Saturday, May 31, 2014

Loading IP Test Data Into Postgres

Recently, I was trolling around the internet looking for some IP address data to play with.  Fortunately, I stumbled across MaxMind's Geolite Database, which is available for free.    All I have to do is include this notice:

This product includes GeoLite2 data created by MaxMind, available from <a href="http://www.maxmind.com">http://www.maxmind.com</a>.

That's easy enough.  The next question is how to get this into my local Postgres database.  A bit over a year ago, I very happily gave up on my Dell computer and opted for a Mac.  One downside to a Mac is that SQL Server doesn't run on it (obviously my personal opinion).  Happily, Postgres does and it is extremely easy to install by going to postgresapp.com.   An interface similar enough to SQL Server Management Studio (called pgadmin3) is almost as easy to install by going here.

So, the next problem is getting the MaxMind data into Postgres.  Getting the two tables into Postgres is easy, using the copy command.  The challenge is IPV6 versus IPV4 addresses.  The data is in IPV6 format with a subnet mask to represent ranges.  Most of us who are familiar with IP addresses are familiar with IPV4 addresses.  These are 32 bits and look something like this:  173.194.121.17 (this happens to be an address for www.google.com attained by running ping www.google.com in a terminal window).  Alas, the data from MaxMind uses IPV6 values rather than IPV4.

In IPV6, the above would look like: ::ffff:173.194.121.17 (to be honest, this is a hybrid format for representing IPV4 addresses in IPV6 address space).  And the situation is a bit worse, because these records contain address ranges.  So the address range is really:  ::ffff:173.194.0.0/112.

The "112" is called a subnet mask.  And, IPV4 also uses them.  In IPV4, they represent the initial range of digits, so they range from 1-32, with a number like "24" being very common.  "112" doesn't make sense in a 32-bit addressing scheme.   To fast forward, the "112" subnet mask for IPV6 corresponds to 16 in the IPV4 world.  This means that the first 16 bits are for the main network and the last 16 bits are for the subnet.  That is, the addresses range from 173.194.0.0 to 173.194.255.255.  The relationship between the subnet mask for IPV6 and IPV4 is easy to express:  the IPV4 subnet mask is the IPV6 subnet mask minus 96.

I have to credit this blog for helping me understand this, even though it doesn't give the exact formula.  Here, I am going to shamelessly reproduce a figure from that blog (along with its original attribution):
ipv6-address
Courtesy of ls-a.org

This figure says  the following.  The last 64 bits are new to IPV6, so they can be automatically subtracted out of the subnet mask.  Then, bits 0-32 also seem to be new, so they can also be subtracted out.  That totals 96 bits in the new version not in the old version.  To be honest, I am not 100% positive about my interpretation.  But it does seem to work.  Google does indeed own exactly this address range.

The Postgres code for creating the table then goes as follows:

create table ipcity_staging (
    network_start_ip varchar(255),
    network_mask_length int,
    geoname_id int,
    registered_country_geoname_id int,
    represented_country_geoname_id int,
    postal_code varchar(255),
    latitude decimal(15, 10),
    longitude decimal(15, 10),
    is_anonymous_proxy int,
    is_satellite_provider int
);

copy public.ipcity_staging
    from '...data/MaxMind IP/GeoLite2-City-CSV_20140401/GeoLite2-City-Blocks.csv'
    with CSV HEADER;

create table ipcity (
    IPCityId serial not null,
    IPStart int not null,
    IPEnd int not null,
    IPStartStr varchar(255) not null,
    IPEndStr varchar(255) not null,
    GeoNameId int,
    GeoNameId_RegisteredCountry int,
    GeoNameId_RepresentedCountry int,
    PostalCode varchar(255),
    Latitude decimal(15, 10),
    Longitude decimal(15, 10),
    IsAnonymousProxy int,
    IsSatelliteProvider int,
    unique (IPStart, IPEnd),
    unique (IPStartStr, IPEndStr)
);

insert into ipcity(IPStart, IPEnd, IPStartStr, IPEndStr, GeoNameId, GeoNameId_RegisteredCountry, GeoNameId_RepresentedCountry,
                   PostalCode, Latitude, Longitude, IsAnonymousProxy, IsSatelliteProvider
                  ) 
    select IPStart, IPEnd, IPStartStr, IPEndStr, GeoName_Id, registered_country_geoname_id, represented_country_geoname_id,
           Postal_Code, Latitude, Longitude, Is_Anonymous_Proxy, Is_Satellite_Provider
    from (select network_mask_length - 96,
                 hostmask(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96)) ,
                 inet(host(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96) )) |
                 hostmask(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96)
                ) as ipend_inet,
                substr(network_start_ip, 8) || '/' || network_mask_length - 96,
                ((split_part(IPStartStr, '.', 1)::int << 24) +
                 (split_part(IPStartStr, '.', 2)::int << 16) +
                 (split_part(IPStartStr, '.', 3)::int << 8) +
                 (split_part(IPStartStr, '.', 4)::int)
                ) as IPStart,
                ((split_part(IPEndStr, '.', 1)::int << 24) +
                 (split_part(IPEndStr, '.', 2)::int << 16) +
                 (split_part(IPEndStr, '.', 3)::int << 8) +
                 (split_part(IPEndStr, '.', 4)::int)
                ) as IPEnd,
                st.*
          from (select st.*,
                       host(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96)) as IPStartStr,
                       host(inet(host(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96) )) |
                            hostmask(inet (substr(network_start_ip, 8) || '/' || network_mask_length - 96))
                           ) as IPEndStr
                from ipcity_staging st 
                where network_start_ip like '::ffff:%'
               ) st
         ) st;



Sunday, May 18, 2014

Armed Bandits: A Statistical Approach

This is a continuation of my previous post on multi-armed bandits.  And, I'm guessing there will be at least one more after this.

The Multi-Armed Bandit problem is a seemingly simple problem.  A gambler is faced with a row of slot machines, each of which returns a different winning.  S/he need to devise a strategy to find the winningest slot machine as quickly as possible and then just play that one.

Most of the strategies for doing this are based on a greedy-algorithm approach.  They are some variation on:  randomly (or round robinly) choose slot machines until some threshold has been reached.  Then continue playing the winningest one.  These actually work pretty well.  But I am interested in applying basic statistics to this.

Before doing that, let me explain why I am interested.  Imagine that I have a web site and I have an ad space to fill.  Here are different things I might put there:

  • A run of network ad that will make some amount of money per impression.
  • A click-through ad that will make some amount of money if someone clicks on it.
  • A partner ad that will make some amount of money if someone signs up for something.
The Multi-Armed Bandit provides an automated means of testing all three of these at once, along with variations that may, or may not, prove better than business-as-usual.   I think of it as automated champion-challenger models.

Here is a "statistical" approach to this problem.  Let me assume that there are N campaigns being run.   Each campaign has a payout distribution.  I can calculate the average payout for each campaign.  In the end, I want to choose the campaign that has the largest average payout.  Note that I'm make assumptions here that the the campaigns perform consistently across time and across the visitor population.  Those are other issues I discussed earlier.  Let's focus on the basic problem here.


By the Central Limit Theorem, we know that we can estimate the average based on a sample of data.  This estimate of the average has an average and a standard deviation, which (once there are enough samples) gets smaller and smaller, meaning that the average is better and better.

The idea is then simple.  At the beginning, give each campaign the same estimate with a wide confidence interval.  The intervals all overlap completely, so the choice of best campaign is random.  Initially, we might want to round-robin the data to get some initial values.  Relatively quickly, though, we should get estimates for each of the campaigns; these will be inaccurate but they will have wide confidence intervals.

At each iteration, we need to update the average and standard deviation.  Fortunately, there are easy incremental algorithms for both, so all the historical data does not need to be saved.  This article discusses various algorithms for calculating variance, and hence standard deviation.

The question is:  if we have multiple averages and standard errors, how do we choose the appropriate campaign at each step.  We can run a fast simulation to get the best campaign.  For each campaign, generate a random number based on the estimated average and standard error.  Choose the campaign that has the largest number.

What happens over time is that the campaign with the best payout should become more and more confident, as well as having the highest average.   Its confidence interval will shift way from the others, further increasing the odds of that campaign being chosen.  This is a positive feedback mechanism.  Note that I am using the term "confidence interval" as an aid to visualizing what is happening; this method is not actually using any p-values generated from the confidence interval.

One nice feature about this method is that it can adapt to the chosen solution getting worse.  If so, the average will decrease (but not the standard error) and other campaigns might be chosen.  Getting this to work involves a bit more effort, because you probably want to keep the sample size fixed -- otherwise the learning rate would be too small.

A note about distributions.  This solution is depending onto the distribution of the sample average, not the distribution of the original payout.  The sample average should (in the limit) have a  normal distribution, characterized by the average and standard error.  This is not a statement about the original data distribution, only about the average.  And, in the end, we want to choose the campaign that has the best average.  This is handy, because the three examples that I gave earlier are very different.  One has a constant (but low) payout and the other two are biased toward zero payouts.

I do believe that this method will produce reasonable results in practice.  However, it does bring up subtle issues about how the underlying distributions of the payouts affect the averages.  On the surface, it seems pretty sound, and it should work pretty well in practice.