NOTE: This page has been revised for Winter 2021, but may undergo further edits.

Geography 4/595: Geographic Data Analysis
Winter 2021

Exercise 5: Some data wrangling and matrix algebra
Finish by Friday, February 12

1. Introduction

The first aim of this exercise is to illustrate the idea of “data wrangling” or the reshaping or restructuring of input data into the “tidy” form (of a rectangular data set) with variables in columns and observations or cases in rows. The second part of the exercise consists of a few examples that illustrate the features and application of matrix algebra.

2. Data and packages

A the concept of data wrangling, or the reshaping of non-rectangular data set into a rectangular one, can be illustrated using a small sample of monthly climate data for Eugene. These data are not currently part of the geog495.RData workspace file (because they may be read in in different ways–as data frames or “tibbles”), but they can be downloaded here:

(Download these to your current working directory, which can be found using getwd().)

Also install the tidyverse package, which in turn installs a number of individual packages that are used in reshaping data.

# install the "tidyverse" suite of packages
install.packages("tidyverse")
# library
library(tidyverse)

Read in a typical “tidy” .csv file, that has variables in columns and observations in rows. This can be done in the usual way using the read.csv() function, which creates a standard data frame, or, if the reader packages has been loaded by library(tidyverse), using the read_csv() function, which creates a “tibble”. The data here are a “short” three-year-long subset of Eugene monthly climate data.

# read a .csv file using the readr package
csv_file <- "/Users/bartlein/Documents/geog495/data/csv/EugeneClim-short.csv"
eugclim

(In the above, you would substitute the path to your working directory.)

Produce a few plots to look at the time series of individual variables, and to look at the annual cycle of each. The first use of plot() below plots the time series of monthly average temperature (tavg), while the second illustrates what the annual cycle looks like.

# time-series plot
plot(eugclim$tavg ~ eugclim$yrmn, type="o", pch=16, xaxp=c(2013, 2016, 3))
# by month
plot(eugclim$mon, eugclim$tavg, pch=16, xaxp=c(1, 12, 1))

Repeat the plots for some other variables, in particular prcp (monthly total precipitation).

(See (see https://pjbartlein.github.io/GeogDataAnalysis/lec08.html#variables for a listing of variables)

Q1. Describe the annual cycles of the temperature and moisture-related variables. When during the year is it colder and when is it warmer, and when is it wetter and when is it drier?

3. Transforming (reshaping) an alternatively shaped table of data

An alternative layout for the data table (of just the precipitation-related variables) has the data arranged with variables in rows and months in columns. Read those data in:

# alternative layout of precipitation data
csv_file <- "/Users/bartlein/Documents/geog495/data/csv/EugeneClim-short-alt-pvars.csv"
eugclim_alt

Q2 Describe the different form of the two tables (eugclim and eugclim_alt). Can you think of a way to produce a time-series plot of precipitation using the data in eugclim_alt? (If so, show the code for doing that, and if not, why not?)

Now use the gather() and spread() functions from the tidyr package to reshape the data. This is done here in two steps:

# reshape by gathering and spreading
# 1) gather
eugclim_alt2 <- gather(eugclim_alt, 1:12, key="month", value="cases")
eugclim_alt2$month <- as.integer(eugclim_alt2$month)
eugclim_alt2
# 2) spread
eugclim_alt3

Plot the reshaped data (eugclim_alt3) to verify that they indeed have been reshaped correctly.

# plot the reshaped data
eugclim_alt3$yrmn <- eugclim_alt3$year + (as.integer(eugclim_alt3$month)-1)/12 plot(eugclim_alt3$prcp ~ eugclim_alt3$yrmn, type="o", pch=16, col="blue", xaxp=c(2013, 2016, 3)) Q3: Compare eugclim_alt2 and eugclim_alt3. What did the application of gather() do in creating eugclim_alt2, and what did the application of spread() do in creating eugclim_alt3? Q4: What is the benefit of reshaping the data in R as opposed to simply doing that in Excel or a text editor? 4. A little matrix algebra Create three matrices, A, B, and C: # create three matrices # default fill method: byrow = FALSE A <- matrix(c(6, 9, 12, 13, 21, 5), nrow=3, ncol=2) A # same elements, but byrow = TRUE B <- matrix(c(6, 9, 12, 13, 21, 5), nrow=3, ncol=2, byrow=TRUE) B # a third matrix C <- matrix(c(1,2,3,4,5,6,7,8,9), nrow=3, ncol=3) C Q5 Describe the shapes of the three matrices. (Note that the dim() function applied to a matrix (e.g. dim(A)) displays the number of rows and the number of columns in the matrix.) Matrix addition Add A and B: # matrix addition F <- A + B F Now try adding A and C: G <- A + C Q6: What happend? Can A and C be added? Why not? (Again, the dim() function might be useful.) Matrix multiplication Matrix multiplication (as distinct from element-by-element multiplication) produces a new matrix whose elements are sums of squares and cross products of the elements of matrices being multiplied (see matrix.pdf). Matrix multiplication uses %*% as the operator. “Postmultiply” the matrix C by A: # matrix multiplication Q <- C %*% A Q … and try to postmultiply A by B (e.g. T <- A %*% B) Q7: What happens here? What are the dimensions of C? What does the message non-conformable arguments imply about the shapes of A and B? Matrix inversion To illustrate matrix inversion (i.e. the matrix algebra version of scalar division), a realistic matrix can be used, in this case the correlation matrix of the temperature variables in the orstationc data set: # a realistic matrix, orstationc temperature-variable correlation matrix R <- cor(cbind(orstationc$tjan, orstationc$tjul, orstationc$tann))
R

Get the inverse of R:

# matrix inversion
Rinv <- solve(R)
Rinv

One property of the inverse matrix is that when pre- or postmultiplied by the original matrix, the identity matrix, I should be produced.

Q8: Check to see if Rinv is indeed the inverse of R. (Show the results of the check.)