## Notes from the Kölner R meeting, 12 December 2014

Last week's Cologne R user group meeting was the best attended so far, and it was a remarkable event - I believe not a single line of R code was shown. Still, it was an R user group meeting with two excellent talks, and you will understand shortly why not much R code needed to be displayed.### Introduction to Julia for R Users

Download slides |

Interestingly, as Hans Werner pointed out, Julia is by far not the only new language around the block. Indeed, over the last decade nearly every year saw the announcement of a new language. Also big tech companies such as Microsoft, Google, Mozilla and Apple are trying to push their own programming languages: F# (2005), Go (2009), Rust (2010) and Swift (2014) respectively.

Over the more recent years we notice a movement towards the use of LLVM (Low Level Virtual Machine), on which Julia is based as well and which makes it fast. The just in time compilation demands a little mind shift if you come from R, where the mantra for speed is: vectorise - remove all for-loops. Well, the opposite is true for Julia, because your code will be compiled. For-loops are much easier to understand for the underlying compiler. Hans Werner's slides provide some good examples to get you started and pointers to further resources.

### Dynamic Linear Models and Kalman Filtering

Download slides |

Nowadays, his challenges can be very different, sometimes only a few data points from a non-stationary process are available, and yet he is still expected to predict the future.

Dynamic linear models (dlm) can provide a remedy in those situations. In their simplest version a dlm links system and observational equations in the following way:

\[

y_t = F \theta_t + \nu_t\quad\mbox{observation eq. }\\

\theta_t = G \theta_{t-1} + \omega_t\quad\mbox{system eq.}

\] with \(\nu_t, \omega_t\) mutually independent random variables. A special case of dynamic linear models is the well known Kalman filter. In the more general case \(y_t\) and \(\theta_t\) are vectors and \(F_t, G_t\) are time variant matrices.

Holger explained that a dlm can principally be used for three purposes:

*Filtering:*Estimate of the current value of the state/system variable.*Smoothing:*Estimate of past values of the state/system variable, i.e., estimating at time \(t\) given measurements up to time \(t' > t\).*Forecasting:*Forecasting future observations or values of the state/system variable.

### Next Kölner R meeting

The next meeting is scheduled for 6 March 2015.Please get in touch if you would like to present and share your experience, or indeed if you have a request for a topic you would like to hear more about. For more details see also our Meetup page.

Thanks again to Bernd Weiß for hosting the event and Revolution Analytics for their sponsorship.

## Next Kölner R User Meeting: Friday, 12 December 2014

The next Cologne R user group meeting is scheduled for this Friday, 12 December 2014.

We have an exciting agenda with two talks on Julia and Dynamic Linear Models:

### Introduction to Julia for R Users

*Hans Werner Borchers*

Julia is a high-performance dynamic programming language for scientific computing, with a syntax that is familiar to users of other technical computing environments (Matlab, Python, R, etc.). It provides a sophisticated compiler, high performance with numerical accuracy, and extensive mathematical function libraries.

Some of the highlights of Julia are an implementation of automated differentiation, an optimisation modelling language, also integrating some of the optimisation solvers available from the COIN-OR project, and user-contributed packages for time series, statistics and machine learning, or operations research.

### Dynamic Linear Models and Kalman Filtering

*Holger Zien*

One of the problems most commonly presented to everybody working in statistics is forecasting time series. The textbook answer is to fit an ARMA model to the data and to use the model for prediction. This approach works well for long and stationary time series. However, often the actual time series one is given to analyse are more complicated. They are ridiculously short and clearly non-stationary. Dynamic Linear Models (DLM) and Kalman Filtering go one step beyond ARMA models and may be applied to these more complicated data. I will give a brief introduction into their mathematical background and will talk about my experience using them in practice.

### Drinks and Networking

The event will be followed by drinks and schnitzel at the Lux.For further details visit our KölnRUG Meetup site. Please sign up if you would like to come along. Notes from past meetings are available here.

The organisers, Bernd Weiß and Markus Gesmann, gratefully acknowledge the sponsorship of Revolution Analytics, who support the Cologne R user group as part of their vector programme.

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## Measuring temperature with my Arduino

It is really getting colder in London - it is now about 5°C outside. The heating is on and I have got better at measuring the temperature at home as well. Or, so I believe.Last week's approach of me guessing/feeling the temperature combined with an old thermometer was perhaps too simplistic and too unreliable. This week's attempt to measure the temperature with my Arduino might be a little OTT (over the top), but at least I am using the micro-controller again.

Yet, I have to overcome similar challenges again as last week. The Arduino with the temperature sensor (TMP36) should be a better device to measure the temperature than my thermometer, but it isn't perfect either and the readings will have some uncertainty as well. But, it will be easy to make frequent measurements and with a little trick I should be able to filter out the noise and get a much better view of the 'true' temperature.

The trick is Bayesian filtering, or as it turns out, a simplified Kalman filter (univariate, no control actions, no transition matrices).

I can't observe the temperature directly, I can only try to measure it. I think of this scenario as a series of Bayesian networks (or Hidden Markov Model). The hidden or latent variable is the 'true' temperature and the observable variable is the reading of my Arduino sensor.

I start with a prior guess of the Temperature at time \(k=0\), just like last week (note, here I will use \(x_k\) instead of \(\mu_k\)). My initial prior is \(N(x_0, \sigma_0^2)\). Now I predict the state (temperature) \(\hat{x}_1\) at the next time step. I expect the temperature to be largely unchanged and therefore my prediction is just \(\hat{x}_1 = x_0 + \varepsilon_P\) with \(\varepsilon_P \sim N(0, \sigma_P^2)\) describing the additional process variance going from one time step to the next. The process variance is independent from the uncertainty of \(x_0\), hence I can add them up and thus my prior for \(x_1\) is \(N(x_0, \sigma_0^2+\sigma_P^2)\).

The measurement of the temperature should reflect the 'true' temperature with some measurement uncertainty \(\sigma_M\). My prediction of the measurement is \(\hat{z}_1 = \hat{x}_1 + \varepsilon_M\) with \(\varepsilon_M \sim N(0, \sigma_M^2)\). The best estimator for \(\hat{z}_1\) is the actual measurement \(z_1\) and therefore the likelihood distribution of \(z_1|x_1\) is \(\,N(z_1, \sigma_M)\). Applying the formulas for Bayesian conjugates again gives the hyper-parameters for the posterior normal distribution \(x_1|\,z_1, x_0\), the temperature given the measurement and my prediction from the previous time step:

\[

x_1=\left.\left(\frac{x_0}{\sigma_0^2+\sigma_P^2} + \frac{z_1}{\sigma_M^2}\right)\right/\left(\frac{1}{\sigma_0^2+\sigma_P^2} + \frac{1}{\sigma_M^2}\right) \\

\sigma_1^2=\left(\frac{1}{\sigma_0^2+\sigma_P^2} + \frac{1}{\sigma_M^2}\right)^{-1}

\]With \(K_1 := \frac{\sigma_0^2+\sigma_P^2}{\sigma_M^2+\sigma_0^2+\sigma_P^2} \) this simplifies to:

\[

x_1 = K_1\, z_1 + (1 - K_1)\, x_0\\

\sigma_1^2 = (1 - K_1) (\sigma_0^2 + \sigma_P^2)

\]

This gives the simple iterative updating procedure (or Kalman filter):

\[

K_{k+1} = \frac{\sigma_k^2+\sigma_P^2}{\sigma_M^2+\sigma_k^2+\sigma_P^2} \\

\sigma_{k+1}^2 = (1 - K_{k+1}) (\sigma_k^2 + \sigma_P^2)\\

x_{k+1} = K_{k+1}\, z_{k+1} + (1 - K_{k+1})\, x_k

\]

In my code below I set the process variance to \(\sigma_P^2=0.01\) and the measurement variance to \(\sigma_M^2=0.5\), which led to a fairly stable reading of 21.5°C.

Changing the parameters \(\sigma_P\) and \(\sigma_M\) influence how quickly the filter reacts to changes. Here is an example where I briefly touch the temperature sensor with the above parameters. The time axis in those screen shots is about 6 seconds.

I have to do a bit more reading on the Kalman filter. It appears to be an immensely powerful tool to extract the signal from the noise. Well, it helped to put a man on the moon.

### Code

This is the Processing and Arduino code I used in this post. You may have to change the port number in line 28 to your own settings.
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