- Reminder: please keep the phones out of sight and sound.
- We'll be having our first exam Friday of next week. I'd
like to make the Keeling report due on Wednesday of
next week.
So we'll have finished three projects by then:
- Cross-Country
- Log Ruler
- Keeling
- Odd announcement: I'm doing a "School Strike for Climate" on Friday,
so I won't be teaching that day.
I will be upstairs, by the elevators, and you're free to come see me. I'll have some information about climate chaos. Kind of an informal teach-in....
- We talked about confidence intervals for
parameters: we provide a box, rather than a single
value, often these are expressed in one of two
formats: with 95% confidence,
$a \in [338.07, 340.07]$ or
$a = 339.07 \pm 0.997$ (these were the values of $a$ from the quadratic model to Stewart's Keeling data).
- Confidence intervals are related to p-values, but perhaps
more informative).
- The standard errors of the parameters pop out of the inverse
matrix we compute, multiplied by the mean SSE. Once we have those,
we have everything we need for confidence intervals.
- We started to talk to about linearization of non-linear models.
- So let's continue talking about transformations of the data that
permit us to continue using linear regression. We've already seen one:
- if one does a log-log transformation of both $x$ and $y$, and
the data look linear, then you'll be expecting a power model: $y=\alpha
x^\beta$.
(Riegel's model)
- If you log-transform only the $y$, and get a linear looking cloud,
then you'll be looking for an exponential model: $y=\alpha e^{b x}$.
(I've asked you for an exponential model for CO2. You log-transform the trend data, and plot. Then linear regress, if appropriate.)
- What about the oscillations in Keeling's model? We typically model
those with sinusoidal functions. Let's write out the model and
see what we need to do.... Hint: a pure sinusoidal variation in a
model can be represented, in all generality, as
\[
A \sin[2\pi\frac{t-t_0}{T}]
\]
where
- $A$ is the amplitude
- $T$ is the period of the oscillation (frequency $f=1/T$)
- $t_0$ is the phase shift
- Let's take an example (Ad Expenditures). We'll see some examples where these linear techniques aren't working...
- if one does a log-log transformation of both $x$ and $y$, and
the data look linear, then you'll be expecting a power model: $y=\alpha
x^\beta$.
- "The
IPCC stands by its equation of 450 ppm = 2 degrees C [of
warming]....": For years scientists have said that if
atmospheric levels of carbon dioxide reach 450 parts per million (ppm)
the planet would heat up by an average of 2 degrees Celsius above
pre-industrial levels. They have also said that if the world crosses
that threshold, ecosystems worldwide would suffer serious damage.
[ael: by the way -- ecosystems worldwide are already suffering serious damage.... The world won't "suddenly heat up" if we reach 450 ppm -- it's heating up right now, and we're seeing bumblebees struggle with the heat, for example. They pollinate our food -- think about it....]