Output from scatter3d R script - how to read the equation - r

Question

I am using scatter3d to find a fit in my R script. I did so, and here is the output:
Call:
lm(formula = y ~ (x + z)^2 + I(x^2) + I(z^2))
Residuals:
Min 1Q Median 3Q Max
-0.78454 -0.02302 -0.00563 0.01398 0.47846
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.051975 0.003945 -13.173 < 2e-16 ***
x 0.224564 0.023059 9.739 < 2e-16 ***
z 0.356314 0.021782 16.358 < 2e-16 ***
I(x^2) -0.340781 0.044835 -7.601 3.46e-14 ***
I(z^2) 0.610344 0.028421 21.475 < 2e-16 ***
x:z -0.454826 0.065632 -6.930 4.71e-12 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05468 on 5293 degrees of freedom
Multiple R-squared: 0.6129, Adjusted R-squared: 0.6125
F-statistic: 1676 on 5 and 5293 DF, p-value: < 2.2e-16
Based on this, what is the equation of the best fit line? I'm not really sure how to read this? Can someone explain? thanks!

Answer 1

This is a basic regression output table. The parameter estimates ("Estimate" column) are the best-fit line coefficients corresponding to the different terms in your model. If you aren't familiar with this terminology, I would suggest reading up on some linear model and regression tutorial. There are thousands around the web. I would also encourage you to play with some simpler 2D simulations.
For example, let's make some data with an intercept of 2 and a slope of 0.5:
# Simulate data
set.seed(12345)
x = seq(0, 10, len=50)
y = 2 + 0.5 * x + rnorm(length(x), 0, 0.1)
data = data.frame(x, y)
Now when we look at the fit, you'll see that the Estimate column shows these same values:
# Fit model
fit = lm(y ~ x, data=data)
summary(fit)
> summary(fit)
Call:
lm(formula = y ~ x, data = data)
Residuals:
Min 1Q Median 3Q Max
-0.26017 -0.06434 0.02539 0.06238 0.20008
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.011759 0.030856 65.20 <2e-16 ***
x 0.501240 0.005317 94.27 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1107 on 48 degrees of freedom
Multiple R-squared: 0.9946, Adjusted R-squared: 0.9945
F-statistic: 8886 on 1 and 48 DF, p-value: < 2.2e-16
Pulling these out, we can then plot the best-fit line:
# Make plot
dev.new(width=4, height=4)
plot(x, y, ylim=c(0,10))
abline(fit$coef[1], fit$coef[2])

Answer 2

It's not a plane but rather a paraboloid surface (and using 'y' as the third dimension since you used 'z' already):
y = -0.051975 + x * 0.224564 + z * 0.356314 +
-x^2 * -0.340781 + z^2 * 0.610344 - x * z * 0.454826