The stated aim of the Piwik project is to provide an alternative to Google Analytics. Using PHP and MySQL, it provides the usual family of statistics about your website with a handsome user interface that's customizable using drag-and-drop.
Its plugin-based architecture makes it flexible for all sorts of uses; and it makes the analytic data easily available in a variety of formats: XML, JSON, serialized PHP. The data can also be gotten at in the form of embeddable widgets and iframes.
Piwik is still very much a work in progress, but a cool one. Try it out on your site, or check out the live demo.
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