# Using an inequality to select elements from a list and then plotting the results

Okay, here is the information that I have.

closingValues =
FinancialData["^GSPC", "Close", {"Jan 1., 2000", "Jan 1., 2012"},
"Value"];
logr = Differences[Log[closingValues]];

x = Tally[Sort[Round[logr, .001]]];


What I want to do is this:

(logr >= x) / Length[logr]

I want to count all the values in logr that are greater than or equal to x (there are about 110 or so values in x).

I also need help on plotting a graph where x is on the x-axis, and the resulting number above is on the y-axis.

Any help or advice, or leading me to any direction of solving my problem is greatly appreciated. Thank you.

-
What elements of "logr" and "x" are you trying to compare? The list x already consists of elements in logr, just rounded to the nearest 0.001. Are you trying to do an element by element compare? –  cartonn Dec 5 '12 at 1:02
What I'm trying to do is find typical values of logr, which i denoted x. And then count which values of logr are greater than x. I didn't know how to find "typical values" so I tallied all of the rounded values and am going to use those. –  user4341 Dec 5 '12 at 1:09
If you are looking for values in logr that are greater than the max element in x (hence all elements in x), all you will find are the ones that were rounded down to the max value in x. Am I right? –  cartonn Dec 5 '12 at 1:12
Right. So let's say the max value was .05, then I want to find count all values in logr that are greater than .05, and then divide how ever many numbers I found that were greater than .05 by Length[logr]. But instead of using just the max value, I want to do it for all listed values in x. –  user4341 Dec 5 '12 at 1:18

Perhaps use UnitStep and Mean. This should be pretty fast.

f[logr_, x_] := Mean[UnitStep[logr - x]]

f[logr, 0]

(*1597/3018*)


Now to plot it.

Plot[f[logr, t], {t, -.1, .1}, Exclusions -> None]


-
Andy, I don't think this is what I'm trying to do, but I greatly appreciate you trying to help. What I wanted to do is find all values in logr that is greater than the first value in x (so I essentially count however many values are larger than x), and then divide by the length of logr (like a probability). And then repeat with the next value in x. –  user4341 Dec 5 '12 at 1:43
This counts all values in logr that are greater than a given x value and divides by the number of elements in logr. Is that not what you wanted? –  Andy Ross Dec 5 '12 at 1:49
Actually, this does what you want, much more easily than what I was doing. Hats off! –  cartonn Dec 5 '12 at 1:54
Ooh, my confusion lies in the := Mean[UnitStep[logr - x]] part. Could you explain what it is doing for me? –  user4341 Dec 5 '12 at 1:58
UnitStep is listable and logr-x gives a list of numbers. If they are negative UnitStep gives 0 otherwise it gives 1. Mean of a list of zero's and 1's is the same as counting the non-negative values and dividing by the number of elements. –  Andy Ross Dec 5 '12 at 2:00

(Cannot connect to FinancialData, so I used fakeReturns instead of logr.)

 fakeReturns = RandomVariate[NormalDistribution[], 600];

Plot[SurvivalFunction[HistogramDistribution[fakeReturns], x], {x, -3, 3},
Exclusions -> None]


Or, Histogram with Round[logr,.001] ("typical values") as bin delimiters and "SF" (i.e., SurvivalFunction) as the height function gives:

Histogram[fakeReturns, {Union@Round[fakeReturns, 0.1]}, "SF", PlotRange -> Full]


or, with a different specification of "typical" values:

 Histogram[fakeReturns, {Union@Round[fakeReturns, 0.01]}, "SF", PlotRange -> Full]


-
OOH! That seems simple also! Now what if I wanted to take the log of the resulting values and plot it with the log of the x values? How would I do that? –  user4341 Dec 5 '12 at 3:21
@user4341, fakeReturns data has negative values as it is intended to represent your logr (Differences[Log[closingPrices]], that is, daily returns). With positive-valued data, you can use Histogram option ScalingFunctions -> {"Log", "Log"} get both axes in log scale. –  kguler Dec 5 '12 at 3:46