Minimize distance between two lists

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP

up vote
5
down vote

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1

Writing:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

On the other hand, if I write:

k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4} k;
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

where it is clear that, compared to the previous case, in some bars the gap has decreased and in others it has increased.

Question: How can I determine the best value of k to get the smallest possible gap?


Writing:

h = -0.35;
k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = h + k {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

Question 2: is it possible to determine the pair of values h, k that minimize the gap?

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  • This is a related question: How to find the distance of two lists?
    – Artes
    Nov 29 at 16:03

up vote
5
down vote

favorite

1

Writing:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

On the other hand, if I write:

k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4} k;
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

where it is clear that, compared to the previous case, in some bars the gap has decreased and in others it has increased.

Question: How can I determine the best value of k to get the smallest possible gap?


Writing:

h = -0.35;
k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = h + k {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

Question 2: is it possible to determine the pair of values h, k that minimize the gap?

share|improve this question

  • This is a related question: How to find the distance of two lists?
    – Artes
    Nov 29 at 16:03

up vote
5
down vote

favorite

1

up vote
5
down vote

favorite

1
1

Writing:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

On the other hand, if I write:

k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4} k;
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

where it is clear that, compared to the previous case, in some bars the gap has decreased and in others it has increased.

Question: How can I determine the best value of k to get the smallest possible gap?


Writing:

h = -0.35;
k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = h + k {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

Question 2: is it possible to determine the pair of values h, k that minimize the gap?

share|improve this question

Writing:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

On the other hand, if I write:

k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4} k;
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

where it is clear that, compared to the previous case, in some bars the gap has decreased and in others it has increased.

Question: How can I determine the best value of k to get the smallest possible gap?


Writing:

h = -0.35;
k = 0.83;
expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = h + k {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

I get:

enter image description here

Question 2: is it possible to determine the pair of values h, k that minimize the gap?

mathematical-optimization charts

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share|improve this question

edited Nov 28 at 21:27

asked Nov 28 at 20:19

TeM

1,777619

1,777619

  • This is a related question: How to find the distance of two lists?
    – Artes
    Nov 29 at 16:03

  • This is a related question: How to find the distance of two lists?
    – Artes
    Nov 29 at 16:03

This is a related question: How to find the distance of two lists?
– Artes
Nov 29 at 16:03

This is a related question: How to find the distance of two lists?
– Artes
Nov 29 at 16:03

2 Answers
2

active

oldest

votes

up vote
4
down vote

accepted

Update: Using two parameters:

lmf2 = LinearModelFit[data, t, t];
Normal@lmf2

1.76563 + 0.546875 t

lmf2["BestFitParameters"]

{1.76563, 0.546875}

Fit[data, {1, t}, t]

1.76563 + 0.546875 t

ClearAll[h, k]
NMinimize[Total[Subtract[expectedresults, h + k achievedresults]^2], {h, k}]

{43.3594, {h -> 1.76562, k -> 0.546875}}

N @ LeastSquares[Thread[{1, achievedresults}], expectedresults]

{1.76563, 0.546875}

Original answer:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
data = Transpose[{ achievedresults,expectedresults}];

You can use LinearModelFit or Fit or NMinimize or LeastSquares to get the value of k that minimizes the sum of squared distances between expectedresults and k achievedresults:

lmf = LinearModelFit[data, t, t, IncludeConstantBasis -> False]

Normal@lmf

0.828025 t

Normal @ LinearModelFit[{Transpose[{achievedresults}], expectedresults}]

0.828025 #1

Fit[data, {t}, t]

0.828025 t

ClearAll[k]
NMinimize[Total[Subtract[expectedresults, k achievedresults]^2], k]

{49.7134, {k -> 0.828025}}

N@LeastSquares[Thread[{achievedresults}], expectedresults]

{0.828025}

k = lmf["BestFitParameters"][[1]]

0.828025

p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[k achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

enter image description here

BarChart[Transpose@{expectedresults, achievedresults,  k achievedresults}, 
 ChartStyle -> {Blue, Red, Green}, ChartLayout -> "Grouped", 
 ChartLegends -> {"expectedresults", "achievedresults", "k achievedresults"}]

enter image description here

share|improve this answer

  • For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
    – David G. Stork
    Nov 28 at 20:42

  • @TeM, please see the update.
    – kglr
    Nov 28 at 21:47

  • Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
    – TeM
    Nov 28 at 21:48

  • 1

    @TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
    – kglr
    Nov 28 at 21:54

up vote
7
down vote

{k, h} = PseudoInverse[{#, 1} & /@ achievedresults].expectedresults

{35/64, 113/64}

share|improve this answer

  • 2

    Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
    – MeMyselfI
    Nov 28 at 20:56

  • 1

    @MeMyselfI Nice! Even better
    – Chris
    Nov 28 at 21:05

  • Really great!!!
    – TeM
    Nov 28 at 21:50

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote

accepted

Update: Using two parameters:

lmf2 = LinearModelFit[data, t, t];
Normal@lmf2

1.76563 + 0.546875 t

lmf2["BestFitParameters"]

{1.76563, 0.546875}

Fit[data, {1, t}, t]

1.76563 + 0.546875 t

ClearAll[h, k]
NMinimize[Total[Subtract[expectedresults, h + k achievedresults]^2], {h, k}]

{43.3594, {h -> 1.76562, k -> 0.546875}}

N @ LeastSquares[Thread[{1, achievedresults}], expectedresults]

{1.76563, 0.546875}

Original answer:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
data = Transpose[{ achievedresults,expectedresults}];

You can use LinearModelFit or Fit or NMinimize or LeastSquares to get the value of k that minimizes the sum of squared distances between expectedresults and k achievedresults:

lmf = LinearModelFit[data, t, t, IncludeConstantBasis -> False]

Normal@lmf

0.828025 t

Normal @ LinearModelFit[{Transpose[{achievedresults}], expectedresults}]

0.828025 #1

Fit[data, {t}, t]

0.828025 t

ClearAll[k]
NMinimize[Total[Subtract[expectedresults, k achievedresults]^2], k]

{49.7134, {k -> 0.828025}}

N@LeastSquares[Thread[{achievedresults}], expectedresults]

{0.828025}

k = lmf["BestFitParameters"][[1]]

0.828025

p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[k achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

enter image description here

BarChart[Transpose@{expectedresults, achievedresults,  k achievedresults}, 
 ChartStyle -> {Blue, Red, Green}, ChartLayout -> "Grouped", 
 ChartLegends -> {"expectedresults", "achievedresults", "k achievedresults"}]

enter image description here

share|improve this answer

  • For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
    – David G. Stork
    Nov 28 at 20:42

  • @TeM, please see the update.
    – kglr
    Nov 28 at 21:47

  • Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
    – TeM
    Nov 28 at 21:48

  • 1

    @TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
    – kglr
    Nov 28 at 21:54

up vote
4
down vote

accepted

Update: Using two parameters:

lmf2 = LinearModelFit[data, t, t];
Normal@lmf2

1.76563 + 0.546875 t

lmf2["BestFitParameters"]

{1.76563, 0.546875}

Fit[data, {1, t}, t]

1.76563 + 0.546875 t

ClearAll[h, k]
NMinimize[Total[Subtract[expectedresults, h + k achievedresults]^2], {h, k}]

{43.3594, {h -> 1.76562, k -> 0.546875}}

N @ LeastSquares[Thread[{1, achievedresults}], expectedresults]

{1.76563, 0.546875}

Original answer:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
data = Transpose[{ achievedresults,expectedresults}];

You can use LinearModelFit or Fit or NMinimize or LeastSquares to get the value of k that minimizes the sum of squared distances between expectedresults and k achievedresults:

lmf = LinearModelFit[data, t, t, IncludeConstantBasis -> False]

Normal@lmf

0.828025 t

Normal @ LinearModelFit[{Transpose[{achievedresults}], expectedresults}]

0.828025 #1

Fit[data, {t}, t]

0.828025 t

ClearAll[k]
NMinimize[Total[Subtract[expectedresults, k achievedresults]^2], k]

{49.7134, {k -> 0.828025}}

N@LeastSquares[Thread[{achievedresults}], expectedresults]

{0.828025}

k = lmf["BestFitParameters"][[1]]

0.828025

p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[k achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

enter image description here

BarChart[Transpose@{expectedresults, achievedresults,  k achievedresults}, 
 ChartStyle -> {Blue, Red, Green}, ChartLayout -> "Grouped", 
 ChartLegends -> {"expectedresults", "achievedresults", "k achievedresults"}]

enter image description here

share|improve this answer

  • For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
    – David G. Stork
    Nov 28 at 20:42

  • @TeM, please see the update.
    – kglr
    Nov 28 at 21:47

  • Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
    – TeM
    Nov 28 at 21:48

  • 1

    @TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
    – kglr
    Nov 28 at 21:54

up vote
4
down vote

accepted

up vote
4
down vote

accepted

Update: Using two parameters:

lmf2 = LinearModelFit[data, t, t];
Normal@lmf2

1.76563 + 0.546875 t

lmf2["BestFitParameters"]

{1.76563, 0.546875}

Fit[data, {1, t}, t]

1.76563 + 0.546875 t

ClearAll[h, k]
NMinimize[Total[Subtract[expectedresults, h + k achievedresults]^2], {h, k}]

{43.3594, {h -> 1.76562, k -> 0.546875}}

N @ LeastSquares[Thread[{1, achievedresults}], expectedresults]

{1.76563, 0.546875}

Original answer:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
data = Transpose[{ achievedresults,expectedresults}];

You can use LinearModelFit or Fit or NMinimize or LeastSquares to get the value of k that minimizes the sum of squared distances between expectedresults and k achievedresults:

lmf = LinearModelFit[data, t, t, IncludeConstantBasis -> False]

Normal@lmf

0.828025 t

Normal @ LinearModelFit[{Transpose[{achievedresults}], expectedresults}]

0.828025 #1

Fit[data, {t}, t]

0.828025 t

ClearAll[k]
NMinimize[Total[Subtract[expectedresults, k achievedresults]^2], k]

{49.7134, {k -> 0.828025}}

N@LeastSquares[Thread[{achievedresults}], expectedresults]

{0.828025}

k = lmf["BestFitParameters"][[1]]

0.828025

p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[k achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

enter image description here

BarChart[Transpose@{expectedresults, achievedresults,  k achievedresults}, 
 ChartStyle -> {Blue, Red, Green}, ChartLayout -> "Grouped", 
 ChartLegends -> {"expectedresults", "achievedresults", "k achievedresults"}]

enter image description here

share|improve this answer

Update: Using two parameters:

lmf2 = LinearModelFit[data, t, t];
Normal@lmf2

1.76563 + 0.546875 t

lmf2["BestFitParameters"]

{1.76563, 0.546875}

Fit[data, {1, t}, t]

1.76563 + 0.546875 t

ClearAll[h, k]
NMinimize[Total[Subtract[expectedresults, h + k achievedresults]^2], {h, k}]

{43.3594, {h -> 1.76562, k -> 0.546875}}

N @ LeastSquares[Thread[{1, achievedresults}], expectedresults]

{1.76563, 0.546875}

Original answer:

expectedresults = {4, 8, 5, 1, 4, 6, 4, 1, 9, 3};
achievedresults = {3, 6, 4, 2, 10, 7, 2, 4, 8, 4};
data = Transpose[{ achievedresults,expectedresults}];

You can use LinearModelFit or Fit or NMinimize or LeastSquares to get the value of k that minimizes the sum of squared distances between expectedresults and k achievedresults:

lmf = LinearModelFit[data, t, t, IncludeConstantBasis -> False]

Normal@lmf

0.828025 t

Normal @ LinearModelFit[{Transpose[{achievedresults}], expectedresults}]

0.828025 #1

Fit[data, {t}, t]

0.828025 t

ClearAll[k]
NMinimize[Total[Subtract[expectedresults, k achievedresults]^2], k]

{49.7134, {k -> 0.828025}}

N@LeastSquares[Thread[{achievedresults}], expectedresults]

{0.828025}

k = lmf["BestFitParameters"][[1]]

0.828025

p1 = BarChart[expectedresults, ChartStyle -> Directive[Opacity[0.1], Blue]];
p2 = BarChart[k achievedresults, ChartStyle -> Directive[Opacity[0.1], Red]];
Show[p1, p2]

enter image description here

BarChart[Transpose@{expectedresults, achievedresults,  k achievedresults}, 
 ChartStyle -> {Blue, Red, Green}, ChartLayout -> "Grouped", 
 ChartLegends -> {"expectedresults", "achievedresults", "k achievedresults"}]

enter image description here

share|improve this answer

share|improve this answer

share|improve this answer

edited Nov 28 at 21:52

answered Nov 28 at 20:29

kglr

175k9197402

175k9197402

  • For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
    – David G. Stork
    Nov 28 at 20:42

  • @TeM, please see the update.
    – kglr
    Nov 28 at 21:47

  • Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
    – TeM
    Nov 28 at 21:48

  • 1

    @TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
    – kglr
    Nov 28 at 21:54

  • For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
    – David G. Stork
    Nov 28 at 20:42

  • @TeM, please see the update.
    – kglr
    Nov 28 at 21:47

  • Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
    – TeM
    Nov 28 at 21:48

  • 1

    @TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
    – kglr
    Nov 28 at 21:54

For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
– David G. Stork
Nov 28 at 20:42

For general data you can always find several values of $k$ that eliminate the difference between whichever bars you like.
– David G. Stork
Nov 28 at 20:42

@TeM, please see the update.
– kglr
Nov 28 at 21:47

@TeM, please see the update.
– kglr
Nov 28 at 21:47

Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
– TeM
Nov 28 at 21:48

Perfect, mathematically it is clear to me! But I wonder if so “improve” the minimization or less than before!
– TeM
Nov 28 at 21:48

1

1

@TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
– kglr
Nov 28 at 21:54

@TeM, If you compare the NMinimize result adding the intercept parameter improves the squared loss from 49.7134 to 43.3594.
– kglr
Nov 28 at 21:54

up vote
7
down vote

{k, h} = PseudoInverse[{#, 1} & /@ achievedresults].expectedresults

{35/64, 113/64}

share|improve this answer

  • 2

    Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
    – MeMyselfI
    Nov 28 at 20:56

  • 1

    @MeMyselfI Nice! Even better
    – Chris
    Nov 28 at 21:05

  • Really great!!!
    – TeM
    Nov 28 at 21:50

up vote
7
down vote

{k, h} = PseudoInverse[{#, 1} & /@ achievedresults].expectedresults

{35/64, 113/64}

share|improve this answer

  • 2

    Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
    – MeMyselfI
    Nov 28 at 20:56

  • 1

    @MeMyselfI Nice! Even better
    – Chris
    Nov 28 at 21:05

  • Really great!!!
    – TeM
    Nov 28 at 21:50

up vote
7
down vote

up vote
7
down vote

{k, h} = PseudoInverse[{#, 1} & /@ achievedresults].expectedresults

{35/64, 113/64}

share|improve this answer

{k, h} = PseudoInverse[{#, 1} & /@ achievedresults].expectedresults

{35/64, 113/64}

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edited Nov 28 at 21:45

answered Nov 28 at 20:36

Chris

54116

54116

  • 2

    Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
    – MeMyselfI
    Nov 28 at 20:56

  • 1

    @MeMyselfI Nice! Even better
    – Chris
    Nov 28 at 21:05

  • Really great!!!
    – TeM
    Nov 28 at 21:50

  • 2

    Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
    – MeMyselfI
    Nov 28 at 20:56

  • 1

    @MeMyselfI Nice! Even better
    – Chris
    Nov 28 at 21:05

  • Really great!!!
    – TeM
    Nov 28 at 21:50

2

2

Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
– MeMyselfI
Nov 28 at 20:56

Why do you add a zero column? I think PseudoInverse[Transpose[{achievedresults}]].expectedresults will do
– MeMyselfI
Nov 28 at 20:56

1

1

@MeMyselfI Nice! Even better
– Chris
Nov 28 at 21:05

@MeMyselfI Nice! Even better
– Chris
Nov 28 at 21:05

Really great!!!
– TeM
Nov 28 at 21:50

Really great!!!
– TeM
Nov 28 at 21:50

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