Processing multi-dimensional data visually

In an earlier post I discussed the challenges we face currently at CEDL when we look at Big Data (http://blog.camenergydatalab.com/2014/10/big-data-crunching.html). This has been complex already, but we love new challenges here at CEDL. So let’s talk about multi-dimensionality.

If we presume that our data is arranged in a table like this:

ID	location	date	temperature	humidity
1	London	2015/02/12	4	68
2	Cambridge	2015/02/12	2.5	55

then aspects of big data refer loosely to the number of rows and multi-dimensionality of the data refers to the number of columns. Basically, we do not only have a lot of data (rows) but it is also complex due to the high number of features (columns).

Understandably, it is very challenging to extract information from such complex data in particular when we do not know what we are looking for. As part of the data exploration a data scientist will look for patterns or clusters that might tell us more about the processes which shape the data.

Of course, a data scientist wants to use the best tools available to find patterns and clusters in the data and as it turns out the most powerful machine for pattern detection is the visual cortex! The brain is your very personal supercomputer. The challenge in utilising the brain for detecting patterns in multidimensional data sets does not, thankfully, come down to brain surgery. Nevertheless, a problem still remains: how to interface the visual cortex with the data set? The only and best working interface are of course the eyes. All what is required is to transform the data set into a representation suitable for the eyes -> visual cortex interface. You might wonder why this sounds rather like an engineering problem than the typical task of a data scientist.  Unfortunately, the role of a data scientist is commonly misunderstood. In fact, with today’s challenges the task is not so much about calculating statistics but to engineer a way to access and consume data.

Usually, this happens in the form of charts and plots and it is up to the data scientist to find a suitable data representation for the problem at hand:  to explore data, find answers, and to communicate them.

For example, a fantastic way to represent multidimensional data are Parallel Coordinates [1] if you want to utilise the pattern recognition abilities of the brain’s visual cortex.

In a chart with parallel coordinates each column of the table is a vertical axis and each row becomes a line in the chart. Here follows an example:

This type of chart works with both discrete and continuous data. Additionally, colour and line types can be used to add some additional context to the data. Obviously, this chart is very simple but it can help us understand how parallel coordinates work. For example, looking at the location axis we can note that we have three records with location London and three with location Cambridge while looking at the axis data we note that we have two records for each day.

This chart shows you parallel coordinates in full action:

The chart shows the visualization of the mtcars dataset [2]. The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of car design and performance for 32 cars (1973–74 models).

A way to explore data in parallel coordinates is called “brushing”: Simply select a range over one or multiple axis and explore how the data segregates.

For example compare the models with better fuel economy versus models with less miles per gallon (mpg):

Whereas the cars with low fuel economy don’t seem to show any specific segregation, the cars with good fuel economy are light cars with 4 cylinders and small displacement.

[1] http://web.cs.ucdavis.edu/~ma/ECS289H/papers/Inselberg1997.pdf

[2] http://stat.ethz.ch/R-manual/R-devel/library/datasets/html/mtcars.html

Renewable energy in Europe: how far are we from the targets?

In 2009, the European Union set mandatory targets for renewable energy use that every member state has to reach by the year 2020. In this post we will analyse the progress of each member state using the latest estimates released by Eurostat.

Shares of renewable energy in 2012

In the figure below we have the shares of gross final renewable energy consumption for each member state and how far the states are from their target: Here we note that Sweden, Estonia and Bulgaria already reached their targets while Malta Luxembourg and the UK have the lowest shares of renewable energy in gross final energy consumption. Also, Norway is the country with the highest share of renewable energy. Netherland, France and the UK are the countries furthest from their targets.

Increase since 2006

In the following chart we compare the increase of shares from 2006 to 2012 of each country: From this chart we note that all the member states increased their share of renewable energy since 2006. Another interesting fact we note here is that the three states with the highest increases are, in order, Malta, the UK and Belgium, which are also some of the countries furthest from the achievement of their targets.

Evolution of the shares from 2004 to 2012

In this figure we compare the trend of the shares of renewable energy among the biggest European countries excluding the Scandinavian ones: We can observe that Italy and the UK had the fastest growth of renewable energy shares, but while the UK share has never been comparable to the ones of the other countries, Italy was able to overtake France and Germany in 2011. We can also see that the German share had the slowest growth and that Spain is the country with the highest share since 2009.

Feed in tariffs: Small scale solar PV cost

In our previous posts we focused on the excess energy that can be generated using photovoltaic (PV) panels and on on its trends. In this post we will focus on the price of the solar PV installation through two interactive visualizations based on the latest data provided by the UK government about the cost per kW of PV deployments by month.

Cost per kW installed by size band

In the following chart we can compare the cost per kW of solar deployments on a monthly basis for different size bands: Each point of this chart represents the median of the costs in a particular month while the error bars represent the 95% confidence interval of the cost in a given month. Which means that we can be 95% certain that the mean cost per kW lies within the two values reported by the error bar assuming that the cost data entered is a non-biased sample. We can note that the cost per kW installed has remained almost static over the 12 months. With the median of the cost per kW ranging from £1830 in June 2013 to £2010 in October 2013. The chart also shows that in the higher size bands the cost per kW is lower, but since the confidence intervals are wider, it is also more changeable.

Number of PV installations

In this figure we compare the number of installations over a monthly period. As in the previous chart, we differentiated deployments into three different size bands: Here we note that in correspondence of the decreases of the deployment costs, June 2013 and March 2014, the number of installations substantially increased. The month with the lowest number of installation is July 2013 while the month with the highest is March 2014. We also have an increasing trend of installation from June 2013 to November 2013.

Big Data Crunching

There has been much talk about Big Data in the last years and the word cloud shows terms commonly related to the definition of Big Data. First and foremost, the most important attribute of Big Data comes as no surprise: its volume! Big Data is, as the name suggests, BIG. What big actually means in regards to bytes or number of records is circumstantial. It becomes big when your traditional way of data processing hits a wall and becomes unfeasible.

The first symptom will be that your data does not fit into memory. In the beginning you might simply beef up your computer with some extra memory. This is commonly called to scale up. A more sophisticated solution would be to load only partial data into memory as disk size is much less of a bottleneck. This is how a database operates. A join operation on two massive tables, e.g. in Postgres, will load and write many chunks of intermediate data but will eventually succeed even though all the data never fit into memory at once. Relational Databases and scaling up to more powerful computers was the gold standard for tackling growing data volumes. Things changed in particular after 2004 with Google's publication of "MapReduce: Simplified Data Processing on Large Clusters"[1].

Instead of running huge databases on expensive supercomputers the trend went to massive parallelisation on clusters of cheap hardware. With this came new challenges which MapReduce successfully addresses:

• parallelisation must be easy
• automatic distribution of data between the workers of the cluster
• fault tolerance

If you process data on a big cluster of cheap hardware the chances are quite high that one of the computers breaks down. In the ACID world of RDBs (all or nothing transactions) this would mean we never get any results.

So what exactly is MapReduce doing differently?

The MapReduce Paradigm

Let's discuss a simple example inspired by a common task in processing genetic data: imagine you have vast amount of strings and you want to trim the last 5 letters of the strings.

In such a task we have to process each single record. This means the task is of linear order:

\[ O(n) \]

where the computational effort grows linearly with the number of records.

However, each record can be processed independently from the other records which allows to scale out the task over multiple processes, cores, or computers.

Let \(k\) be the number of processes, cores, or computers available, the order of our task becomes

\[ O \left ( \frac{n}{k} \right ). \]

This is much better for the case of Big Data when \(n\) is very large as we can control the computational effort easily by increasing \(k\). Additionally, if one subtask fails we only have to rerun that specific subtask. A complete rollback of the transaction is not required as it would be the case in ACID conform RDBs.

Let’s consider a slightly more complex task: the "hello world" program in the world of  MapReduce is the counting of word frequencies in a very big number of documents. As it was the case in the previous task, the word count of a single document is independent from the other documents, this makes the task perfectly suitable for scaling out:

A common pattern is emerging here: we use a function which maps each document to a list of independent word counts. The result of this map is a distributed list of word counts. So far our MapReduce programme comprises the following steps:

1. distribute the documents over multiple computers in a cluster
2. apply a word count function on each computer
3. generate a distributed list of word counts

The next step is to aggregate the distributed list of word counts. However, we want to scale out the aggregation again over multiple computers:

This aggregation step is also called reduce. The steps involved are as follows

1. send the same words to the same computer for aggregation (called shuffle)
2. apply a sum function to generate the word count over the complete set of documents (reduce)

And there we have the complete MapReduce paradigm:

Some tasks might be more difficult to translate into map and reduce steps and can require multiple rounds of mapreduce. However, the mapreduce ecosystem is growing steadily with new libraries implementing now even complex machine learning algorithms in mapreduce [3,4,5].

Last but not least, comparing mapreduce to RDB we see that mapreduce is using schema at read, which is ideal for messy and inconsistent data, and RDB is traditionally using schemas at write. In the world of Big Data the schema at read approach has the following advantages:

• the flexibility to store data of any kind including unstructured or semi-structured data
• it allows flexible data consumption
• it allows the storage of raw data for future processing and changing objectives
• it removes the cost of data formatting at the moment of data creation which results in faster data availability
• it allows you to experiment with the data at low risk as the raw data can be kept to correct mistakes

There is always the elephant in the room when speaking about MapReduce: Hadoop!

Most importantly, Hadoop is not MapReduce it is just one implementation of the mapreduce framework! Hadoop is quite a beast and targets the really BIG Big Data. An alternative implementation we are using here at Cambridge Energy Data Lab is Disco.

The main reason we use Disco over Hadoop: Disco jobs are written in Python and Hadoop jobs are mainly written in Java. (Strictly speaking you can also use other languages with Hadoop). Also Disco is much lighter and easier to administrate. [2]

The word count example in Disco is as simple as the underlying problem itself:

  from disco.core import Job, result_iterator

   def map(line, params):
       for word in line.split():
           yield word, 1

   def reduce(iter, params):
       from disco.util import kvgroup
       for word, counts in kvgroup(sorted(iter)):
           yield word, sum(counts)

   if __name__ == '__main__':
       input = ["http://discoproject.org/media/text/chekhov.txt"]
       job = Job().run(input=input, map=map, reduce=reduce)
       for word, count in result_iterator(job.wait()):
           print word, count

I leave it to you to compare this with the Java version for Hadoop: https://wiki.apache.org/hadoop/WordCount

6 screens for monitoring with Raspberry Pi cluster

Do you monitor?

Key performance indicators (KPI) , project progresses, server status, user logs (and many more) are constantly changing and the amount of data collected is growing bigger everyday. However it is hard to monitor these values, and sharing them with all the members of the team.

Instead, let's constantly display it.

We had too many things to display on a single monitor. Our solution was to buy 6 monitors!

6 screens for monitoring.

In order to handle 6 monitors, we would normally have to buy a massive desktop PC and a top-end graphic board. The solution we found is using Raspberry Pi! Raspberry Pi is one of the smallest computers in the world and it was born in Cambridge UK. Each Pi's performance is not perfect but it can be powerful enough if we assemble them together to form a cluster.

We immediately bought 6 Raspberry Pis. The main issue was to find a rack to store the 6 Pis. It's common to build a case from lego, however, we didn't have any Lego laying around. Our solution we found is using a shoebox!

Raspberry Pis in a Nike shoe case.

Just Do It!

It looks cool. Cambridge style monitoring environment. Try it like us!

Challenge for Excess Generation

Do you consider to buy a photovoltaic (PV) installation for your house? Or do you already have some solar panels on your roof and you are looking for ways to maximise your return from this investment? Read on as we have some important advice for you:

So far the PV industry's main growth is driven by the government's support with subsidies. Unfortunately, solar panel installations have not yet achieved a competitive advantage due to their still limited efficiency and high cost. This is bound to change in the future. However, if you consider investing into a domestic PV installation today, the governmentally backed financial incentives are key to your investment's return! Here is the secret.

What is Excess Generation?

Sometimes excess generation is also referred to as  surplus generation, excess electricity, or exported energy among others. Excess generation is defined as the amount of electricity generated by your rooftop panels (1. Total generation) minus your daytime electricity consumption (2. Electricity used). It is such excess generation which is available for export to a grid system (3. Export energy).

Feed-in-tariff Incentive Scheme. 

Feed-in tariffs (FITs) are the most widely used policy in the world for accelerating renewable energy (RE) deployment, accounting for a greater share of RE development than either tax incentives or renewable portfolio standard (RPS) policies. In the European Union (EU), FIT policies have led to the deployment of more than 15,000 MW of solar photovoltaic (PV) power and more than 55,000 MW of wind power between 2000 and the end of 2009. In total, FITs are responsible for approximately 75% of the global PV deployment.

In a grid connected rooftop photovoltaic power station, the generated electricity can be sold to the grid at a higher price than what the grid charges for the consumers. This arrangement provides a secure return for the installer’s investment. Many consumers from across the world are switching to this mechanism due to the revenue yielded. However, the details of the financial mechanism varies depending on countries as illustrated by two examples as follows:

Case Study 1: Desincentive for Excess Generation (UK)

In the UK, consumers have a stronger  incentive to minimise Excess Generation, by using the majority of their generated electricity on sunny days. The UK customers receive a guaranteed Feed-in-tariff for all electricity generation (10-14 p/kWh) , plus an 'Export tariff' (4.77 p/kWh) for their excess generation, which, however, is much smaller than the average electricity bill (12-15 p/kWh) . Therefore, customers should consume their generated electricity rather than export it to the grid.

As goes the theory.

However, in reality the ratio of excess generation is fixed to 50% of PV generation due to a lack of smart-meters. Thus, the importance of 'Excess Generation' will definitely emerge with the rollout of smart-meters in the near future.

Case Study 2: Incentive for Excess Generation (JAPAN)

In Japan, FiTs are only paid for 'Excess Generation', not "Total Generation' as it is the case in the UK. FiT’s price is currently much higher (38-42 JPY per kWh) than the average electricity bill (20-25 JPY per kWh), so customers have a strong financial incentive to maximise their amount of 'Excess Generation'. Therefore, customers are willing to change their consumption behaviour by shifting the usage of electricity-heavy appliances, such as dishwashers and wasching machines to the nighttime with cheaper electricity tariff. This individual behavioral change is expected to contribute to a nationwide peak-reduction in the future.

Our Challenge for Excess Generation

We are developing Eneberg, a Domestic PV Generation Forecasting and Trading Software. Eneberg is mainly dealing with aggregated "Excess Generation". Whilst there is a vast body of research and models dealing with  PV Generation and Energy Demand, "Excess Generation" is still an open frontier. It is our aim to pioneer in that new field of Excess Generation.

In-depth excess generation analysis is already covered by this post: Energy Surplus Trends from Domestic UK Solar Panels in October 2013 to January 2014

Artificial Neural Networks

An artificial neural network (ANN) is a computational model inspired by the information processing functionality of the brain. But how does the brain compute?

 Generally, the central elements of computation are processing, transmission, and storage. Within the brain the neuron is the central computing element. Neurons receive signals and produce responses. The transmission of information at the neural level involves electrical signals – so called action potentials – based broadly on ions and semi-permeable membranes, and chemical signals at the synapses. In the brain the storage of information corresponds to learning which occurs at the synapses. These synapses are at the interface between neurons and regulate the transmission of information from neuron to neuron.

 An ANN widely corresponds to the processing paradigm of neural networks with the nodes of the ANN being the central computing element similar to the neuron. In fact, ANNs are nothing but networks of primitive functions where the chain of function compositions transforms an input to an output. The composition of the computational model is contained implicitly in the interconnections of the nodes and is referred to as the network function. Each node comprises a primitive function transforming its input into an output:

Typically, the inputs of a node have an associated weight w by which the input x i is multiplied. The node integrates all its inputs – usually by adding the different inputs – followed by the evaluation of its primitive function f. The primitive function f computed in the node can be any function but common choices are differentiable functions such as the sigmoid function. Models of ANNs mainly differ in their choice of the primitive function, the topology of the network, and rarely in the timing of the evaluation of the primitive function. In feed-forward ANNs the network is composed of distinctive layers where each neuron only receives input from neurons of the previous layer. Accordingly, a feed-forward network has a distinct input and output layer with the intermediate layers being referred to as hidden layers:

(A second class of ANNs are recurrent networks where connections between nodes form directed cycles.)

The network function of an ANN can be understood as a universal function approximation. However, the difference between ANNs and a Taylor or Fourier series is that the function to be approximated is given not  explicitly but implicitly, through a representative set of input-output examples. It will be the task of the learning algorithm to adjust the parameters of the ANN to reflect the input-output examples and to extrapolate to new input patterns in an optimal manner. The learning algorithm is an adaptive method by which the network self-organises to reflect the function to be approximated. The computational effort directly relates to the number of parameters and therefore to the topology of the network and increases substantially for more complicated ANNs. It was not until the proposal of back-propagation as a learning algorithm [Werbos, 1974] that the application of ANNs gained momentum and it has been the most widely used algorithm for neural network learning ever since.

 The back-propagation algorithm uses gradient descent on the error function of an ANN in weight space. Thus, the weights of an ANN which minimise its error function are considered to be the solution of the learning problem. As a precondition for gradient descent the error function of an ANN needs to be continuous and differentiable. Since the ANN is simply the composition of its primitive functions the error function becomes differentiable if the networks primitive functions are differentiable themselves.
In the back-propagation algorithm an ANN is initialised randomly with weights. Next, the gradient of the error function is computed recursively and the weights of the ANN are adjusted accordingly using gradient descent. Because an ANN is a complex chain of a sequential function composition the chain rule plays a most important role in calculating the gradient of the network function's error. The back-propagation algorithm implements the chain rule for  the recursive calculation of the gradient of the error function in weight space in a very efficient manner.

Learning in an ANN with back-propagation consists of two stages: in the first stage – the feed-forward step – the information progresses form the input layer throughout the network towards the output layer.
Each node of the network evaluates its primitive function \(f_j(e)\) and emits the result \(y_j\) to the connected  nodes in the subsequent layer. Additionally, each node calculates and stores the derivative of its primitive function \(df_j(e)/de\).

The second stage -- the back-propagation step -- consists in reversing the flow of information throughout the network whereby a unit input propagates from the output layer towards the input layer with the activation of each neuron now being the back-propagation term \(\delta_j\).
At each node the back-propagation term \(\delta_j\) is multiplied by the stored derivative of the node's primitive function from the previous feed-forward step which gives the gradient in weight space \((d f_j(e)/de) \delta_j\).

Finally, the weights are updated using gradient descent as given by
$$
w'_{i,j} = w_{i,j} + \alpha y_{i} \frac{d f_j(e)}{de} \delta_j
$$
with \(\alpha\) being the learning rate and \(w_{i,j}\) being the weight of the feed-forward connection from neuron \(i\) in the previous layer to neuron \(j\) in the subsequent layer.


[Werbos, 1974] Beyond regression: New tools for prediction and analysis in the behavioural sciences, Pd.D. Thesis, Harvard University (1974).
[Gurney, 1997] An introduction to neural networks, UCL Press (1997).
[Montavon, 1998] Neural Networks: Tricks of the Trade, Springer (1998).