I am trying to re-create the factor scores computed by AMOS for a latent construct. However, I get a very different answer when using the "Estimates" for the variables than when AMOS computes the factor scores (using the Impute method). So for example, AMOS reports the weights of the variables as:
V1 ~ Latent_var1 = 1
V2 ~ Latent_var1 = .75
V3 ~ Latent_var1 = .67
V4 ~ Latent_var1 = .45
If I simply multiple the actual scores from any respondent for V1, V2, and V3 by these coefficients, it will not equal the "Factor Score" reported by AMOS (not even close), for that same respondent. If this were standard linear regression, it would work just fine: constant+beta*_value + beta2*_value + beta3*_value, etc.
Does anyone know why?
For those familiar with AMOS, here's snippet from the model output. the Latent construct Personality estimates vp1 - vp6. If I take the actual variable values and multiple them by these coefficients, I do not get the same factor score.
Related
I implemented a cosine-theta function, which calculates the relation between two articles. If two articles are very similar then the words should contain quite some overlap. However, a cosine theta score of 0.54 does not mean "related" or "not related". I should end up with a definitive answer which is either 0 for 'not related' or 1 for 'related'.
I know that there are sigmoid and softmax functions, yet I should find the optimal parameters to give to such functions and I do not know if these functions are satisfactory solutions. I was thinking that I have the cosine theta score, I can calculate the percentage of overlap between two sentences two (e.g. the amount of overlapping words divided by the amount of words in the article) and maybe some more interesting things. Then with the data, I could maybe write a function (what type of function I do not know and is part of the question!), after which I can minimize the error via the SciPy library. This means that I should do some sort of supervised learning, and I am willing to label article pairs with labels (0/1) in order to train a network. Is this worth the effort?
# Count words of two strings.
v1, v2 = self.word_count(s1), self.word_count(s2)
# Calculate the intersection of the words in both strings.
v3 = set(v1.keys()) & set(v2.keys())
# Calculate some sort of ratio between the overlap and the
# article length (since 1 overlapping word on 2 words is more important
# then 4 overlapping words on articles of 492 words).
p = min(len(v1), len(v2)) / len(v3)
numerator = sum([v1[w] * v2[w] for w in v3])
w1 = sum([v1[w]**2 for w in v1.keys()])
w2 = sum([v2[w]**2 for w in v2.keys()])
denominator = math.sqrt(w1) * math.sqrt(w2)
# Calculate the cosine similarity
if not denominator:
return 0.0
else:
return (float(numerator) / denominator)
As said, I would like to use variables such as p, and the cosine theta score in order to produce an accurate discrete binary label, either 0 or 1.
As said, I would like to use variables such as p, and the cosine theta score in order to produce an accurate discrete binary label, either 0 or 1.
Here it really comes down to what you mean by accuracy. It is up to you to choose how the overlap affects whether or not two strings are "matching" unless you have a labelled data set. If you have a labelled data set (I.e., a set of pairs of strings along with a 0 or 1 label), then you can train a binary classification algorithm and try to optimise based on that. I would recommend something like a neural net or SVM due to the potentially high dimensional, categorical nature of your problem.
Even the optimisation, however, is a subjective measure. For example, in theory let's pretend you have a model which out of 100 samples only predicts 1 answer (Giving 99 unknowns). Technically if that one answer is correct, that is a model with 100% accuracy, but which has a very low recall. Generally in machine learning you will find a trade off between recall and accuracy.
Some people like to go for certain metrics which combine the two (The most famous of which is the F1 score), but honestly it depends on the application. If I have a marketing campaign with a fixed budget, then I care more about accuracy - I would only want to target consumers who are likely to buy my product. If however, we are looking to test for a deadly disease or markers for bank fraud, then it's feasible for that test to be accurate only 10% of the time - if its recall of true positives is somewhere close to 100%.
Finally, if you have no labelled data, then your best bet is just to define some cut off value which you believe indicates a good match. This is would then be more analogous to a binary clustering problem, and you could use some more abstract measure such as distance to a centroid to test which cluster (Either the "related" or "unrelated" cluster) the point belongs to. Note however that here your features feel like they would be incredibly hard to define.
I have recently started experimenting with OneClassSVM ( using Sklearn ) for unsupervised learning and I followed
this example .
I apologize for the silly questions But I’m a bit confused about two things :
Should I train my svm on both regular example case as well as the outliers , or the training is on regular examples only ?
Which of labels predicted by the OSVM and represent outliers is it 1 or -1
Once again i apologize for those questions but for some reason i cannot find this documented anyware
As this example you reference is about novelty-detection, the docs say:
novelty detection:
The training data is not polluted by outliers, and we are interested in detecting anomalies in new observations.
Meaning: you should train on regular examples only.
The approach is based on:
Schölkopf, Bernhard, et al. "Estimating the support of a high-dimensional distribution." Neural computation 13.7 (2001): 1443-1471.
Extract:
Suppose you are given some data set drawn from an underlying probability distribution P and you want to estimate a “simple” subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specied value between 0 and 1.
We propose a method to approach this problem by trying to estimate a function f that is positive on S and negative on the complement.
The above docs also say:
Inliers are labeled 1, while outliers are labeled -1.
This can also be seen in your example code, extracted:
# Generate some regular novel observations
X = 0.3 * np.random.randn(20, 2)
X_test = np.r_[X + 2, X - 2]
...
# all regular = inliers (defined above)
y_pred_test = clf.predict(X_test)
...
# -1 = outlier <-> error as assumed to be inlier
n_error_test = y_pred_test[y_pred_test == -1].size
I have a few distance functions which return distance between two images , I want to combine these distance into a single distance, using weighted scoring e.g. ax1+bx2+cx3+dx4 etc i want to learn these weights automatically such that my test error is minimised.
For this purpose i have a labeled dataset which has various triplets of images such that (a,b,c) , a has less distance to b than it has to c.
i.e. d(a,b)<d(a,c)
I want to learn such weights so that this ordering of triplets can be as accurate as possible.(i.e. the weighted linear score given is less for a&b and more for a&c).
What sort of machine learning algorithm can be used for the task,and how the desired task can be achieved?
Hopefully I understand your question correctly, but it seems that this could be solved more easily with constrained optimization directly, rather than classical machine learning (the algorithms of which are often implemented via constrained optimization, see e.g. SVMs).
As an example, a possible objective function could be:
argmin_{w} || e ||_2 + lambda || w ||_2
where w is your weight vector (Oh god why is there no latex here), e is the vector of errors (one component per training triplet), lambda is some tunable regularizer constant (could be zero), and your constraints could be:
max{d(I_p,I_r)-d(I_p,I_q),0} <= e_j for jth (p,q,r) in T s.t. d(I_p,I_r) <= d(I_p,I_q)
for the jth constraint, where I_i is image i, T is the training set, and
d(u,v) = sum_{w_i in w} w_i * d_i(u,v)
with d_i being your ith distance function.
Notice that e is measuring how far your chosen weights are from satisfying all the chosen triplets in the training set. If the weights preserve ordering of label j, then d(I_p,I_r)-d(I_p,I_q) < 0 and so e_j = 0. If they don't, then e_j will measure the amount of violation of training label j. Solving the optimization problem would give the best w; i.e. the one with the lowest error.
If you're not familiar with linear/quadratic programming, convex optimization, etc... then start googling :) Many libraries exist for this type of thing.
On the other hand, if you would prefer a machine learning approach, you may be able to adapt some metric learning approaches to your problem.
I developed a image processing program that identifies what a number is given an image of numbers. Each image was 27x27 pixels = 729 pixels. I take each R, G and B value which means I have 2187 variables from each image (+1 for the intercept = total of 2188).
I used the below gradient descent formula:
Repeat {
θj = θj−α/m∑(hθ(x)−y)xj
}
Where θj is the coefficient on variable j; α is the learning rate; hθ(x) is the hypothesis; y is real value and xj is the value of variable j. m is the number of training sets. hθ(x), y are for each training set (i.e. that's what the summation sign is for). Further the hypothesis is defined as:
hθ(x) = 1/(1+ e^-z)
z= θo + θ1X1+θ2X2 +θ3X3...θnXn
With this, and 3000 training images, I was able to train my program in just over an hour and when tested on a cross validation set, it was able to identify the correct image ~ 67% of the time.
I wanted to improve that so I decided to attempt a polynomial of degree 2.
However the number of variables jumps from 2188 to 2,394,766 per image! It takes me an hour just to do 1 step of gradient descent.
So my question is, how is this vast number of variables handled in machine learning? On the one hand, I don't have enough space to even hold that many variables for each training set. On the other hand, I am currently storing 2188 variables per training sample, but I have to perform O(n^2) just to get the values of each variable multiplied by another variable (i.e. the polynomial to degree 2 values).
So any suggestions / advice is greatly appreciated.
try to use some dimensionality reduction first (PCA, kernel PCA, or LDA if you are classifying the images)
vectorize your gradient descent - with most math libraries or in matlab etc. it will run much faster
parallelize the algorithm and then run in on multiple CPUs (but maybe your library for multiplying vectors already supports parallel computations)
Along with Jirka-x1's answer, I would first say that this is one of the key differences in working with image data than say text data for ML: high dimensionality.
Second... this is a duplicate, see How to approach machine learning problems with high dimensional input space?
I want to classify documents (composed of words) into 3 classes (Positive, Negative, Unknown/Neutral). A subset of the document words become the features.
Until now, I have programmed a Naive Bayes Classifier using as a feature selector Information gain and chi-square statistics. Now, I would like to see what happens if I use Odds ratio as a feature selector.
My problem is that I don't know hot to implement Odds-ratio. Should I:
1) Calculate Odds Ratio for every word w, every class:
E.g. for w:
Prob of word as positive Pw,p = #positive docs with w/#docs
Prob of word as negative Pw,n = #negative docs with w/#docs
Prob of word as unknown Pw,u = #unknown docs with w/#docs
OR(Wi,P) = log( Pw,p*(1-Pw,p) / (Pw,n + Pw,u)*(1-(Pw,n + Pw,u)) )
OR(Wi,N) ...
OR(Wi,U) ...
2) How should I decide if I choose or not the word as a feature ?
Thanks in advance...
Since it took me a while to independently wrap my head around all this, let me explain my findings here for the benefit of humanity.
Using the (log) odds ratio is a standard technique for filtering features prior to text classification. It is a 'one-sided metric' [Zheng et al., 2004] in the sense that it only discovers features which are positively correlated with a particular class. As a log-odds-ratio for the probability of seeing a feature 't' given the class 'c', it is defined as:
LOR(t,c) = log [Pr(t|c) / (1 - Pr(t|c))] : [Pr(t|!c) / (1 - Pr(t|!c))]
= log [Pr(t|c) (1 - Pr(t|!c))] / [Pr(t|!c) (1 - Pr(t|c))]
Here I use '!c' to mean a document where the class is not c.
But how do you actually calculate Pr(t|c) and Pr(t|!c)?
One subtlety to note is that feature selection probabilities, in general, are usually defined over a document event model [McCallum & Nigam 1998, Manning et al. 2008], i.e., Pr(t|c) is the probability of seeing term t one or more times in the document given the class of the document is c (in other words, the presence of t given the class c). The maximum likelihood estimate (MLE) of this probability would be the proportion of documents of class c that contain t at least once. [Technically, this is known as a Multivariate Bernoulli event model, and is distinct from a Multinomial event model over words, which would calculate Pr(t|c) using integer word counts - see the McCallum paper or the Manning IR textbook for more details, specifically on how this applies to a Naive Bayes text classifier.]
One key to using LOR effectively is to smooth these conditional probability estimates, since, as #yura noted, rare events are problematic here (e.g., the MLE of Pr(t|!c) could be zero, leading to an infinite LOR). But how do we smooth?
In the literature, Forman reports smoothing the LOR by "adding one to any zero count in the denominator" (Forman, 2003), while Zheng et al (2004) use "ELE [Expected Likelihood Estimation] smoothing" which usually amounts to adding 0.5 to each count.
To smooth in a way that is consistent with probability theory, I follow standard practices in text classification with a Multivariate Bernoulli event model. Essentially, we assume that we have seen each presence count AND each absence count B extra times. So our estimate for Pr(t|c) can be written in terms of #(t,c): the number of times we've seen t and c, and #(t,!c): the number of times we've seen t without c, as follows:
Pr(t|c) = [#(t,c) + B] / [#(t,c) + #(t,!c) + 2B]
= [#(t,c) + B] / [#(c) + 2B]
If B = 0, we have the MLE. If B = 0.5, we have ELE. If B = 1, we have the Laplacian prior. Note this looks different than smoothing for the Multinomial event model, where the Laplacian prior leads you to add |V| in the denominator [McCallum & Nigam, 1998]
You can choose 0.5 or 1 as your smoothing value, depending on which prior work most inspires you, and plug this into the equation for LOR(t,c) above, and score all the features.
Typically, you then decide on how many features you want to use, say N, and then choose the N highest-ranked features based on the score.
In a multi-class setting, people have often used 1 vs All classifiers and thus did feature selection independently for each classifier and thus each positive class with the 1-sided metrics (Forman, 2003). However, if you want to find a unique reduced set of features that works in a multiclass setting, there are some advanced approaches in the literature (e.g. Chapelle & Keerthi, 2008).
References:
Zheng, Wu, Srihari, 2004
McCallum & Nigam 1998
Manning, Raghavan & Schütze, 2008
Forman, 2003
Chapelle & Keerthi, 2008
Odd ratio is not good measure for feature selection, because it is only shows what happen when feature present, and nothing when it is not. So it will not work for rare features and almost all features are rare so it not work for almost all features. Example feature with 100% confidence that class is positive which present in 0.0001 is useless for classification. Therefore if you still want to use odd ratio add threshold on frequency of feature, like feature present in 5% of cases. But I would recommend better approach - use Chi or info gain metrics which automatically solve those problems.