I am trying to compare the forecast accuracy of a number of methods using MAPE across different commodity markets, such as corn, wheat, soybeans, coffee, cotton. Obviously the relative MAPE’s area impacted by the relative volatilities of each commodity: a high MAPE for wheat may simply reflect a volatile market, not necessarily a poor forecast.
I am wondering how to correct for this: some kind of vol-adjusted MAPE I suppose, but I cannot find any literature on this. Alternatively, I was thinking of comparing the MAPE of a certain forecast method with the MAPE of a naïve forecast…this should also correct for the vol difference somewhat, I suppose.
Any further suggestions/comments are greatly appreciated.
I'm not aware of any measures that directly incorporates volatility, in order to enable comparison across. I would also question the relevance of directly comparing accuracy measures across like that, as the accuracy would depend - as you also points out - on the volatility/signal-to-noise ratio of the time series.
I approach a problem like this by what you also suggest - create a naïve forecast, and have that as the lowest acceptable accuracy for that series, and also an initial measure of the forecastability of the series.
Note: i follow the definition of a naïve forecast as: one which is a very simple forecast model, could be naive1, naive2, moving average or combination of those - where no further work needs to be done on parameters.
Try to have a look at the work of Michael Gilliland on FVA for inspiration
I have come across a problem.
I’m not asking for help how to construct what I’m searching for, but only to guide me to what I’m looking for! 😊
The thing I want to create is some sort of ‘Sorting Algorithm/Mechanism’.
Example:
Imagine I have a database with over 1000 pictures of different vehicles.
A person sees a vehicle, he now tries to get as much information and details about that vehicle, such as:
Shape
number of wheels
number and shape of windows
number and shape of light(s)
number and shape of exhaust(s)
Etc…
He then gives me all information about that vehicle he saw. BUT! Without telling me anything about:
Make and model.
…
I will now take that information and tell my database to sort out every vehicle so that it arranges all 1000 vehicle by best match, based by the description it have been given.
But it should NOT exclude any vehicle!
So…
If the person tells me that the vehicle only has 4 wheels, but in reality it has 5 (he might not have seen the fifth wheel) it should just get a bad score in the # of wheels.
But if every other aspect matches that vehicle perfect it will still get a high score.
That way we don’t exclude the vehicle that he has seen, and we still have a change to find the correct vehicle.
The whole aspect of this mechanism is to, as said, sort out the most, so instead of looking through 1000 vehicles we only need to sort through the best matches which is 10 to maybe 50 vehicles out of a 1000 (hopefully).
I tried to describe it the best I could in a language that isn’t ‘my father’s tongue’. So bear with me.
Again, I’m not looking for anybody telling me how to make this algorithm, I’m pretty sure nobody even wants of have the time to do that for me, without getting paid somehow...
But I just need to know where to look regarding learning and understanding how to create this mess of a mechanism.
Kind regards
Gent!
Assuming that all your pictures have been indexed with the relevant fields (number of wheels, window shapes...), and given that they are not too numerous (a thousand is peanuts for a computer), you can proceed as follows:
for every criterion, weight the possible discrepancies (e.g. one wheel too much costs 5, one wheel too few costs 10, bad window shape costs 8...). Make this in a coherent way so that the costs of the criteria are well balanced.
to perform a search, evaluate the total discrepancy cost of every car, and sort the values increasingly. Report the first ten.
Technically, what you are after is called a "nearest neighbor search" in a high dimensional space. This problem has been well studied. There are fast solutions but they are extremely complex, and in your case are absolutely not worth using.
The default way of doing this for example in artificial intelligence is to encode all properties as a vector and applying certain weights to each property. The distance can then be calculated using any metric you like. In your case manhatten-distance should be fine. So in pseudocode:
distance(first_car, second_car):
return abs(first_car.n_wheels - second_car.n_wheels) * wheels_weight+ ... +
abs(first_car.n_windows - second_car.n_windows) * windows_weight
This works fine for simple properties like the number of wheels. For more complex properties like the shape of a window you'll probably need to split it up into multiple attributes depending on your requirements on similarity.
Weights are usually picked in such a way as to normalize all values, if their range is known. Optionally an additional factor can be multiplied to increase the impact of a specific attribute on the overall distance.
This is more of a challenge question than something I urgently need, so don't spend all day on it guys.
I built a dating site (long gone) back in 2000 or so, and one of the challenges was calculating the distance between users so we could present your "matches" within an X mile radius. To just state the problem, given the following database schema (roughly):
USER TABLE
UserId
UserName
ZipCode
ZIPCODE TABLE
ZipCode
Latitude
Longitude
With USER and ZIPCODE being joined on USER.ZipCode = ZIPCODE.ZipCode.
What approach would you take to answer the following question: What other users live in Zip Codes that are within X miles of a given user's Zip Code.
We used the 2000 census data, which has tables for zip codes and their approximate lattitude and longitude.
We also used the Haversine Formula to calculate distances between any two points on a sphere... pretty simple math really.
The question, at least for us, being the 19 year old college students we were, really became how to efficiently calculate and/store distances from all members to all other members. One approach (the one we used) would be to import all the data and calculate the distance FROM every zip code TO every other zip code. Then you'd store and index the results. Something like:
SELECT User.UserId
FROM ZipCode AS MyZipCode
INNER JOIN ZipDistance ON MyZipCode.ZipCode = ZipDistance.MyZipCode
INNER JOIN ZipCode AS TheirZipCode ON ZipDistance.OtherZipCode = TheirZipCode.ZipCode
INNER JOIN User AS User ON TheirZipCode.ZipCode = User.ZipCode
WHERE ( MyZipCode.ZipCode = 75044 )
AND ( ZipDistance.Distance < 50 )
The problem, of course, is that the ZipDistance table is going to have a LOT of rows in it. It isn't completely unworkable, but it is really big. Also it requires complete pre-work on the whole data set, which is also not unmanageable, but not necessarily desireable.
Anyway, I was wondering what approach some of you gurus might take on something like this. Also, I think this is a common issue programmers have to tackle from time to time, especially if you consider problems that are just algorithmically similar. I'm interested in a thorough solution that includes at least HINTS on all the pieces to do this really quickly end efficiently. Thanks!
Ok, for starters, you don't really need to use the Haversine formula here. For large distances where a less accurate formula produces a larger error, your users don't care if the match is plus or minus a few miles, and for closer distances, the error is very small. There are easier (to calculate) formulas listed on the Geographical Distance Wikipedia article.
Since zip codes are nothing like evenly spaced, any process that partitions them evenly is going to suffer mightily in areas where they are clustered tightly (east coast near DC being a good example). If you want a visual comparison, check out http://benfry.com/zipdecode and compare the zipcode prefix 89 with 07.
A far better way to deal with indexing this space is to use a data structure like a Quadtree or an R-tree. This structure allows you to do spatial and distance searches over data which is not evenly spaced.
Here's what an Quadtree looks like:
To search over it, you drill down through each larger cell using the index of smaller cells that are within it. Wikipedia explains it more thoroughly.
Of course, since this is a fairly common thing to do, someone else has already done the hard part for you. Since you haven't specified what database you're using, the PostgreSQL extension PostGIS will serve as an example. PostGIS includes the ability to do R-tree spatial indexes which allow you to do efficient spatial querying.
Once you've imported your data and built the spatial index, querying for distance is a query like:
SELECT zip
FROM zipcode
WHERE
geom && expand(transform(PointFromText('POINT(-116.768347 33.911404)', 4269),32661), 16093)
AND
distance(
transform(PointFromText('POINT(-116.768347 33.911404)', 4269),32661),
geom) < 16093
I'll let you work through the rest of the tutorial yourself.
http://unserializableone.blogspot.com/2007/02/using-postgis-to-find-points-of.html
Here are some other references to get you started.
http://www.bostongis.com/PrinterFriendly.aspx?content_name=postgis_tut02
http://www.manning.com/obe/PostGIS_MEAPCH01.pdf
http://postgis.refractions.net/docs/ch04.html
I'd simply just create a zip_code_distances table and pre-compute the distances between all 42K zipcodes in the US which are within a 20-25 mile radius of each other.
create table zip_code_distances
(
from_zip_code mediumint not null,
to_zip_code mediumint not null,
distance decimal(6,2) default 0.0,
primary key (from_zip_code, to_zip_code),
key (to_zip_code)
)
engine=innodb;
Only including zipcodes within a 20-25 miles radius of each other reduces the number of rows you need to store in the distance table from it's maximum of 1.7 billion (42K ^ 2) - 42K to a much more manageable 4 million or so.
I downloaded a zipcode datafile from the web which contained the longitudes and latitudes of all the official US zipcodes in csv format:
"00601","Adjuntas","Adjuntas","Puerto Rico","PR","787","Atlantic", 18.166, -66.7236
"00602","Aguada","Aguada","Puerto Rico","PR","787","Atlantic", 18.383, -67.1866
...
"91210","Glendale","Los Angeles","California","CA","818","Pacific", 34.1419, -118.261
"91214","La Crescenta","Los Angeles","California","CA","818","Pacific", 34.2325, -118.246
"91221","Glendale","Los Angeles","California","CA","818","Pacific", 34.1653, -118.289
...
I wrote a quick and dirty C# program to read the file and compute the distances between every zipcode but only output zipcodes that fall within a 25 mile radius:
sw = new StreamWriter(path);
foreach (ZipCode fromZip in zips){
foreach (ZipCode toZip in zips)
{
if (toZip.ZipArea == fromZip.ZipArea) continue;
double dist = ZipCode.GetDistance(fromZip, toZip);
if (dist > 25) continue;
string s = string.Format("{0}|{1}|{2}", fromZip.ZipArea, toZip.ZipArea, dist);
sw.WriteLine(s);
}
}
The resultant output file looks as follows:
from_zip_code|to_zip_code|distance
...
00601|00606|16.7042215574185
00601|00611|9.70353520976393
00601|00612|21.0815707704904
00601|00613|21.1780461311929
00601|00614|20.101431539283
...
91210|90001|11.6815708119899
91210|90002|13.3915723402714
91210|90003|12.371251171873
91210|90004|5.26634939906721
91210|90005|6.56649623829871
...
I would then just load this distance data into my zip_code_distances table using load data infile and then use it to limit the search space of my application.
For example if you have a user whose zipcode is 91210 and they want to find people who are within a 10 mile radius of them then you can now simply do the following:
select
p.*
from
people p
inner join
(
select
to_zip_code
from
zip_code_distances
where
from_zip_code = 91210 and distance <= 10
) search
on p.zip_code = search.to_zip_code
where
p.gender = 'F'....
Hope this helps
EDIT: extended radius to 100 miles which increased the number of zipcode distances to 32.5 million rows.
quick performance check for zipcode 91210 runtime 0.009 seconds.
select count(*) from zip_code_distances
count(*)
========
32589820
select
to_zip_code
from
zip_code_distances
where
from_zip_code = 91210 and distance <= 10;
0:00:00.009: Query OK
You could shortcut the calculation by just assuming a box instead of a circular radius. Then when searching you simply calculate the lower/upper bound of lat/lon for a given point+"radius", and as long as you have an index on the lat/lon columns you could pull back all records that fall within the box pretty easily.
I know that this post is TOO old, but making some research for a client I've found some useful functionality of Google Maps API and is so simple to implement, you just need to pass to the url the origin and destination ZIP codes, and it calculates the distance even with the traffic, you can use it with any language:
origins = 90210
destinations = 93030
mode = driving
http://maps.googleapis.com/maps/api/distancematrix/json?origins=90210&destinations=93030&mode=driving&language=en-EN&sensor=false%22
following the link you can see that it returns a json. Remember that you need an API key to use this on your own hosting.
source:
http://stanhub.com/find-distance-between-two-postcodes-zipcodes-driving-time-in-current-traffic-using-google-maps-api/
You could divide your space into regions of roughly equal size -- for instance, approximate the earth as a buckyball or icosahedron. The regions could even overlap a bit, if that's easier (e.g. make them circular). Record which region(s) each ZIP code is in. Then you can precalculate the maximum distance possible between every region pair, which has the same O(n^2) problem as calculating all the ZIP code pairs, but for smaller n.
Now, for any given ZIP code, you can get a list of regions that are definitely within your given range, and a list of regions that cross the border. For the former, just grab all the ZIP codes. For the latter, drill down into each border region and calculate against individual ZIP codes.
It's certainly more complex mathematically, and in particular the number of regions would have to be chosen for a good balance between the size of the table vs. the time spent calculating on the fly, but it reduces the size of the precalculated table by a good margin.
I would use latitude and longitude. For example, if you have a latitude of 45 and a longitude of 45 and were asked to find matches within 50 miles, then you could do it by moving 50/69 ths up in latitude and 50/69 ths down in latitude (1 deg latitude ~ 69 miles). Select zip codes with latitudes in this range. Longitudes are a little different, because they get smaller as you move closer to the poles.
But at 45 deg, 1 longitude ~ 49 miles, so you could move 50/49ths left in latitude and 50/49ths right in latitude, and select all zip codes from the latitude set with this longitude. This gives you all zip codes within a square with lengths of a hundred miles. If you wanted to be really precise, you could then use the Haversine formula witch you mentioned to weed out zips in the corners of the box, to give you a sphere.
Not every possible pair of zip codes are going to be used. I would build zipdistance as a 'cache' table. For each request calculate the distance for that pair and save it in the cache. When a request for a distance pair comes, first look in the cache, then compute if it's not available.
I do not know the intricacies of distance calculations, so I would also check whether computing on the fly is cheaper than looking up (also taking into consideration how often you have to compute).
I have the problem running great, and pretty much everyone's answer got used. I was thinking about this in terms of the old solution instead of just "starting over." Babtek gets the nod for stating in in simplest terms.
I'll skip the code because I'll provide references to derive the needed formulas, and there is too much to cleanly post here.
Consider Point A on a sphere, represented by latitude and longitude. Figure out North, South, East, and West edges of a box 2X miles across with Point A at the center.
Select all point within the box from the ZipCode table. This includes a simple WHERE clause with two Between statements limiting by Lat and Long.
Use the haversine formula to determine the spherical distance between Point A and every point B returned in step 2.
Discard all points B where distance A -> B > X.
Select users where ZipCode is in the remaining set of points B.
This is pretty fast for > 100 miles. Longest result was ~ 0.014 seconds to calculate the match, and trivial to run the select statement.
Also, as a side note, it was necessary to implement the math in a couple of functions and call them in SQL. Once I got past a certain distance the matching number of ZipCodes was too large to pass back to SQL and use as an IN statement, so I had to use a temp table and join the resulting ZipCodes to User on the ZipCode column.
I suspect that using a ZipDistance table will not provide a long-term performance gain. The number of rows just gets really big. If you calculate the distance from every zip to to every other zip code (eventually) then the resultant row count from 40,000 zip codes would be ~ 1.6B. Whoah!
Alternately, I am interested in using SQL's built in geography type to see if that will make this easier, but good old int/float types served fine for this sample.
So... final list of online resources I used, for your easy reference:
Maximum Difference, Latitude and Longitude.
The Haversine Formula.
Lengthy but complete discussion of the whole process, which I found from Googling stuff in your answers.
I am writing an application which is recording some 'basic' stats -- page views, and unique visitors. I don't like the idea of storing every single view, so have thought about storing totals with a hour/day resolution. For example, like this:
Tuesday 500 views 200 unique visitors
Wednesday 400 views 210 unique visitors
Thursday 800 views 420 unique visitors
Now, I want to be able to query this data set on chosen time periods -- ie, for a week. Calculating views is easy enough: just addition. However, adding unique visitors will not give the correct answer, since a visitor may have visited on multiple days.
So my question is how do I determine or estimate unique visitors for any time period without storing each individual hit. Is this even possible? Google Analytics reports these values -- surely they don't store every single hit and query the data set for every time period!?
I can't seem to find any useful information on the net about this. My initial instinct is that I would need to store 2 sets of values with different resolutions (ie day and half-day), and somehow interpolate these for all possible time ranges. I've been playing with the maths, but can't get anything to work. Do you think I may be on to something, or on the wrong track?
Thanks,
Brendon.
If you are OK with approximations, I think tom10 is onto something, but his notion of random subsample is not the right one or needs a clarification. If I have a visitor that comes on day1 and day2, but is sampled only on day2, that is going to introduce a bias in the estimation. What I would do is to store full information for a random subsample of users (let's say, all users whose hash(id)%100 == 1). Then you do the full calculations on the sampled data and multiply by 100. Yes tom10 said about just that, but there are two differences: he said "for example" sample based on the ID and I say that's the only way you should sample because you are interested in unique visitors. If you were interested in unique IPs or unique ZIP codes or whatever you would sample accordingly. The quality of the estimation can be assessed using the normal approximation to the binomial if your sample is big enough. Beyond this, you can try and use a model of user loyalty, like you observe that over 2 days 10% of visitors visit on both days, over three days 11% of visitors visit twice and 5% visit once and so forth up to a maximum number of day. These numbers unfortunately can depend on time of the week, season and even modeling those, loyalty changes over time as the user base matures, changes in composition and the service changes as well, so any model needs to be re-estimated. My guess is that in 99% of practical situations you'd be better served by the sampling technique.
You could store a random subsample of the data, for example, 10% of the visitor IDs, then compare these between days.
The easiest way to do this is to store a random subsample of each day for future comparisons, but then, for the current day, temporarily store all your IDs and compare them to the subsampled historical data and determine the fraction of repeats. (That is, you're comparing the subsampled data to a full dataset for a given day and not comparing two subsamples -- it's possible to compare two subsamples and get an estimate for the total but the math would be a bit trickier.)
You don't need to store every single view, just each unique session ID per hour or day depending on the resolution you need in your stats.
You can keep these log files containing session IDs sorted to count unique visitors quickly, by merging multiple hours/days. One file per hour/day, one unique session ID per line.
In *nix, a simple one-liner like this one will do the job:
$ sort -m sorted_sid_logs/2010-09-0[123]-??.log | uniq | wc -l
It counts the number of unique visitors during the first three days of September.
You can calculate the uniqueness factor (UF) on each day and use it to calculate the composite (week by example) UF.
Let's say that you counted:
100 visits and 75 unique session id's on monday (you have to store the sessions ID's at least for a day, or the period you use as unit).
200 visits and 100 unique session id's on tuesday.
If you want to estimate the UF for the period Mon+Tue you can do:
UV = UVmonday + UVtuesday = TVmonday*UFmonday + TVtuesday*UFtuesday
being:
UV = Unique Visitors
TV = Total Visits
UF = Uniqueness Factor
So...
UV = (Sum(TVi*UFi))
UF = UV / TV
TV = Sum(TVi)
I hope it helps...
This math counts two visits of the same person as two unique visitors. I think it's ok if the only way you have to identify somebody is via the session ID.
I love to work on AI optimization software (Genetic Algorithms, Particle Swarm, Ant Colony, ...). Unfortunately I have run out of interesting problems to solve. What problem would you like to have solved?
This list of NP complete problems should keep you busy for a while...
How about the Hutter Prize?
From the entry on Wikipedia:
The Hutter Prize is a cash prize
funded by Marcus Hutter which rewards
data compression improvements on a
specific 100 MB English text file.
[...]
The goal of the Hutter Prize is to
encourage research in artificial
intelligence (AI). The organizers
believe that text compression and AI
are equivalent problems.
Basically the idea is that in order to make a compressor which is able to compress data most efficiently, the compressor must be, in Marcus Hutter's words, "smarter". For more information on the relation between artificial intelligence and compression, see the Motivation and FAQ sections of the Hutter Prize website.
Does the Netflix Prize count?
I would like my bank balance optimised so that there is as much money as possible left at the end of the month, instead of the other way round.
What about the Go Game ?
Here's an interesting practical problem I came up while tinkering with color quantization and image compression.
The basic idea is that I would like a program to which I give a picture and it reduces the amount of colors is it as much as possible without me noticing it. Since every person has a different sensitivity of the eye (and eyes have different sensitivity of red/green/blue intensities), it should be possible to specify this sensitivity threshold in some way.
In other words, in a truecolor picture, replace every pixel's color with another color so that:
The total count of different colors in a picture would be the smallest possible; and
Every new pixel would have it's color no further from the original color than some user-specified value D.
The D can be defined in different ways, pick your favorite. For example:
Separate red, green and blue components for specifying the maximum possible deviation for each of them (for every pixel you get a rectangular cuboid of valid replacement values);
A real number which would represent the maximum allowable distance in the RGB cube (for every pixel you get a sphere of valid replacement values);
Something inbetween or completely different.
Most efficient solution to a given set of Sudoku puzzles. (excluding brute-force methods)