Map Math and StaMsMcs

Transcription

1 Map Math and StaMsMcs A. Michelle Lawing Ecosystem Science and Management Texas A&M University College StaMon, TX

2 ObjecMves What is "Map Algebra or Map Math Images are data DescripMve vs. inferenmal stamsmcs Comparing two qualitamve data sets Comparing two quanmtamve data sets

3 What is Map Math and how does it work? High- level computamon used for performing analysis using raster data. Math applied to rasters. An image acts as a variable in a mathemamcal equamon. The same equamon is applied to each cell. Cell by cell transformamon

4 OperaMons and funcmons Map math uses math- like expressions containing operators and funcmons. Expressions are entered and evaluated using raster calculator. Operators Arithmetic Relational Boolean Logical Combinatorial Bitwise Functions Aspect Con Isnull Mean Merge Nibble + 100s more

5 Map Operators ArithmeMc add, subtract, mulmply, divide RelaMonal logical tests, true (1) and false (0) [=, <, >] Boolean and, or, not [&,,!]

6 Map Operators Logical logical tests on a cell- by- cell basis, but are implemented with specific rules [DIFF, IN, Over] Combinatorial combines the anributes of mulmple input rasters [CAND, COR, CXOR] Bitwise funcmons such as right shio, leo shio, bitwise complement [^^, >>, <<]

7 Map funcmons FuncMons are programs the perform specific tasks, such as calculamng slope, deriving hillshade, etc. There are 100s of Map algebra funcmons.

8 Map funcmons CON performs one or more condimonal if/else evaluamons INT converts input floamng- point values to integer values through truncamon ISNULL returns 1 if the input value is NODATA, and 0 if it is not

9 Map funcmons MEAN uses mulmple input grids to determine the mean value MOSAIC merges mulmple adjacent conmnuous grids and performs interpolamon in the overlapping areas NIBBLE replaces areas in a grid corresponding to a mask, with the values of the nearest neighbors SETNULL returns NODATA if the evaluamon of the input condimon is TRUE ; if it is FALSE, returns the value specified by the second input argument

10 What funcmon have you used in ArcMap?

11 Puvng expressions to work Logical Values CondiMonal Processing Working with NODATA values Merging mulmple rasters

12 Logical Values Non- zero value = logical true Zero values = logical false

13 CondiMonal processing You specify which acmon to take, depending on condimon(s). This is useful for creamng analysis masks.

14 NoData NoData values influence the evaluamon of expressions. The NoData value is the only non- zero value that is not interpreted as a true condimon.

15 Merging rasters Combining individual rasters to create a single, larger raster

16 Combining operamons Some mathemamcal models require that many operamons are performed in a row Example: convermng a temperature map from Fahrenheit to Celsius units involves two operamons: TempC = (TempF - 32)/1.8

17 Combining operamons Example: Convert distance measurements in miles to Mme measurements in minutes if the speed of movement is 5 km per hour (walking speed): 1. to convert miles into km, mulmply each cell by to convert distance to Mme, divide each cell by speed of walking 5 km/h 3. to convert hours to minutes, mulmply the result by 60 TimH = (DistM * 1.61)/5 * 60

18 Combining operamons Example: In the USLE (Universal Soil Loss EquaMon) we have A = K * C * R * LS * P, where A = soil loss in metric tons per hectare per year K = soil erodibility factor C = vegetamve cover factor R = rainfall and run- off factor LS = slope and slope length factor P = conservamon pracmces factor

19 Combining operamons

20 StaMsMcs Important in GIS Formal analysis of quanmtamve and qualitamve data DescripMve stamsmcs Assess the nature of the dataset DistribuMon or frequency of values Commonly, mean, standard deviamon, frequency (hist) InferenMal stamsmcs Determine relamonship between variables and its strength

21 Normal distribumon 99.7% 95.4% 68.2%

22 DescripMve stamsmcs Example of populamon density data SD provides a measure of the range of the data (different from just knowing the min and max values). Using SD assumes that the data is described by a normal (bell shape) curve. Mean value is 49. Minimum is 18 and Maximum is 105.

23 DescripMve stamsmcs Example of populamon density data SD provides a measure of the range of the data (different from just knowing the min and max values). Using SD assumes that the data is described by a normal (bell shape) curve. Mean value is 49. Minimum is 18 and Maximum is SD is a distance of 7 from the mean in either direcmon. 1SD describes the interval 42 to 56 (68% of all data is in this interval).

24 DescripMve stamsmcs Example of populamon density data SD provides a measure of the range of the data (different from just knowing the min and max values). Using SD assumes that the data is described by a normal (bell shape) curve. Mean value is 49. Minimum is 18 and Maximum is SD is a distance of 7 from the mean in either direcmon. 1SD describes the interval 42 to 56 (68% of all data is in this interval). 2 SD is 35 to 63 (95% of all data is in this interval). 3 SD is 28 to 70 (99% of all data is in this interval).

25 InferenMal stamsmcs EsMmate the relamonship or associamon between different variables and data sets (as images or values). Different stamsmcs are applied to qualitamve and quanmtamve data.

26 Ecosystem Science and Management Texas A&M University (c) 2015, A. Michelle Lawing ESSM/GEOG 462: Advanced GIS QualitaMve data Categorical data Cross- tabulamon For example, change in land use between two dates for the same territory (expressed as two land use images).

27 QualitaMve data analysis CHI squared esmmates the likelihood that a relamonship between the two variables (the landuse75 and landuse80 images) exists. Cramer s V - - esmmates the strength of the associamon (0-1).

28 QualitaMve data analysis K (Kappa) index an index of the agreement between two images. Ranges from - 1 to +1 1 = full agreement - 1 = full disagreement (the images are opposite, complete transformamon in a consistent manner) 0 = no correlamon (change is random)

29 Ecosystem Science and Management Texas A&M University (c) 2015, A. Michelle Lawing ESSM/GEOG 462: Advanced GIS Kappa K = (Po Pe)/(1- Pe) Where Po is the propormon of cells not changed (sum of all the diagonal values divided by the total number of cells)

30 Ecosystem Science and Management Texas A&M University (c) 2015, A. Michelle Lawing Kappa K = (Po Pe)/(1- Pe) Pe is the expected change due to chance Pe = ((2845/171186)*(2849/171186)) + ((41682/171186)*(32471/171186)) + ((73415/171186)*(73573/171186)) + ((53244/171186)*(62291/171186)) ESSM/GEOG 462: Advanced GIS

31 Ecosystem Science and Management Texas A&M University (c) 2015, A. Michelle Lawing ESSM/GEOG 462: Advanced GIS Kappa Kappa indices can be calculated for each category as well as. Individual indices tell us which categories changed and which did not.