Sample Selection Method
| Selection Method | Simple Random Sampling |
|---|
Mars Craters - Looking for major events
Can crater’s records help to understand Mars’ history?
Angelo Klin
21 February 2016
Course Details
| Specialisation | Data Analysis and Interpretation |
| Course | Machine Learning for Data Analysis |
| Education Institution | Wesleyan University |
| Publisher | Coursera |
| Assignment | Running a Lasso Regression Analysis |
Introduction
Besides the historical fascination on Mars due to its proximity and some similarities with Earth, the collection of facts allow scientists to put together a big jigsaw puzzle.
An announcement made by NASA just recently about evidence of flowing liquid water on the surface of Mars just adds to all that is known and the curiosity, or even need, to find out much more.
Summary
When trying to identify an association of the physical and geographical characteristics of a crater with its morphology using a classification tree algorithm it is possible to identify a good number for some particular and more common formations. The identification does vary depending on the kind of formation and there are some characteristics that are common to some formations.
The Data Set
With all the talks about Mars, including both Science and Fiction (a Ridley Scott’s movie called The Martian, based on the book with the same name, by Andy Weir was released on early Oct/15), the data set chosen for this assignment is the Mars Craters.
The Mars Craters Study, presents a global database that includes 378,540 Mars craters, with diameter of 1 km or larger, that were created between 4.2 and 3.8 billion years ago during a period of heavy bombardment (i.e. impacts of asteroids, proto-planets, and comets).
The data set was made available by Wesleyan University/Coursera as part of the Data Analysis and Interpretation Specialisation, from the Ph.D. Thesis Planetary Surface Properties, Cratering Physics, and the Volcanic History of Mars from a New Global Martian Crater Database (2011) by Robbins, S.J., University of Colorado at Boulder.
Topic of Interest
The data set provides a catalogue of craters on Mars. The initial thoughts are about checking for patterns that could identify specific major events that might have happened and that would have significant impact on Mars’ geology, climate and life as a planetary body.
Codebook
As the initial data set has only nine variables, they could all the relevant to formulate hypothesis and help in leading to a conclusion, so all the variables will be kept for this assignment.
Variables
- CRATER_ID: crater ID for internal sue, based upon the region of the planet (\({1 \over 16}\)), the “pass” under which the crate was identified, ad the order in which it was identified
- LATITUDE_CIRCLE_IMAGE: latitude from the derived centre of a non-linear least-squares circle fit to the vertices selected to manually identify the crater rim (units are decimal degrees North)
- LONGITUDE_CIRCLE_IMAGE: longitude from the derived centre of a non-linear least-squares circle fit to the vertices selected to manually identify the crater rim (units are decimal degrees East)
- DIAM_CIRCLE_IMAGE: diameter from a non-linear least squares circle fit to the vertices selected to manually identify the crater rim (units are km)
- DEPTH_RIMFLOOR_TOPOG: average elevation of each of the manually determined N points along (or inside) the crater rim (units are km)
- Depth Rim: Points are selected as relative topographic highs under the assumption they are the least eroded so most original points along the rim
- Depth Floor: Points were chosen as the lowest elevation that did not include visible embedded craters
- MORPHOLOGY_EJECTA_1: ejecta morphology classified.
- If there are multiple values, separated by a “/”, then the order is the inner-most ejecta through the outer-most, or the top-most through the bottom-most
- MORPHOLOGY_EJECTA_2: the morphology of the layer(s) itself/themselves. This classification system is unique to this work.
- MORPHOLOGY_EJECTA_3: overall texture and/or shape of some of the layer(s)/ejecta that are generally unique and deserve separate morphological classification.
- NUMBER_LAYERS: the maximum number of cohesive layers in any azimuthal direction that could be reliably identified
Extra Variables
MorphoE1_RD is a categorical variable that has value “Yes” is the primary morphology is classified as “Rd” (Radial), and “No” otherwise.
Quadrangle is a variable derived from both LATITUDE_CIRCLE_IMAGE and LONGITUDE_CIRCLE_IMAGE variables. (see below a definition from Wikipedia)
Each Quadrangle has approximately from one to five percent of the recorded craters, being MC-16, Memnonia the one with the most observations (20455 = 5.32%), and MC-10: Lunae Palus the one with the lower number of records (3478 = 0.90%).
List of quadrangles on Mars (Wikipedia):
The surface of Mars has been divided into 30 quadrangles by the United States Geological Survey, so named because their borders lie along lines of latitude and longitude and so maps appear rectangular. Martian quadrangles are named after local features and are numbered with the prefix “MC” for “Mars Chart”. West longitude is used.
The following imagemap of the planet Mars is divided into 30 linked quadrangles. North is at the top; 0°N 180°W is at the far left on the equator. The map images were taken by the Mars Global Surveyor.
From Wikipedia, Source: http://photojournal.jpl.nasa.gov/catalog/PIA03467
Running a Lasso Regression Analysis
Is the morphology of a crater strongly associate with its physical and geographical characteristics?
The Test
The variables Quadrangle (Categorical), DIAM_CIRCLE_IMAGE, DEPTH_RIMFLOOR_TOPOG and NUMBER_LAYERS (Quantitative) were used to classify the main characteristic of MORPHOLOGY_EJECTA_1 as “Rd” (Radial) or not (coded as 1-Yes or 0-No in the variable MorphoE1_RD).
The variable Quadrangle was split by SAS in its 30 categories.
The Results
The number of valid observations was 44625. Using a seed of 6587 the data set was divided in 31238 observations on Training and 13387 observations on Testing.
Using Cross Validation, the number of steps was 26 (including the Intercept). A visual and numeric analysis could arguably reduce it to five. Other methods would show similar split numbers and also similar charts supporting a smaller number.
The Mean Square Error for training and testing are visually superimposed, with no clear indication on what would be the best cut point. A visual inspection also supports a smaller number of variables, for the sake of simplicity.
Using the 25 variables, the model can explain about 57% (R-Square = 0.5773).
Using SAS
SAS Output
Results: W03-Running a Lasso Regression Analysis.sas
The SURVEYSELECT Procedure
The SURVEYSELECT Procedure
Sample Selection Summary
| Input Data Set | WORK |
|---|---|
| Random Number Seed | 6587 |
| Sampling Rate | 0.7 |
| Sample Size | 31238 |
| Selection Probability | 0.700011 |
| Sampling Weight | 0 |
| Output Data Set | TRAINTEST |
The GLMSELECT Procedure
The GLMSELECT Procedure
Model Information
| Data Set | WORK.TRAINTEST |
|---|---|
| Dependent Variable | MorphoE1_RD |
| Selection Method | LAR |
| Stop Criterion | None |
| Choose Criterion | Cross Validation |
| Cross Validation Method | Random |
| Cross Validation Fold | 10 |
| Effect Hierarchy Enforced | None |
| Random Number Seed | 6587 |
Number of Observations
| Number of Observations Read | 44625 |
|---|---|
| Number of Observations Used | 44625 |
| Number of Observations Used for Training | 31238 |
| Number of Observations Used for Testing | 13387 |
Class Level Information
| Class Level Information | ||
|---|---|---|
| Class | Levels | Values |
| MorphoE1_RD | 2 | 0 1 |
| Quadrangle | 30 | MC-01: Mare Boreum (North Pole) MC-02: Diacria MC-03: Arcadia MC-04: Mare Acidalium MC-05: Ismenius Lacus MC-06: Casius MC-07: Cebrenia MC-08: Amazonis MC-09: Tharsis MC-10: Lunae Palus MC-11: Oxia Palus MC-12: Arabia MC-13: Syrtis Major … |
Dimensions
| Dimensions | |
|---|---|
| Number of Effects | 5 |
| Number of Effects after Splits | 34 |
| Number of Parameters | 34 |
The GLMSELECT Procedure
Model Building Summary
LAR Selection Summary
| LAR Selection Summary | |||||
|---|---|---|---|---|---|
| Step |
Effect Entered |
Number Effects In |
ASE | Test ASE | CV PRESS |
| * Optimal Value of Criterion | |||||
| 0 | Intercept | 1 | 0.2386 | 0.2387 | 7454.6787 |
| 1 | NUMBER_LAYERS | 2 | 0.1090 | 0.1088 | 3304.3110 |
| 2 | DIAM_CIRCLE_IMAGE | 3 | 0.1051 | 0.1049 | 3199.1954 |
| 3 | DEPTH_RIMFLOOR_TOPOG | 4 | 0.1022 | 0.1020 | 3177.3827 |
| 4 | Quadrangle_MC-01: Mare Boreum (North Pole) | 5 | 0.1020 | 0.1019 | 3174.4301 |
| 5 | Quadrangle_MC-07: Cebrenia | 6 | 0.1018 | 0.1017 | 3170.1046 |
| 6 | Quadrangle_MC-05: Ismenius Lacus | 7 | 0.1018 | 0.1017 | 3166.7408 |
| 7 | Quadrangle_MC-18: Coprates | 8 | 0.1018 | 0.1016 | 3164.3432 |
| 8 | Quadrangle_MC-21: Iapygia | 9 | 0.1016 | 0.1015 | 3162.4652 |
| 9 | Quadrangle_MC-04: Mare Acidalium | 10 | 0.1015 | 0.1014 | 3161.6088 |
| 10 | Quadrangle_MC-17: Phoenicis Lacus | 11 | 0.1014 | 0.1013 | 3160.5560 |
| 11 | Quadrangle_MC-08: Amazonis | 12 | 0.1014 | 0.1013 | 3159.3801 |
| 12 | Quadrangle_MC-02: Diacria | 13 | 0.1014 | 0.1013 | 3158.2723 |
| 13 | Quadrangle_MC-24: Phaethontis | 14 | 0.1013 | 0.1012 | 3157.4473 |
| 14 | Quadrangle_MC-20: Sinus Sabaeus | 15 | 0.1013 | 0.1012 | 3156.7711 |
| 15 | Quadrangle_MC-12: Arabia | 16 | 0.1012 | 0.1011 | 3155.9961 |
| 16 | Quadrangle_MC-06: Casius | 17 | 0.1012 | 0.1011 | 3154.5947 |
| 17 | Quadrangle_MC-19: Margaritifer Sinus | 18 | 0.1012 | 0.1011 | 3154.5210 |
| 18 | Quadrangle_MC-16: Memnonia | 19 | 0.1011 | 0.1011 | 3154.0017 |
| 19 | Quadrangle_MC-11: Oxia Palus | 20 | 0.1011 | 0.1010 | 3153.6842 |
| 20 | Quadrangle_MC-10: Lunae Palus | 21 | 0.1010 | 0.1010 | 3153.7239 |
| 21 | Quadrangle_MC-09: Tharsis | 22 | 0.1010 | 0.1009 | 3153.4905 |
| 22 | Quadrangle_MC-29: Eridania | 23 | 0.1010 | 0.1009 | 3153.3422 |
| 23 | Quadrangle_MC-03: Arcadia | 24 | 0.1009 | 0.1009 | 3153.1830 |
| 24 | Quadrangle_MC-28: Hellas | 25 | 0.1009 | 0.1009 | 3153.0932 |
| 25 | Quadrangle_MC-14: Amenthes | 26 | 0.1009 | 0.1009 | 3153.0284* |
| 26 | Quadrangle_MC-30: Mare Australe (South Pol | 27 | 0.1008 | 0.1009 | 3153.0699 |
| 27 | Quadrangle_MC-26: Argyre | 28 | 0.1008 | 0.1008 | 3153.2531 |
| 28 | Quadrangle_MC-22: Mare Tyrrhenum | 29 | 0.1008 | 0.1008 | 3153.2018 |
| 29 | Quadrangle_MC-23: Aeolis | 30 | 0.1008 | 0.1008 | 3153.1941 |
| 30 | Quadrangle_MC-15: Elysium | 31 | 0.1007 | 0.1008 | 3153.2564 |
| 31 | Quadrangle_MC-25: Thaumasia | 32 | 0.1007 | 0.1008 | 3153.3774 |
| 32 | Quadrangle_MC-13: Syrtis Major | 33 | 0.1007 | 0.1008 | 3153.5994 |
Stop Reason
| Selection stopped because the change of the maximum absolute correction is tiny. |
Coefficient Plot
Criterion Panel
ASE Plot
Selected Model
The GLMSELECT Procedure
Selected Model
The selected model, based on Cross Validation, is the model at Step 25.
Selected Effects
| Effects: | Intercept Quadrangle_MC-01: Mare Boreum (North Pole) Quadrangle_MC-02: Diacria Quadrangle_MC-03: Arcadia Quadrangle_MC-04: Mare Acidalium Quadrangle_MC-05: Ismenius Lacus Quadrangle_MC-06: Casius Quadrangle_MC-07: Cebrenia Quadrangle_MC-08: Amazonis Quadrangle_MC-09: Tharsis Quadrangle_MC-10: Lunae Palus Quadrangle_MC-11: Oxia Palus Quadrangle_MC-12: Arabia Quadrangle_MC-14: Amenthes Quadrangle_MC-16: Memnonia Quadrangle_MC-17: Phoenicis Lacus Quadrangle_MC-18: Coprates Quadrangle_MC-19: Margaritifer Sinus Quadrangle_MC-20: Sinus Sabaeus Quadrangle_MC-21: Iapygia Quadrangle_MC-24: Phaethontis Quadrangle_MC-28: Hellas Quadrangle_MC-29: Eridania DIAM_CIRCLE_IMAGE DEPTH_RIMFLOOR_TOPOG NUMBER_LAYERS |
|---|
ANOVA
| Analysis of Variance | ||||
|---|---|---|---|---|
| Source | DF |
Sum of Squares |
Mean Square |
F Value |
| Model | 25 | 4303.36664 | 172.13467 | 1705.22 |
| Error | 31212 | 3150.72571 | 0.10095 | |
| Corrected Total | 31237 | 7454.09236 | ||
Fit Statistics
| Root MSE | 0.31772 |
|---|---|
| Dependent Mean | 0.60666 |
| R-Square | 0.5773 |
| Adj R-Sq | 0.5770 |
| AIC | -40368 |
| AICC | -40368 |
| SBC | -71391 |
| ASE (Train) | 0.10086 |
| ASE (Test) | 0.10087 |
| CV PRESS | 3153.02838 |
Parameter Estimates
| Parameter Estimates | ||
|---|---|---|
| Parameter | DF | Estimate |
| Intercept | 1 | 0.826756 |
| Quadrangle_MC-01: Mare Boreum (North Pole) | 1 | -0.038410 |
| Quadrangle_MC-02: Diacria | 1 | 0.027574 |
| Quadrangle_MC-03: Arcadia | 1 | 0.005809 |
| Quadrangle_MC-04: Mare Acidalium | 1 | -0.034143 |
| Quadrangle_MC-05: Ismenius Lacus | 1 | 0.062124 |
| Quadrangle_MC-06: Casius | 1 | 0.024366 |
| Quadrangle_MC-07: Cebrenia | 1 | 0.053741 |
| Quadrangle_MC-08: Amazonis | 1 | -0.019499 |
| Quadrangle_MC-09: Tharsis | 1 | -0.007935 |
| Quadrangle_MC-10: Lunae Palus | 1 | 0.022122 |
| Quadrangle_MC-11: Oxia Palus | 1 | -0.017943 |
| Quadrangle_MC-12: Arabia | 1 | 0.024276 |
| Quadrangle_MC-14: Amenthes | 1 | 0.000678 |
| Quadrangle_MC-16: Memnonia | 1 | -0.010887 |
| Quadrangle_MC-17: Phoenicis Lacus | 1 | -0.022572 |
| Quadrangle_MC-18: Coprates | 1 | -0.060352 |
| Quadrangle_MC-19: Margaritifer Sinus | 1 | -0.011520 |
| Quadrangle_MC-20: Sinus Sabaeus | 1 | 0.013952 |
| Quadrangle_MC-21: Iapygia | 1 | 0.028199 |
| Quadrangle_MC-24: Phaethontis | 1 | 0.019741 |
| Quadrangle_MC-28: Hellas | 1 | 0.001184 |
| Quadrangle_MC-29: Eridania | 1 | -0.003254 |
| DIAM_CIRCLE_IMAGE | 1 | 0.002834 |
| DEPTH_RIMFLOOR_TOPOG | 1 | 0.090837 |
| NUMBER_LAYERS | 1 | -0.512287 |
SAS Code
1 /* Use Course's Library */
2 LIBNAME mydata "/courses/d1406ae5ba27fe300" ACCESS = readonly;
3
4 DATA WORK;
5 SET mydata.marscrater_pds;
6
7 WHERE MORPHOLOGY_EJECTA_1 NE " ";
8
9 /* Collapse the Morphology of Eject 1 to its Main Feature, to reduce the output */
10 IF (INDEX(MORPHOLOGY_EJECTA_1, "/") = 0)
11 THEN MorphoE1 = MORPHOLOGY_EJECTA_1;
12 ELSE MorphoE1 = SUBSTR(MORPHOLOGY_EJECTA_1, 1, INDEX(MORPHOLOGY_EJECTA_1, "/") - 1);
13 MorphoE1 = UPCASE(TRIM(MorphoE1));
14
15 /* Does the the Morphology 1 equals to "RD" */
16 IF MorphoE1 = "RD"
17 THEN MorphoE1_RD = 1;
18 ELSE MorphoE1_RD = 0;
19
20 /* convert coordinates to Quadrangles: https://en.wikipedia.org/wiki/List_of_quadrangles_on_Mars */
21 LA = LATITUDE_CIRCLE_IMAGE;
22 LO = LONGITUDE_CIRCLE_IMAGE + 180;
23 IF LA >= 65 AND LA <= 90 AND LO >= 0 AND LO <= 360 THEN Quadrangle = "MC-01: Mare Boreum (North Pole)";
24 IF LA >= 30 AND LA < 65 AND LO >= 120 AND LO < 180 THEN Quadrangle = "MC-02: Diacria";
25 IF LA >= 30 AND LA < 65 AND LO >= 60 AND LO < 120 THEN Quadrangle = "MC-03: Arcadia";
26 IF LA >= 30 AND LA < 65 AND LO >= 0 AND LO < 60 THEN Quadrangle = "MC-04: Mare Acidalium";
27 IF LA >= 30 AND LA < 65 AND LO >= 300 AND LO <= 360 THEN Quadrangle = "MC-05: Ismenius Lacus";
28 IF LA >= 30 AND LA < 65 AND LO >= 240 AND LO < 300 THEN Quadrangle = "MC-06: Casius";
29 IF LA >= 30 AND LA < 65 AND LO >= 180 AND LO < 240 THEN Quadrangle = "MC-07: Cebrenia";
30 IF LA >= 0 AND LA < 30 AND LO >= 135 AND LO < 180 THEN Quadrangle = "MC-08: Amazonis";
31 IF LA >= 0 AND LA < 30 AND LO >= 90 AND LO < 135 THEN Quadrangle = "MC-09: Tharsis";
32 IF LA >= 0 AND LA < 30 AND LO >= 45 AND LO < 90 THEN Quadrangle = "MC-10: Lunae Palus";
33 IF LA >= 0 AND LA < 30 AND LO >= 0 AND LO < 45 THEN Quadrangle = "MC-11: Oxia Palus";
34 IF LA >= 0 AND LA < 30 AND LO >= 315 AND LO <= 360 THEN Quadrangle = "MC-12: Arabia";
35 IF LA >= 0 AND LA < 30 AND LO >= 270 AND LO < 315 THEN Quadrangle = "MC-13: Syrtis Major";
36 IF LA >= 0 AND LA < 30 AND LO >= 225 AND LO < 270 THEN Quadrangle = "MC-14: Amenthes";
37 IF LA >= 0 AND LA < 30 AND LO >= 180 AND LO < 225 THEN Quadrangle = "MC-15: Elysium";
38 IF LA >= -30 AND LA < 0 AND LO >= 135 AND LO < 180 THEN Quadrangle = "MC-16: Memnonia";
39 IF LA >= -30 AND LA < 0 AND LO >= 90 AND LO < 135 THEN Quadrangle = "MC-17: Phoenicis Lacus";
40 IF LA >= -30 AND LA < 0 AND LO >= 45 AND LO < 90 THEN Quadrangle = "MC-18: Coprates";
41 IF LA >= -30 AND LA < 0 AND LO >= 0 AND LO < 45 THEN Quadrangle = "MC-19: Margaritifer Sinus";
42 IF LA >= -30 AND LA < 0 AND LO >= 315 AND LO <= 360 THEN Quadrangle = "MC-20: Sinus Sabaeus";
43 IF LA >= -30 AND LA < 0 AND LO >= 270 AND LO < 315 THEN Quadrangle = "MC-21: Iapygia";
44 IF LA >= -30 AND LA < 0 AND LO >= 225 AND LO < 270 THEN Quadrangle = "MC-22: Mare Tyrrhenum";
45 IF LA >= -30 AND LA < 0 AND LO >= 180 AND LO < 225 THEN Quadrangle = "MC-23: Aeolis";
46 IF LA >= -65 AND LA < -30 AND LO >= 120 AND LO < 180 THEN Quadrangle = "MC-24: Phaethontis";
47 IF LA >= -65 AND LA < -30 AND LO >= 60 AND LO < 120 THEN Quadrangle = "MC-25: Thaumasia";
48 IF LA >= -65 AND LA < -30 AND LO >= 0 AND LO < 60 THEN Quadrangle = "MC-26: Argyre";
49 IF LA >= -65 AND LA < -30 AND LO >= 300 AND LO <= 360 THEN Quadrangle = "MC-27: Noachis";
50 IF LA >= -65 AND LA < -30 AND LO >= 240 AND LO < 300 THEN Quadrangle = "MC-28: Hellas";
51 IF LA >= -65 AND LA < -30 AND LO >= 180 AND LO < 240 THEN Quadrangle = "MC-29: Eridania";
52 IF LA >= -90 AND LA < -65 AND LO >= 0 AND LO <= 360 THEN Quadrangle = "MC-30: Mare Australe (South Pole)";
53
54 LABEL Quadrangle = "Quadrangle"
55 DIAM_CIRCLE_IMAGE = "Diameter"
56 DEPTH_RIMFLOOR_TOPOG = "Depth"
57 MorphoE1_RD = "Morphology 1-RD"
58 NUMBER_LAYERS = "Layers";
59
60 RUN;
61
62 ODS GRAPHICS ON;
63
64 /* Split data randomly into test and training data */
65 PROC SURVEYSELECT DATA = WORK
66 OUT = TRAINTEST
67 METHOD = SRS
68 SAMPRATE = 0.7
69 SEED = 6587
70 OUTALL;
71 RUN;
72
73 /* lasso multiple regression with lars algorithm k=10 fold validation */
74 PROC GLMSELECT DATA = TRAINTEST
75 PLOTS = ALL
76 SEED = 6587;
77 PARTITION ROLE = SELECTED(TRAIN = '1'
78 TEST = '0');
79 CLASS MorphoE1_RD
80 Quadrangle;
81 MODEL MorphoE1_RD = Quadrangle
82 DIAM_CIRCLE_IMAGE
83 DEPTH_RIMFLOOR_TOPOG
84 NUMBER_LAYERS
85 / SELECTION = LAR(CHOOSE = CV
86 STOP = NONE)
87 CVMETHOD = RANDOM(10);
88 TITLE "Mars' Craters - LASSO Regression - Morphology = RD";
89 run;
90
91 TITLE;