Reinaldo R. Rosa Laboratório de Computação & Matemática Aplicada

Instituto Nacional de Pesquisas Espaciais (INPE)

reinaldo.rosa@inpe.br

PPG CAP-C&T ESPACIAIS

(Matemática Computacional e Ciência de Dados)

Paulo H. Barchi

Rubens A. Sautter

Diego Stalder

Neelakshi Joshi

Igor Kolesnikov

1999 2018

Asymmetry Patterns: Bilateral Asymmetry Patterns:

Where do I find

asymmetry

patterns?

Gradient Pattern Analysis (GPA)

Starting from

Canonical Symmetric Patterns

• THE GRADIENT MOMENTS

4 Rosa et al.

F igur e 3. T he applicat ion of GPA, for calculat ing G1 , over two

galaxies selected from our sample - a spiral on the left and an ellip-

t ical on the right . T he respect ive gradients containing VA vect ors

are shown in the intermediate panels and the t riangulat ion fields,

containing TA edges, are shown in the bot tom panels. Images are

in reverse color cont rast for bet ter viewing.

metrical vector sum and |vi | is the i t h asymmet rical vector

norm. Not ice that for misaligned vectors, the vectorial sum

tends to zero. More formally, we can write |VAi vi | = 0,

then according to equat ion 1, G2 = 2VA

N. Whereas if K vec-

tors with same moduli are aligned, |VAi vi | = K |vi |, and

VAi |vi | = K |vi |. Therefore, G2 = VA

N2 −

K |v i |

K |v i |= VA

N.

This means that this operator considers the proport ion of

asymmetrical vectors, and also, without using explicit ly the

phases (GP3), the correspondent alignment rate. Higher G2

values means that the gradient lat t ice has many misaligned

asymmetrical vectors and then a high diversity for the values

in the lat t ice GP2.

The calculat ion of G2 on canonical mat rices demon-

st rates its ability to characterise the basic asymmet ry con-

jectures invest igated in Rosa et al. (1999) using G1 . However

with less sensit ivity to the noise due to phase fluctuat ions

imprinted in the t riangulat ion field. For the example in Fig-

ure 2, the result for G2 = 0.656 is invariant considering both

mat rices.

The operat ion for comput ing G2 , via Eq. 2, presents the

following improvements compared to G1 : (i) For the same

type of gradient pat tern, the value of G2 is invariant to the

size of the mat rix; ( i i) More appropriately, it does not con-

sider the elements of the mat rix border for calculat ing the

F igur e 4. Histograms for the six morphological paramet ers used

to classify galaxies. Early-type (red) and late-t ype (blue).

gradient ; (i i i ) Can be applied, without loss of generality, to

rectangular mat rices; and ( iv) I t is less sensit ive to noise and

faster because it avoids t riangulat ion.

4 G A L A X Y M OR P H OM ET RY U SI N G G PA

The morphological analysis of the 54,896 objects, as classi-

fied by Galaxy Zoo, has been done on the basis of the pa-

rameters G1 , G2 , C , H , A and S. The respect ive histograms

are shown in Figure 3 where the red (blue) line refers to

early(late)-type galaxies.

Here, we present a comparat ive analysis between well

established parameters, C, A, S, and H, used in several

galaxy morphology studies (e.g. Conselice et al. 2000; Fer-

rari et al. 2015), and those proposed in this invest igat ion, G1

and G2 . A given parameter is considered a useful morpholog-

ical indicator when it separates early and late type galaxies

the best possible. Therefore, it is of paramount importance

to object ively define separat ion. In our case, we est imate

how far apart two histograms are (see Figure 4), using the

index δG H S which is calculated from the GHS (Geomet ric

Histogram Separat ion) algorithm (Saut ter & Barchi 2017).

This algorit hm determines separat ion using only the geo-

met ric characterist ics of a binomial proport ion represented

by histograms. The GHS input values are: AB (blue his-

togram area), AR (red histogram area), AB R (intersect ion

area between AB and AR ), and the respect ive heights for

AB , AR and AB R : hB , hR , and hB R . The separat ion is then

defined as:

δG H S =(1 − A B R

A B + A R + A B R)1/ 2 + (

h a + h b− 2h c

h a + h b)

2(3)

MNRAS 000, 1–5 (2017)

NxN <VA> <TA> < G1> σ

• Color and Magnitudes

• Sersic Parameters

• CAS + H + GPA

• Geometrical Moments

• Luminosity Profile

https://doi.org/10.1093/mnrasl/sly054

Gradient Pattern Analysis Applied to Galaxy Morphology

R.R.Rosa et al. MNRASL, 2018.

Background Segmentation Resampling star masking stamp analysis

Op1

Op2

Op3

Op4

Op1

Op2

Op3

Op4

μ1

μ2

μ3

μ4

. . .

. . .

μ1

μ2

μ3

μ4

• Discriminant Analysis from Big Data 1

2

3

4

4 Rosa et al.

F igur e 3. T he applicat ion of GPA, for calculat ing G1 , over two

galaxies selected from our sample - a spiral on the left and an ellip-

t ical on the right . T he respect ive gradients containing VA vect ors

are shown in the intermediate panels and the t riangulat ion fields,

containing TA edges, are shown in the bot tom panels. Images are

in reverse color cont rast for bet ter viewing.

metrical vector sum and |vi | is the i t h asymmet rical vector

norm. Not ice that for misaligned vectors, the vectorial sum

tends to zero. More formally, we can write |VAi vi | = 0,

then according to equat ion 1, G2 = 2VA

N. Whereas if K vec-

tors with same moduli are aligned, |VAi vi | = K |vi |, and

VAi |vi | = K |vi |. Therefore, G2 = VA

N2 −

K |v i |

K |v i |= VA

N.

This means that this operator considers the proport ion of

asymmetrical vectors, and also, without using explicit ly the

phases (GP3), the correspondent alignment rate. Higher G2

values means that the gradient lat t ice has many misaligned

asymmetrical vectors and then a high diversity for the values

in the lat t ice GP2.

The calculat ion of G2 on canonical mat rices demon-

st rates its ability to characterise the basic asymmet ry con-

jectures invest igated in Rosa et al. (1999) using G1 . However

with less sensit ivity to the noise due to phase fluctuat ions

imprinted in the t riangulat ion field. For the example in Fig-

ure 2, the result for G2 = 0.656 is invariant considering both

mat rices.

The operat ion for comput ing G2 , via Eq. 2, presents the

following improvements compared to G1 : (i) For the same

type of gradient pat tern, the value of G2 is invariant to the

size of the mat rix; ( i i) More appropriately, it does not con-

sider the elements of the mat rix border for calculat ing the

F igur e 4. Histograms for the six morphological paramet ers used

to classify galaxies. Early-type (red) and late-t ype (blue).

gradient ; (i i i ) Can be applied, without loss of generality, to

rectangular mat rices; and ( iv) I t is less sensit ive to noise and

faster because it avoids t riangulat ion.

4 G A L A X Y M OR P H OM ET RY U SI N G G PA

The morphological analysis of the 54,896 objects, as classi-

fied by Galaxy Zoo, has been done on the basis of the pa-

rameters G1 , G2 , C , H , A and S. The respect ive histograms

are shown in Figure 3 where the red (blue) line refers to

early(late)-type galaxies.

Here, we present a comparat ive analysis between well

established parameters, C, A, S, and H, used in several

galaxy morphology studies (e.g. Conselice et al. 2000; Fer-

rari et al. 2015), and those proposed in this invest igat ion, G1

and G2 . A given parameter is considered a useful morpholog-

ical indicator when it separates early and late type galaxies

the best possible. Therefore, it is of paramount importance

to object ively define separat ion. In our case, we est imate

how far apart two histograms are (see Figure 4), using the

index δG H S which is calculated from the GHS (Geomet ric

Histogram Separat ion) algorithm (Saut ter & Barchi 2017).

This algorit hm determines separat ion using only the geo-

met ric characterist ics of a binomial proport ion represented

by histograms. The GHS input values are: AB (blue his-

togram area), AR (red histogram area), AB R (intersect ion

area between AB and AR ), and the respect ive heights for

AB , AR and AB R : hB , hR , and hB R . The separat ion is then

defined as:

δG H S =(1 − A B R

A B + A R + A B R)1/ 2 + (

h a + h b− 2h c

h a + h b)

2(3)

MNRAS 000, 1–5 (2017)

4 Rosa et al.

F igur e 3. T he applicat ion of GPA, for calculat ing G1 , over two

galaxies selected from our sample - a spiral on the left and an ellip-

t ical on the right . T he respect ive gradients containing VA vect ors

are shown in the intermediate panels and the t riangulat ion fields,

containing TA edges, are shown in the bot tom panels. Images are

in reverse color cont rast for bet ter viewing.

metrical vector sum and |vi | is the i t h asymmet rical vector

norm. Not ice that for misaligned vectors, the vectorial sum

tends to zero. More formally, we can write |VAi vi | = 0,

then according to equat ion 1, G2 = 2VA

N. Whereas if K vec-

tors with same moduli are aligned, |VAi vi | = K |vi |, and

VAi |vi | = K |vi |. Therefore, G2 = VA

N2 −

K |v i |

K |v i |= VA

N.

This means that this operator considers the proport ion of

asymmetrical vectors, and also, without using explicit ly the

phases (GP3), the correspondent alignment rate. Higher G2

values means that the gradient lat t ice has many misaligned

asymmetrical vectors and then a high diversity for the values

in the lat t ice GP2.

The calculat ion of G2 on canonical mat rices demon-

st rates its ability to characterise the basic asymmet ry con-

jectures invest igated in Rosa et al. (1999) using G1 . However

with less sensit ivity to the noise due to phase fluctuat ions

imprinted in the t riangulat ion field. For the example in Fig-

ure 2, the result for G2 = 0.656 is invariant considering both

mat rices.

The operat ion for comput ing G2 , via Eq. 2, presents the

following improvements compared to G1 : (i) For the same

type of gradient pat tern, the value of G2 is invariant to the

size of the mat rix; ( i i) More appropriately, it does not con-

sider the elements of the mat rix border for calculat ing the

F igur e 4. Histograms for the six morphological paramet ers used

to classify galaxies. Early-type (red) and late-t ype (blue).

gradient ; (i i i ) Can be applied, without loss of generality, to

rectangular mat rices; and ( iv) I t is less sensit ive to noise and

faster because it avoids t riangulat ion.

4 G A L A X Y M OR P H OM ET RY U SI N G G PA

The morphological analysis of the 54,896 objects, as classi-

fied by Galaxy Zoo, has been done on the basis of the pa-

rameters G1 , G2 , C , H , A and S. The respect ive histograms

are shown in Figure 3 where the red (blue) line refers to

early(late)-type galaxies.

Here, we present a comparat ive analysis between well

established parameters, C, A, S, and H, used in several

galaxy morphology studies (e.g. Conselice et al. 2000; Fer-

rari et al. 2015), and those proposed in this invest igat ion, G1

and G2 . A given parameter is considered a useful morpholog-

ical indicator when it separates early and late type galaxies

the best possible. Therefore, it is of paramount importance

to object ively define separat ion. In our case, we est imate

how far apart two histograms are (see Figure 4), using the

index δG H S which is calculated from the GHS (Geomet ric

Histogram Separat ion) algorithm (Saut ter & Barchi 2017).

This algorit hm determines separat ion using only the geo-

met ric characterist ics of a binomial proport ion represented

by histograms. The GHS input values are: AB (blue his-

togram area), AR (red histogram area), AB R (intersect ion

area between AB and AR ), and the respect ive heights for

AB , AR and AB R : hB , hR , and hB R . The separat ion is then

defined as:

δG H S =(1 − A B R

A B + A R + A B R)1/ 2 + (

h a + h b− 2h c

h a + h b)

2(3)

MNRAS 000, 1–5 (2017)

Spiral

Elliptical

? ?

Physical size x redshift

Angular size x redshift

Age x redshift

Morphology x redshift

Hierarchical Merging? Dark halo?

Gas Infall? Isothermal spheres?

Nongaussianity?

From large surveys we need to know more about ...

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!

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!!/87@:6!MM!N!!\6]!(R 67AR !D87856E!; H58D6!6!B6: 58: !D; !U#&-T0 !!C; R !B6D: P; !D6!; >D6!U~" -1&e a! \H]! %! C; : : A4B; >DA>5A! R 65: 8I ! ; H58D6! 6BS4! 6! 6BE8C6OP; ! D; ! B: ; CA446R A>5; ! D6!8R 67AR !F86!; !R SD@E; !%!D; !6E7; : 85R ; !R ; 45: 6D; !>; !%BY>D8CA!Za!!

9 0 80 251 255 255 228 170 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 61 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 45 0 1 0 0 0 0 2 0 49 238 255 255 236 181 60 0 1 0 0 0 0 0 0

9 0 79 251 255 255 227 169 44 0 1 0 0 0 0 2 0 48 238 255 255 235 181 60 0 1 0 0 0 0 0 0

11 0 82 252 255 255 230 172 47 0 1 0 0 0 0 2 0 51 239 255 255 238 184 63 0 1 0 0 0 0 0 0

(a) (b)

COSMOLOGY

GRADIENT PATTERN ANALISYS IN STRUCTURE FORMATION

Z = 0 Z =

0.35 Z = 1 Z = 2 Z = 5 Z = 10

Random Patterns: > 90%

Typical range for Turbulence Patterns: 60-85%

Quasi-Symmetric Patterns: 1-40%

Gradient Asymmetry Andrade, Ribeiro, Rosa, Physica D 223(2006)139-145

Rosa & Zaniboni, Nonlinear Analysis (in press)

Concluding Remarks GPA Theory & Applications

Santiago, 2018

{G1, G2, G3, G4}

GA = [ NC – NV ] / NV

Part A: Reminiscences & Theory

• The G1 Matrix Operation

GPA Theory & Applications

Santiago, 2018

9 (16-9)/9=0.778 =0.78 (17-9)/9=0.889 =0.89

GA

Nv

I = TA

L= VA

Gradient Pattern Analysis

Ex.: Elementar ramp

GA=(16-9)/9=0.7777

G1=0.778

G1=1.222

Part A: Reminiscences & Theory

• The GPA for Time Series

GPA Theory & Applications

Santiago, 2018

• Color and Magnitudes

• Sersic Parameters

• CAS + H + GPA

• Geometrical Moments

• Luminosity Profile

https://doi.org/10.1093/mnrasl/sly054

Gradient Pattern Analysis Applied to Galaxy Morphology

R.R.Rosa et al. MNRASL, 2018.

Background Segmentation Resampling star masking stamp analysis

Op1

Op2

Op3

Op4

Op1

Op2

Op3

Op4

μ1

μ2

μ3

μ4

. . .

. . .

μ1

μ2

μ3

μ4

• Discriminant Analysis from Big Data 1

2

3

4

4 Rosa et al.

F igur e 3. T he applicat ion of GPA, for calculat ing G1 , over two

galaxies selected from our sample - a spiral on the left and an ellip-

t ical on the right . T he respect ive gradients containing VA vect ors

are shown in the intermediate panels and the t riangulat ion fields,

containing TA edges, are shown in the bot tom panels. Images are

in reverse color cont rast for bet ter viewing.

metrical vector sum and |vi | is the i t h asymmet rical vector

norm. Not ice that for misaligned vectors, the vectorial sum

tends to zero. More formally, we can write |VAi vi | = 0,

then according to equat ion 1, G2 = 2VA

N. Whereas if K vec-

tors with same moduli are aligned, |VAi vi | = K |vi |, and

VAi |vi | = K |vi |. Therefore, G2 = VA

N2 −

K |v i |

K |v i |= VA

N.

This means that this operator considers the proport ion of

asymmetrical vectors, and also, without using explicit ly the

phases (GP3), the correspondent alignment rate. Higher G2

values means that the gradient lat t ice has many misaligned

asymmetrical vectors and then a high diversity for the values

in the lat t ice GP2.

The calculat ion of G2 on canonical mat rices demon-

st rates its ability to characterise the basic asymmet ry con-

jectures invest igated in Rosa et al. (1999) using G1 . However

with less sensit ivity to the noise due to phase fluctuat ions

imprinted in the t riangulat ion field. For the example in Fig-

ure 2, the result for G2 = 0.656 is invariant considering both

mat rices.

The operat ion for comput ing G2 , via Eq. 2, presents the

following improvements compared to G1 : (i) For the same

type of gradient pat tern, the value of G2 is invariant to the

size of the mat rix; ( i i) More appropriately, it does not con-

sider the elements of the mat rix border for calculat ing the

F igur e 4. Histograms for the six morphological paramet ers used

to classify galaxies. Early-type (red) and late-t ype (blue).

gradient ; (i i i ) Can be applied, without loss of generality, to

rectangular mat rices; and ( iv) I t is less sensit ive to noise and

faster because it avoids t riangulat ion.

4 G A L A X Y M OR P H OM ET RY U SI N G G PA

The morphological analysis of the 54,896 objects, as classi-

fied by Galaxy Zoo, has been done on the basis of the pa-

rameters G1 , G2 , C , H , A and S. The respect ive histograms

are shown in Figure 3 where the red (blue) line refers to

early(late)-type galaxies.

Here, we present a comparat ive analysis between well

established parameters, C, A, S, and H, used in several

galaxy morphology studies (e.g. Conselice et al. 2000; Fer-

rari et al. 2015), and those proposed in this invest igat ion, G1

and G2 . A given parameter is considered a useful morpholog-

ical indicator when it separates early and late type galaxies

the best possible. Therefore, it is of paramount importance

to object ively define separat ion. In our case, we est imate

how far apart two histograms are (see Figure 4), using the

index δG H S which is calculated from the GHS (Geomet ric

Histogram Separat ion) algorithm (Saut ter & Barchi 2017).

This algorit hm determines separat ion using only the geo-

met ric characterist ics of a binomial proport ion represented

by histograms. The GHS input values are: AB (blue his-

togram area), AR (red histogram area), AB R (intersect ion

area between AB and AR ), and the respect ive heights for

AB , AR and AB R : hB , hR , and hB R . The separat ion is then

defined as:

δG H S =(1 − A B R

A B + A R + A B R)1/ 2 + (

h a + h b− 2h c

h a + h b)

2(3)

MNRAS 000, 1–5 (2017)

4 Rosa et al.

F igur e 3. T he applicat ion of GPA, for calculat ing G1 , over two

galaxies selected from our sample - a spiral on the left and an ellip-

t ical on the right . T he respect ive gradients containing VA vect ors

are shown in the intermediate panels and the t riangulat ion fields,

containing TA edges, are shown in the bot tom panels. Images are

in reverse color cont rast for bet ter viewing.

metrical vector sum and |vi | is the i t h asymmet rical vector

norm. Not ice that for misaligned vectors, the vectorial sum

tends to zero. More formally, we can write |VAi vi | = 0,

then according to equat ion 1, G2 = 2VA

N. Whereas if K vec-

tors with same moduli are aligned, |VAi vi | = K |vi |, and

VAi |vi | = K |vi |. Therefore, G2 = VA

N2 −

K |v i |

K |v i |= VA

N.

This means that this operator considers the proport ion of

asymmetrical vectors, and also, without using explicit ly the

phases (GP3), the correspondent alignment rate. Higher G2

values means that the gradient lat t ice has many misaligned

asymmetrical vectors and then a high diversity for the values

in the lat t ice GP2.

The calculat ion of G2 on canonical mat rices demon-

st rates its ability to characterise the basic asymmet ry con-

jectures invest igated in Rosa et al. (1999) using G1 . However

with less sensit ivity to the noise due to phase fluctuat ions

imprinted in the t riangulat ion field. For the example in Fig-

ure 2, the result for G2 = 0.656 is invariant considering both

mat rices.

The operat ion for comput ing G2 , via Eq. 2, presents the

following improvements compared to G1 : (i) For the same

type of gradient pat tern, the value of G2 is invariant to the

size of the mat rix; ( i i) More appropriately, it does not con-

sider the elements of the mat rix border for calculat ing the

F igur e 4. Histograms for the six morphological paramet ers used

to classify galaxies. Early-type (red) and late-t ype (blue).

gradient ; (i i i ) Can be applied, without loss of generality, to

rectangular mat rices; and ( iv) I t is less sensit ive to noise and

faster because it avoids t riangulat ion.

4 G A L A X Y M OR P H OM ET RY U SI N G G PA

The morphological analysis of the 54,896 objects, as classi-

fied by Galaxy Zoo, has been done on the basis of the pa-

rameters G1 , G2 , C , H , A and S. The respect ive histograms

are shown in Figure 3 where the red (blue) line refers to

early(late)-type galaxies.

Here, we present a comparat ive analysis between well

established parameters, C, A, S, and H, used in several

galaxy morphology studies (e.g. Conselice et al. 2000; Fer-

rari et al. 2015), and those proposed in this invest igat ion, G1

and G2 . A given parameter is considered a useful morpholog-

ical indicator when it separates early and late type galaxies

the best possible. Therefore, it is of paramount importance

to object ively define separat ion. In our case, we est imate

how far apart two histograms are (see Figure 4), using the

index δG H S which is calculated from the GHS (Geomet ric

Histogram Separat ion) algorithm (Saut ter & Barchi 2017).

This algorit hm determines separat ion using only the geo-

met ric characterist ics of a binomial proport ion represented

by histograms. The GHS input values are: AB (blue his-

togram area), AR (red histogram area), AB R (intersect ion

area between AB and AR ), and the respect ive heights for

AB , AR and AB R : hB , hR , and hB R . The separat ion is then

defined as:

δG H S =(1 − A B R

A B + A R + A B R)1/ 2 + (

h a + h b− 2h c

h a + h b)

2(3)

MNRAS 000, 1–5 (2017)

Concluding Remarks GPA Theory & Applications

Santiago, 2018

{G1, G2, G3, G4}

Future Challenges:

• GPA FOR HYPERCUBES (FROM PIXEL TO VOXEL)

• GPA OpenACC for Fast Machine Learning Applications

reinaldo.rosa@inpe.br

Part A: Reminiscences & Theory

• The Need for Spatiotemporal Analytic Tools (Matrix Operation)

GPA Theory & Applications

Santiago, 2018

... ...

UMD, USA 1994

Part A: Reminiscences & Theory

• The Need for Spatiotemporal Analytic Tools (Matrix Operation)

GPA Theory & Applications

Santiago, 2018

Examples for Modeling Validation

(a)

(b)

(c)

(d)

(e)

(f)

EX. de Sistema Simples

Equação da Membrana

(Coordenadas Polares) Att = (1/c2) (Arr + r-1 Ar + r-2 A )

A(t,r,) é um deslocamento uma solução (EDO) para a variável radial

Prova-se, analiticamente, que existe uma família de funções que admitem

Soluções do tipo:

Sn,k (t,r,) = Jn(bn,k) cos(nt) (Funções Especiais de Bessel)

Onde n indica a quantidade de diâmetros nodais e k a ordem da função.

• Spatiotemporal Data: from simple to complex