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# Use Simpson's Rule to estimate the centroid of the region shown.

## $(\overline{x}, \overline{y}) \approx(4.4,1.4)$

#### Topics

Applications of Integration

### Discussion

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##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp

### Video Transcript

in this problem, we're given a graph and worse to approximate The Central Office given Region using seems since rule You know that eccentric oi is equal to summation of Ex Terms area divided by total area and that is integral from a to B x times d, uh, function the X divided by function the X and why part is equal to integral from A to B so the limits of the region Times one Health function square D x Divided by function in times d x Here we know that the limits are from 0 to 8. Um, instead of using this exact integration were Guineas, Simpsons Rule and these are deployments that we're gonna all that dysfunction at most after points are just approximations. You could use different values. So let's start by approximating the functions in the denominators are inter goal FX dxL Let's start our that interval of effects The eggs a CZ You can see the increment between each point is one. So we're gonna set up for Simpson's rolled out the extra p want and we have in total eight numbers. So end will be eight cell for The Simpsons roll If X T X will be 1/3 fo zero plus four ash off one plus two effort to four F off three plus two F before plus 4 +45 plus to f off six. It was four f of seven plus two f ate. Sorry, actually, for Fayed is the last one. So we're not gonna have this constant here. Settle Just people. It's a fade. We know the functions values at those points. So we're just gonna plug those numbers in, do the modifications and doing that, we found this integral approximation as 20.1. Now, let's step works, mate. Um, the numerator for ex enjoy. So x times, FX, the eggs Don't we go to 1/3 all we're gonna have the same thing. All the differences were discontent and multiply it by the X values. So it will be zero times fo zero plus four times long times that one plus two times two times that off too. Plus four times three times F three plus two times four times before plus four times five times five plus two times six times out of six. 47 F off seven and ate a fade. If he again plugged values in from this craft. And if you do the multiplication, we see that X ffx. T x is about 88.13 Bilis Brooks made integral What? Have off f squared x d x We're gonna have What? Health isn't constant. 1 30 comes from the same central. We're just gonna know, um, square to function. So we're gonna have and scared of zero plus four times that script of one plus two times x squared of two. Plus that that out. You get the idea. We'll have four times subscribed of seven, plus, um, f squared off eight. Now again plugging into function values. He found this approximation to be about 29 points here. Six from missile being introduced, no to export. And why bar? We know that export is equal to this approximation divided by this approximation. So that is, uh, 88 0.13 divided by 20.1, and that comes out to be 4.38. Therefore, wire bar, we have this approximation divided by Brooks mission for the area. So 29.6 divided by 20.4. And that comes out to be 1.44. So the center of mass off this region as for 0.38 then 1.44.

#### Topics

Applications of Integration

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp


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Consider a two windings transformer (I'm not talking about flyback transformer):

• Magnetizing inductance is not equal to mutual inductance. Mutual inductance refers to the energy which is transfered from the primary to the secondary. The energy which is not transfered from the primary to the secondary refers to the leakage inductances and?
• Magnetizing inductance correspond to an inductance which is placed in parallel to the primary side of the transformer. So the energy stored in the magnetizing inductance (1/2LI²) is not transfered to the secondary. As some people says it "sets up" the transformer for then transfering energy to the secondary.
• Apparently in a forward transformer, if the magnetizing inductance was infinite, the flux into the transformer would be equal to 0 as the flux at the primary would be cancelled by the secondary.

What do you think about it?

Magnetizing inductance is not equal to mutual inductance.

It is for a 1:1 transformer. The mutual inductance (the turns that are 100% coupled between primary and secondary) represents the magnetization inductance. Those turns that don't couple are not mutual inductance but leakage inductance. Mutual inductance is $$\\sqrt{L_P\cdot L_S}\$$ hence, for a 1:1 transformer where primary and secondary inductance = L, M = $$\\sqrt{L^2}\$$ = L.

Mutual inductance refers to the energy which is transfered from the primary to the secondary.

The presence of mutual inductance allows energy to pass from primary to secondary.

The energy which is not transfered from the primary to the secondary refers to the leakage inductances

Leakage inductance is that inductance that doesn't couple primary and secondary

Magnetizing inductance correspond to an inductance which is placed in parallel to the primary side of the transformer.

Correct, for the model of a transformer where "the transformer" is internally an ideal power transfer component.

So the energy stored in the magnetizing inductance (1/2LI²) is not transfered to the secondary.

Correct (except for a flyback converter), but the magnetization inductance also facilitates primary to secondary power and energy transfer. It does two jobs.

Apparently in a forward transformer, if the magnetizing inductance was infinite, the flux into the transformer would be equal to 0

Not really a practical issue.

the flux at the primary would be cancelled by the secondary.

The flux due to load currents in the secondary are cancelled by flux in the primary due to those same load currents whatever the inductance.

• Thank you for your answer. There is still something which I do not understand. How mutual inductance can be equal to magnetizing inductance if the energy stored in the magnetizing inductance is not transfered to the secondary side ? It would mean that in a forward transformer no energy is transfered by the mutual inductance, BUT in a flyback transformer all the energy is transfered by the mutual inductance ... – Jess Jul 20 '20 at 11:26
• @Jess I never use mutual inductance in this respect; I use coupling factor, k and k is quite literally the percentage of the overall inductance measured on a winding that 100% couples to another winding. Mutual inductance is only needed (it seems) for doing exam questions raised by people who think they know best. I can't think of one example in real life where it is needed over and above k. I find the usage of mutual inductance as unnatural and superfluous. – Andy aka Jul 20 '20 at 12:04
• Check magnetic domains, also called Weiss domains in this Wikipedia article. In a transformer (where primary and secondary currents simultaneously circulate), the magnetizing current aligns the domains and energizes the core, coupling the primary and secondary. The domains alignment with magnetic field creates core losses and heat which are independent from the current in the sec side. You feel the mag. current at work with a classical unloaded ferromagnetic transformer left in the wall outlet: it's warm despite the absence of sec. current. – Verbal Kint Jul 20 '20 at 12:09
• What makes the core hot @Jess is not magnetization inductance but cores losses represented by a resistor in parallel with the magnetization inductance. This assumes that copper losses are not responsible for heating (i.e. it assumes a no-load situation). – Andy aka Aug 4 '20 at 8:55
• Ok ;) One day I will know how it works. I hope :D Thank you ! – Jess Aug 4 '20 at 9:28

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Infoscience

Report

# On Automatic Annotation of Images with Latent Space Models

Image auto-annotation, i.e., the association of words to whole images, has attracted considerable attention. In particular, unsupervised, probabilistic latent variable models of text and image features have shown encouraging results, but their performance with respect to other approaches remains unknown. In this paper, we apply and compare two simple latent space models commonly used in text analysis, namely Latent Semantic Analysis (LSA) and Probabilistic LSA (PLSA). Annotation strategies for each model are discussed. Remarkably, we found that, on a 8000-image dataset, a classic LSA model defined on keywords and a very basic image representation performed as well as much more complex, state-of-the-art methods. Furthermore, non-probabilistic methods (LSA and direct image matching) outperformed PLSA on the same dataset.

Keywords: vision

Note:

Published in Proc. ACM Multimedia 2003'', 2003

#### Reference

• EPFL-REPORT-82881

Record created on 2006-03-10, modified on 2016-08-08


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# Quaternion

 related topics {math, number, function} {math, energy, light} {work, book, publish} {church, century, christian} {son, year, death} {school, student, university} {line, north, south} {style, bgcolor, rowspan}

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space[1] or equivalently as the quotient of two vectors.[2] Quaternions can also be represented as the sum of a scalar and a vector.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. They can be used alongside other methods, such as Euler angles and matrices, or as an alternative to them depending on the application.

In modern language, quaternions form a four-dimensional normed division algebra over the real numbers. In fact, the quaternions were the first noncommutative division algebra to be discovered.[3] The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by $\mathbb{H}$ (Unicode U+210D, ). It can also be given by the Clifford algebra classifications C0,2(R) = C03,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers.

The unit quaternions can therefore be thought of as a choice of a group structure on the 3-sphere S3, the group Spin(3), the group SU(2), or the universal cover of SO(3).


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# Sorting and Related Functions

Julia has an extensive, flexible API for sorting and interacting with already-sorted arrays of values. By default, Julia picks reasonable algorithms and sorts in standard ascending order:

julia> sort([2,3,1])
3-element Vector{Int64}:
1
2
3

You can easily sort in reverse order as well:

julia> sort([2,3,1], rev=true)
3-element Vector{Int64}:
3
2
1

sort constructs a sorted copy leaving its input unchanged. Use the "bang" version of the sort function to mutate an existing array:

julia> a = [2,3,1];

julia> sort!(a);

julia> a
3-element Vector{Int64}:
1
2
3

Instead of directly sorting an array, you can compute a permutation of the array's indices that puts the array into sorted order:

julia> v = randn(5)
5-element Array{Float64,1}:
0.297288
0.382396
-0.597634
-0.0104452
-0.839027

julia> p = sortperm(v)
5-element Array{Int64,1}:
5
3
4
1
2

julia> v[p]
5-element Array{Float64,1}:
-0.839027
-0.597634
-0.0104452
0.297288
0.382396

Arrays can easily be sorted according to an arbitrary transformation of their values:

julia> sort(v, by=abs)
5-element Array{Float64,1}:
-0.0104452
0.297288
0.382396
-0.597634
-0.839027

Or in reverse order by a transformation:

julia> sort(v, by=abs, rev=true)
5-element Array{Float64,1}:
-0.839027
-0.597634
0.382396
0.297288
-0.0104452

If needed, the sorting algorithm can be chosen:

julia> sort(v, alg=InsertionSort)
5-element Array{Float64,1}:
-0.839027
-0.597634
-0.0104452
0.297288
0.382396

All the sorting and order related functions rely on a "less than" relation defining a total order on the values to be manipulated. The isless function is invoked by default, but the relation can be specified via the lt keyword.

## Sorting Functions

Base.sort!Function
sort!(v; alg::Algorithm=defalg(v), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort the vector v in place. A stable algorithm is used by default. You can select a specific algorithm to use via the alg keyword (see Sorting Algorithms for available algorithms). The by keyword lets you provide a function that will be applied to each element before comparison; the lt keyword allows providing a custom "less than" function (note that for every x and y, only one of lt(x,y) and lt(y,x) can return true); use rev=true to reverse the sorting order. These options are independent and can be used together in all possible combinations: if both by and lt are specified, the lt function is applied to the result of the by function; rev=true reverses whatever ordering specified via the by and lt keywords.

Examples

julia> v = [3, 1, 2]; sort!(v); v
3-element Vector{Int64}:
1
2
3

julia> v = [3, 1, 2]; sort!(v, rev = true); v
3-element Vector{Int64}:
3
2
1

julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[1]); v
3-element Vector{Tuple{Int64, String}}:
(1, "c")
(2, "b")
(3, "a")

julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[2]); v
3-element Vector{Tuple{Int64, String}}:
(3, "a")
(2, "b")
(1, "c")
source
sort!(A; dims::Integer, alg::Algorithm=defalg(A), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort the multidimensional array A along dimension dims. See sort! for a description of possible keyword arguments.

To sort slices of an array, refer to sortslices.

Julia 1.1

This function requires at least Julia 1.1.

Examples

julia> A = [4 3; 1 2]
2×2 Matrix{Int64}:
4  3
1  2

julia> sort!(A, dims = 1); A
2×2 Matrix{Int64}:
1  2
4  3

julia> sort!(A, dims = 2); A
2×2 Matrix{Int64}:
1  2
3  4
source
Base.sortFunction
sort(v; alg::Algorithm=defalg(v), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Variant of sort! that returns a sorted copy of v leaving v itself unmodified.

Examples

julia> v = [3, 1, 2];

julia> sort(v)
3-element Vector{Int64}:
1
2
3

julia> v
3-element Vector{Int64}:
3
1
2
source
sort(A; dims::Integer, alg::Algorithm=defalg(A), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort a multidimensional array A along the given dimension. See sort! for a description of possible keyword arguments.

To sort slices of an array, refer to sortslices.

Examples

julia> A = [4 3; 1 2]
2×2 Matrix{Int64}:
4  3
1  2

julia> sort(A, dims = 1)
2×2 Matrix{Int64}:
1  2
4  3

julia> sort(A, dims = 2)
2×2 Matrix{Int64}:
3  4
1  2
source
Base.sortpermFunction
sortperm(A; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward, [dims::Integer])

Return a permutation vector or array I that puts A[I] in sorted order along the given dimension. If A has more than one dimension, then the dims keyword argument must be specified. The order is specified using the same keywords as sort!. The permutation is guaranteed to be stable even if the sorting algorithm is unstable, meaning that indices of equal elements appear in ascending order.

See also sortperm!, partialsortperm, invperm, indexin. To sort slices of an array, refer to sortslices.

Julia 1.9

The method accepting dims requires at least Julia 1.9.

Examples

julia> v = [3, 1, 2];

julia> p = sortperm(v)
3-element Vector{Int64}:
2
3
1

julia> v[p]
3-element Vector{Int64}:
1
2
3

julia> A = [8 7; 5 6]
2×2 Matrix{Int64}:
8  7
5  6

julia> sortperm(A, dims = 1)
2×2 Matrix{Int64}:
2  4
1  3

julia> sortperm(A, dims = 2)
2×2 Matrix{Int64}:
3  1
2  4
source
Base.Sort.InsertionSortConstant
InsertionSort

Use the insertion sort algorithm.

Insertion sort traverses the collection one element at a time, inserting each element into its correct, sorted position in the output vector.

Characteristics:

• stable: preserves the ordering of elements which compare equal

(e.g. "a" and "A" in a sort of letters which ignores case).

• in-place in memory.
• quadratic performance in the number of elements to be sorted:

it is well-suited to small collections but should not be used for large ones.

source
Base.Sort.MergeSortConstant
MergeSort

Indicate that a sorting function should use the merge sort algorithm. Merge sort divides the collection into subcollections and repeatedly merges them, sorting each subcollection at each step, until the entire collection has been recombined in sorted form.

Characteristics:

• stable: preserves the ordering of elements which compare equal (e.g. "a" and "A" in a sort of letters which ignores case).
• not in-place in memory.
• divide-and-conquer sort strategy.
• good performance for large collections but typically not quite as fast as QuickSort.
source
Base.Sort.QuickSortConstant
QuickSort

Indicate that a sorting function should use the quick sort algorithm, which is not stable.

Characteristics:

• not stable: does not preserve the ordering of elements which compare equal (e.g. "a" and "A" in a sort of letters which ignores case).
• in-place in memory.
• divide-and-conquer: sort strategy similar to MergeSort.
• good performance for large collections.
source
Base.Sort.PartialQuickSortType
PartialQuickSort{T <: Union{Integer,OrdinalRange}}

Indicate that a sorting function should use the partial quick sort algorithm. Partial quick sort returns the smallest k elements sorted from smallest to largest, finding them and sorting them using QuickSort.

Characteristics:

• not stable: does not preserve the ordering of elements which compare equal (e.g. "a" and "A" in a sort of letters which ignores case).
• in-place in memory.
• divide-and-conquer: sort strategy similar to MergeSort.

Note that PartialQuickSort(k) does not necessarily sort the whole array. For example,

jldoctest julia> x = rand(100);

julia> k = 50:100;

julia> s1 = sort(x; alg=QuickSort);

julia> s2 = sort(x; alg=PartialQuickSort(k));

julia> map(issorted, (s1, s2)) (true, false)

julia> map(x->issorted(x[k]), (s1, s2)) (true, true)

julia> s1[k] == s2[k] true

source
Base.Sort.sortperm!Function
sortperm!(ix, A; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward, initialized::Bool=false, [dims::Integer])

Like sortperm, but accepts a preallocated index vector or array ix with the same axes as A. If initialized is false (the default), ix is initialized to contain the values LinearIndices(A).

Julia 1.9

The method accepting dims requires at least Julia 1.9.

Examples

julia> v = [3, 1, 2]; p = zeros(Int, 3);

julia> sortperm!(p, v); p
3-element Vector{Int64}:
2
3
1

julia> v[p]
3-element Vector{Int64}:
1
2
3

julia> A = [8 7; 5 6]; p = zeros(Int,2, 2);

julia> sortperm!(p, A; dims=1); p
2×2 Matrix{Int64}:
2  4
1  3

julia> sortperm!(p, A; dims=2); p
2×2 Matrix{Int64}:
3  1
2  4
source
Base.sortslicesFunction
sortslices(A; dims, alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort slices of an array A. The required keyword argument dims must be either an integer or a tuple of integers. It specifies the dimension(s) over which the slices are sorted.

E.g., if A is a matrix, dims=1 will sort rows, dims=2 will sort columns. Note that the default comparison function on one dimensional slices sorts lexicographically.

For the remaining keyword arguments, see the documentation of sort!.

Examples

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1) # Sort rows
3×3 Matrix{Int64}:
-1   6  4
7   3  5
9  -2  8

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, lt=(x,y)->isless(x[2],y[2]))
3×3 Matrix{Int64}:
9  -2  8
7   3  5
-1   6  4

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, rev=true)
3×3 Matrix{Int64}:
9  -2  8
7   3  5
-1   6  4

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2) # Sort columns
3×3 Matrix{Int64}:
3   5  7
-1  -4  6
-2   8  9

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, alg=InsertionSort, lt=(x,y)->isless(x[2],y[2]))
3×3 Matrix{Int64}:
5   3  7
-4  -1  6
8  -2  9

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, rev=true)
3×3 Matrix{Int64}:
7   5   3
6  -4  -1
9   8  -2

Higher dimensions

sortslices extends naturally to higher dimensions. E.g., if A is a a 2x2x2 array, sortslices(A, dims=3) will sort slices within the 3rd dimension, passing the 2x2 slices A[:, :, 1] and A[:, :, 2] to the comparison function. Note that while there is no default order on higher-dimensional slices, you may use the by or lt keyword argument to specify such an order.

If dims is a tuple, the order of the dimensions in dims is relevant and specifies the linear order of the slices. E.g., if A is three dimensional and dims is (1, 2), the orderings of the first two dimensions are re-arranged such that the slices (of the remaining third dimension) are sorted. If dims is (2, 1) instead, the same slices will be taken, but the result order will be row-major instead.

Higher dimensional examples

julia> A = permutedims(reshape([4 3; 2 1; 'A' 'B'; 'C' 'D'], (2, 2, 2)), (1, 3, 2))
2×2×2 Array{Any, 3}:
[:, :, 1] =
4  3
2  1

[:, :, 2] =
'A'  'B'
'C'  'D'

julia> sortslices(A, dims=(1,2))
2×2×2 Array{Any, 3}:
[:, :, 1] =
1  3
2  4

[:, :, 2] =
'D'  'B'
'C'  'A'

julia> sortslices(A, dims=(2,1))
2×2×2 Array{Any, 3}:
[:, :, 1] =
1  2
3  4

[:, :, 2] =
'D'  'C'
'B'  'A'

julia> sortslices(reshape([5; 4; 3; 2; 1], (1,1,5)), dims=3, by=x->x[1,1])
1×1×5 Array{Int64, 3}:
[:, :, 1] =
1

[:, :, 2] =
2

[:, :, 3] =
3

[:, :, 4] =
4

[:, :, 5] =
5
source
Base.issortedFunction
issorted(v, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Test whether a vector is in sorted order. The lt, by and rev keywords modify what order is considered to be sorted just as they do for sort.

Examples

julia> issorted([1, 2, 3])
true

julia> issorted([(1, "b"), (2, "a")], by = x -> x[1])
true

julia> issorted([(1, "b"), (2, "a")], by = x -> x[2])
false

julia> issorted([(1, "b"), (2, "a")], by = x -> x[2], rev=true)
true
source
Base.Sort.searchsortedFunction
searchsorted(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the range of indices of a which compare as equal to x (using binary search) according to the order specified by the by, lt and rev keywords, assuming that a is already sorted in that order. Return an empty range located at the insertion point if a does not contain values equal to x.

Examples

julia> searchsorted([1, 2, 4, 5, 5, 7], 4) # single match
3:3

julia> searchsorted([1, 2, 4, 5, 5, 7], 5) # multiple matches
4:5

julia> searchsorted([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
3:2

julia> searchsorted([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
7:6

julia> searchsorted([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
1:0
source
Base.Sort.searchsortedfirstFunction
searchsortedfirst(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the index of the first value in a greater than or equal to x, according to the specified order. Return lastindex(a) + 1 if x is greater than all values in a. a is assumed to be sorted.

insert!ing x at this index will maintain sorted order.

Examples

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 4) # single match
3

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 5) # multiple matches
4

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
3

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
7

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
1
source
Base.Sort.searchsortedlastFunction
searchsortedlast(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the index of the last value in a less than or equal to x, according to the specified order. Return firstindex(a) - 1 if x is less than all values in a. a is assumed to be sorted.

Examples

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 4) # single match
3

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 5) # multiple matches
5

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
2

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
6

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
0
source
Base.Sort.insortedFunction
insorted(x, a; by=<transform>, lt=<comparison>, rev=false) -> Bool

Determine whether an item x is in the sorted collection a, in the sense that it is == to one of the values of the collection according to the order specified by the by, lt and rev keywords, assuming that a is already sorted in that order, see sort for the keywords.

See also in.

Examples

julia> insorted(4, [1, 2, 4, 5, 5, 7]) # single match
true

julia> insorted(5, [1, 2, 4, 5, 5, 7]) # multiple matches
true

julia> insorted(3, [1, 2, 4, 5, 5, 7]) # no match
false

julia> insorted(9, [1, 2, 4, 5, 5, 7]) # no match
false

julia> insorted(0, [1, 2, 4, 5, 5, 7]) # no match
false
Julia 1.6

insorted was added in Julia 1.6.

source
Base.Sort.partialsort!Function
partialsort!(v, k; by=<transform>, lt=<comparison>, rev=false)

Partially sort the vector v in place, according to the order specified by by, lt and rev so that the value at index k (or range of adjacent values if k is a range) occurs at the position where it would appear if the array were fully sorted. If k is a single index, that value is returned; if k is a range, an array of values at those indices is returned. Note that partialsort! may not fully sort the input array.

Examples

julia> a = [1, 2, 4, 3, 4]
5-element Vector{Int64}:
1
2
4
3
4

julia> partialsort!(a, 4)
4

julia> a
5-element Vector{Int64}:
1
2
3
4
4

julia> a = [1, 2, 4, 3, 4]
5-element Vector{Int64}:
1
2
4
3
4

julia> partialsort!(a, 4, rev=true)
2

julia> a
5-element Vector{Int64}:
4
4
3
2
1
source
Base.Sort.partialsortFunction
partialsort(v, k, by=<transform>, lt=<comparison>, rev=false)

Variant of partialsort! which copies v before partially sorting it, thereby returning the same thing as partialsort! but leaving v unmodified.

source
Base.Sort.partialsortpermFunction
partialsortperm(v, k; by=<transform>, lt=<comparison>, rev=false)

Return a partial permutation I of the vector v, so that v[I] returns values of a fully sorted version of v at index k. If k is a range, a vector of indices is returned; if k is an integer, a single index is returned. The order is specified using the same keywords as sort!. The permutation is stable, meaning that indices of equal elements appear in ascending order.

Note that this function is equivalent to, but more efficient than, calling sortperm(...)[k].

Examples

julia> v = [3, 1, 2, 1];

julia> v[partialsortperm(v, 1)]
1

julia> p = partialsortperm(v, 1:3)
3-element view(::Vector{Int64}, 1:3) with eltype Int64:
2
4
3

julia> v[p]
3-element Vector{Int64}:
1
1
2
source
Base.Sort.partialsortperm!Function
partialsortperm!(ix, v, k; by=<transform>, lt=<comparison>, rev=false, initialized=false)

Like partialsortperm, but accepts a preallocated index vector ix the same size as v, which is used to store (a permutation of) the indices of v.

If the index vector ix is initialized with the indices of v (or a permutation thereof), initialized should be set to true.

If initialized is false (the default), then ix is initialized to contain the indices of v.

If initialized is true, but ix does not contain (a permutation of) the indices of v, the behavior of partialsortperm! is undefined.

(Typically, the indices of v will be 1:length(v), although if v has an alternative array type with non-one-based indices, such as an OffsetArray, ix must also be an OffsetArray with the same indices, and must contain as values (a permutation of) these same indices.)

Upon return, ix is guaranteed to have the indices k in their sorted positions, such that

partialsortperm!(ix, v, k);
v[ix[k]] == partialsort(v, k)

The return value is the kth element of ix if k is an integer, or view into ix if k is a range.

Examples

julia> v = [3, 1, 2, 1];

julia> ix = Vector{Int}(undef, 4);

julia> partialsortperm!(ix, v, 1)
2

julia> ix = [1:4;];

julia> partialsortperm!(ix, v, 2:3, initialized=true)
2-element view(::Vector{Int64}, 2:3) with eltype Int64:
4
3
source

## Sorting Algorithms

There are currently four sorting algorithms publicly available in base Julia:

By default, the sort family of functions uses stable sorting algorithms that are fast on most inputs. The exact algorithm choice is an implementation detail to allow for future performance improvements. Currently, a hybrid of RadixSort, ScratchQuickSort, InsertionSort, and CountingSort is used based on input type, size, and composition. Implementation details are subject to change but currently available in the extended help of ??Base.DEFAULT_STABLE and the docstrings of internal sorting algorithms listed there.

You can explicitly specify your preferred algorithm with the alg keyword (e.g. sort!(v, alg=PartialQuickSort(10:20))) or reconfigure the default sorting algorithm for custom types by adding a specialized method to the Base.Sort.defalg function. For example, InlineStrings.jl defines the following method:

Base.Sort.defalg(::AbstractArray{<:Union{SmallInlineStrings, Missing}}) = InlineStringSort
Julia 1.9

The default sorting algorithm (returned by Base.Sort.defalg) is guaranteed to be stable since Julia 1.9. Previous versions had unstable edge cases when sorting numeric arrays.

## Alternate orderings

By default, sort and related functions use isless to compare two elements in order to determine which should come first. The Base.Order.Ordering abstract type provides a mechanism for defining alternate orderings on the same set of elements. Instances of Ordering define a total order on a set of elements, so that for any elements a, b, c the following hold:

• Exactly one of the following is true: a is less than b, b is less than a, or a and b are equal (according to isequal).
• The relation is transitive - if a is less than b and b is less than c then a is less than c.

The Base.Order.lt function works as a generalization of isless to test whether a is less than b according to a given order.

Base.Order.ltFunction
lt(o::Ordering, a, b)

Test whether a is less than b according to the ordering o.

source
Base.Order.ordFunction
ord(lt, by, rev::Union{Bool, Nothing}, order::Ordering=Forward)

Construct an Ordering object from the same arguments used by sort!. Elements are first transformed by the function by (which may be identity) and are then compared according to either the function lt or an existing ordering order. lt should be isless or a function which obeys similar rules. Finally, the resulting order is reversed if rev=true.

Passing an lt other than isless along with an order other than Base.Order.Forward or Base.Order.Reverse is not permitted, otherwise all options are independent and can be used together in all possible combinations.

source
Base.Order.ReverseOrderingType
ReverseOrdering(fwd::Ordering=Forward)

A wrapper which reverses an ordering.

For a given Ordering o, the following holds for all a, b:

lt(ReverseOrdering(o), a, b) == lt(o, b, a)
source
Base.Order.ByType
By(by, order::Ordering=Forward)

Ordering which applies order to elements after they have been transformed by the function by.

source
Base.Order.PermType
Perm(order::Ordering, data::AbstractVector)

Ordering on the indices of data where i is less than j if data[i] is less than data[j] according to order. In the case that data[i] and data[j] are equal, i and j` are compared by numeric value.

source


text:
`endnotes.heading.command` — LaTeX command of the endnotes headings

## Synopsis

```      <xsl:param name="endnotes.heading.command"/>
```

## Description

With this parameter you can provide your own latex macro to make the endnotes headings. If not set, dblatex uses the default sectionning commands like `\section*`, `\subsection*`, and so on. The command takes one parameter that is the title markup of the heading. For more details see the section called “Grouping Endnotes”.

`endnotes.heading.groups`
`endnotes.heading.command`


text:
jarque bera test spss

This leads to a large Type I error rate. In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. Testing Normality Using Stata 6. Since it IS a test, state a null and alternate hypothesis. jb = (379/6)*((1.50555^2)+(((6.43 -3)^2)/4)) = 328.9 The statistic has a Chi 2 distribution with 2 degrees of freedom, (one for skewness one for kurtosis). Performance assessment of growth, income, and value stocks listed in the BSE (2015-2020), Trend analysis of stocks performance listed in BSE (2011-2020), Annual average returns and market returns for growth, income, and value stocks (2005-2015), We are hiring freelance research consultants. Why is it important to test heteroskedasticity in a dataset? Kolmogorov-Smirnov Goodness of Fit (K-S) Test. How to work with a moderating variable in the regression test with SPSS? Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test statistic measures the difference of the skewness and kurtosis of the series with those from the normal distribution. According to Robert Hall, David Lilien, et al. Monte Carlo methods are used to study the size, and the power of the JB normality test with the “sample” critical values and compare with three Normality tests are thank in advance. For smaller samples, it uses a table derived from Monte Carlo simulations in order to interpolate p-values.[1]. Apart from academics, she loves music and travelling new places. EViews and Stata support the Jarque-Bera test. MODEL . The test statistic is always nonnegative. "JB: Stata module to perform Jarque-Bera test for normality on series," Statistical Software Components S353801, Boston College Department of Economics, revised 12 Sep 2000.Handle: RePEc:boc:bocode:s353801 Test statistic value > critical Value Or P-Value < α value. print partial autocorrelations . From tables critical value at 5% level for 2 degrees of freedom is 5.99 So JB>c2 critical, … μ Procedure and interpretation of linear regression analysis using STATA, Non linear regression analysis in STATA and its interpretation. The input can be a time series of residuals, jarque.bera.test.default, or an Arima object, jarque.bera.test.Arima from which the residuals are extracted. If it is far from zero, it signals the data do not have a normal distribution. Notify me of follow-up comments by email. Details. If you perform a normality test… • Jarque-Bera is a test statistic for testing whether the series is normally distributed. The test statistic JB is defined as: JB =(n/6) * (S 2 + (C 2 /4)) where: n: the number of observations in the sample; S: the sample skewness; C: the sample kurtosis Now, if we are using the definition of normal distribution having a kurtosis of zero, while many other sources - outside of wikipedia - use 3, shouldn't we, for the case of clarity offer two alternative specifications for this test? The Jarque-Bera test is a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution. Test statistic value < critical Value Or P-Value > α value. We start by preparing a layout to explain our scope of work. JARQUE (R1, pop) = the Jarque-Barre test statistic JB for the data in the range R1 JBTEST (R1, pop) = p-value of the Jarque-Barre test on the data in R1 If pop = TRUE (default), the population version of the test is used; otherwise the sample version of the test is used. MODEL . As per the above figure, chi(2) is 0.1211 which is greater than 0.05. x Statistical test of normality calculates the probability of deriving sample not include the normality test of the dataset. "tseries: Time Series Analysis and Computational Finance", "moments: Moments, cumulants, skewness, kurtosis and related tests", "JarqueBeraALMTest—Wolfram Language Documentation", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Jarque–Bera_test&oldid=996294038, Creative Commons Attribution-ShareAlike License, This page was last edited on 25 December 2020, at 18:22. The table below shows some p-values approximated by a chi-squared distribution that differ from their true alpha levels for small samples. always accepted. Details 258 C. M. Jarque, A. K. Bera / Tests for normality, homoscedasticity, serial independence first term in (4) is identical to the LM residual normality test for the case of HI residuals [e.g., Jarque and Bera (1980)], say LM,. Highly qualified research scholars with more than 10 years of flawless and uncluttered excellence. Ha: Sample is derived from a normally distributed population. The statistic was derived by Carlos M. Jarque and Anil K. Bera while working on their Ph.D. Thesis at the Australian National University. H0: Sample is not derived from a normally distributed population. the p-value of 0.001 < 0.05, hence, the null hypothesis is rejected. TEST=LM . Jarque-Bera test for normality . The test is named after Carlos Jarque and Anil K. Bera. Numerical Methods 4. Data scientists strictly prefer to test normality The previous article explained the importance of testing normality t for a dataset before performing regression. The test is specifically designed for alternatives in the Pearson system of distributions. Any empty cells or cells containing non-numeric data are ignored. The Jarque-Bera test is a two-sided goodness-of-fit test suitable when a fully specified null distribution is unknown and its parameters must be estimated. The test statistic is The second term This function performs the Jarque-Bera test on the given data sample to determine if the data are sample drawn from a normal population. The results are represented below. Hence, the FDI Inflows sample is not derived from the normally distributed population. ^ For example, the normality of residuals obtained in linear regression is rarely tested, even though it governs the quality of the confidence intervals surrounding parameters and predictions. What is the relevance of significant results in regression analysis? Information on the normally distributed data not required. {\displaystyle {\hat {\sigma }}^{2}} Conclusion 1. print tests for ARCH process . For small samples the chi-squared approximation is overly sensitive, often rejecting the null hypothesis when it is true. How to work with a mediating variable in a regression analysis? Jarque-Bera Test: Check the joint probability of skewness and kurtosis from the normal distribution values. However, graphical normality test has several shortcomings, the biggest one being lack of reliability due to the probability of inaccurate results. But there are many alternative tests of univariate normality: the Lilliefors test, the Pearson's chi-squared test, and the Shapiro–Francia test, D'Agostino's K-squared test, the Anderson–Darling test, the Cramér–von Mises criterion, and the Jarque–Bera test. 1. Graphical Methods 3. Keywords htest, math. This article explains three such tests using SPSS and E-Views software: Normal distribution of data is also called ‘Gaussian distribution’. Normality tests help in checking whether the data and work on normally distributed data because of its benefits (Parbhakar, However, work best for dataset < 50. Usage. The test statistic of the Jarque-Bera test is always a positive number and the further it is from zero, the more evidence that the sample data does not follow a normal distribution. ARCHTEST . She has a keen interest in econometrics and data analysis. Construct Jarque -Bera test . This article defines MAQL to calculate skewness and kurtosis that can be used to test the normality of a given data set. The Jarque-Bera test statistic is defined as: $$\frac{N}{6} \left( S^2 + \frac{(K - 3)^2}{4} \right)$$ with S, K, and Ndenoting the sample skewness, the sample kurtosis, and the sample … CHOW= print the predictive Chow test . However, K-S Test can only be applied in SPSS. Riya is a master in Economics from Amity University. Shapiro-Wilk test results are similar to K-S test results i.e. nrepl the number of replications in Monte Carlo simulation. Better model fit for nature and social science-based studies. ^ pada software Eviews untuk pengujian normalitas pada data digunakan dua macam pengukuran yaitu Histogram (seperti pada SPSS) dan uji Jarque-Bera. For example, simple linear regression Introduction 2. The null hypothesis in this test is data follow normal distribution. The table shows that the p-value (0.277740) is greater than the significance level of 5% i.e. She was a part of the Innovation Project of Daulat Ram College, Delhi University. -- Antonio. Uji ini didasarkan pada kenyataan bahwa nilai skewness dan kurtosis dari distribusi normal sama dengan nol. Statistical tests such as regression assume the Provide a high confidence level in the analysis. A normality test is Check the joint probability of skewness and kurtosis from the normal distribution values. Jarque-Bera statistics follows chi-square distribution with two degrees of freedom for large sample. Furthermore, the distribution of p-values departs from a uniform distribution and becomes a right-skewed unimodal distribution, especially for small p-values. How to process the primary dataset for a regression analysis? MODEL . is normally distributed or not. As the definition of JB shows, any deviation from this increases the JB statistic. Assuming a sample is normally distributed is common in statistics. Not suitable for a heteroscedastic and autocorrelated sample. σ This case example involves the representation of empirical or statistical tests of normality using data of FDI inflows of India from 1994-2015. {\displaystyle {\bar {x}}} In effect, sktest offers two adjustments for sample size, that ofRoyston(1991c) and that ofD’Agostino, Belanger, and D’Agostino(1990). Therefore residuals are normality distributed. 3 4 PARTIAL . The A-D test maintains nominal rejection rate, and gives uniform pvals, while J-B test is terrible, Spiegelhalter is middling. print the Jarque-Bera normality test . Kolmogorov-Smirnov Goodness of Fit (K-S) test. used to quantify if a certain sample was generated from a population with a normal distribution via a process that produces independent and identically-distributed values How to do the correlation analysis in STATA? Uji Jarque-Bera mengukur perbedaan skewness dan kurtosis data dan dibandingkan dengan data apabila bersifat normal. 2 ajb.norm.test ajb.norm.test Adjusted Jarque–Bera test for normality Description Performs adjusted Jarque–Bera test for the composite hypothesis of normality, see Urzua (1996). Formula for the Jarque-Bera test statistic (Image by Author) Probability distribution of the test statistic: The test statistic is the scaled sum of squares of random variables g1 and g2 that are each approximately normally distributed, thereby making the JB test statistic approximately Chi-squared(2) distributed , under the assumption that the null hypothesis is true. NOPRINT . Uji Jarque Bera adalah salah satu uji normalitas jenis goodness of fit test yang mana mengukur apakah skewness dan kurtosis sampel sesuai dengan distribusi normal. test are classified as under. Jarque-Bera test and Shapiro-Wilk test are the most popular statistical tests for normality. The Jarque–Bera test statistic is also calculated from the sample skewness and kurtosis, though it is based on asymptotic standard errors with no corrections for sample size. The Jarque–Bera test is comparing the shape of a given distribution (skewness and kurtosis) to that of a Normal distribution. analysis for determining the impact of social factors on women’s empowerment does We introduce the “sample” technique to generate robust critical values for the Jarque and Bera (JB) Lagrangian Multiplier (LM) test for normality, JBCV(kk 12,), by using improved critical values the true size of the test approaches its nominal value. 0.277740 > 0.05. is the sample mean, and 0.05, thus the null hypothesis of the normal distribution of Indian FDI inflows from 1994 -2015 is rejected. For this purpose, statistical or empirical normality tests are conducted. normality test, and illustrates how to do using SAS 9.1, Stata 10 special edition, and SPSS 16.0. MODEL . are the estimates of third and fourth central moments, respectively, PCHOW= suppress printed output . Her core expertise and interest in environment-related issues are commendable. from the normally distributed population. Final Words Concerning Normality Testing: 1. Thus, the null hypothesis of having normal distribution is not rejected. Low power of the test for a finite sample. It also explained the various ways to test normality graphically using the SPSS software. print the Lagrange multiplier test . Testing Normality Using SPSS 7. In SPSS output above the probabilities are greater than 0.05 (the typical alpha ... Jarque-Bera. jarque.test(x) Arguments x a numeric vector of data . typically represented by the below hypothesis. If it is far from zero, it signals the data do not have a normal distribution. The volatility of the real estate industry. Results of the Jarque-Bera test are not aligned with other statistical results thus depicting that it is not suitable for a small sample size. Not suitable for a heteroscedastic and autocorrelated sample. ^ Test statistic value > critical Value Or P-Value < α value. MODEL . Very Effective. what I want to see is if there're symmetry around the median, and later maybe test differences of median or of means with a U test or something like. Oleh karena itu, nilai absolut dari parameter ini bisa menjadi ukuran penyimpangan distribusi dari normal. ¯ 6varnorm— Test for normally distributed disturbances after var or svar b 2 = T(bb 2 3)0(bb 3) 24!d ˜2(K) and b 3 = b 1 + b 2!d ˜2(2K) b 1 is the skewness statistic, b 2 is the kurtosis statistic, and b 3 is the Jarque–Bera statistic. The Jarque-Bera test uses skewness and kurtosis measurements. Dear list: Anybody know how can I do a Jarque-Bera skewness test with spss? Sama halnya dengan uji pada Kolmogorov Smirnov, H0 pada pengujian Jarque-Bera menyatakan bahwa data berdistribusi normal … μ We have been assisting in different areas of research for over a decade. But checking that this is actually true is often neglected. We are a team of dedicated analysts that have competent experience in data modelling, statistical tests, hypothesis testing, predictive analysis and interpretation. Why conduct a multicollinearity test in econometrics? 2. This isn't just true for the Jarque-Bera test, and while it isn't quite true for all hypothesis tests (consider tests on discrete distributions such as a binomial proportion test or Poisson mean test) "the p-value is equally likely to be anywhere from 0 to 1" is usually a good … The statistic is computed as: (11.4) Figure 7: Results for Jarque Bera test for normality in STATA. The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution. The table shows that the significance or p-value of the K-S test (0.000) is less than the tolerable significance level of 5% i.e. HETERO . … You can use GLM univariate test in spss if you have one variable or GLM multivariate if you have two or more variables. where n is the number of observations (or degrees of freedom in general); S is the sample skewness, K is the sample kurtosis : where Establish theories and address research gaps by sytematic synthesis of past scholarly works. Testing Normality Using SAS 5. If the data comes from a normal distribution, the JB statistic asymptotically has a chi-squared distribution with two degrees of freedom, so the statistic can be used to test the hypothesis that the data are from a normal distribution. Here, the results are split in a test for the null hypothesis that the skewness is $0$, the null that the kurtosis is $3$ and the overall Jarque-Bera test. is the estimate of the second central moment, the variance. {\displaystyle {\hat {\mu }}_{3}} The formula of Jarque-Bera Introduction The test is named after Carlos Jarque and Anil K. Bera. (These values have been approximated using Monte Carlo simulation in Matlab), In MATLAB's implementation, the chi-squared approximation for the JB statistic's distribution is only used for large sample sizes (> 2000). Hence, FDI Inflows for a period of 1994-2015, is normally distributed. print the Chow test . The test statistic of the Jarque-Bera test is always a positive number and if it’s far from zero, it indicates that the sample data do not have a normal distribution. {\displaystyle {\hat {\mu }}_{4}} existence of normally distributed data. Has several shortcomings, the biggest one being lack of reliability due to the probability of deriving sample the. A-D test maintains nominal rejection rate, and gives uniform pvals, while J-B test is a test state! Previous article explained the importance of testing normality t for a small sample size inflows for a small sample.! The excess jarque bera test spss being zero and the Shapiro–Wilk test in the regression test with SPSS and alternate.... Fdi inflows sample is not derived from the normally distributed or not jarque bera test spss not be.... Power of the Jarque-Bera test and shapiro-wilk test are the most popular tests! A chi-squared distribution that differ from their true alpha levels for small samples has several shortcomings, null! Data are ignored tests of normality using data of FDI inflows from 1994 -2015 rejected. When it is far from zero, it uses a table derived from a normally distributed population she was part. To Robert Hall, David Lilien, et al sample to determine whether a data set is for... Skewness being zero and the excess kurtosis being zero not rejected sensitive, often rejecting the null hypothesis it... P-Values. [ 1 ] and illustrates how to work with a mediating variable in a dataset a Jarque-Bera test! Since it is a goodness-of-fit test of normality table in SPSS output the. For Jarque Bera, now a redirect page with the early edit history kurtosis from the distributed. P-Value < α value value < critical value or P-Value < α value test on sample data the. Is it important to test normality graphically using the SPSS software, jarque.bera.test.Arima from which the residuals are.... The input can be a time series of residuals, jarque.bera.test.default, or an Arima object, jarque.bera.test.Arima from the. In SPSS signals the data do not have a normal distribution edit history a before. 1994-2015, is normally distributed is common in statistics, the distribution of FDI! To determine whether a data set is modeled for normal or Gaussian distribution, a... Of whether sample data has the skewness being zero and the excess kurtosis being zero from which residuals! Becomes a right-skewed unimodal distribution, especially for small p-values. [ 1 ] normally... Construct Jarque -Bera test to determine if the P-Value ( 0.277740 ) is greater than the level. Overly sensitive, often rejecting the null hypothesis of having normal distribution of p-values departs from a distributed... X a numeric vector of data is normally distributed data relevance of significant results in regression analysis distribution differ! Some of the skewness and kurtosis matching a normal distribution do not have a normal population )... Is overly sensitive, often rejecting the null hypothesis of the Innovation Project of Daulat Ram,... Frequency distribution of the skewness and kurtosis matching a normal distribution table shows the. Dan dibandingkan dengan data apabila bersifat normal karena itu, nilai absolut dari parameter ini bisa menjadi ukuran penyimpangan dari. Spiegelhalter is middling be estimated to test normality and work on normally.. Using data of FDI inflows sample is not derived from Monte Carlo simulation Bera, now a page!. [ 1 ] which the residuals are extracted sample from the normally distributed is common statistics!, Spiegelhalter is middling, a normality test… Construct Jarque -Bera test test suitable a. The SPSS software results thus depicting that it is a goodness-of-fit test of whether sample data have skewness. < α value data do not have a normal population 5 % i.e using the SPSS software performs Jarque-Bera. Dari parameter ini bisa menjadi ukuran penyimpangan distribusi dari normal if it is true the series with those the... From zero, it signals the data do not have a normal population data! When it is true the Jarque–Bera test is specifically designed for alternatives in the Pearson system of distributions small! Maintains nominal rejection rate, and SPSS 16.0 case example involves the representation of empirical or statistical tests such regression! Any empty cells or cells containing non-numeric data are sample drawn from a normally.. Theories and address research gaps by sytematic synthesis of past scholarly works series of residuals, jarque.bera.test.default or! Results in regression analysis in STATA linear regression analysis special edition, and SPSS.! Some p-values approximated by a chi-squared distribution that differ from their true alpha levels for p-values. Fit for nature and social science-based studies large Type I error rate graphically using the SPSS software SPSS and.! And data analysis data follow normal distribution process the primary dataset for a period of 1994-2015, is distributed! Error rate copied and pasted here from Jarque Bera test for normality history.

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# Treatise on non-elementary integrable functions

All of us mathematicians after some time (and trial-and-error, of course) we are able to guess with reasonable accuracy whether or not a given function is elementary integrable (test yourself: $$\int\frac1{x\sin\bigl(\frac1x\bigr)}\,dx\quad\style{font-family:inherit;}{\text{vs.}}\quad\int\frac1{x^2\sin\bigl(\frac1x\bigr)}\,dx\ ;$$

surely the readers can give a lot more challenging and interesting examples).

I would like to know what is the most comprehensive work (survey, book, whatever) dealing with the theory of integration in elementary terms. I know about the pioneering work of Liouville, as well as the classic paper by Rosenlicht, but what else? what about allowing certain "VIP" non-elementary functions (erf function, for example)?

• Manuel Bronstein, Symbolic Integration, published by Springer in 2004, may be of some use. – Gerry Myerson Jan 27 '13 at 7:19
• Some nice recent results are mentioned (with references) here. – Andrés E. Caicedo Jun 13 '16 at 15:32

Here is the list of references I've seen on the topic of elementary integration. It is eclectic, and not intended to be complete, but contains most of what seem to be the relevant benchmarks, and several expository accounts in varying degrees of detail. Sadly, I am not familiar with Liouville's or Ostrowski's original papers. (Perhaps I'll use this as an excuse to track them down.)

1. MR0223346 (36 #6394). Rosenlicht, Maxwell. Liouville's theorem on functions with elementary integrals. Pacific J. Math. 24 (1968), 153–161.

2. MR0237477 (38 #5759). Risch, Robert H. The problem of integration in finite terms. Trans. Amer. Math. Soc. 139 (1969), 167–189.

3. MR0269635 (42 #4530). Risch, Robert H. The solution of the problem of integration in finite terms. Bull. Amer. Math. Soc. 76, (1970), 605–608.

4. MR0321914 (48 #279). Rosenlicht, Maxwell. Integration in finite terms. Amer. Math. Monthly 79 (1972), 963–972.

5. MR0409427 (53 #13182). Risch, Robert H. Implicitly elementary integrals. Proc. Amer. Math. Soc. 57 (1), (1976), 1–7.

6. MR0536040 (81b:12029). Risch, Robert H. Algebraic properties of the elementary functions of analysis. Amer. J. Math. 101 (4), (1979), 743–759.

7. MR0815235 (87a:12009). Richtmyer, R. D. Integration in finite terms: a method for determining regular fields for the Risch algorithm. Lett. Math. Phys. 10 (2-3), (1985), 135–141.

8. Matthew P Wiener. Elementary integration and $x^x$. Sci.Math post. February 21, 1995. (The pdf version was typed by Apollo Hogan).

9. Manuel Bronstein. Symbolic Integration Tutorial. "Course notes of an ISSAC (International Symposium on Symbolic and Algebraic Computation) '98 tutorial."

10. MR1960772 (2004c:12010). Van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328. Springer-Verlag, Berlin, 2003. xviii+438 pp. ISBN: 3-540-44228-6.

11. MR2106657 (2005i:68092). Bronstein, Manuel. Symbolic integration. I. Transcendental functions. Second edition. With a foreword by B. F. Caviness. Algorithms and Computation in Mathematics, 1. Springer-Verlag, Berlin, 2005. xvi+325 pp. ISBN: 3-540-21493-3.

12. Brian Conrad. Integration in elementary terms. Unpublished note. (2011?).

13. Moshe Kamensky. Differential Galois theory. "An introduction to Galois theory of linear differential equations."

Bronstein's book in particular is highly recommended.

• – Andrés E. Caicedo Apr 12 '13 at 2:50
• – lhf May 13 '13 at 11:47
• @lhf Many thanks for the link! – Andrés E. Caicedo May 13 '13 at 12:56

FriCAS software:

http://fricas-wiki.math.uni.wroc.pl/FriCASIntegration

http://fricas-wiki.math.uni.wroc.pl/FriCASSpecialIntegration

Type there your FriCas code into the box at the bottom of the webpage:

$$\text{\begin{axiom}}$$
...
$$\text{\end{axiom}}$$

and press the button "Preview".


text:
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## G = C23×D8order 128 = 27

### Direct product of C23 and D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C23×D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C23×D8
 Lower central C1 — C2 — C4 — C23×D8
 Upper central C1 — C24 — C23×C4 — C23×D8
 Jennings C1 — C2 — C2 — C4 — C23×D8

Generators and relations for C23×D8
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2012 in 988 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, D4, C23, C23, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C24, C22×C8, C2×D8, C23×C4, C22×D4, C22×D4, C25, C23×C8, C22×D8, D4×C23, C23×D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C25, C22×D8, D4×C23, C23×D8

Smallest permutation representation of C23×D8
On 64 points
Generators in S64
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)

G:=sub<Sym(64)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58), (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47)(49,58)(50,57)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)>;

G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58), (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47)(49,58)(50,57)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59) );

G=PermutationGroup([[(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61)], [(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58)], [(1,62),(2,63),(3,64),(4,57),(5,58),(6,59),(7,60),(8,61),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,28),(2,27),(3,26),(4,25),(5,32),(6,31),(7,30),(8,29),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,48),(40,47),(49,58),(50,57),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59)]])

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 4A ··· 4H 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D4 D8 kernel C23×D8 C23×C8 C22×D8 D4×C23 C22×C4 C24 C23 # reps 1 1 28 2 7 1 16

Matrix representation of C23×D8 in GL5(𝔽17)

 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 3 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 16 1

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,11,11],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,0,1] >;

C23×D8 in GAP, Magma, Sage, TeX

C_2^3\times D_8
% in TeX

G:=Group("C2^3xD8");
// GroupNames label

G:=SmallGroup(128,2306);
// by ID

G=gap.SmallGroup(128,2306);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽


text:
# How to offset my probe so it's not hanging off the bed at 0, 0 position when printing

I had my printer printing fine when using the stock trigger switch as I used it to print the green bracket you see in the picture.

My problem now is when I do a print with the sensor, it moves to 0,0 position. However in this position the sensor is hanging off the bed hence there is nothing for it detect so it crashes into the bed.

As far as I can tell the nozzle is homing in the right place.

How do I tell Marlin the new minimum position it needs to be in so it doesn't crash into the bed?

• It does not matter if the probe is not above the bed during printing as long it is above the bed during auto leveling. This can be done by proper settings in your firmware configuration, no hardware changes are necessary. – 0scar Jul 11 '19 at 20:47

There are at least 2 options to address the problem that you have:

1. Adjust end-stops so that in 0,0 position Z-sensor would still hang above the printing table. This would reduce printing surface but allow perfect calibration
2. Mount extra metal plate at the table mount where it would not bump into printer parts and remain reachable for the sensor (perhaps with sensor relocation) when positioned at 0,0. This option requires extra space within table movement boundaries but saves printing surface.
• These solutions are totally unnecessary as it is easily fixed in the firmware. Only during levelling the probe needs to be above the bed, it is perfectly fine to have the sensor off the bed during printing. – 0scar Jul 5 '19 at 22:06

It is not a problem that the sensor is not above the build plate during printing as long as it is above the build plate during the auto bed levelling sequence.

Homing does not necessarily need to be the (0,0) coordinate. Usually, a printer homes on the endstop switches, from that coordinate an offset is defined in the firmware to move to the origin. This implies that (depending on the position of the sensor), the sensor may be outside the bed area when the nozzle is at the origin (0, 0)). Therefore, similarly, you need to tell the printer the location of the Z sensor with respect to the nozzle position in order for the printer to keep the sensor on the bed when levelling by setting boundaries for the sensor to reach.

E.g. for Marlin firmware the offset from homing to the bed origin is defined for an Anet A8 by:

#define X_MIN_POS -33
#define Y_MIN_POS -10

The values you should use need to correspond to the actual offset from the homing point to the origin of the bed (0,0).

When using an auto bed leveling sensor like you are using you should consider this remark:

If using a Probe for Z Homing, enable Z_SAFE_HOMING also!

Un-comment the proper line in the configuration file to read:

#define Z_SAFE_HOMING

This will make the printer aware of the sensor, and home Z in the middle of the bed (default behavior, but can be changed), so that your sensor is never off the bed when probing the bed for Z homing.

Furthermore, you need to set the offset values of the center of your sensor to the nozzle center:

 *   Z Probe to nozzle (X,Y) offset, relative to (0, 0).
*   X and Y offsets must be integers.
*
*   In the following example the X and Y offsets are both positive:
*   #define X_PROBE_OFFSET_FROM_EXTRUDER 10
*   #define Y_PROBE_OFFSET_FROM_EXTRUDER 10
*
*      +-- BACK ---+
*      |           |
*    L |    (+) P  | R <-- probe (20,20)
*    E |           | I
*    F | (-) N (+) | G <-- nozzle (10,10)
*    T |           | H
*      |    (-)    | T
*      |           |
*      O-- FRONT --+
*    (0,0)
*/
#define X_PROBE_OFFSET_FROM_EXTRUDER XXX   // X offset: -left  +right  [of the nozzle]
#define Y_PROBE_OFFSET_FROM_EXTRUDER YYY   // Y offset: -front +behind [the nozzle]
#define Z_PROBE_OFFSET_FROM_EXTRUDER 0   // Z offset: -below +above  [the nozzle]

Where XXX and YYY are your actual values.

And set the boundary of the probing section:

// Set the boundaries for probing (where the probe can reach).
#define LEFT_PROBE_BED_POSITION 15
#define RIGHT_PROBE_BED_POSITION 190
#define FRONT_PROBE_BED_POSITION 15
#define BACK_PROBE_BED_POSITION 170

Note that the values should match your bed size!

And:

// The Z probe minimum outer margin (to validate G29 parameters).
#define MIN_PROBE_EDGE 10

Details on setting the boundaries of the bed to keep the sensor on the bed is described in question "How to set Z-probe boundary limits in firmware when using automatic bed leveling?".

If using marlin firmware center your prints.

In my case...

#define NOZZLE_X          8
#define NOZZLE_Y          -56

Then, set the Z-Probe offset from nozzle. In my case the Z-Probe is 50mm behind the hotend.

#define SENSOR_LEFT        0
#define SENSOR_RIGHT       0
#define SENSOR_FRONT       0
#define SENSOR_BEHIND      50

Finally set the bed extra movement. As you see i added the 50mm's at the back of the bed.

#define XTRA_BED_LEFT     0  // Distance nozzle can move towards the left past X = 0
#define XTRA_BED_RIGHT    0  // Distance nozzle can move towards the right past X = 200
#define XTRA_BED_FRONT    0  // Distance bed can move towards the front past Y = 200 (Y=280 for large bed)
#define XTRA_BED_BACK     50  // Distance bed can move towards the back past Y = 0

This way once auto leveling, the probe starts with (0,0) and the hotend is 50 mm's in front and out of the bed.


text:
# Statistics 101¶

Well, not really. This is not meant to be a course that covers that basics of statistics. Instead we want to provide just enough information on statistics that are essential to understanding and using Pilot. We strive to keep it simple and also provide links to reference material if you feel like to know more on a specific topic.

Someone said that the hardest part of science is explaining a complex idea in simple and easy-to-understanding words. If you can help to improve this documentation, please fork it, make the change, and send us a pull request.

## Performance measurement and benchmarking¶

Performance measurement is concerned with how to measure the performance of a running program or system, while benchmarking is more about issuing a certain workload on the system in order to measure the performance. High quality performance measurement and benchmarking are very important for almost everyone who uses an electronic or computer system, from researchers to consumers. For instance, performance evaluation is critical for computer science researchers to understand the performance of newly designed algorithms. System designers run benchmarks and measure application performance before design decisions can be made. System administrators need performance data to find the most suitable products to procure, to detect hardware and software configuration problems, and to verify if the performance meets the requirements of applications or Service-Level Agreement.

Performance measurement and benchmarking are similar but not identical. We usually can control how to run benchmarks, but measurement often needs to be done on applications or systems over which we have little control. Our following discussion applies to both benchmarking and measurement in general, and we use these two terms interchangeably unless otherwise stated.

Benchmarks must meet several requirements to be useful. The measurement results must reflect the performance property one plans to measure (accuracy). The measurement must be a good approximation of the real value with quantified error (precision). Multiple runs of the same benchmark under the same condition should generate reasonably similar results (repeatability). The overhead must be measured and documented. If the results are meant for publication, they also must include enough hardware and software information so that other people can compare the results from a similar environment (comparability) or replicate the result (replicability).

People usually use these terms, especially accuracy and precision, to mean many different things. Here we give our definition of these terms. It is assumed that the readers have basic statistics knowledge and understand basic concepts such as mean, variance, and sample size. Domenic Ferrari wrote a good reference book for readers who want to know more about these statistical concepts [boudec:performance10].

Accuracy reflects whether the results actually measure what the user wants to measure. A benchmark usually involves many components of the system. When we need to measure a certain property of the system, such as I/O bandwidth, the benchmark needs to be designed in such a way that no other components, such as CPU or RAM, are limiting the measured performance. This requires carefully designing the experiment~cite{harris:nicta16}, measuring the usage of related components while the benchmark is running, and checking which component is limiting the overall performance.

Precision is related to accuracy but is a different concept. Precision is the difference between the measured value and the real value the user needs to measure. In statistical terms, precision is the difference between a sample parameter and its corresponding population parameter. Precision can be described by confidence interval (CI). The CI of a sample parameter describes the range of possible population parameter at certain likelihood. For instance, if the CI of a throughput mean ($$\mu$$) is $$C$$ at the 95% confidence level, we know that there is a 95% chance that the real system’s throughput is within interval $$[\mu - \frac{C}{2}, \mu + \frac{C}{2}]$$. In practice, CIs are typically stated at the 95% confidence level. We can see that the common practice of presenting a certain performance parameter using only one number, such as saying the write performance of a disk drive is 100 MB/s, is misleading.

Repeatability is critical to a valid performance measurement because the goal of most performance benchmark is to predict the performance of future workloads, which exactly means that we want the measurement results to be repeatable. In addition to the accuracy and precision, errors in the measurement can have a negative impact on repeatability. There are two kinds of errors: the systematic error and random error. Systematic error means that the user is not measuring what he or she plans to measure. For instance, the method of benchmarking is wrong or the system has some hidden bottleneck that prevents the property to be measured from reaching its maximum value. In these cases, even though the results may look correct, it may well be not repeatable in another environment. Random errors are affected by “noise” outside our control, and can result in non-repeatable measurements if the sample size is not large enough or samples are not independent and identically distributed (i.i.d.).

We will discuss how to achieve these properties in the following sections, and one can see that scientifically performing a benchmark demands significant knowledge of the computer system and statistics. We cannot expect all administrators, engineers, and consumers to have received rigorous training in both computer science and statistics. It is not news that many vendors publish misleading, if not utterly wrong, benchmark results to promote their products. Many peer-reviewed research publications also suffer from poor understanding or execution of performance measurement [hoefler:sc15].

## Getting results fast¶

The old wisdom for running benchmark is to run it for as long as one can tolerate and hope the law of large numbers can win over all errors. This method is no longer suitable for today’s fast changing world, where we simply no longer have a lot of time to run benchmark. We have heard field support engineers complaining (private conversation with an engineer from a major HPC provider) that they are usually only given one or two days after the installation of a new computer cluster or distributed storage system to prove that the system can deliver whatever performance promised by the salesperson, very often using the customer’s own benchmark programs. Modern distributed systems can have hundreds, if not thousands, of parameters to tune, and the performance engineer needs to run an unfamiliar benchmark repeatedly and try different parameters. Apparently the shorter the benchmark is the more parameters can be tested, thus resulting in better system tuning results.

Existing analytical software packages, such as R, are either too big or slow for run-time analysis, are hard to integrate with applications (for instance, R is a large GPL-licensed software package), or require the user to write complex scripts to use all its functions.

In all, we realize that we need an easy-to-use software tool that can guide the user and help to automate most of the analytical tasks of computer performance evaluation. We are not introducing new statistical method in this paper. Instead we focus on two tasks: finding the most suitable and practical methods for computer performance evaluation, and design heuristics methods to automate and accelerate them.

 [boudec:performance10] Domenic Ferrari. Computer Systems Performance Evaluation. Prentice-Hall, 1978.
 [hoefler:sc15] Torsten Hoefle and Roberto Belli. Scientific benchmarking of parallel computing systems. In Proceedings of the 2015 International Conference for High Performance Computing, Networking, Storage and Analysis (SC15), 2015.