How do I get the amplitude and phase for sine wave from lm() summary?

Question

A simple sine curve could be written as $\text{amplitude}\cdot\sin(x+\text{phase})$. It can be also written in linear form as $a \cdot \sin(x) + b \cdot \cos(x)$.

I run my analysis with R as:

 fit.lm2 <- lm(temperature~sin(2*pi*Time/366) + cos(2*pi*Time/366))
 summary(fit.lm2)
Coefficients:
                       Estimate Std. Error t value Pr(>|t|)
(Intercept)             26.9188     0.1005  267.87  < 2e-16
sin(2 * pi * Time/366)   1.7468     0.1390   12.56  < 2e-16
cos(2 * pi * Time/366)   1.2077     0.1485    8.13 6.94e-11

The general form of the equation is $y = b_0 + b_1x_1 + b_2x_2$, thus, in my case, it can be written as $y = 26.9188x_0 + 1.7468x_1 + 1.2077x_2$.

If I were to write it back to the simple sine form, $\text{amplitude}\cdot\sin(x+\text{phase})$, is it correct to say that:

$\text{amplitude} = b_0 = 26.9188$

$\text{phase} = \arctan\left(\frac{b_1}{b_2}\right)$

Is this the correct way how to do it?

Answer 1

The fit is

$$y = 26.9188 + 1.7468\sin(x) + 1.2077\cos(x).$$

Consider a general (non-zero) linear combination $\alpha \sin(x) + \beta\cos(x).$ Viewing $(\alpha, \beta)$ as a vector and writing it in polar coordinates $(r, \phi)$ yields

$$\alpha = r \cos(\phi),\quad \beta = r \sin(\phi), \quad r = \sqrt{\alpha^2+\beta^2}$$

whence

$$\alpha\sin(x) + \beta\cos(x) = r\cos(\phi)\sin(x) + r\sin(\phi)\cos(x) = r\sin(x+\phi).$$

$r$ is the amplitude and $\phi$ is the phase. In the present case $\alpha=1.7468$ and $ \beta=1.2077$ entailing

$$r = \sqrt{ 1.7468^2+1.2077^2 } = 2.123641$$

and

$$\phi = \arctan(\beta, \alpha) = 0.6049163.$$

Consequently

$$y = 26.9188 + 2.123641 \sin(x + 0.6049163).$$

This can be checked by plotting. Here is R code to do it:

b0 <- coef(fit.lm2)[1]
alpha <- coef(fit.lm2)[2]
beta <- coef(fit.lm2)[3]

r <- sqrt(alpha^2 + beta^2)
phi <- atan2(beta, alpha)

par(mfrow=c(1,2))
curve(b0 + r * sin(x + phi), 0, 2*pi, lwd=3, col="Gray",
      main="Overplotted Graphs", xlab="x", ylab="y")
curve(b0 + alpha * sin(x) + beta * cos(x), lwd=3, lty=3, col="Red", add=TRUE)

curve(b0 + r * sin(x + phi) - (b0 + alpha * sin(x) + beta * cos(x)), 
      0, 2*pi, n=257, lwd=3, col="Gray", main="Difference", xlab="x", y="")

The two formulas agree to sixteen significant figures in double-precision arithmetic. The difference reflects pseudo-random floating point errors. (Because my data are not exactly the same as the original data, the "difference" plot will differ in its details but will still exhibit only tiny variations.)