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--- estimated_standard_deviation [2025/08/12 19:57] – [수학적 증명] hkimscil
+++ estimated_standard_deviation [2025/08/12 20:00] (current) – [R 에서 SS값이 최소인 v값을 찾기] hkimscil
@@ Line 293: / Line 293: @@
+====== R 에서 SS값이 최소인 v값을 찾기 ======
+<code>
+#library(ggplot2)
+#library(ggpmisc)
-{{tag>"research methods" "조사방법론" "표준편차" "n-1" "자유도" "degrees of freedom" "n-1" "표준오차"}}
+rm(list=ls())
+rnorm2 <- function(n,mean,sd){
+  mean+sd*scale(rnorm(n))
+}
-====== R 에서 SS값이 최소인 v값을 찾기 ======
+# set.seed(191)
+nx <- 20
+mx <- 50
+sdx <- mx * 0.15
+x <- rnorm2(nx, mx, sdx)
+mean(x)
+sd(x)
+length(x)
+hist(x)
+x.span <- seq(from = mean(x)-6*sd(x),
+              to = mean(x)+6*sd(x),
+              by = 0.1)
+residuals <- function(x, v) {
+  return(x - v)
+}
+ssr <- function(x, v) {
+  residuals <- (x - v)
+  return(sum(residuals^2))
+}
+msr <- function(x, v) {
+  residuals <- (x - v)
+#  return((sum(residuals^2))/(length(x)-1))
+  return((mean(residuals^2)))
+}
+srs <- c() # sum of residuals
+ssrs <- c() # sum of square residuals
+msrs <- c() # mean square residuals = variance
+vs <- c() # the value of v in (x - v)
+for (i in x.span) {
+  res.x <- residuals(x,i)
+  srs.x <- sum(res.x)
+  ssr.x <- ssr(x,i)
+  msr.x <- msr(x,i)
+  srs <- append(srs, srs.x)
+  ssrs <- append(ssrs, ssr.x)
+  msrs <- append(msrs, msr.x)
+  vs <- append(vs, i)
+}
+plot(srs)
+plot(msrs)
+plot(srs)
+min(msrs)
+min.pos.msrs <- which(msrs == min(msrs))
+min.pos.msrs
+print(vs[min.pos.msrs])
+plot(vs, msrs)
+plot(vs, srs)
+# the above no gradient
+# mse 값으로 계산 rather than sse
+# 후자는 값이 너무 커짐
+gradient <- function(x, v){
+  residuals = x - v
+  dx = -2 * mean(residuals)
+  return(list("ds" = dx))
+} # function returns ds value
+residuals <- function(x, v) {
+  return(x - v)
+}
+ssr <- function(x, v) {
+  residuals <- (x - v)
+  return(sum(residuals^2))
+}
+msr <- function(x, v) {
+  residuals <- (x - v)
+  return((sum(residuals^2))/(length(x)-1))
+#  return(mean(residuals^2))
+}
+# pick one random v in (x-v)
+v <- rnorm(1)
+# Train the model with scaled features
+learning.rate = 1e-1
+ssrs <- c()
+msrs <- c()
+mres <- c()
+vs <- c()
+# Record Loss for each epoch:
+zx <- (x-mean(x))/sd(x)
+nlen <- 100
+for (epoch in 1:nlen) {
+  residual <- residuals(zx, v)
+  ssr.x <- ssr(zx, v)
+  msr.x <- msr(zx, v)
+  ssrs <- append(ssrs, ssr.x)
+  msrs <- append(msrs, msr.x)
+  grad <- gradient(zx, v)
+  step.v <- grad$ds * learning.rate
+  v <- v - step.v
+  vs <- append(vs, v)
+}
+tail(srs)
+tail(msrs)
+tail(ssrs)
+tail(vs)
+plot(srs)
+plot(msrs)
+plot(ssrs)
+plot(vs)
+# scaled
+v
+v.orig <- (v*sd(x))+mean(x)
+v.orig
+</code>
+====== output ======
+<code>
+> #library(ggplot2)
+> #library(ggpmisc)
+>
+> rm(list=ls())
+> rnorm2 <- function(n,mean,sd){
++   mean+sd*scale(rnorm(n))
++ }
+>
+> # set.seed(191)
+> nx <- 20
+> mx <- 50
+> sdx <- mx * 0.15
+> x <- rnorm2(nx, mx, sdx)
+>
+> mean(x)
+[1] 50
+> sd(x)
+[1] 7.5
+> length(x)
+[1] 20
+> hist(x)
+>
+> x.span <- seq(from = mean(x)-6*sd(x),
++               to = mean(x)+6*sd(x),
++               by = 0.1)
+>
+> residuals <- function(x, v) {
++   return(x - v)
++ }
+>
+> ssr <- function(x, v) {
++   residuals <- (x - v)
++   return(sum(residuals^2))
++ }
+>
+> msr <- function(x, v) {
++   residuals <- (x - v)
++ #  return((sum(residuals^2))/(length(x)-1))
++   return((mean(residuals^2)))
++ }
+>
+> srs <- c() # sum of residuals
+> ssrs <- c() # sum of square residuals
+> msrs <- c() # mean square residuals = variance
+> vs <- c() # the value of v in (x - v)
+>
+> for (i in x.span) {
++   res.x <- residuals(x,i)
++   srs.x <- sum(res.x)
++   ssr.x <- ssr(x,i)
++   msr.x <- msr(x,i)
++   srs <- append(srs, srs.x)
++   ssrs <- append(ssrs, ssr.x)
++   msrs <- append(msrs, msr.x)
++   vs <- append(vs, i)
++ }
+> plot(srs)
+> plot(msrs)
+> plot(srs)
+>
+> min(msrs)
+[1] 53.4375
+> min.pos.msrs <- which(msrs == min(msrs))
+> min.pos.msrs
+[1] 451
+> print(vs[min.pos.msrs])
+[1] 50
+>
+> plot(vs, msrs)
+> plot(vs, srs)
+>
+>
+>
+> # the above no gradient
+> # mse 값으로 계산 rather than sse
+> # 후자는 값이 너무 커짐
+>
+> gradient <- function(x, v){
++   residuals = x - v
++   dx = -2 * mean(residuals)
++   return(list("ds" = dx))
++ } # function returns ds value
+>
+> residuals <- function(x, v) {
++   return(x - v)
++ }
+>
+> ssr <- function(x, v) {
++   residuals <- (x - v)
++   return(sum(residuals^2))
++ }
+>
+> msr <- function(x, v) {
++   residuals <- (x - v)
++   return((sum(residuals^2))/(length(x)-1))
++ #  return(mean(residuals^2))
++ }
+>
+> # pick one random v in (x-v)
+> v <- rnorm(1)
+> # Train the model with scaled features
+> learning.rate = 1e-1
+>
+> ssrs <- c()
+> msrs <- c()
+> mres <- c()
+> vs <- c()
+> # Record Loss for each epoch:
+> zx <- (x-mean(x))/sd(x)
+>
+> nlen <- 100
+> for (epoch in 1:nlen) {
++   residual <- residuals(zx, v)
++   ssr.x <- ssr(zx, v)
++   msr.x <- msr(zx, v)
++   ssrs <- append(ssrs, ssr.x)
++   msrs <- append(msrs, msr.x)
++
++   grad <- gradient(zx, v)
++
++   step.v <- grad$ds * learning.rate
++   v <- v - step.v
++   vs <- append(vs, v)
++ }
+>
+> tail(srs)
+[1] -890 -892 -894 -896 -898 -900
+> tail(msrs)
+[1] 1 1 1 1 1 1
+> tail(ssrs)
+[1] 19 19 19 19 19 19
+> tail(vs)
+[1] 2.936258e-11 2.349006e-11 1.879204e-11 1.503363e-11 1.202690e-11 9.621523e-12
+>
+> plot(srs)
+> plot(msrs)
+> plot(ssrs)
+> plot(vs)
+> # scaled
+> v
+[1] 9.621523e-12
+> v.orig <- (v*sd(x))+mean(x)
+> v.orig
+[1] 50
+>
+>
+</code>
+====== tags ======
+{{tag>"research methods" "조사방법론" "표준편차" "n-1" "자유도" "degrees of freedom" "n-1" "표준오차"}}