Weights: Initially, we have to pass some random values as values to the weights and these values get automatically … It takes an input, aggregates it (weighted sum) and returns 1 only if the aggregated sum is more than some threshold else returns 0. The numpy random.rand() function generates an array of length input_dim populated with random values distributed over the interval [0, 1). Training a Perceptron Model The Train() function is presented in Listing 2. Let us see the terminology of the above diagram. Gradient Descent minimizes the cost function by gradually updating the weight values. The training data set includes input values and corresponding output values. https://www.allaboutcircuits.com/technical-articles/understanding-simple-neural-network-training-and-learning/, How to Train a Basic Perceptron Neural Network. https://www.allaboutcircuits.com/technical-articles/understanding-simple-neural-network-training-and-learning/, That file contains data that are specific to this project. I use the pandas library to import training data from an Excel spreadsheet. Finally, the outputs are received at the output layer(nodes in green color). Don't have an AAC account? if negative, the sample is a female sample(Y = -1). So if we increase the weight on the neurons that lead to correct predictions. Training_Data = np.asarray(Training_Data) training_count = len(Training_Data[:,0]) ¶. The source of all motivation was from mother nature. ¶. Here, a good choice for ΔW would be -η*X (negative value), i.e., Note: η is called the learning rate (usually greater than 0). The maximum number of passes over the training data (aka epochs). The demo associates a sum of exactly 0.0 to class +1. Create one now. The perceptron model is a more general computational model than McCulloch-Pitts neuron. The discussion became a bit more interesting when I presented a network that created its own weights through the procedure known as training: In the rest of this article, we’ll examine the Python code that I used to obtain these weights. The demo program uses a variation of perceptron training called average perceptron. If you want to replicate this exact project, you can generate your own training data using the information in the following article. This is how I implemented the learning rule in Python: You now have code that you can use for training a single-layer, single-output-node Perceptron. There’s an optimization algorithm, called the Gradient Descent. It does this by using a cost/loss function, that penalizes/tells us the loss in case of misclassification. A Perceptron is a neural network unit that does certain computations to … The training data set includes input values and corresponding output values. Why Gradient Descent ? In other words, we will loop through all the inputs n_iter times training … After many years of research, Artificial Neural Networks were invented vaguely inspired from the biological neural networks inside our brain. ax+by+c = 0 — — — eqn (1) When we substitute the point P (x,y) in the equation, ax+by+c, it will give a value of 0 (Since P lies on the line). What can a perceptron do? Feel free to post your responses down in the response section below. That's included in the following article. That’s all well and good, but how does a perceptron “learn?” A perceptron “learns” through a process called “supervised learning” or “training.” Here’s how it works: First, you need a large set of example inputs (training data) for which you already know the correct output (active/inactive). Training error is the measure of how accurate a perceptron’s classification is for a specific training data sample. How to Use a Simple Perceptron Neural Network Example to Classify Data, Understanding Simple Neural Network Training, An Introduction to Training Theory for Neural Networks, Understanding Learning Rate in Neural Networks, Advanced Machine Learning with the Multilayer Perceptron, The Sigmoid Activation Function: Activation in Multilayer Perceptron Neural Networks, How to Train a Multilayer Perceptron Neural Network, Understanding Training Formulas and Backpropagation for Multilayer Perceptrons, Neural Network Architecture for a Python Implementation, How to Create a Multilayer Perceptron Neural Network in Python, Signal Processing Using Neural Networks: Validation in Neural Network Design, Training Datasets for Neural Networks: How to Train and Validate a Python Neural Network, theory and practice of neural-network training, Communicating Over Billions of Miles: Long Distance Communications in the Voyager Spacecraft, How to Use a Rotary Encoder in an MCU-Based Project, Leveraging the LPC55S16-EVK for Industrial Applications, Op-Amps as Low-Pass and High-Pass Active Filters. But, there’s one problem with this cost function, when the output is correctly classified, Cost, J = -Y(X.W) = “Some negative value”…. However, you can continue optimizing the weights by training the network multiple times using the same data set—the benefits of training don’t disappear simply because the network has already seen these training data. The inputs are propagated from the input layer to the hidden layer (nodes in blue color). Because a lazy person will find an easy way to do it.”. w0 = c ; w1 = a ; w2 = b ; x1 = x ; x2 = y. This aspect will be discussed in depth in subsequent articles. Similarly, there were efforts made to replicate the human brain. But, this is just a toy data, in real life applications, data is humongous, and we humans are too lazy to sit and go through each and every data point to find the equation of the decision boundary. Provide perceptron with inputs and known answer. The dimensionality is adjustable. We’ll discuss learning rate in a future article. If the output is correct then the next training example is presented to perceptron. I convert the training data set, which is currently a pandas data structure, into a numpy array and then look at the length of one of the columns to determine how many data points are available for training. Simple Perceptron Training Algorithm:Explained. Build a perceptron classifier. Where do I found your file “3D_data.xlsx”? If we want our model to train on non-linear data sets too, its better to go with neural networks. ¶ Step 1. ... A higher learning rate may increase training speed. 2017. What if a machine could do all this stuff? Today, lets build a perceptron model, which is nothing but a single node of a neural network. Lets understand the perceptron model with a simple classification problem. If the activation function or the underlying process being modeled by the perceptron is nonlinear, alternative learning algorithms such as the delta rule can be used as long as the activation function is differentiable. We’ll explore more details about the theory and practice of neural-network training in the next article. Train the Perceptron. Follow me for more such Machine learning and Deep Learning articles. Our input data, if you recall, consists of three-dimensional coordinates, so we need three input nodes. It only impacts the behavior in the fit method, and not the partial_fit method. 1. Hope that made everything clear. An if-else statement applies the unit-step activation function: if the summation is less than zero, the value generated by the output node is 0; if the summation is equal to or greater than zero, the output value is one. Many efforts were made, many inventions were invented, and eventually aeroplanes came into existence that enabled us to fly from one place to another. An artificial neural network looks something like this. It is a bad name because its most fundamental piece, the training algorithm, is completely different from the one in the perceptron. So, let us assume our cost function(J) as. The mathematical relationship required for this task was so simple that I was able to design the network just by thinking about how a certain set of weights would allow the output node to correctly categorize the input data. New in version 0.19. tol float, default=1e-3. However, the initial weight values influence the final weight values produced by the training procedure, so if you want to evaluate the effects of other variables (such as training-set size or learning rate), you can remove this confounding factor by setting all the weights to a known constant instead of a randomly generated number. Not to say the game is simple. The next step should be to create a step function. For multilayer perceptrons, where a hidden layer exists, more sophisticated algorithms such as backpropagation must be used. Hence, we’ll use the perceptron model that’ll find the equation of the decision boundary for us. SUV Purchase Prediction Using Logistic Regression, Building a Sentiment Analyzer With Naive Bayes, Predict your Wine Quality using Deep Learning with PyTorch, SFU Professional Master’s Program in Computer Science, Neural storytelling: Skip thought vectors and Visual attention modelling. Now, let us consider the following basic steps of training logistic regression − The weights are initialized with random values at the beginning of the training. Perceptron Preprocessing After fetching the X and Y variables, we will perform Min-Max scaling to bring all the features in the range 0 — … It essentially measures “how bad” the perceptron is performing and helps … The concepts behind a neural network have been distilled to their essence in this idle simulation. A perceptron consists of one or more inputs, a processor, and a single output. You can just go through my previous post on the perceptron model (linked above) but I will assume that you won’t. The type of training and the optimization algorithm determine which training options are available. The general equation of a straight line is. The inputs to the neural network are fed to the input layer(the nodes in red color). Training a Perceptron. It can identify objects, recognize patterns, classify things, and much more. How to Perform Classification Using a Neural Network: What Is the Perceptron? Content created by webstudio Richter alias Mavicc on March 30. The perceptron can be used for supervised learning. but the cost function can’t be negative, so we’ll define our cost functions as follows. Multilayer Perceptron. Perceptrons are especially suited for simple problems in pattern classification. It is used to update the weights in case of misclassification. Keep reading to find out. Perceptron This is a simple binary perceptron demo. Truth be told, “multilayer perceptron” is a terrible name for what Rumelhart, Hinton, and Williams introduced in the mid-‘80s. Training examples are presented to perceptron one by one from the beginning, and its output is observed for each training example. The first instruction separates the output values and stores them in a separate array, and the next instruction removes the output values from the training data set. In the Multilayer Perceptron dialog box, click the Training tab. The perceptron algorithm • One of the oldest algorithm in machine learning introduced by Rosenblatt in 1958 • the perceptron algorithm is an online algorithm for learning a linear classiﬁer • an online algorithm is an iterative algorithm that takes a single paired example at -iteration, and computes the updated iterate according to some rule How did we get ΔW = Y*(η*X)? Here, a good choice for ΔW would be η*X (positive value), i.e.. Each complete pass through the entire training set is called an epoch. Lets can combine eqns (4) and (5) as follows. Perceptron models can only learn on linearly separable data. So here goes, a perceptron is not the Sigmoid neuron we use in ANNs or any deep learning networks today. When we substitute the point P(x,y) in the equation, ax+by+c, it will give a value of 0(Since P lies on the line). Humans saw birds flying and wanted to invent something so that they could fly too. ... Perceptron is a classification algorithm which shares the same underlying implementation with SGDClassifier. The next article will go into more detail on the training data. The previous article demonstrated that a single-layer Perceptron simply cannot produce the sort of performance that we expect from a modern neural-network architecture. Training a perceptron is based on the idea, that the weights between the input and the output neurons. If the resultant value is positive, the sample belongs to class Male(Y = 1). The output node must sum the values delivered by the three input nodes. Let’s take a closer look at these instructions. Step 2: Ask perceptron to guess an answer. Are going to influence the decision, that is made by that threshold logic unit. 3. Related information. “I choose a lazy person to do a hard job. Rewriting the threshold as sho… This value does not matter much in the case of a single perceptron, but in more compex neural networks, the algorithm may diverge if the learning rate is … Each node in a neural network has some function associated with it, each connection/edge has some weight value. A comprehensive description of the functionality of a perceptron is out of scope here. Would be nice if there was a photo of a row or two of the training data to show what was imported, what a row looks like. The perceptron needs supervised learning so the training set will consist of objects from the space X labelled as belonging or not to the binary class or category we look into. Perceptron is a new incremental game which is based around the idea of building and training a neural network. A number of researchers tried to understand the working of a human brain. All we have to do is feed the input and output data for the model to train. Check out my github repository to see Perceptron training algorithm in action!! We have the values of x1 and x2. Create a Step Function. In our previous post, we discussed about training a perceptron using The Perceptron Training Rule. We humans are so enthusiastic that we look at different things in nature and try to replicate it in our own way. Using this intuition, we can classify any point by substituting its value in the line equation. The activation function in the output node is the unit step: $f(x)=\begin{cases}0 & x < 0\\1 & x \geq 0\end{cases}$. We initialize the perceptron class with a learning rate of 0.1 and we will run 15 training iterations. we need to update the Weights in such a way that. Perceptron Training Rule (Rosenblatt’s Rule): Works well when training samples are linearly separable. 2. The sample is said to be misclassified if. The procedure contained in this loop occurs one time for each row in the training set, where “row” refers to a group of input data values and the corresponding output value (in our case, an input group consists of three numbers representing x, y, and z components of a point in three-dimensional space). In Chapter 2, Training Simple Machine Learning Algorithms for Classification, you learned about two related learning algorithms for classification, the perceptron rule and Adaline, which we implemented in Python and NumPy by ourselves. Our motive is to fit a decision boundary(a line) that separates all the male samples from the female samples. From eqn(7), we have the misclassification condition. Which means that “ -Y*(X.W) ” gives us a positive value for misclassification of input X. The delta rule works both if the perceptron's output is too large and if it is too small. If you're looking to start from the beginning for background or jump ahead, check out the rest of the articles here: The previous article introduced a straightforward classification task that we examined from the perspective of neural-network-based signal processing. For mathematical convenience, lets vectorize eqn(2) as follows, we can define eqn(2) as dot product of vectors W and X, If we successfully train our model and obtain optimum values of vector W, then eqn(3) should make classifications as follows…. It can solve binary linear classification problems. Then of course, that is going to influence the decision made by the perceptron. For example, our training set may consist of 100 fruits represented by their prices and weights and labelled as ‘watermelons” or “not watermelons”. Like their biological counterpart, ANN’s are built upon simple signal processing elements that are connected together into a large mesh. Implementation of Perceptron Algorithm Python Example Import the Libraries. Perceptron models(with slight modifications), when connected with each other, form a neural network. The length of one training session is governed by the number of training data available. A Perceptron in just a few Lines of Python Code. w0 + w1 * x1 + w2 * x2 = 0 — — — eqn (2) w0 * 1 + w1 * … Invented in 1957 by Frank Rosenblatt at the Cornell Aeronautical Laboratory , a perceptron is the simplest neural network possible: a computational model of a single neuron. The first instruction separates the output values and stores them in a separate array, and the next instruction removes the output values from the training data set. The training technique used is called the perceptron learning rule. Until then, don’t forget to feed your curiosity!! Sure, it starts simple with only nodes, training, and data, but soon balloons into a complex idle game with prestige and upgrades. Similarly, when we substitute the point Q(x,y) in the equation, ax+by+c, it will give us a value greater than 0(Since Q lies above the line)., and. As we have discussed earlier, the perceptron training rule works for the training… Have you ever wondered why there are tasks that are dead simple for any human but incredibly difficult for computers?Artificial neural networks(short: ANN’s) were inspired by the central nervous system of humans. Damn, now we got 2 constraints to satisfy(eqns 4 and 5). The perceptron generated great interest due to its ability to generalize from its training vectors and learn from initially randomly distributed connections. Wouldn’t that be cool? If we define delta ($$\delta$$) as ($$output_{expected} - output_{calculated}$$), we can rewrite this as, $w_{new} = w+(\alpha\times\delta\times input)$. Note that we need to calculate the partial derivative of the cost function(J), with respect to weights W. Substituting the partial derivatives in gradient descent algorithm, if, -Y(X.W) < 0 , (Correct Classification). The following diagram is the architecture of perceptron for multiple output classes. Many AI models are invented that could classify things, predict future, play games better than humans, and even communicate with us. Gradient descent updates the weights as shown above. Where n represents the total number of features and X represents the value of the feature. When the first output calculation is complete, we have weight values, but they don’t help us achieve classification because they are randomly generated. Weight modification is accomplished by applying the following learning rule for each row in the training set: $w_{new} = w+(\alpha\times(output_{expected}-output_{calculated})\times input)$, The symbol $$\alpha$$ denotes the learning rate. Human brain is really an amazing thing. This section provides a brief introduction to the Perceptron algorithm and the Sonar dataset to which we will later apply it. If the output is incorrect then the weights are modified as per the following formula. Let us summarize what we have learned in this lesson: An artificial neuron is a mathematical function conceived as a model of biological neurons, that is, a neural network. Today, as in 2018, we have come a long way in Artificial Intelligence. In this blog, we will learn about The Gradient Descent and The Delta Rule for training a perceptron and its implementation using python. Alright, So we can conclude that our model correctly classifies the sample X if. How to Train Artificial Neural Networks (ANN) Single layer neural network (or perceptrons) can be trained using either the Perceptron training rule or the Adaline rule. The new Weight i is found simply by adding the change for Weight i to the current value of Weight i.. Interestingly, if you graph the possible inputs on different axes of a mathematical graph, with pluses for where the perceptron fires and minuses where the perceptron doesn't, the weights … The best example to illustrate the single layer perceptron is through representation of “Logistic Regression”. We need the values of w0, w1, w2. A neural network is a collection of neurons/nodes interconnected with each other through synaptic connections. The Training tab is used to specify how the network should be trained. Hence, that’s how we got “W = W + η * (Y*X)” for cases of misclassification. For perceptron classification, a sum of products of 0.0 must be arbitrarily associated to either class -1 or class +1. Parent topic: Multilayer Perceptron. Below is an example of a learning algorithm for a single-layer perceptron. Training Datasets for Neural Networks: How to Train and Validate a Python Neural Network . Input: All the features of the model we want to train the neural network will be passed as the input to it, Like the set of features [X1, X2, X3…..Xn]. Now we will take a look at the scikit-learn API, which, as mentioned, combines a user-friendly and consistent interface with a highly … Now, to start off, we’ll randomly initialize the Weight vector W and for each misclassification we’ll update the weights as follows. Training Algorithm for Multiple Output Units. Since we are training the perceptron with stochastic gradient descent (rather than the perceptron learning rule) it is necessary to intialise the weights with non-zero random values rather than initially set them to zero. We turn the neural network into an effective classifying system by repeatedly modifying the weights such that they gradually reflect the mathematical relationship between the input data and the desired output values. This program does not support multiple output nodes, but we’ll incorporate adjustable output dimensionality into a future experiment. To use our perceptron class, we will now run the below code that will train our model. when we substitute the point R(x,y) in the equation ax+by+c, it will give us a value less than 0(Since R lies below the line). Nonetheless, the learning algorithm described in the steps below will often work, even for multilayer perceptrons with nonlinear activation functions. Welcome to AAC's series on Perceptron neural networks. Logical functions are a great starting point since they will bring us to a natural development of the theory behind the perceptron and, as a consequence, neural networks. I am introducing some examples of what a perceptron can implement with its capacity (I will talk about this term in the following parts of this series!). Well, we can do it by hand, try to find the equation of a line that separates both the classes. 4. Finally, to summarize Perceptron training algorithm. What Is a Multilayer Perceptron Neural Network? Thus, to calculate a new weight value, we multiply the corresponding input value by the learning rate and by the difference between the expected output (which is provided by the training set) and the calculated output, and then the result of this multiplication is added to the current weight value. My Python implementation does this by first performing an element-wise multiplication of the Training_Data array and the Weights array and then calculating the summation of the elements in the array produced by that multiplication. Weights are typically initialized to random values. where ΔW is a small change that we will make in W. Let’s Examine each misclassification case. Logic unit introduction to the neural network are fed to the perceptron to hidden. Next training example more general computational model than McCulloch-Pitts neuron learning networks today ” gives us a positive value,... To see perceptron training Rule works both if the output is too small Rule for training a and... Train our model down in the following formula example is presented to perceptron t negative! Hand, try to replicate this exact project, you can generate your training! The training… Train the perceptron a closer look at these instructions misclassification case the. Perceptron generated great interest due to its ability to generalize from its training vectors and from... Layer perceptron is a female sample ( Y = -1 ) together a. 4 ) and ( 5 ) as follows fundamental piece, the learning algorithm a... Found your file “ 3D_data.xlsx ” are connected together into a future article Ask perceptron to guess answer! Correct predictions model with a learning algorithm described in the line equation illustrate single... Games better than humans, and its implementation using Python only impacts the behavior the... But a single node of a neural network dialog box, click the training tab used... Future article ( the nodes in red color ) will make in W. Let ’ Examine! Got 2 constraints to satisfy ( eqns 4 and 5 ) the source of all motivation was from nature! It, each connection/edge has some function associated with it, each connection/edge has weight. Through synaptic connections run the below code that will Train our model classifies! Examples are presented to perceptron representation of “ Logistic Regression ” ll more! Me for more such machine learning and deep learning articles with SGDClassifier the one in next! The sample is a collection of neurons/nodes interconnected with each other through synaptic connections your file “ 3D_data.xlsx?. Any deep learning networks today ; x1 = X ; x2 = Y from a modern neural-network architecture do ”! The Sigmoid neuron we use in ANNs or any deep learning networks today great interest due its... Its ability to generalize from its training vectors and learn from initially randomly distributed connections work even! Next article a learning algorithm for a specific training data ( aka epochs ) in ANNs or any deep articles. For each training example, each connection/edge has some function associated with it, connection/edge! Examine each misclassification case from its training vectors and learn from initially randomly distributed connections w1, w2 different. Substituting its value in the response section below, you can generate own. Are built upon simple signal processing elements that are specific to this project a closer look at these.. Mother nature with it, each connection/edge has some function associated with,. Class male ( Y = 1 ) post your responses down in the multilayer dialog. Functions as follows an example of a learning rate in a neural network perceptrons nonlinear... And ( 5 ) Let us see the terminology of the above diagram the algorithm. Will make in W. Let ’ s are built upon simple signal processing elements that connected. Rule ): works well when training samples are linearly separable data to guess an.. Can only learn on linearly separable can conclude that our model correctly classifies the belongs... Higher learning rate in a future experiment class male ( Y = -1 ) could. That our model to Train and Validate a Python neural network have been distilled to their in. Through the entire training set is called the perceptron training called average perceptron classification algorithm shares... The concepts behind a neural network: What is the measure of how a! Δw would be η * X ( positive value for misclassification of X! A single output, called the Gradient Descent minimizes the cost function ( J ) follows. Of all motivation was from mother nature the terminology of the decision made by the three nodes..., ANN ’ s Examine each misclassification case we get ΔW = Y it... Training a perceptron using the information in the next article will go into detail. Will learn about the theory and practice of neural-network training in the perceptron is through representation of “ Logistic ”. Of features and X represents the total number of researchers tried to understand the perceptron understand working... Used is called the perceptron generated great interest due to its ability to generalize from training! When training samples are linearly separable data perceptrons are especially suited for simple problems in pattern classification ;! Expect from a modern neural-network architecture misclassification of input X learning algorithm for a specific training available... Simply can not produce the sort of performance that we will later it! Measure of how accurate a perceptron ’ s take a closer look at these instructions function can ’ t to! Demonstrated that a single-layer training a perceptron simply can not produce the sort of that! Cost/Loss function, that penalizes/tells us the loss in case of misclassification replicate this exact project, you generate! Need the values of w0, w1, w2 learning articles ability to generalize its... That “ -Y * ( X.W ) ” gives us a positive value for misclassification of input.. Sample is a neural network class, we ’ ll define our cost function by gradually updating weight! Suited for simple problems in pattern classification = -1 ) ( X.W ) ” gives a... Content created by webstudio Richter alias Mavicc on March 30 where do found. Many years of research, Artificial neural networks: how to Train and Validate a Python network. W1, w2 only learn on linearly separable line ) that separates both the classes got constraints. Humans saw birds flying and wanted to invent something so that they could fly too male! Enthusiastic that we will run 15 training iterations: What is the measure of how accurate a perceptron is neural... Data, if you want to replicate the human brain alias Mavicc on 30! Function, that penalizes/tells us the loss in case of misclassification by one from the,... How to Train both the classes depth in subsequent articles a single of. Should be to create a step function than McCulloch-Pitts neuron because a lazy will! Validate a Python neural network unit that does certain computations to … Let us assume cost. To create a step function below code that will Train our model correctly the! Our own way, recognize patterns, classify things, and its output is correct the. Does not support multiple output nodes, but we ’ ll use the perceptron with... 0.1 and we will run 15 training iterations efforts made to replicate it in our own way course that! Neurons that lead to correct predictions Descent minimizes the cost function ( J ) as follows class male ( =. Following article by hand, try to replicate it in our previous post, we will about! Initialize the perceptron algorithm Python example Import the Libraries lazy person will find easy! Models ( with slight modifications ), when connected with each other through synaptic.! The response section below cost function by gradually updating the weight on the neurons that lead correct... About the Gradient Descent minimizes the cost function can ’ t be negative, the sample to... Train on non-linear data sets too, its better to go with networks... We can classify any point by substituting its value in the steps below will work. Got 2 constraints to satisfy ( eqns 4 and 5 ) the network should to! Sum the values delivered by the three input nodes perceptron to guess an answer perceptron the... So that they could fly too ; w2 = b ; x1 = ;... To fit a decision boundary ( a line that separates all the male samples the. In red color ) is through representation of “ Logistic Regression ” elements that are specific to this project technique. Functionality of a learning algorithm described in the steps below will often work, even for multilayer perceptrons with activation. Of one or more inputs, a perceptron is out of scope.. ( with slight modifications ), i.e for misclassification of input X model correctly classifies sample. 0.1 and we will now run the below code that will Train our model correctly classifies the belongs... A classification algorithm which shares the same underlying implementation with SGDClassifier the total number of researchers tried understand... Represents the value of the above diagram to … Let us assume cost... Large mesh Y = -1 ) class with a learning algorithm described in the fit method, and the! Out my github repository to see perceptron training algorithm, called the perceptron input and output data for the to! Presented to perceptron one by one from the female samples use the pandas to. The neurons that lead to correct predictions network: What is the architecture of perceptron Rule. Are built upon simple signal processing elements that are connected together into a mesh! Perceptron 's output is incorrect then the weights in such a way that brain. To influence the decision boundary for us and not the partial_fit method all we have a! On March 30 that file contains data that are specific to this project perceptron ’ s classification for! And wanted to invent something so that they could fly too a lazy person will find an easy to. Of three-dimensional coordinates, so we ’ ll find the equation of the functionality of a line ) that both...

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## The Life Underground

### ## Cooling Expectations for Copenhagen Nov.16.09 | Comments (0)As the numbers on the Copenhagen Countdown clock continue to shrink, so too do e ...

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