I have a dataset where a process is described as a time series made of ~2000 points and 1500 dimensions.
I would like to quantify how much each dimension is correlated with another time series measured by another method.
What is the appropriate way to do this (eventually done in python) ? I have heard that Pearson is not well suited for this task, at least without data preparation. What are your thoughts about that?
Many thanks!
A general good rule in data science is to first try the easy thing. Only when the easy thing fails should you move to something more complicated. With that in mind, here is how you would compute the Pearson correlation between each dimension and some other time series. The key function here being pearsonr:
import numpy as np
from scipy.stats import pearsonr
# Generate a random dataset using 2000 points and 1500 dimensions
n_times = 2000
n_dimensions = 1500
data = np.random.rand(n_times, n_dimensions)
# Generate another time series, also using 2000 points
other_time_series = np.random.rand(n_times)
# Compute correlation between each dimension and the other time series
correlations = np.zeros(n_dimensions)
for dimension in range(n_dimensions):
# The Pearson correlation function gives us both the correlation
# coefficient (r) and a p-value (p). Here, we only use the coefficient.
r, p = pearsonr(data[:, dimension], other_time_series)
correlations[dimension] = r
# Now we have, for each dimension, the Pearson correlation with the other time
# series!
len(correlations)
# Print the first 5 correlation coefficients
print(correlations[:5])
If Pearson correlation doesn't work well for you, you can try swapping out the pearsonr function with something else, like:
spearmanr Spearman rank-order correlation coefficient.
kendalltau Kendall’s tau, a correlation measure for ordinal data.
Related
I want to check if a clustering would be helpful or not on my coordinates.
I'm dealing with trajectories and want to check if all of them are starting on a same area (the trajectories are different). Thus the aim here is to characterise the most frequent departure points.
However, sometimes there is no need for clustering. I'm using K-means here. I had thought of using the Silhouette Score but I don't see if it is mathematically correct for the case where there is only one cluster. DBScan will not be a good clustering as density are not similar in the clusters I wanted to build.
Would you have an idea to create a kind of check between k=1 and k=3, which would be the best split for my data? I'm dealing here with data with coordinates (latitude/longitude) where the starting point is not 100% fixed but can vary within 2km around a kind of barycentre.
Simple extract with k=2 :
from pyspark.ml.feature import VectorAssembler
vecAssembler = VectorAssembler(inputCols=["lat", "lon"], outputCol="features")
df1= vecAssembler.transform(df)
from pyspark.ml.clustering import KMeans
from pyspark.ml.evaluation import ClusteringEvaluator
# Loads data.
# Trains a k-means model.
kmeans = KMeans().setK(2).setSeed(1)
model = kmeans.fit(df1.select('features'))
# Make predictions
transformed = model.transform(df1)
evaluator = ClusteringEvaluator(predictionCol='prediction', featuresCol='features', \
metricName='silhouette', distanceMeasure='squaredEuclidean')
evaluator.evaluate(transformed)
Is there a way to compute in pySpark a case with k=1 ? in order to derive Elbow or gap statistics ?
I have 38 variables, like oxygen, temperature, pressure, etc and have a task to determine the total yield produced every day from these variables. When I calculate the regression coefficients and intercept value, they seem to be abnormal and very high (Impractical). For example, if 'temperature' coefficient was found to be +375.456, I could not give a meaning to them saying an increase in one unit in temperature would increase yield by 375.456g. That's impractical in my scenario. However, the prediction accuracy seems right. I would like to know, how to interpret these huge intercept( -5341.27355) and huge beta values shown below. One other important point is that I removed multicolinear columns and also, I am not scaling the variables/normalizing them because I need beta coefficients to have meaning such that I could say, increase in temperature by one unit increases yield by 10g or so. Your inputs are highly appreciated!
modl.intercept_
Out[375]: -5341.27354961415
modl.coef_
Out[376]:
array([ 1.38096017e+00, -7.62388829e+00, 5.64611255e+00, 2.26124164e-01,
4.21908571e-01, 4.50695302e-01, -8.15167717e-01, 1.82390184e+00,
-3.32849969e+02, 3.31942553e+02, 3.58830763e+02, -2.05076898e-01,
-3.06404757e+02, 7.86012402e+00, 3.21339318e+02, -7.00817205e-01,
-1.09676321e+04, 1.91481734e+00, 6.02929848e+01, 8.33731416e+00,
-6.23433431e+01, -1.88442804e+00, 6.86526274e+00, -6.76103795e+01,
-1.11406021e+02, 2.48270706e+02, 2.94836048e+01, 1.00279016e+02,
1.42906659e-02, -2.13019683e-03, -6.71427100e+02, -2.03158515e+02,
9.32094007e-03, 5.56457014e+01, -2.91724945e+00, 4.78691176e-01,
8.78121854e+00, -4.93696073e+00])
It's very unlikely that all of these variables are linearly correlated, so I would suggest that you have a look at simple non-linear regression techniques, such as Decision Trees or Kernel Ridge Regression. These are however more difficult to interpret.
Going back to your issue, these high weights might well be due to there being some high amount of correlation between the variables, or that you simply don't have very much training data.
If you instead of linear regression use Lasso Regression, the solution is biased away from high regression coefficients, and the fit will likely improve as well.
A small example on how to do this in scikit-learn, including cross validation of the regularization hyper-parameter:
from sklearn.linear_model LassoCV
# Make up some data
n_samples = 100
n_features = 5
X = np.random.random((n_samples, n_features))
# Make y linear dependent on the features
y = np.sum(np.random.random((1,n_features)) * X, axis=1)
model = LassoCV(cv=5, n_alphas=100, fit_intercept=True)
model.fit(X,y)
print(model.intercept_)
If you have a linear regression, the formula looks like this (y= target, x= features inputs):
y= x1*b1 +x2*b2 + x3*b3 + x4*b4...+ c
where b1,b2,b3,b4... are your modl.coef_. AS you already realized one of your bigges number is 3.319+02 = 331 and the intercept is also quite big with -5431.
As you already mentioned the coeffiecient variables means how much the target variable changes, if the coeffiecient feature changes with 1 unit and all others features are constant.
so for your interpretation, the higher the absoult coeffienct, the higher the influence of your analysis. But it is important to note that the model is using a lot of high coefficient, that means your model is not depending only of one variable
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'm building Kmeans in pytorch using gradient descent on centroid locations, instead of expectation-maximisation. Loss is the sum of square distances of each point to its nearest centroid. To identify which centroid is nearest to each point, I use argmin, which is not differentiable everywhere. However, pytorch is still able to backprop and update weights (centroid locations), giving similar performance to sklearn kmeans on the data.
Any ideas how this is working, or how I can figure this out within pytorch? Discussion on pytorch github suggests argmax is not differentiable: https://github.com/pytorch/pytorch/issues/1339.
Example code below (on random pts):
import numpy as np
import torch
num_pts, batch_size, n_dims, num_clusters, lr = 1000, 100, 200, 20, 1e-5
# generate random points
vector = torch.from_numpy(np.random.rand(num_pts, n_dims)).float()
# randomly pick starting centroids
idx = np.random.choice(num_pts, size=num_clusters)
kmean_centroids = vector[idx][:,None,:] # [num_clusters,1,n_dims]
kmean_centroids = torch.tensor(kmean_centroids, requires_grad=True)
for t in range(4001):
# get batch
idx = np.random.choice(num_pts, size=batch_size)
vector_batch = vector[idx]
distances = vector_batch - kmean_centroids # [num_clusters, #pts, #dims]
distances = torch.sum(distances**2, dim=2) # [num_clusters, #pts]
# argmin
membership = torch.min(distances, 0)[1] # [#pts]
# cluster distances
cluster_loss = 0
for i in range(num_clusters):
subset = torch.transpose(distances,0,1)[membership==i]
if len(subset)!=0: # to prevent NaN
cluster_loss += torch.sum(subset[:,i])
cluster_loss.backward()
print(cluster_loss.item())
with torch.no_grad():
kmean_centroids -= lr * kmean_centroids.grad
kmean_centroids.grad.zero_()
As alvas noted in the comments, argmax is not differentiable. However, once you compute it and assign each datapoint to a cluster, the derivative of loss with respect to the location of these clusters is well-defined. This is what your algorithm does.
Why does it work? If you had only one cluster (so that the argmax operation didn't matter), your loss function would be quadratic, with minimum at the mean of the data points. Now with multiple clusters, you can see that your loss function is piecewise (in higher dimensions think volumewise) quadratic - for any set of centroids [C1, C2, C3, ...] each data point is assigned to some centroid CN and the loss is locally quadratic. The extent of this locality is given by all alternative centroids [C1', C2', C3', ...] for which the assignment coming from argmax remains the same; within this region the argmax can be treated as a constant, rather than a function and thus the derivative of loss is well-defined.
Now, in reality, it's unlikely you can treat argmax as constant, but you can still treat the naive "argmax-is-a-constant" derivative as pointing approximately towards a minimum, because the majority of data points are likely to indeed belong to the same cluster between iterations. And once you get close enough to a local minimum such that the points no longer change their assignments, the process can converge to a minimum.
Another, more theoretical way to look at it is that you're doing an approximation of expectation maximization. Normally, you would have the "compute assignments" step, which is mirrored by argmax, and the "minimize" step which boils down to finding the minimizing cluster centers given the current assignments. The minimum is given by d(loss)/d([C1, C2, ...]) == 0, which for a quadratic loss is given analytically by the means of data points within each cluster. In your implementation, you're solving the same equation but with a gradient descent step. In fact, if you used a 2nd order (Newton) update scheme instead of 1st order gradient descent, you would be implicitly reproducing exactly the baseline EM scheme.
Imagine this:
t = torch.tensor([-0.0627, 0.1373, 0.0616, -1.7994, 0.8853,
-0.0656, 1.0034, 0.6974, -0.2919, -0.0456])
torch.argmax(t).item() # outputs 6
We increase t[0] for some, δ close to 0, will this update the argmax? It will not, so we are dealing with 0 gradients, all the time. Just ignore this layer, or assume it is frozen.
The same is for argmin, or any other function where the dependent variable is in discrete steps.
The current tf.contrib.metrics.streaming_accuracy is only able to calculate the top 1 accuracy, and not the top k. As a workaround, this is what I've been using:
tf.reduce_mean(tf.cast(tf.nn.in_top_k(predictions=predictions, targets=labels, k=5), tf.float32))
However, this does not give me a way to calculate the streaming accuracies averaged across each batch, which would be useful in getting a stable evaluation accuracy. I am currently manually calculating this streaming top 5 accuracy through using its numpy output, but this means I won't be able to visualize this metric on tensorboard.
Is there a way to have a simpler implementation by creating an accuracy_update function, or is there an existing function that already does this?
Thank you.
You could replace your use of tf.contrib.metrics.streaming_accuracy by the lower-level tf.metrics.mean, which is by the way ultimately used by streaming_accuracy -- you will find a similarity in their respective documentations.
E.g. (not tested)
tf.metrics.mean(tf.nn.in_top_k(predictions=predictions, targets=labels, k=5))
For top-k accuracy per batch, this also works.
k_val=3
accs = []
for each_bach in range(batch_size):
acc = tf.keras.metrics.top_k_categorical_accuracy(y_true=tf_class1[each_bach], y_pred=tf_class2[each_bach], k=k_val)
accs.append(acc)
acc_data_per_batch = tf.reduce_mean(accs)
tf.keras.metrics.top_k_categorical_accuracy returns K.mean(
nn.in_top_k(y_pred, math_ops.argmax(y_true, axis=-1), k), axis=-1) per batch