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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Simulation Code
The code below will generate a test folume and populate it with a set of clusters.
Step1:  Cluster Extraction Validation
First, be sure that our cluster extraction algorithm works as intended on the test volume
Step2:  The nonzero xor between the volumes proves that clustering the volume maintains the integrity of the cluster lists. This test is run 5 times in a row below to validate that the 0 difference metric is valid across trials
Step3:  The set of 5 zeros validates that the error in the F1 code is not the extraction of the cluster lists
Precision Recall F1 2.0
Step4:  Precision Recall F1 2.0 Testing
First, the algorithm will be tested on a case where it should get 100% precision and 100% recall
Step5:  The 1 precision, recall and f1 metrics show that the algorithm performs as expected on the case of labels being their own predicions
Next, the recall will be tested by randomly selecting a percentile of the true clusters to be passed to the prediction. This should modulate only the recall, and not the precision, of the data, as all clusters being passed are still "correct" predictions. If the algorithm works as expected, I should see a constant precision metric and an upwardly sloping recall metric with slope of 10% That is, for every additional 10% of the labels included in the predictions, the recall sould increase by 10%
Step6:  The linearly increasing nature of recall plot demonstrates that the recall portion of the code indeed corresponds to the ratio of true labels passed in.
Additionally, the precision plot being constant shows that modulating only the recall has no affect on the precision of the data
Next, the data will be diluted such that it contains a portion of noise clusters. This should change the precision, but not the recall, of the data. If this algorithm works as expected, it will produce a constant recall and a downward sloping precision curve with a slope of 10%. That is, for every additional 10% of noise added to the predictions, the precision should drop by 10%
Step7:  As these plots demonstrate, adding noise to the list of all true clusters delivers the expected result, that the precision drops, and the recall remains constant.
In our data, there is not a guarantee that the predicted clusters and the actual clusters will exactly overlap. In fact, this is likely not the case. However, we would not like to consider a cluster a false positive if it only differs from the true cluster by one pixel. For this reason, I have included an overlapRatio parameter to vary how much overlap between a prediction and a true cluster must exist for the prediction to be considered correct
In the following simulation, the cluster labels will be evenly divided and then eroded between 10% and 100%. I will then run the precision recall code against them with an ever increasing percent overlap metric. If the code works, I expect both the precision and the recall to drop by about 10% for every 10% increase in the percent overlap metric.
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Python Code:
def generatePointSet():
    center = (rand(0, 9), rand(0, 999), rand(0, 999))
    toPopulate = []
    for z in range(-3, 2):
        for y in range(-3, 2):
            for x in range(-3, 2):
                curPoint = (center[0]+z, center[1]+y, center[2]+x)
                #only populate valid points
                valid = True
                for dim in range(3):
                    if curPoint[dim] < 0 or curPoint[dim] >= 1000:
                        valid = False
                if valid:
                    toPopulate.append(curPoint)
    return set(toPopulate)
    
def generateTestVolume():
    #create a test volume
    volume = np.zeros((10, 1000, 1000))
    myPointSet = set()
    for _ in range(rand(1000, 2000)):
        potentialPointSet = generatePointSet()
        #be sure there is no overlap
        while len(myPointSet.intersection(potentialPointSet)) > 0:
                potentialPointSet = generatePointSet()
        for elem in potentialPointSet:
            myPointSet.add(elem)
    #populate the true volume
    for elem in myPointSet:
        volume[elem[0], elem[1], elem[2]] = rand(40000, 60000)
    #introduce noise
    noiseVolume = np.copy(volume)
    for z in range(noiseVolume.shape[0]):
        for y in range(noiseVolume.shape[1]):
            for x in range(noiseVolume.shape[2]):
                if not (z, y, x) in myPointSet:
                    toPop = rand(0, 10)
                    if toPop == 5:
                        noiseVolume[z][y][x] = rand(0, 60000)
    return volume, noiseVolume
Explanation: Simulation Code
The code below will generate a test folume and populate it with a set of clusters.
End of explanation
myTestVolume, _ = generateTestVolume()
#passed arbitrarily large param for upper bound to get all clusters
testList = clusterThresh(myTestVolume, 0, 100000)
#generate an annotation volume from the test list
annotations = mv.generateAnnotations(testList, *(myTestVolume.shape))
print np.count_nonzero(np.logical_xor(annotations, myTestVolume))
Explanation: Cluster Extraction Validation
First, be sure that our cluster extraction algorithm works as intended on the test volume
End of explanation
for i in range(5):
    myTestVolume,  _ = generateTestVolume()
    testList = clusterThresh(myTestVolume, 0, 100000)
    annotations = mv.generateAnnotations(testList, *(myTestVolume.shape))
    print np.count_nonzero(np.logical_xor(annotations, myTestVolume))
Explanation: The nonzero xor between the volumes proves that clustering the volume maintains the integrity of the cluster lists. This test is run 5 times in a row below to validate that the 0 difference metric is valid across trials
End of explanation
def precision_recall_f1(labels, predictions, overlapRatio):
   
    if len(predictions) == 0:
        print 'ERROR: prediction list is empty'
        return 0., 0., 0.
    
    labelFound = np.zeros(len(labels))
    truePositives = 0
    falsePositives = 0
    for prediction in predictions:
        #casting to set is ok here since members are uinque
        predictedMembers = set([tuple(elem) for elem in prediction.getMembers()])
        detectionCutoff = overlapRatio * len(predictedMembers)
        found = False
        
        for idx, label in enumerate(labels):
            labelMembers = set([tuple(elem) for elem in label.getMembers()])
            #if the predictedOverlap is over the detectionCutoff ratio
            if len(predictedMembers & labelMembers) >= detectionCutoff:
                truePositives +=1
                found=True
                labelFound[idx] = 1
        
        if not found:
            falsePositives +=1
            
    precision = truePositives/float(truePositives + falsePositives)
    recall = np.count_nonzero(labelFound)/float(len(labels))
    f1 = 0
    try:
        f1 = 2 * (precision*recall)/(precision + recall)
        
    except ZeroDivisionError:
        f1 = 0
        
    return precision, recall, f1
Explanation: The set of 5 zeros validates that the error in the F1 code is not the extraction of the cluster lists
Precision Recall F1 2.0
End of explanation
myTestVolume,  _ = generateTestVolume()
testList = clusterThresh(myTestVolume, 0, 100000)
#run the code on a test volume identical labels
precision, recall, f1 = precision_recall_f1(testList, testList, 1)
print precision, recall, f1
Explanation: Precision Recall F1 2.0 Testing
First, the algorithm will be tested on a case where it should get 100% precision and 100% recall
End of explanation
statList = []
for i in range(1, 11):
    percentile = i/10.
    predictions = np.random.choice(testList, int(percentile * len(testList)), replace=False)
    precision, recall, f1 = precision_recall_f1(testList, predictions, 1)
    statList.append([percentile, precision, recall])
    print 'Percentile: ', percentile
    print '\t precision: ', precision
    print '\t recall: ', recall
fig = plt.figure()
elemwiseStats = zip(*(statList))
plt.title('Recall vs Percentile Passed')
plt.scatter(elemwiseStats[0], elemwiseStats[2])
plt.show()
fig = plt.figure()
plt.title('Precision vs Percentile Passed')
plt.scatter(elemwiseStats[0], elemwiseStats[1], c='r')
plt.show()
Explanation: The 1 precision, recall and f1 metrics show that the algorithm performs as expected on the case of labels being their own predicions
Next, the recall will be tested by randomly selecting a percentile of the true clusters to be passed to the prediction. This should modulate only the recall, and not the precision, of the data, as all clusters being passed are still "correct" predictions. If the algorithm works as expected, I should see a constant precision metric and an upwardly sloping recall metric with slope of 10% That is, for every additional 10% of the labels included in the predictions, the recall sould increase by 10%
End of explanation
statList = []
#get list points in data that I can populate noise clusters with
noCluster = zip(*(np.where(myTestVolume == 0)))
for i in range(0, 10):
    #get the number of noise clusters that must be added for data to be diluted
    #to target percent
    percentile = i/10.
    numNoise = int(percentile * len(testList)/float(1-percentile))
    
    #generate the prediction + noise list
    noiseList =[]
    for j in range(numNoise):
        badPoint = noCluster[rand(0, len(noCluster)-1)]
        noiseList.append(Cluster([list(badPoint)]))
    predictions = testList + noiseList
    
    precision, recall, f1 = precision_recall_f1(testList, predictions, 1)
    statList.append([percentile, precision, recall])
    print 'Percentile: ', percentile
    print '\t precision: ', precision
    print '\t recall: ', recall
fig = plt.figure()
elemwiseStats = zip(*(statList))
plt.title('Recall vs Percentile of Data that is Not Noise')
plt.scatter(elemwiseStats[0], elemwiseStats[2])
plt.show()
fig = plt.figure()
plt.title('Precision vs Percent of Data that is Not Noise')
plt.scatter(elemwiseStats[0], elemwiseStats[1], c='r')
plt.show()
Explanation: The linearly increasing nature of recall plot demonstrates that the recall portion of the code indeed corresponds to the ratio of true labels passed in.
Additionally, the precision plot being constant shows that modulating only the recall has no affect on the precision of the data
Next, the data will be diluted such that it contains a portion of noise clusters. This should change the precision, but not the recall, of the data. If this algorithm works as expected, it will produce a constant recall and a downward sloping precision curve with a slope of 10%. That is, for every additional 10% of noise added to the predictions, the precision should drop by 10%
End of explanation
statList = []
#generate the list of eroded clusters
erodedList = []
for idx, cluster in enumerate(testList):
    percentile = (idx%10)/10. + .1
    members = cluster.getMembers()
    erodedList.append(Cluster(members[:int(len(members)*percentile)]))
for i in range(1, 11):
    percentile = i/10.   
    precision, recall, f1 = precision_recall_f1(erodedList, testList, percentile)
    statList.append([percentile, precision, recall])
    print 'Percentile: ', percentile
    print '\t precision: ', precision
    print '\t recall: ', recall
fig = plt.figure()
elemwiseStats = zip(*(statList))
plt.title('Recall vs Percent Required Overlap')
plt.scatter(elemwiseStats[0], elemwiseStats[2])
plt.show()
fig = plt.figure()
elemwiseStats = zip(*(statList))
plt.title('Precision vs Percent Required Overlap')
plt.scatter(elemwiseStats[0], elemwiseStats[1])
plt.show()
Explanation: As these plots demonstrate, adding noise to the list of all true clusters delivers the expected result, that the precision drops, and the recall remains constant.
In our data, there is not a guarantee that the predicted clusters and the actual clusters will exactly overlap. In fact, this is likely not the case. However, we would not like to consider a cluster a false positive if it only differs from the true cluster by one pixel. For this reason, I have included an overlapRatio parameter to vary how much overlap between a prediction and a true cluster must exist for the prediction to be considered correct
In the following simulation, the cluster labels will be evenly divided and then eroded between 10% and 100%. I will then run the precision recall code against them with an ever increasing percent overlap metric. If the code works, I expect both the precision and the recall to drop by about 10% for every 10% increase in the percent overlap metric.
End of explanation
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