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Archive for November 2013

Estimation of Exponential Ornstein-Uhlenbeck process

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In the previous article about process calibration we derived the following updating formula:
P_t - P_{t-1} = \left(\theta(\mu - \ln P_{t-1}) + \frac{1}{2} \sigma^2 \right) P_{t-1} \Delta t + \sigma P_{t-1} \sqrt{\Delta t} W_{t-1}
Now we can rearrange equation’s parts as follows:
\frac{P_t - P_{t-1}}{P_{t-1}} = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t - \theta\Delta t \ln P_{t-1} + \sigma \sqrt{\Delta t} W_{t-1}
Thus we can equate it against simple regression formula:
y = a + bx + \epsilon

Therefore, we obtain:
y = \frac{P_t - P_{t-1}}{P_{t-1}}
x = \ln P_{t-1}
b = - \theta\Delta t
a = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t
\epsilon = \sigma \sqrt{\Delta t} W_{t-1}
And this gives us the following OLS estimates:
\theta = - \frac{b}{\Delta t}
a = - b \mu + \frac{1}{2} \sigma^2 \Delta t
As it was in the previous articles:
\sigma^2 = \frac{\sigma^2_{\epsilon}}{\Delta t}
a = \frac{1}{2} \sigma^2_{\epsilon} - b \mu
And finally the estimation of \mu is:
\mu = \frac{\frac{1}{2} \sigma^2_{\epsilon} - a}{b}

Thus we obtained the estimation of exponential Ornstein-Uhlenbeck process.

Written by fxpaul

November 20, 2013 at 13:04

Posted in Uncategorized

Recommender systems – Intro

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Recommender System (RS) provides suggestions for items to be of use to a user. As the definition says we should define three pieces, i.e. item, user and useful suggestion.

Item is a general term that describes everything what RS recommends to users. It might be stocks, books, videos or anything else.

User of RS is usually an individual who doesn’t have enough experience or competence to make a decision of item choice. For instance, popular on-line book store Amazon provides recommendations based on what user bought and viewed in past.

To be useful suggestion (or recommendation) should support some decision-making process, like what book to buy, what news to read or what to do in a spare time. In the simplest form, it might be a ranked list of items. This ranking is way to predict what the most suitable items are for user behaviour.

RS should track how users interact with items. For instance, on the book store one might view a book title, look inside of the book and/or buy it. Viewing a title can be considered as an implicit sign of preference. Thus the usual recommender system has to deal with User, Item and User-to-Item actions (transactions).

For RS implementation to be successful it should achieve one or more business goals:

  • Sell more items
  • Sell more diverse items
  • Increase user satisfaction and fidelity
  • Understand user behaviour and habits

And those could be approached by implementation of few tasks:

  • Find some good items: Create a ranked list of items along with predictions of how much user would like them. This is the main task of many RS.
  • Find all good items: Create a complete ranked list of items. Usually it is required when number of items is small and user can benefit from ranking information. Such RS are quite common in financial application. They usually need to examine and to rank all possible scenarios.
  • Annotate items in context: Given an existing context, emphasise items based on long-term user preferences. For instance, such RS might emphasise TV shows in EPG based on previous user behaviour.
  • Recommend a bundle: Suggest a group of items that fits well together. You’ll find such bundles at cable internet providers, travelling agencies etc. For instance, airlines are starting to recommend accommodation and car hire during ticket purchase.
  • Recommend a sequence: Recommend a sequence of items that is pleasing as a whole. For instance, a recommended track of courses at the university might depend not only on chosen major, but also on the absolved courses.
  • Browsing: RS should help the user to browse items that are more likely in the user’s interest in this browsing session.
  • Improve user profile: This task is all-time task of RS. It collects information about user’s actions to provide more personalised recommendations.

That is pretty much what one can expect from such thing as recommender system. In next posts I plan to cover:

  • Overview of basic techniques
  • Clustering
  • Content-based RS
  • Collaborative filtering in RS

Written by fxpaul

November 6, 2013 at 16:11

Posted in Uncategorized

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