A few weeks ago I was doing some research on web analytics tools. On the search for a open source alternative to the commercial web analytics packages I came across Piwik, which is the successor of PHPmyVisites.
It is still beta and there is still a lot room for improvement, but it works quite well for me and is pretty stable. It looks like a really promising open source alternative to Google Analytics. Google offers its web analytics tool for free, but the data will be stored on their servers and you do not have access to the database. For some it might be interesting to get some data from the database to connect it with other databases, etc.
I will follow the development of Piwik. While the name sound quite weird 
I see a lot of potential for Piwik. I hope they will get the heatmap tool from PHPmyVisites back into the tool and they really should consider to offer click paths, so that you can see how visitors navigate through your website. It can give you a lot of feedback about the usability of your website, for example.
As soon as I have a little more time I will write a more comprehensive review about Piwik. So far I can only suggest to download the latest version (0.2.3) and give it a try. It is well worth it.
You might want to read the following posts as well:- Webanalytics Tools
- New Google Analytics
- Impact of Wordpress theme on search engine results
- Wordpress-Plugin: Google Analyticator
- Open Source Business: Groupware Tine 2.0
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