I am asking because I have observed sometimes in neuroimaging that a brain region might have different average activation between two experimental conditions, but sometimes an SVM classifier somehow can't distinguish the patterns of activation between the two conditions.
My intuition is that this might happen in cases where the within-class variance is far greater than the between-class variance. For example, suppose we have two classes, A and B, and that for simplicity our data consists just of integers (rather than vectors). Let the data falling under class A be 0,0,0,0,0,10,10,10,10,10. Let the data falling under class B be 1,1,1,1,1,11,11,11,11,11. Here, A and B are clearly different on average, yet there's no decision boundary that would allow A and B to be distinguished. I believe this logic would hold even if our data consisted of vectors, rather than integers.
Is this a special case of some broader range of cases where an SVM would fail to distinguish two classes that are different on average? Is it possible to delineate the precise conditions under which an SVM classifier would fail to distinguish two classes that differ on average?
EDIT: Assume a linear SVM.
As described in the comments - there are no such conditions because SVM will separate data just fine (I am not talking about any generalisation here, just separating training data). For the rest of the answer I am assuming there are no two identical points with different labels.
Non-linear case
For a kernel case, using something like RBF kernel, SVM will always perfectly separate any training set, given that C is big enough.
Linear case
If data is linearly separable then again - with big enough C it will separate data just fine. If data is not linearly separable, cranking up C as much as possible will lead to smaller and smaller training error (of course it will not get 0 since data is not linearly separable).
In particular for the data you provided kernelized SVM will get 100%, and any linear model will get 50%, but it has nothing to do with means being different or variances relations - it is simply a dataset where any linear separator has at most 50% accuracy, literally every decision point, thus it has nothing to do with SVM. In particular it will separate them "in the middle", meaning that the decision point will be somewhere around "5".
Related
Most recent object detection methods rely on a convolutional neural network. They create a feature map by running input data through a feature extraction step. They then add more convolutional layers to output a set of values like so (this set is from YOLO, but other architectures like SSD differ slightly):
pobj: probability of being an object
c1, c2 ... cn: indicating which class the object belongs to
x, y, w, h: bounding box of the object
However, one particular box cannot be multiple objects. As in, wouldn't having a high value for, say, c1 mean that the values for all the others c2 ... cn would be low? So why use different values for c1, c2 ... cn? Couldn't they all be represented by a single value, say 0-1, where each object has a certain range within the 0-1, say 0-0.2 is c1, 0.2-0.4 is c2 and so on...
This would reduce the dimension of the output from NxNx(5+C) (5 for the probability and bounding box, +C one for each class) to NxNx(5+1) (5 same as before and 1 for the class)
Thank you
Short answer, NO! That is almost certainly not an acceptable solution. It sounds like your core question is: Why is a a single value in the range [0,1] not a sufficient, compact output for object classification? As a clarification, I'd say this doesn't really have to do with single-shot detectors; the outputs from 2-stage detectors and most all classification networks follows this same 1D embedding structure. As a secondary clarification, I'd say that many 1-stage networks also don't output pobj in their original implementations (YOLO is the main one that does but Retinanet and I believe SSD does not).
An object's class is a categorical attribute. Assumed within a standard classification problem is that the set of possible classes is flat (i.e. no class is a subclass of any other), mutually exclusive (each example falls into only a single class), and unrelated (not quite the right term here but essentially no class is any more or less related to any other class).
This assumed attribute structure is well represented by an orthonormal encoding vector of the same length as the set of possible attributes. A vector [1,0,0,0] is no more similar to [0,1,0,0] than it is to [0,0,0,1] in this space.
(As an aside, a separate branch of ML problems called multilabel classification removes the mutual exclusivity constrain (so [0,1,1,0] and [0,1,1,1] would both be valid label predictions. In this space class or label combinations COULD be construed as more or less related since they share constituent labels or "basis vectors" in the orthonormal categorical attribute space. But enough digression..)
A single, continuous variable output for class destroys the assumption that all classes are unrelated. In fact, it assumes that the relation between any two classes is exact and quantifiable! What an assumption! Consider attempting to arrange the classes of, let's say, the ImageNet classification task, along a single dimension. Bus and car should be close, no? Let's say 0.1 and 0.2, respectively in our 1D embedding range of [0,1]. Zebra must be far away from them, maybe 0.8. But should be close to zebra fish (0.82)? Is a striped shirt closer to a zebra or a bus? Is the moon more similar to a bicycle or a trumpet? And is a zebra really 5 times more similar to a zebra fish than a bus is to a car? The exercise is immediately, patently absurd. A 1D embedding space for object class is not sufficiently rich to capture the differences between object classes.
Why can't we just place object classes randomly in the continuous range [0,1]? In a theoretical sense nothing is stopping you, but the gradient of the network would become horrendously, unmanageably non-convex and conventional approaches to training the network would fail. Not to mention the network architecture would have to encode extremely non-linear activation functions to predict the extremely hard boundaries between neighboring classes in the 1D space, resulting in a very brittle and non-generalizable model.
From here, the nuanced reader might suggest that in fact, some classes ARE related to one another (i.e. the unrelated assumption of the standard classification problem is not really correct). Bus and car are certainly more related than bus and trumpet, no? Without devolving into a critique on the limited usefulness of strict ontological categorization of the world, I'll simply suggest that in many cases there is an information embedding that strikes a middle ground. A vast field of work has been devoted to finding embedding spaces that are compact (relative to the exhaustive enumeration of "everything is its own class of 1") but still meaningful. This is the work of principal component analysis and object appearance embedding in deep learning.
Depending on the particular problem, you may be able to take advantage of a more nuanced embedding space better suited towards the final task you hope to accomplish. But in general, canonical deep learning tasks such as classification / detection ignore this nuance in the hopes of designing solutions that are "pretty good" generalized over a large range of problem spaces.
For object classification head, usually cross-entropy loss function is used which operates on the probability distribution to compute the difference between ground-truth(a one hot encoded vector) and prediction class scores.
On the otherhand, you are proposing a different way of encoding the ground-truth class labels which can be further used with certain custom loss function say L1/l2 loss function, which looks theoretically correct but it might not be as good as cross-entropy function in terms of model convergence/optimization.
I am new to using linear mixed-effects models. I have a dataset where participants (ID, N = 7973) completed two experimental conditions (A and B). A subset of participants are siblings and thus nested in families (famID, N = 6908).
omnibus_model <- lmer(Outcome ~ Var1*Var2*Cond + (Cond|ID) + (1|famID), data=df)
The omnibus model converges and indicates a significant three way interaction between Var1, Var2 and Cond. As a post-hoc, to better understand what is driving the omnibus model effect, I subsetted the data so that there is only one observation per ID.
condA <- df[which(df$condition=='A'),]
condA_model <- lmer(Outcome ~ Var1*Var2 + (1|famID), data=condA)
condB <- df[which(df$condition=='B'),]
condB_model <- lmer(Outcome ~ Var1*Var2 + (1|famID), data=condB)
condA_model converges; condB_model does not. In condB_model "famID (Intercept)" variance is estimated at 0. In the condA_model, I get a small, but non-zero estimate (variance=0.001479). I know I could get an estimate of the fixed effect of interest in condition A versus B by a different method(such as randomly selecting one sibling per family for the analysis and not using random effects), but I am concerned that this differential convergence pattern may indicate differences between the conditions that would influence the interpretation of the omnibus model effect.
What difference in the two conditions could causing the model in one subset not to converge? How would I test for the possible differences in my data? Shouldn't the random effect of famID be identical in both subsets and thus equally able to be estimated in both post-hoc models?
As a post-hoc, to better understand what is driving the omnibus model effect, I subsetted the data so that there is only one observation per ID.
This procedure does not make sense.
What difference in the two conditions could causing the model in one subset not to converge?
There are many reasons. For one thing, these reduced datasets are, well, reduced, ie smaller, so there is far less statistical power to detect the "effects" that you are interested in, such as a variance of a random effect. In such cases, it may be estimated as zero and result in a singular fit.
Shouldn't the random effect of famID be identical in both subsets and thus equally able to be estimated in both post-hoc models?
No, these are completely different models, since the underlying data are different. There is no reason to expect the same estimates from both models.
I am training my own embedding vectors as I'm focused on an academic dataset (WOS); whether the vectors are generated via word2vec or fasttext doesn't particularly matter. Say my vectors are 150 dimensions each. I'm wondering what the desired distribution of weights within a vector ought to be, if you averaged across an entire corpus's vectors?
I did a few experiments while looking at the distributions of a sample of my vectors and came to these conclusions (uncertain as to how absolutely they hold):
If one trains their model with too few epochs then the vectors don't change significantly from their initiated values (easy to see if you start you vectors as weight 0 in every category). Thus if my weight distribution is centered around some point (typically 0) then I've under-trained my corpus.
If one trains their model with too few documents/over-trains then the vectors show significant correlation with each other (I typically visualize a random set of vectors and you can see stripes where all the vectors have weights that are either positive or negative).
What I imagine is a single "good" vector has various weights across the entire range of -1 to 1. For any single vector it may have significantly more dimensions near -1 or 1. However, the weight distribution of an entire corpus would balance out vectors that randomly have more values towards one end of the spectrum or another, so that the weight distribution of the entire corpus is approximately evenly distributed across the entire corpus. Is this intuition correct?
I'm unfamiliar with any research or folk wisdom about the desirable "weights of the vectors" (by which I assume you mean the individual dimensions).
In general, since the individual dimensions aren't strongly interpretable, I'm not sure you could say much about how any one dimension's values should be distributed. And remember, our intuitions from low-dimensional spaces (2d, 3d, 4d) often don't hold up in high-dimensional spaces.
I've seen two interesting, possibly relevant observations in research:
some have observed that the raw trained vectors for words with singular meanings tend to have a larger magnitude, and those with many meanings have smaller magnitudes. A plausible explanation for this would be that word-vectors for polysemous word-tokenss are being pulled in different directions for the multiple contrasting meanings, and thus wind up "somewhere in the middle" (closer to the origin, and thus of lower magnitude). Note, though, that most word-vector-to-word-vector comparisons ignore the magnitudes, by using cosine-similarity to only compare angles (or largely equivalently, by normalizing all vectors to unit length before comparisons).
A paper "All-but-the-Top: Simple and Effective Postprocessing for Word Representations" by Mu, Bhat, & Viswanath https://arxiv.org/abs/1702.01417v2 has noted that the average of all word-vectors that were trained together tends to biased in a certain direction from the origin, but that removing that bias (and other commonalities in the vectors) can result in improved vectors for many tasks. In my own personal experiments, I've observed that the magnitude of that bias-from-origin seems correlated with the number of negative samples chosen - and that choosing the extreme (and uncommon) value of just 1 negative sample makes such a bias negligible (but might not be best for overall quality or efficiency/speed of training).
So there may be useful heuristics about vector quality from looking at the relative distributions of vectors, but I'm not sure any would be sensitive to individual dimensions (except insofar as those happen to be the projections of vectors onto a certain axis).
I have a trading strategy on the foreign exchange market that I am attempting to improve upon.
I have a huge table (100k+ rows) that represent every possible trade in the market, the type of trade (buy or sell), the profit/loss after that trade closed, and 10 or so additional variables that represent various market measurements at the time of trade-opening.
I am trying to find out if any of these 10 variables are significantly related to the profits/losses.
For example, imagine that variable X ranges from 50 to -50.
The average value of X for a buy order is 25, and for a sell order is -25.
If most profitable buy orders have a value of X > 25, and most profitable sell orders have a value of X < -25 then I would consider the relationship of X-to-profit as significant.
I would like a good starting point for this. I have installed RapidMiner 5 in case someone can give me a specific recommendation for that.
A Decision Tree is perhaps the best place to begin.
The tree itself is a visual summary of feature importance ranking (or significant variables as phrased in the OP).
gives you a visual representation of the entire
classification/regression analysis (in the form of a binary tree),
which distinguishes it from any other analytical/statistical
technique that i am aware of;
decision tree algorithms require very little pre-processing on your data, no normalization, no rescaling, no conversion of discrete variables into integers (eg, Male/Female => 0/1); they can accept both categorical (discrete) and continuous variables, and many implementations can handle incomplete data (values missing from some of the rows in your data matrix); and
again, the tree itself is a visual summary of feature importance ranking
(ie, significant variables)--the most significant variable is the
root node, and is more significant than the two child nodes, which in
turn are more significant than their four combined children. "significance" here means the percent of variance explained (with respect to some response variable, aka 'target variable' or the thing
you are trying to predict). One proviso: from a visual inspection of
a decision tree you cannot distinguish variable significance from
among nodes of the same rank.
If you haven't used them before, here's how Decision Trees work: the algorithm will go through every variable (column) in your data and every value for each variable and split your data into two sub-sets based on each of those values. Which of these splits is actually chosen by the algorithm--i.e., what is the splitting criterion? The particular variable/value combination that "purifies" the data the most (i.e., maximizes the information gain) is chosen to split the data (that variable/value combination is usually indicated as the node's label). This simple heuristic is just performed recursively until the remaining data sub-sets are pure or further splitting doesn't increase the information gain.
What does this tell you about the "importance" of the variables in your data set? Well importance is indicated by proximity to the root node--i.e., hierarchical level or rank.
One suggestion: decision trees handle both categorical and discrete data usually without problem; however, in my experience, decision tree algorithms always perform better if the response variable (the variable you are trying to predict using all other variables) is discrete/categorical rather than continuous. It looks like yours is probably continuous, in which case in would consider discretizing it (unless doing so just causes the entire analysis to be meaningless). To do this, just bin your response variable values using parameters (bin size, bin number, and bin edges) meaningful w/r/t your problem domain--e.g., if your r/v is comprised of 'continuous values' from 1 to 100, you might sensibly bin them into 5 bins, 0-20, 21-40, 41-60, and so on.
For instance, from your Question, suppose one variable in your data is X and it has 5 values (10, 20, 25, 50, 100); suppose also that splitting your data on this variable with the third value (25) results in two nearly pure subsets--one low-value and one high-value. As long as this purity were higher than for the sub-sets obtained from splitting on the other values, the data would be split on that variable/value pair.
RapidMiner does indeed have a decision tree implementation, and it seems there are quite a few tutorials available on the Web (e.g., from YouTube, here and here). (Note, I have not used the decision tree module in R/M, nor have i used RapidMiner at all.)
The other set of techniques i would consider is usually grouped under the rubric Dimension Reduction. Feature Extraction and Feature Selection are two perhaps the most common terms after D/R. The most widely used is PCA, or principal-component analysis, which is based on an eigen-vector decomposition of the covariance matrix (derived from to your data matrix).
One direct result from this eigen-vector decomp is the fraction of variability in the data accounted for by each eigenvector. Just from this result, you can determine how many dimensions are required to explain, e.g., 95% of the variability in your data
If RapidMiner has PCA or another functionally similar dimension reduction technique, it's not obvious where to find it. I do know that RapidMiner has an R Extension, which of course let's you access R inside RapidMiner.R has plenty of PCA libraries (Packages). The ones i mention below are all available on CRAN, which means any of the PCA Packages there satisfy the minimum Package requirements for documentation and vignettes (code examples). I can recommend pcaPP (Robust PCA by Projection Pursuit).
In addition, i can recommend two excellent step-by-step tutorials on PCA. The first is from the NIST Engineering Statistics Handbook. The second is a tutorial for Independent Component Analysis (ICA) rather than PCA, but i mentioned it here because it's an excellent tutorial and the two techniques are used for the similar purposes.
I want to use a linear regression model, but I want to use ordinary least squares, which I think it is a type of linear regression. The software I use is SPSS. It only has linear regression, partial least squares and 2-stages least squares. I have no idea which one is ordinary least squares (OLS).
Yes, although 'linear regression' refers to any approach to model the relationship between one or more variables, OLS is the method used to find the simple linear regression of a set of data.
Linear regression is a vast term that just says we are finding a relationship between the dependent and independent variable(s), no matter what technique we are using.
OLS is just one of the technique to do linear reg.
Lets say,
error(e) = (observed value - predicted value)
Observed values - blue dots in picture
predicted values - points on the line(vertically below to the observed values)
The vertical lines below represent 'e'. We square them -> add them and get total err. And we try to reduce this total error.
For OLS, as the name says (ordinary least squared method), here we reduce the sum of all e^2 i.e. we try to make the error least.