Equation 2.5 is a key update rule we will use throughout the
course. This exercise will give you a better hands-on feel for
how it works. This exercise has ten parts. Please label
them clearly (1)-(10). There are two points for plotting each
graph and one point for answering each question. Get a piece of
graph paper (or print this pdf file)
and prepare to plot by hand. Do not use a computer or a
calculator; it is enough to figure out the values approximately and
qualitatively, and then plot them by hand.
(1) (5 pts.) Make a vertical axis one inch high
that runs from 0 to 1 and a horizontal axis from 1 to 18. Suppose
the estimate starts at 0 and the step-size (in the equation) is
0.5. Suppose the
target is 1.0 for 6 steps. Plot both the target and the
estimate points and connect the
points of each type by lines.
Thus, for time step 1, the estimate will be zero and the target will be
1. (The estimate should be zero on the first time step, and there
should be no time step 0.) To check that you have the timing right,
look at any time step; the vertical distance between the target and the
estimate on that time step should correspond to the error used to
produce the estimate on the next time step.
How
close
is the estimate
to 1.0 after 6 steps?
Without plotting or using a calculator, how
close
would it
be after 10 steps (assuming the target continues to be 1) (symbolic
answer preferred)? After
20? Suppose instead, after the 6 steps, the
target changes to 0. Plot the target and the estimate for 6 more
steps.
Now suppose the target alternates between 0 and 1 (starting with 1) for
the next 6
steps. Plot the target and then the trajectory of the estimate
over this time (qualitatively). This case is similar to that of a
noisy target.
(2) (2 pts.) Start over with a new graph and the estimate again at
zero, this
time
with a step size of 1/8. Repeat the target trajectory as
above.
(3) (3 pts.) Make a third graph with a step size of 1.0 and
repeat. Which step size produces estimates of smaller error when
the target is alternating?
(4) (4 pts.) Make a fourth graph with a step size of 1/t (i.e., the
first step
size
is 1, the second is 1/2, the third is 1/3, etc.) Repeat the
target
trajectory. Based on these graphs, why is the 1/t step size
appealing? Why is it not always the right choice?
(5)-(8) (9 pts.) You might think that the step size should be between 0
and
1.
Make plots for the first 6 steps with step sizes of -1/2, 1.5, 2.0, and
2.5 (adjust the range of the vertical axis appropriately). What
is the safe range for the step size?
(9)-(10) (5 pts.) Finally, suppose the step size is 1, and the target
is defined
as 1
plus half the current estimate. Plot the estimate for 6
steps. Repeat with a step size of 1.5. Which of these two
cases approaches the asymptotic value faster?
Disclaimer: I'm neither a professor nor
a TA, so take my advice at your own risk. :)
StepSize
is just a scale on how much we adjust our old estimate toward (or away
from) matching the target. A useful way to think about it is "how
big of a step should we take toward the target's value?" As for
comparing the graphs, the common scale they share is the *number* of
steps taken (18 steps for graphs 1-4, and 6 for graphs 5-10). So,
I suggest plotting the graphs as (Target or Estimate) vs. Step Number.
StepSize is just a scale on how much we adjust our old estimate toward (or away from) matching the target. A useful way to think about it is "how big of a step should we take toward the target's value?" As for comparing the graphs, the common scale they share is the *number* of steps taken (18 steps for graphs 1-4, and 6 for graphs 5-10). So, I suggest plotting the graphs as (Target or Estimate) vs. Step Number.