I have a dataframe where A and B is used to predict C
df = df[['A','B','C']]
array = df.values
X = array[:,0:-1]
Y = array[:,-1]
# Feature Importance
model = GradientBoostingClassifier()
model.fit(X, Y)
print ("Importance:")
print((model.feature_importances_)*100)
#PCA
pca = PCA(n_components=len(df.columns)-1)
fit = pca.fit(X)
print("Explained Variance")
print(fit.explained_variance_ratio_)
This prints
Importance:
[ 53.37975706 46.62024294]
Explained Variance
[ 0.98358394 0.01641606]
However when I changed the dataframe position swapping A and B, only the importance changed, but the Explain variance remains, why did the explained variance not change according to [0.01641606 0.98358394]?
df = df[['B','A','C']]
Importance:
[ 46.40771024 53.59228976]
Explained Variance
[ 0.98358394 0.01641606]
Explained variance does not refer to A or B or any columns of your dataframe. It refers to the principal components identified by the PCA, which are some linear combinations of the columns. These components are sorted in the order of decreasing variance as the documentation says:
components_ : array, shape (n_components, n_features)
Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by explained_variance_.
explained_variance_ : array, shape (n_components,)
The amount of variance explained by each of the selected components.
Equal to n_components largest eigenvalues of the covariance matrix of X.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
So, the order of features does not affect the order of components returned. It does affect the array components_ which is a matrix that can be used to map principal components to the feature space.
Related
I am doing a project on multiclass semantic segmentation. I have formulated a model that outputs pretty descent segmented images by decreasing the loss value. However, I cannot evaluate the model performance in metrics, such as meanIoU or Dice coefficient.
In case of binary semantic segmentation it was easy just to set the threshold of 0.5, to classify the outputs as an object or background, but it does not work in the case of multiclass semantic segmentation. Could you please tell me how to obtain model performance on the aforementioned metrics? Any help will be highly appreciated!
By the way, I am using PyTorch framework and CamVid dataset.
If anyone is interested in this answer, please also look at this issue. The author of the issue points out that mIoU can be computed in a different way (and that method is more accepted in literature). So, consider that before using the implementation for any formal publication.
Basically, the other method suggested by the issue-poster is to separately accumulate the intersections and unions over the entire dataset and divide them at the final step. The method in the below original answer computes intersection and union for a batch of images, then divides them to get IoU for the current batch, and then takes a mean of the IoUs over the entire dataset.
However, this below given original method is problematic because the final mean IoU would vary with the batch-size. On the other hand, the mIoU would not vary with the batch size for the method mentioned in the issue as the separate accumulation would ensure that batch size is irrelevant (though higher batch size can definitely help speed up the evaluation).
Original answer:
Given below is an implementation of mean IoU (Intersection over Union) in PyTorch.
def mIOU(label, pred, num_classes=19):
pred = F.softmax(pred, dim=1)
pred = torch.argmax(pred, dim=1).squeeze(1)
iou_list = list()
present_iou_list = list()
pred = pred.view(-1)
label = label.view(-1)
# Note: Following for loop goes from 0 to (num_classes-1)
# and ignore_index is num_classes, thus ignore_index is
# not considered in computation of IoU.
for sem_class in range(num_classes):
pred_inds = (pred == sem_class)
target_inds = (label == sem_class)
if target_inds.long().sum().item() == 0:
iou_now = float('nan')
else:
intersection_now = (pred_inds[target_inds]).long().sum().item()
union_now = pred_inds.long().sum().item() + target_inds.long().sum().item() - intersection_now
iou_now = float(intersection_now) / float(union_now)
present_iou_list.append(iou_now)
iou_list.append(iou_now)
return np.mean(present_iou_list)
Prediction of your model will be in one-hot form, so first take softmax (if your model doesn't already) followed by argmax to get the index with the highest probability at each pixel. Then, we calculate IoU for each class (and take the mean over it at the end).
We can reshape both the prediction and the label as 1-D vectors (I read that it makes the computation faster). For each class, we first identify the indices of that class using pred_inds = (pred == sem_class) and target_inds = (label == sem_class). The resulting pred_inds and target_inds will have 1 at pixels labelled as that particular class while 0 for any other class.
Then, there is a possibility that the target does not contain that particular class at all. This will make that class's IoU calculation invalid as it is not present in the target. So, you assign such classes a NaN IoU (so you can identify them later) and not involve them in the calculation of the mean.
If the particular class is present in the target, then pred_inds[target_inds] will give a vector of 1s and 0s where indices with 1 are those where prediction and target are equal and zero otherwise. Taking the sum of all elements of this will give us the intersection.
If we add all the elements of pred_inds and target_inds, we'll get the union + intersection of pixels of that particular class. So, we subtract the already calculated intersection to get the union. Then, we can divide the intersection and union to get the IoU of that particular class and add it to a list of valid IoUs.
At the end, you take the mean of the entire list to get the mIoU. If you want the Dice Coefficient, you can calculate it in a similar fashion.
I have read an article on data leakage. In a hackathon there are two sets of data, train data on which participants train their algorithm and test set on which performance is measured.
Data leakage helps in getting a perfect score in test data, with out viewing train data by exploiting the leak.
I have read the article, but I am missing the crux how the leakage is exploited.
Steps as shown in article are following:
Let's load the test data.
Note, that we don't have any training data here, just test data. Moreover, we will not even use any features of test objects. All we need to solve this task is the file with the indices for the pairs, that we need to compare.
Let's load the data with test indices.
test = pd.read_csv('../test_pairs.csv')
test.head(10)
pairId FirstId SecondId
0 0 1427 8053
1 1 17044 7681
2 2 19237 20966
3 3 8005 20765
4 4 16837 599
5 5 3657 12504
6 6 2836 7582
7 7 6136 6111
8 8 23295 9817
9 9 6621 7672
test.shape[0]
368550
For example, we can think that there is a test dataset of images, and each image is assigned a unique Id from 0 to N−1 (N -- is the number of images). In the dataframe from above FirstId and SecondId point to these Id's and define pairs, that we should compare: e.g. do both images in the pair belong to the same class or not. So, for example for the first row: if images with Id=1427 and Id=8053 belong to the same class, we should predict 1, and 0 otherwise.
But in our case we don't really care about the images, and how exactly we compare the images (as long as comparator is binary).
print(test['FirstId'].nunique())
print(test['SecondId'].nunique())
26325
26310
So the number of pairs we are given to classify is very very small compared to the total number of pairs.
To exploit the leak we need to assume (or prove), that the total number of positive pairs is small, compared to the total number of pairs. For example: think about an image dataset with 1000 classes, N images per class. Then if the task was to tell whether a pair of images belongs to the same class or not, we would have 1000*N*(N−1)/2 positive pairs, while total number of pairs was 1000*N(1000N−1)/2.
Another example: in Quora competitition the task was to classify whether a pair of qustions are duplicates of each other or not. Of course, total number of question pairs is very huge, while number of duplicates (positive pairs) is much much smaller.
Finally, let's get a fraction of pairs of class 1. We just need to submit a constant prediction "all ones" and check the returned accuracy. Create a dataframe with columns pairId and Prediction, fill it and export it to .csv file. Then submit
test['Prediction'] = np.ones(test.shape[0])
sub=pd.DataFrame(test[['pairId','Prediction']])
sub.to_csv('sub.csv',index=False)
All ones have accuracy score is 0.500000.
So, we assumed the total number of pairs is much higher than the number of positive pairs, but it is not the case for the test set. It means that the test set is constructed not by sampling random pairs, but with a specific sampling algorithm. Pairs of class 1 are oversampled.
Now think, how we can exploit this fact? What is the leak here? If you get it now, you may try to get to the final answer yourself, othewise you can follow the instructions below.
Building a magic feature
In this section we will build a magic feature, that will solve the problem almost perfectly. The instructions will lead you to the correct solution, but please, try to explain the purpose of the steps we do to yourself -- it is very important.
Incidence matrix
First, we need to build an incidence matrix. You can think of pairs (FirstId, SecondId) as of edges in an undirected graph.
The incidence matrix is a matrix of size (maxId + 1, maxId + 1), where each row (column) i corresponds i-th Id. In this matrix we put the value 1to the position [i, j], if and only if a pair (i, j) or (j, i) is present in a given set of pais (FirstId, SecondId). All the other elements in the incidence matrix are zeros.
Important! The incidence matrices are typically very very sparse (small number of non-zero values). At the same time incidence matrices are usually huge in terms of total number of elements, and it is impossible to store them in memory in dense format. But due to their sparsity incidence matrices can be easily represented as sparse matrices. If you are not familiar with sparse matrices, please see wiki and scipy.sparse reference. Please, use any of scipy.sparseconstructors to build incidence matrix.
For example, you can use this constructor: scipy.sparse.coo_matrix((data, (i, j))). We highly recommend to learn to use different scipy.sparseconstuctors, and matrices types, but if you feel you don't want to use them, you can always build this matrix with a simple for loop. You will need first to create a matrix using scipy.sparse.coo_matrix((M, N), [dtype]) with an appropriate shape (M, N) and then iterate through (FirstId, SecondId) pairs and fill corresponding elements in matrix with ones.
Note, that the matrix should be symmetric and consist only of zeros and ones. It is a way to check yourself.
import networkx as nx
import numpy as np
import pandas as pd
import scipy.sparse
import matplotlib.pyplot as plt
test = pd.read_csv('../test_pairs.csv')
x = test[['FirstId','SecondId']].rename(columns={'FirstId':'col1', 'SecondId':'col2'})
y = test[['SecondId','FirstId']].rename(columns={'SecondId':'col1', 'FirstId':'col2'})
comb = pd.concat([x,y],ignore_index=True).drop_duplicates(keep='first')
comb.head()
col1 col2
0 1427 8053
1 17044 7681
2 19237 20966
3 8005 20765
4 16837 599
data = np.ones(comb.col1.shape, dtype=int)
inc_mat = scipy.sparse.coo_matrix((data,(comb.col1,comb.col2)), shape=(comb.col1.max() + 1, comb.col1.max() + 1))
rows_FirstId = inc_mat[test.FirstId.values,:]
rows_SecondId = inc_mat[test.SecondId.values,:]
f = rows_FirstId.multiply(rows_SecondId)
f = np.asarray(f.sum(axis=1))
f.shape
(368550, 1)
f = f.sum(axis=1)
f = np.squeeze(np.asarray(f))
print (f.shape)
Now build the magic feature
Why did we build the incidence matrix? We can think of the rows in this matix as of representations for the objects. i-th row is a representation for an object with Id = i. Then, to measure similarity between two objects we can measure similarity between their representations. And we will see, that such representations are very good.
Now select the rows from the incidence matrix, that correspond to test.FirstId's, and test.SecondId's.
So do not forget to convert pd.series to np.array
These lines should normally run very quickly
rows_FirstId = inc_mat[test.FirstId.values,:]
rows_SecondId = inc_mat[test.SecondId.values,:]
Our magic feature will be the dot product between representations of a pair of objects. Dot product can be regarded as similarity measure -- for our non-negative representations the dot product is close to 0 when the representations are different, and is huge, when representations are similar.
Now compute dot product between corresponding rows in rows_FirstId and rows_SecondId matrices.
From magic feature to binary predictions
But how do we convert this feature into binary predictions? We do not have a train set to learn a model, but we have a piece of information about test set: the baseline accuracy score that you got, when submitting constant. And we also have a very strong considerations about the data generative process, so probably we will be fine even without a training set.
We may try to choose a thresold, and set the predictions to 1, if the feature value f is higer than the threshold, and 0 otherwise. What threshold would you choose?
How do we find a right threshold? Let's first examine this feature: print frequencies (or counts) of each value in the feature f.
For example use np.unique function, check for flags
Function to count frequency of each element
from scipy.stats import itemfreq
itemfreq(f)
array([[ 14, 183279],
[ 15, 852],
[ 19, 546],
[ 20, 183799],
[ 21, 6],
[ 28, 54],
[ 35, 14]])
Do you see how this feature clusters the pairs? Maybe you can guess a good threshold by looking at the values?
In fact, in other situations it can be not that obvious, but in general to pick a threshold you only need to remember the score of your baseline submission and use this information.
Choose a threshold below:
pred = f > 14 # SET THRESHOLD HERE
pred
array([ True, False, True, ..., False, False, False], dtype=bool)
submission = test.loc[:,['pairId']]
submission['Prediction'] = pred.astype(int)
submission.to_csv('submission.csv', index=False)
I want to understand the idea behind this. How we are exploiting the leak from the test data only.
There's a hint in the article. The number of positive pairs should be 1000*N*(N−1)/2, while the number of all pairs is 1000*N(1000N−1)/2. Of course, the number of all pairs is much, much larger if the test set was sampled at random.
As the author mentions, after you evaluate your constant prediction of 1s on the test set, you can tell that the sampling was not done at random. The accuracy you obtain is 50%. Had the sampling been done correctly, this value should've been much lower.
Thus, they construct the incidence matrix and calculate the dot product (the measure of similarity) between the representations of our ID features. They then reuse the information about the accuracy obtained with constant predictions (at 50%) to obtain the corresponding threshold (f > 14). It's set to be greater than 14 because that constitutes roughly half of our test set, which in turn maps back to the 50% accuracy.
The "magic" value didn't have to be greater than 14. It could have been equal to 14. You could have adjusted this value after some leader board probing (as long as you're capturing half of the test set).
It was observed that the test data was not sampled properly; same-class pairs were oversampled. Thus there is a much higher probability of each pair in the training set to have target=1 than any random pair. This led to the belief that one could construct a similarity measure based only on the pairs that are present in the test, i.e., whether a pair made it to the test is itself a strong indicator of similarity.
Using this insight one can calculate an incidence matrix and represent each id j as a binary array (the i-th element representing the presence of i-j pair in test, and thus representing the strong probability of similarity between them). This is a pretty accurate measure, allowing one to find the "similarity" between two rows just by taking their dot product.
The cutoff arrived at is purely by the knowledge of target-distribution found by leaderboard probing.
l would like to average the scores of two different SVMs trained on different samples but same classes
# Data have the smae label x_1[1] has y_1[1] and x_2[1] has y_2[1]
# Where y_2[1] == y_1[1]
Dataset_1=(x_1,y)
Dataset_2=(x_2,y)
test_data=(test_sample,test_labels)
We have 50 classes. Same classes for dataset_1 and dataset_2 :
list(set(y_1))=list(set(y_2))
What l have tried :
from sklearn.svm import SVC
clf_1 = SVC(kernel='linear', random_state=42).fit(x_1, y)
clf_2 = SVC(kernel='linear', random_state=42).fit(x_2, y)
How to average clf_1 and clf_2 scores before doing :
predict(test_sample)
?
What l would like to do ?
Not sure I understand your question; to simply average the scores as in a typical ensemble, you should first get prediction probabilities from each model separately, and then just take their average:
pred1 = clf_1.predict_proba(test_sample)
pred2 = clf_2.predict_proba(test_sample)
pred = (pred1 + pred2)/2
In order to get prediction probabilities instead of hard classes, you should initialize the SVC using the additional argument probability=True.
Each row of pred will be an array of length 50, as many as your classes, with each element representing the probability that the sample belongs to the respective class.
After averaging, simply take the argmax of pred - just be sure that the order of the returned probabilities is OK; according to the docs:
The columns correspond to the classes in sorted order, as they appear in the attribute classes_
As I am not exactly sure what this means, run some checks with predictions on your training set, to be sure that the order is correct.
I am trying to implement Expectation Maximization algorithm(Gaussian Mixture Model) on a data set data=[[x,y],...]. I am using mv_norm.pdf(data, mean,cov) function to calculate cluster responsibilities. But after calculating new values of covariance (cov matrix) after 6-7 iterations, cov matrix is becoming singular i.e determinant of cov is 0 (very small value) and hence it is giving errors
ValueError: the input matrix must be positive semidefinite
and
raise np.linalg.LinAlgError('singular matrix')
Can someone suggest any solution for this?
#E-step: Compute cluster responsibilities, given cluster parameters
def calculate_cluster_responsibility(data,centroids,cov_m):
pdfmain=[[] for i in range(0,len(data))]
for i in range(0,len(data)):
sum1=0
pdfeach=[[] for m in range(0,len(centroids))]
pdfeach[0]=1/3.*mv_norm.pdf(data[i], mean=centroids[0],cov=[[cov_m[0][0][0],cov_m[0][0][1]],[cov_m[0][1][0],cov_m[0][1][1]]])
pdfeach[1]=1/3.*mv_norm.pdf(data[i], mean=centroids[1],cov=[[cov_m[1][0][0],cov_m[1][0][1]],[cov_m[1][1][0],cov_m[0][1][1]]])
pdfeach[2]=1/3.*mv_norm.pdf(data[i], mean=centroids[2],cov=[[cov_m[2][0][0],cov_m[2][0][1]],[cov_m[2][1][0],cov_m[2][1][1]]])
sum1+=pdfeach[0]+pdfeach[1]+pdfeach[2]
pdfeach[:] = [x / sum1 for x in pdfeach]
pdfmain[i]=pdfeach
global old_pdfmain
if old_pdfmain==pdfmain:
return
old_pdfmain=copy.deepcopy(pdfmain)
softcounts=[sum(i) for i in zip(*pdfmain)]
calculate_cluster_weights(data,centroids,pdfmain,soft counts)
Initially, I've passed [[3,0],[0,3]] for each cluster covariance since expected number of clusters is 3.
Can someone suggest any solution for this?
The problem is your data lies in some manifold of dimension strictly smaller than the input data. In other words for example your data lies on a circle, while you have 3 dimensional data. As a consequence when your method tries to estimate 3 dimensional ellipsoid (covariance matrix) that fits your data - it fails since the optimal one is a 2 dimensional ellipse (third dimension is 0).
How to fix it? You will need some regularization of your covariance estimator. There are many possible solutions, all in M step, not E step, the problem is with computing covariance:
Simple solution, instead of doing something like cov = np.cov(X) add some regularizing term, like cov = np.cov(X) + eps * np.identity(X.shape[1]) with small eps
Use nicer estimator like LedoitWolf estimator from scikit-learn.
Initially, I've passed [[3,0],[0,3]] for each cluster covariance since expected number of clusters is 3.
This makes no sense, covariance matrix values has nothing to do with amount of clusters. You can initialize it with anything more or less resonable.
I've read from the relevant documentation that :
Class balancing can be done by sampling an equal number of samples from each class, or preferably by normalizing the sum of the sample weights (sample_weight) for each class to the same value.
But, it is still unclear to me how this works. If I set sample_weight with an array of only two possible values, 1's and 2's, does this mean that the samples with 2's will get sampled twice as often as the samples with 1's when doing the bagging? I cannot think of a practical example for this.
Some quick preliminaries:
Let's say we have a classification problem with K classes. In a region of feature space represented by the node of a decision tree, recall that the "impurity" of the region is measured by quantifying the inhomogeneity, using the probability of the class in that region. Normally, we estimate:
Pr(Class=k) = #(examples of class k in region) / #(total examples in region)
The impurity measure takes as input, the array of class probabilities:
[Pr(Class=1), Pr(Class=2), ..., Pr(Class=K)]
and spits out a number, which tells you how "impure" or how inhomogeneous-by-class the region of feature space is. For example, the gini measure for a two class problem is 2*p*(1-p), where p = Pr(Class=1) and 1-p=Pr(Class=2).
Now, basically the short answer to your question is:
sample_weight augments the probability estimates in the probability array ... which augments the impurity measure ... which augments how nodes are split ... which augments how the tree is built ... which augments how feature space is diced up for classification.
I believe this is best illustrated through example.
First consider the following 2-class problem where the inputs are 1 dimensional:
from sklearn.tree import DecisionTreeClassifier as DTC
X = [[0],[1],[2]] # 3 simple training examples
Y = [ 1, 2, 1 ] # class labels
dtc = DTC(max_depth=1)
So, we'll look trees with just a root node and two children. Note that the default impurity measure the gini measure.
Case 1: no sample_weight
dtc.fit(X,Y)
print dtc.tree_.threshold
# [0.5, -2, -2]
print dtc.tree_.impurity
# [0.44444444, 0, 0.5]
The first value in the threshold array tells us that the 1st training example is sent to the left child node, and the 2nd and 3rd training examples are sent to the right child node. The last two values in threshold are placeholders and are to be ignored. The impurity array tells us the computed impurity values in the parent, left, and right nodes respectively.
In the parent node, p = Pr(Class=1) = 2. / 3., so that gini = 2*(2.0/3.0)*(1.0/3.0) = 0.444..... You can confirm the child node impurities as well.
Case 2: with sample_weight
Now, let's try:
dtc.fit(X,Y,sample_weight=[1,2,3])
print dtc.tree_.threshold
# [1.5, -2, -2]
print dtc.tree_.impurity
# [0.44444444, 0.44444444, 0.]
You can see the feature threshold is different. sample_weight also affects the impurity measure in each node. Specifically, in the probability estimates, the first training example is counted the same, the second is counted double, and the third is counted triple, due to the sample weights we've provided.
The impurity in the parent node region is the same. This is just a coincidence. We can compute it directly:
p = Pr(Class=1) = (1+3) / (1+2+3) = 2.0/3.0
The gini measure of 4/9 follows.
Now, you can see from the chosen threshold that the first and second training examples are sent to the left child node, while the third is sent to the right. We see that impurity is calculated to be 4/9 also in the left child node because:
p = Pr(Class=1) = 1 / (1+2) = 1/3.
The impurity of zero in the right child is due to only one training example lying in that region.
You can extend this with non-integer sample-wights similarly. I recommend trying something like sample_weight = [1,2,2.5], and confirming the computed impurities.