# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
- A time-series is any sequene of observation such that the distribution of a given value depends on the previous values.
- Time is an exogeneous (outside the model) variable that is directional - measuremets only depend on the past.
- Time is an exogeneous (outside the model) variable that is directional - measurements only depend on the past.
- This is a statement of causality.
- However, the exogenous variable can be anything.
- Let'assume to have a set of data extracted from ``y(t) = A \sin(\omega t)`` with homoscedastic variance ``V = \sigma^2 + A^2/2``.
- Let's assume to have a set of data extracted from ``y(t) = A \sin(\omega t)`` with homoscedastic variance ``V = \sigma^2 + A^2/2``.
- This is easy to prove if you compute the variance as ``\sum (y-\lt y \gt)^2 / N``. Since the average value is zero, this turns out to be ``V = \frac{A^2}{N} \sum \sin^2 (\omega t)`` giving the ``A^2/2`` term.
- We can compute the ``\chi^2`` for this toy model:
- `w(t)` is a boxcar window function, which is 1 in the `(0,T)` interval and zero outside. `s(t)' is a series of delta functions at `t_k`, spaced by `T/N`:
- ``w(t)`` is a boxcar window function, which is 1 in the ``(0,T) `` interval and zero outside. ``s(t)`` is a series of delta functions at ``t_k``, spaced by ``T/N``:
$(LocalResource("Pics/windowing.png"))
"""
@@ -715,9 +743,18 @@ md"""
### Exercise about PSD manipulation
***
- Let's generate two arrays of relative timestamps, one 8 seconds long and one 1600 seconds long, with dt = 0.03125 s, and make two signals in units of counts. The signal is a sine wave with amplitude = 300 cts/s, frequency = 2 Hz, phase offset = 0 radians, and mean = 1000 cts/s. We then add Poisson noise to the light curve amd plot the shortest of the two.
- Let's generate two arrays of relative timestamps, one 8 seconds long and one 1600 seconds long, with dt = 0.03125 s, and make two signals in units of counts. The signal is a sine wave with amplitude = 300 cts/s, frequency = 2 Hz, phase offset = 0 radians, and mean = 1000 cts/s. We then add Poisson noise to the light curve and plot the shortest of the two.
