*x2 # Add noises y1<-yt1+rnorm(length(x1),0,0.25) y2<- yt2+rnorm(length(x2),0,0.25) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a L-L regression with a change point between 1.5 and 2.5 # Fit with vrcp with L-L segments and "c0" constraint ans <- vrcp(z,1.5,2.5,"c0","L","L","Common", spline = "TRUE") # Fit with common variance ans # The fitted L-L regression and spline are superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt), color = c("blue"), size=1) + ggtitle("LL-c0-com model: Estimates vs. true model (in blue)") ans <- vrcp(z,1.5,2.5,"c0","L","L","Diff",spline = "TRUE") # Fit with different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt), color = "blue", size=1) + ggtitle("LL-c0-diff model: Estimates vs. true model (in blue)") # compare ## Not run: # Example 2: L-Log model with "c1" change point and common variance. # Simulate the data x1<-seq(0.05,2.05,by=0.05) x2<-seq(2.1,5.05,by=0.05) # The true regression functions yt1 <- 3+1*x1 yt2 <- 3.61+2*log(x2) # Add noises y1<- yt1+rnorm(length(x1),0,0.5) y2<- yt2+rnorm(length(x2),0,0.5) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)<-c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a L-Log regression with a change point between 1.9 and 2.2 # Fit with vrcp with specification of L-Log segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.9,2.2,"c1","L","Log","Common") ans # The fitted L-Log regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-c1-com model: Estimate (in magenta) vs. true model") # Fit with common variance ans <- vrcp(z,1.9,2.5,"c1","L","Log","Diff") ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-c1-diff model: Estimate (in magenta) vs. true model") # Fit with different variance # both fits look good # Check what would look like with misspecification of smoothness at change point ans <- vrcp(z,1.9,2.2,"c0","L","Log","Common") # Fit with common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-c0-com fit to LLog-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"c0","L","Log","Diff") # Fit with different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-c0-diff fit to LLog-c1-com model: Estimate (in magenta) vs. true model") # both look lack of fit, especially at the change point. # Hence, the correct specification of model is important # Example 3: Log-L - Simulated data set is "c1", smooth. # Simulate the data x1<-seq(2,4,by=0.05) x2<-seq(4,7,by=0.05) # The true regression functions yt1 <- 1.6+0.5*log(x1) yt2 <- 1.89+0.1*x2 # Add noises y1<- yt1+rnorm(length(x1),0,0.1) y2<- yt2+rnorm(length(x2),0,0.1) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Log-L regression with a change point between 3.9 and 4.5 # Fit with vrcp with specification of Log-L segments and "c1" options with # and without common variance restriction ans <- vrcp(z,3.9,4.5,"c1","Log","L","Common") # Fit with common variance ans # The fitted Log-L regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogL-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,3.8,4.2,"c1","Log","L","Diff") # Fit with different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogL-c1-diff model: Estimate (in magenta) vs. true model") # results look similar, not bad. # Fit with Log-L segments and "c0" options with and without common variance restriction ans <- vrcp(z,3.5,4.5,"c0","Log","L","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogL-c0-com fit to LLog-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,3.5,4.5,"c0","Log","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogL-c0-diff fit to LLog-c1-com model: Estimate (in magenta) vs. true model") # Little worse than the one with c1 constraint # Fit with Log-L segments and u" options with and without common variance restriction ans <- vrcp(z,3.5,4.5,"u","Log","L","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-u-com fit to LLog-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,3.5,4.5,"u","Log","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LLog-u-diff fit to LLog-c1-diff model: Estimate (in magenta) vs. true model") # Clearly shows lack of fit at the change point. # Again, the correct specification of the model, or use of available information is important. # Example 4: QL-c1-com model, fitted by Q-L and Exp-L models, # with and without a common variance constraint, respectively. # Simulate Q-L data x1<-seq(0,2,by=0.05) x2<-seq(2,5,by=0.05) # The true regression functions yt1 <- 2+2*x1+2*x1^2 yt2 <- -6+10*x2 # Add noises y1<- yt1+rnorm(length(x1),0,3) y2<- yt2+rnorm(length(x2),0,3) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Q-L regression with a change point between 1.8 and 2.5 # Fit with vrcp with specification of Q-L segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.8,2.5,"c1","Q","L","Common") # Common variance ans # The fitted Q-L regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QL-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.8,2.5,"c1","Q","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QL-c1-diff model: Estimate (in magenta) vs. true model") # Fit with vrcp with specification of Exp-L segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"c1","Exp","L","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-c1-com fit to QL-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"c1","Exp","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-c1-diff fit to QL-c1-com model: Estimate (in magenta) vs. true model") # Fit with vrcp with specification of Exp-L segments and "c0" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"c0","Exp","L","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-c0-com fit to QL-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"c0","Exp","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-c0-com fit to QL-c1-com model: Estimate (in magenta) vs. true model") # Fit with vrcp with specification of Exp-L segments and "u" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"u","Exp","L","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-u-com fit to QL-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"u","Exp","L","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpL-u-com fit to QL-c1-com model: Estimate (in magenta) vs. true model") # Exp-L fits surprisingly well in this case. # Example 5: Exp-Exp with "c0" change point and common variance. - No option of smoothness # Simulate the data x1<-seq(0,2,by=0.05) x2<-seq(2.05,5,by=0.05) # The true regression functions yt1 <- 0.916+2*exp(x1) yt2 <- 12+0.5*exp(x2) # Add noises y1<-yt1+rnorm(length(x1),0,5) y2<-yt2+rnorm(length(x2),0,5) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Exp-Exp regression with a change point between 1.5 and 2.5 # Fit with vrcp with specification of Exp-Exp segments and "c0" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"c0","Exp","Exp","Common") # Common variance ## simulation of smooth ans # The fitted Exp-Exp regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpExp-c0-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"c0","Exp","Exp","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpExp-c0-diff model: Estimate (in magenta) vs. true model") # Fit with vrcp with specification of Exp-Exp segments and "u" options with # and without common variance restriction ans <- vrcp(z,1.5,2.2,"u","Exp","Exp","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpExp-u-com fit to ExpExp-c0-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"u","Exp","Exp","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("ExpExp-u-diff fit to ExpExp-c0-com model: Estimate (in magenta) vs. true model") # Unconstraint fits okay, considering information of continuity was not used. # Example 6: Log-Log with "c0" change point and common variance. - No option of smoothness # Simulate the data x1<-seq(2,4,by=0.05) x2<-seq(4.05,7,by=0.05) # The true regression functions yt1 <- 2 - 2*log(x1) yt2 <- 13.1 - 10*log(x2) # Add noises y1<- yt1+rnorm(length(x1),0,0.5) y2<- yt2+rnorm(length(x2),0,0.5) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Log-Log regression with a change point between 3.5 and 4.5 # Fit with vrcp with specification of Log-Log segments and "c0" options with # and without common variance restriction ans <- vrcp(z,3.5,4.5,"c0","Log","Log","Common") # Common variance ans # The fitted Log-Log regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogLog-c0-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,3.5,4.5,"c0","Log","Log","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogLog-c0-diff model: Estimate (in magenta) vs. true model") # Fit with vrcp with specification of Log-Log segments and "u" options with # and without common variance restriction ans <- vrcp(z,3.7,4.5,"u","Log","Log","Common") # Common variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogLog-u-com fit to LogLog-c0-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,3.7,4.5,"u","Log","Log","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LogLog-u-diff fit to LogLog-c0-com model: Estimate (in magenta) vs. true model") # Example 7: Q-Exp with "c1" change point and common variance. # Simulate the data x1<-seq(0,2,by=0.05) x2<-seq(2,5,by=0.05) # The true regression functions yt1 <- .2+.2*x1+.5*x1^2 yt2 <- .3832+.3*exp(x2) # Add noises y1<- yt1+rnorm(length(x1),0,3) y2<- yt2+rnorm(length(x2),0,3) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Q-Exp regression with a change point between 1.5 and 2.2 # Fit with vrcp with specification of Q-Exp segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.5,2.2,"c1","Q","Exp","Common") # Common variance ans # The fitted Q-Exp regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QExp-c0-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.2,"c1","Q","Exp","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QExp-c0-diff model: Estimate (in magenta) vs. true model") # Example 8: Q-Log with "c1" change point and common variance. # Simulate the data x1<-seq(0.05,2.05,by=0.05) x2<-seq(2.1,5.05,by=0.05) # The true regression functions yt1 <- 2+1*x1+5*x1^2 yt2 <- 0+35*log(x2) # Add noises y1<-yt1+rnorm(length(x1),0,4) y2<-yt2+rnorm(length(x2),0,4) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Q-Log regression with a change point between 1.5 and 2.5 # Fit with vrcp with specification of Q-Log segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"c1","Q","Log","Common") # Common variance ans # The fitted Q-Log regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QLog-c1-com model: Estimate (in magenta) vs. true model") ans <- vrcp(z,1.5,2.5,"c1","Q","Log","Diff") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("QLog-c1-diff model: Estimate (in magenta) vs. true model") # Example 9: Q-Q with "c1" change point and common variance. # Simulate the data x1<-seq(0.05,2,by=0.05) x2<-seq(2.05,5,by=0.05) # The true regression functions yt1 <- 2+10*x1-5*x1^2 yt2 <- 30-21*x2+3.5*x2^2 # Add noises y1<-yt1+rnorm(length(x1),0,2) y2<-yt2+rnorm(length(x2),0,2) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a Q-Q regression with a change point between 1.5 and 2.5 # Fit with vrcp with specification of Q-Q segments and "c1" options with # and without common variance restriction ans <- vrcp(z,1.5,2.5,"c1","Q","Q","Common",spline="TRUE") # Common variance ans # The fitted Q-Q regression and spline are superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt), color = "blue", size=1) + ggtitle("QQ-c1-com model: Estimates vs. true model (in blue)") ans <- vrcp(z,1.5,2.5,"c1","Q","Q","Diff",spline="TRUE") # Different variance ans ans\$plot + ggplot2::geom_line(aes(x = x, y = yt), color = "blue", size=1) + ggtitle("QQ-c1-diff model: Estimates vs. true model (in blue)") # vrcp fits better than splines fitting. # Example 10: L-NLExp with "c1" change point and common variance. # Simulate the data x1<- seq(0,2,by=0.05) x2<- seq(2.05,5,by=0.05) # The true regression functions yt1 <- 10-0.8*x1 yt2 <- 8.4 - (-0.78/0.5) + (-0.78/0.5)*(exp(0.5*(x2-2))) # Add noises y1<- yt1+rnorm(length(x1),0,0.5) y2 <- yt2+rnorm(length(x2),0,0.5) z<-data.frame(c(x1,x2),c(y1,y2)) names(z)=c("x","y") # z is the simulated data in data frame. Let's visualize it plot(z) # It looks like a L-NLExp regression with a change point between 1.8 and 2.05 # Fit with smooth L-NLExp, common or different variances tryCatch(vrcp(z,1.8,2.05,"c1","L","NLExp","Common",start="FALSE"), error=function(e) {return("Try different starting values. If this still fails, try a different nonlinear model that might be more suitable to data.")}) ans <- vrcp(z,1.8,2.05,"c1","L","NLExp","Common",start="FALSE") ans # The fitted L-NLExp regression is superimposed on the data # Let's compare it with the true regression x<-z\$x yt<-c(yt1,yt2) ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LNLExp-c1-com model: Estimate (in magenta) vs. true model") tryCatch(vrcp(z,1.8,2.05,"c1","L","NLExp","Diff",start="FALSE"), error=function(e){ return("Try different starting values. If this still fails, try a different nonlinear model that might be more suitable to data.")}) ans <- vrcp(z,1.8,2.05,"c1","L","NLExp","Diff",start="FALSE") ans\$plot + ggplot2::geom_line(aes(x = x, y = yt, colour = c("true")), size=1) + scale_colour_grey(name = "Model") + ggtitle("LNLExp-c1-diff model: Estimate (in magenta) vs. true model") ## End(Not run) ```

vrcp documentation built on May 2, 2019, 12:41 a.m.