1
$\begingroup$

I was trying to plot the OLS loss function as a function of coefficients $\beta_0$, $\beta_1$. As far as I know it should be a convex function with one local minimum which is also a global minimum.

I've expanded $\sum_i^n(y_i-\beta_0 -\beta_1x_i)^2$ $=$

$\beta_1^2\sum_i^nx_i^2+\beta_0^2\sum_i^n1+\sum_i^ny_i^2+2\beta_1\beta_0\sum_i^nx_i-2\beta_1\sum_i^n x_i y_i - 2\beta_0\sum_i^n y_i$

Then computed sums and plot as a function of $\beta_0, \beta_1$. Resulting plot doesn't look like as a convex function with one minimum. Contour lines looks linear and has tiny slope.

This is what I get

enter image description here

Also, plotting function of the form $x^2+y^2+2xy-2x-2y$ gives the same looking graph, however slighltly modifying it to $2x^2+y^2+2xy-2x-2y$ or $x^2+y^2+xy-2x-2y$ gives good looking parabola.

Can anyone please provide some insights, what I'm getting wrong? Is it true that the OLS loss function looks like a parabola?

$\endgroup$
4
  • $\begingroup$ What data, i.e. what $x_i$ and $y_i$ did you use for the plotting? $\endgroup$ May 1, 2019 at 10:13
  • $\begingroup$ Dont' know if you did it in the code, but you haven't squared $y_i$ in the above expression. $\endgroup$
    – gunes
    May 1, 2019 at 10:21
  • $\begingroup$ @COOLSerdash I've generated data myself: x = 1,2...,100, y = 3 + 0.6x + np.random.normal(loc=0, scale=4). Here is a link for ds 1drv.ms/u/s!Arn3_AEhHx7saJ0tADjQEb3Uoos $\endgroup$ May 1, 2019 at 10:24
  • $\begingroup$ @gunes Yes, thanks, I missed it here but not in the code. $\endgroup$ May 1, 2019 at 10:25

1 Answer 1

2
$\begingroup$

It's just difficult to see in your graph because there is a long "ridge". Incidentally, there is an interesting connection to ridge regression and a graphic depiction can be found in this answer. If you plot the contours, it's apparent that there is a minimum. Here is a contour plot using your data:

library(ggplot2)

x <- 1:100
y <- c(-1.06786217, -0.33984722, -1.36759494, 7.56540282, 2.19205822, 
  14.8377512, 13.0539231, 0.63289129, 7.75364681, 6.54673087, 10.8224342, 
  10.1020288, 13.5477161, 15.3642923, 7.36291541, 16.6248658, 15.0631203, 
  25.1800439, 11.2292748, 13.8589657, 10.19495, 9.01691165, 14.7285545, 
  11.4168574, 15.2110969, 18.0854476, 23.2120067, 12.0950959, 16.1230903, 
  15.0810368, 20.2311771, 23.0948634, 15.3996733, 23.6483798, 23.3256991, 
  25.2116597, 29.1645765, 31.5760183, 24.6586337, 24.0144962, 29.7650414, 
  27.5561203, 25.34471, 30.0982008, 25.2412531, 31.6709949, 29.9007839, 
  22.914041, 28.3002482, 26.4310713, 35.7958481, 30.4163521, 28.5912421, 
  37.352515, 37.0315531, 37.5393569, 41.1098306, 36.4876877, 44.9613038, 
  34.9987338, 45.4365697, 39.5746548, 43.7593438, 35.489477, 47.3233672, 
  44.342282, 43.7527713, 50.1770972, 48.2229851, 43.3526442, 48.7265076, 
  51.4778536, 44.1065885, 46.5162551, 47.7805753, 47.8910884, 47.2123014, 
  54.8892224, 49.760496, 43.0211547, 47.3799211, 54.9845947, 50.1267701, 
  50.4283826, 52.674689, 51.8781938, 50.8822024, 50.6212418, 57.2954308, 
  52.4427199, 60.6958874, 52.075629, 53.4571635, 56.3515546, 55.4839035, 
  59.7015534, 55.6301584, 66.8236549, 60.9454023, 67.7678088)

mod <- lm(y~x)

beta0 <- seq(0, 5, length.out = 100)
beta1 <- seq(0.4, 0.8, length.out = 100)

coef_frame <- expand.grid(beta0 = beta0, beta1 = beta1)

ss <- NULL

for(i in 1:dim(coef_frame)[1]) {

  pred <- coef_frame[i, 1] + coef_frame[i, 2]*x
  resid <- y - pred

  ss[i] <- sum(resid^2)

}

coef_frame$ss <- ss

theme_set(theme_bw())
p <- ggplot(data = coef_frame, aes(x = beta0, y = beta1, z = ss)) +
  geom_raster(aes(fill = ss), interpolate = TRUE) +
  stat_contour(breaks = exp(quantile(log(ss), c(0.01, 0.025, 0.05, 0.1, 0.3, 0.5))), colour = "white") +
  scale_fill_gradient(name = expression("Sum of residuals"^2), low = "#08519c", high = "#f7fbff", trans = "log") +
  geom_point(aes(x = coef(mod)[1], y = coef(mod)[2]), size = 4, colour = "#ff8409") +
  geom_segment(aes(x = coef(mod)[1], y = min(beta1), xend = coef(mod)[1], yend = coef(mod)[2]), colour = "#ff8409", size = 0.75, linetype = 2) +
  geom_segment(aes(x = min(beta0), y = coef(mod)[2], xend = coef(mod)[1], yend = coef(mod)[2]), colour = "#ff8409", size = 0.75,  linetype = 2) +
  ylab("Beta 1") +
  xlab("Beta 0") +
  theme(
    axis.title.y=element_text(colour = "black", size = 17, hjust = 0.5, margin=margin(0,12,0,0)),
    axis.title.x=element_text(colour = "black", size = 17),
    axis.text.x=element_text(colour = "black", size=15),
    axis.text.y=element_text(colour = "black", size=15),
    legend.position="none",
    legend.text=element_text(size=12.5),
    panel.grid.minor = element_blank(),
    panel.grid.major = element_blank(),
    legend.key=element_blank(),
    plot.title = element_text(face = "bold"),
    legend.title=element_text(size=15)

  )

p

Contourplot

The least-squares estimates for $\beta_0$ and $\beta_1$ are indicated by an orange dot.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.