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For instance, take X= S 2 × RP 3 and Y= RP 2 × S 3. Then X and Y have the same fundamental group, namely the cyclic group Z/2, and the same universal cover, namely S 2 × S 3; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the Künneth formula); thus, X and Y are not ...
A two-tailed test applied to the normal distribution. A one-tailed test, showing the p-value as the size of one tail. In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test ...
CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. A covering space of a CW complex is also a CW complex. [13] The product of two CW complexes can be made into a CW complex.
Persistent homology has no inherent mechanism to distinguish between low-probability features and high-probability features. One way to apply statistics to topological data analysis is to study the statistical properties of topological features of point clouds. The study of random simplicial complexes offers some insight into statistical topology.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
Z-test tests the mean of a distribution. For each significance level in the confidence interval, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test whose critical values are defined by the sample size (through the corresponding degrees of freedom). Both the Z ...
Two pairs (X 1, A) and (X 2, A) are said to be equivalent, if there is a simple homotopy equivalence between X 1 and X 2 relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X 1 and X 2 with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW-complex A.
In order to consider both the biases, we use a two-tailed test. Note that to do this we cannot simply double the one-tailed p-value unless the probability of the event is 1/2. This is because the binomial distribution becomes asymmetric as that probability deviates from 1/2. There are two methods to define the two-tailed p-value.