I'm trying to implementing a Finite Element Analysis algorithm. I solve K u = f to get the displacement u, and then calculate strain with u, then calculate the stress. Finally, I use the stress to calculate the Von Mises Stress, and visualize this. From the result I find the strain is not continuous between tetrahedrons.
I use 10 nodes tetrahedron as the element, so the displacement is a second-order polynomial in every element. The displacement should be enforced to be continuous between tetrahedrons. And the strain, which is the first order derivatives of the displacements should be continuous inside every tetrahedron. But I'm not sure: is this true across the interface between tetrahedrons?
Only the components of strain tangent to the adjoining face are guaranteed continuous.
This follows from the displacement continuity, when you take derivatives in the direction of the interface they are the same.
Commercial FEM programs typically do some post process averaging to make the other components look continuous. Note the strain components normal to an element boundary are only expected to be continuous if the underlying constitutive model is continuous, so such averaging is not always appropriate.
You should not compute the stress and strain at the nodes but inside the elements. You can choose for example 4 Gauss points and compute the values there. You then have to think about a scheme on how to get the values computed at the Gauss points onto the tet nodes.
There is a Mathematica application example which illustrates this. Unfortunately the web page is no longer available, but the notebooks are here. You'll find the example in the application example section under Finite Element Method, Structural Mechanics 3D (in the old HelpBrowser). If you have difficulties I could convert it to PDF and send it you.
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I am trying to use Reinforcement Learning for traffic signal phase optimization for improving traffic flow at intersections.
I am aware that in practice we won't be able to get the information about all the vehicles in each of the lanes.
If we use a camera for getting information about the queue length then we can get accurate data only upto, say 200 meters.
Should I take this into consideration while defining my observation space or can I directly use the data from sumo?
Furthermore, what should be the ideal observation space for such a task?
sumo_rl allows to use various metrics for reward calucation such as pressure metric, queue length metric, etc. What will be a good choice of rewards for my use case or what factors should I consider while defining my reward?
I have tried getting metrics from the e2 detector's output file such as throughput, lane delay and queue length. For the agent however, I might not be able to use them (as traci/sumo wrappers offer better implementations?) So how do I use traci for getting this modified information?
Yes, you should try to match your observation space as close to the real world as possible. SUMO can also filter the data directly (for instance with an E3 detector).
If you want to maximize flow than the reward should also include the flow metric (throughput). It's quite easy to get it via traci (as you already noticed) but I cannot tell how it integrates with your framework since you did not give details about it.
I have been reading the Deep Learning book by Ian Goodfellow and it mentions in Section 6.5.7 that
The main memory cost of the algorithm is that we need to store the input to the nonlinearity of the hidden layer.
I understand that backprop stores the gradients in a similar fashion to dynamic programming so not to recompute them. But I am confused as to why it stores the input as well?
Backpropagation is a special case of reverse mode automatic differentiation (AD).
In contrast to the forward mode, the reverse mode has the major advantage that you can compute the derivative of an output w.r.t. all inputs of a computation in one pass.
However, the downside is that you need to store all intermediate results of the algorithm you want to differentiate in a suitable data structure (like a graph or a Wengert tape) for as long as you are computing its Jacobian with reverse mode AD, because you're basically "working your way backwards" through the algorithm.
Forward mode AD does not have this disadvantage, but you need to repeat its calculation for every input, so it only makes sense if your algorithm has a lot more output variables than input variables.
I am trying to train a learning model to recognize one specific scene. For example, say I would like to train it to recognize pictures taken at an amusement park and I already have 10 thousand pictures taken at an amusement park. I would like to train this model with those pictures so that it would be able to give a score for other pictures of the probability that they were taken at an amusement park. How do I do that?
Considering this is an image recognition problem, I would probably use a convolutional neural network, but I am not quite sure how to train it in this case.
Thanks!
There are several possible ways. The most trivial one is to collect a large number of negative examples (images from other places) and train a two-class model.
The second approach would be to train a network to extract meaningful low-dimensional representations from an input image (embeddings). Here you can use siamese training to explicitly train the network to learn similarities between images. Such an approach is employed for face recognition, for instance (see FaceNet). Having such embeddings, you can use some well-established methods for outlier detections, for instance, 1-class SVM, or any other classifier. In this case you also need negative examples.
I would heavily augment your data using image cropping - it is the most obvious way to increase the amount of training data in your case.
In general, your success in this task strongly depends on the task statement (are restricted to parks only, or any kind of place) and the proper data.
I have extracted features from many images of isolated characters (such as gradient, neighbouring pixel weight and geometric properties. How can I use HMMs as a classifier trained on this data? All literature I read about HMM refers to states and state transitions but I can't connect it to features and class labeling. The example on JAHMM's home page doesn't relate to my problem.
I need to use HMM not because it will work better than other approaches for this problem but because of constraints on project topic.
There was an answer to this question for online recognition but I want the same for offline and in a little more detail
EDIT: I partitioned each character into a grid with fixed number of squares. Now I am planning to perform feature extraction on each grid block and thus obtain a sequence of features for each sample by moving from left to right and top to bottom.
Would this represent an adequate "sequence" for an HMM i.e. would an HMM be able to guess the temporal variation of the data, even though the character is not drawn from left to right and top to bottom? If not suggest an alternate way.
Should I feed a lot of features or start with a few? how do I know if the HMM is underforming or if the features are bad? I am using JAHMM.
Extracting stroke features is difficult and cant be logically combined with grid features? (since HMM expects a sequence generated by some random process)
I've usually seen neural networks used for this sort of recognition task, i.e. here, here here, and here. Since a simple google search turns up so many hits for neural networks in OCR, I'll assume you are set in using HMMs (a project limitation, correct?) Regardless, these links can offer some insight into gridding the image and obtaining image features.
Your approach for turning a grid into a sequence of observations is reasonable. In this case, be sure you do not confuse observations and states. The features you extract from one block should be collected into one observation, i.e. a feature vector. (In comparison to speech recognition, your block's feature vector is analogous to the feature vector associated with a speech phoneme.) You don't really have much information regarding the underlying states. This is the hidden aspect of HMMs, and the training process should inform the model how likely one feature vector is to follow another for a character (i.e. transition probabilities).
Since this is an off-line process, don't be concerned with the temporal aspects of how characters are actually drawn. For the purposes of your task, you've imposed a temporal order on the sequence of observations with your the left-to-right, top-to-bottom block sequence. This should work fine.
As for HMM performance: choose a reasonable vector of salient features. In speech recog, the dimensionality of a feature vector can be high (>10). (This is also where the cited literature can assist.) Set aside a percentage of the training data so that you can properly test the model. First, train the model, and then evaluate the model on the training dataset. How well does classify your characters? If it does poorly, re-evaluate the feature vector. If it does well on the test data, test the generality of the classifier by running it on the reserved test data.
As for the number of states, I would start with something heuristically derived number. Assuming your character images are scaled and normalized, perhaps something like 40%(?) of the blocks are occupied? This is a crude guess on my part since a source image was not provided. For an 8x8 grid, this would imply that 25 blocks are occupied. We could then start with 25 states - but that's probably naive: empty blocks can convey information (meaning the number of states might increase), but some features sets may be observed in similar states (meaning the number of states might decrease.) If it were me, I would probably pick something like 20 states. Having said that: be careful not to confuse features and states. Your feature vector is a representation of things observed in a particular state. If the tests described above show your model is performing poorly, tweak the number of states up or down and try again.
Good luck.
I don't have much experience in machine learning, pattern recognition, data mining, etc. and in their underlying theory and systems.
I would like to develop an artificial model of the time it takes a human to make a move in a given Sudoku puzzle.
So what I'm looking for as an output from the machine learning process is a model that can give predictions on how long does it take for a target human to make a move in a given Sudoku situation.
Same input doesn't always map to same outcome. It takes different times for the human to make a move with the same situation, but my hypothesis is that there's a tendency in the resulting probability distribution. (My educated guess is that it is ~normal.)
I have ideas about the factors that influence the distribution (like #empty slots) but would preferably leave it to the system to figure these patterns out. Please notice, that I'm not interested in the patterns, just the model.
I can generate sample and test data easily by running sudoku puzzles and measuring the times it takes to make the moves.
What kind of learning algorithm would you suggest to use for this?
I was thinking NNs, but I'm not sure if they can have the desired property of giving weighted random outcomes for the same input.
If I understand this correctly you have an input vector of length 81, which contains 1 if the square is filled in and 0 otherwise. You want to learn a function which returns a probability distribution which models the response time of a human to that board position.
My first response would be that this is a regression problem and you should try straightforward linear regression. This will not provide you with a distribution of response times, but a single 'best-guess' response time.
I'm not clear on why you want to model a distribution of response times. However, if you really want to do want to output a distribution then it sounds like you want to look at Bayesian methods. I'm not really an expert on Bayesian inference, so I can't help you much further here.
However, I don't really think your approach is going to work because I agree with your intuition about features such as the number of empty slots being important. There are also other obvious features, such as the number of empty slots per row/column that are likely to be important. Explicitly putting these features in your representation will probably be much more successful than expecting that the learning algorithm will infer something similar on its own.
The monte carlo method seems like it would work well here but would require a stack of solutions the size of the moon to really do it. And it wouldn't give you the time per person, just the time on average.
My understanding of it, tenuous as it is, is that you have a database with a board position and the time it took a human to make the next move. At the very least you have a starting point for most moves. Even if it's not in the database you could start to calculate how long it would take to make a move based on some algorithm. Though I know you had specified you wanted machine learning to do this it might be worth segmenting the problem into something a little smaller then building on it.
If you have some guesstimate as to what influences the function (# of empty cell, etc), try to train a classifier on a vector of features, and not on the 81 cells vector (0/1 or 0..9, doesn't really matter for my argument).
I think that your claim:
we wouldn't have to necessary know the underlying patterns, the "trained patterns" in a learning system automatically encodes these sometimes quite delicate and subtle patterns inside them -- that's one of their great power
is wrong. you do have to give the network the right domain. for example, when trying to detect object in an image, working in the pixel domain is pointless. you'll only get results if you first run some feature detection to detect edges, corners, etc.
Theoretically, with enough non-linearity (in NN - enough layers in the network) it can detect such things, but in practice, I have never seen that work, without giving the classifier the right features to work with.
I was thinking NNs, but I'm not sure if they can have the desired property of giving weighted random outcomes for the same input.
You're just trying to learn a function from 2^81 or 10^81 (or a much smaller feature space as I suggest) to R (response time between 0 and Inf) or some discretization of that. So NN and other classifiers can do that.