What is Clock offset, fractional frequency offset?

The clock offset $\vartheta(t)$ of a clock is the difference between the time it shows and the reference time at the same instant. Its fractional frequency offset $y_{\mathrm f}(t) = d\vartheta/dt$ is the rate at which the offset grows, a dimensionless number quoted in parts per million (ppm) or per billion (ppb): a clock with $y_{\mathrm f} = 1$ ppm gains a microsecond every second.