Homework 5#
Problem 1: Trend Analysis#
Download the cascades_swe.xlsx data file.
The first column is the water year, and data in the next three columns are values for total precipitation (mm), daily maximum temperature (°C), and daily minimum temperature (°C) averaged from October-March over the Pacific Northwest Cascades in Washington and Oregon. The last column is an estimate of April 1st snow water equivalent (in mm, the water content of the snowpack on this day) from model simulations, averaged over the same domain.
A. Calculate the long-term trend in April 1 SWE from 1916-2003 by fitting a linear model to the data. Estimate the uncertainty in the trend by evaluating a 95% confidence interval around the estimate of B1. That is, report the trend as: Trend = B1 ± t*sB1
B. Is the trend statistically significant with 95% confidence? Can we reject the null hypothesis that the trend is equal to zero?
C. Repeat this analysis (parts A and B) for just the more recent period, 1977-2003. Discuss any similarities or differences in the results of the two time periods.
Problem 2: Correlation, Autocorrelation, Multiple Linear Regression#
In this problem, you will explore the relationship between air temperature, precipitation, and snow water equivalent over time, using observations from a study site in the Washington Cascades. Download the cascades_swe.xlsx dataset for this problem.
A. Begin by making scatterplots of each of these variables vs. all the other variables. Describe any visual patterns you see between each pair of variables.
B. Calculate the correlation (R) between April 1 SWE and the three meteorological variables (precipitation, maximum temperature, and minimum temperature). Then also calculate R between each unique combination of the meteorological variables. Which meteorological variables seem the most correlated with each other? How might it cause problems if we use both of these in a multi-linear regression?
C. Calculate the autocorrelation in precipitation, maximum temperature, and minimum temperature over the timeseries. Can we consider each of these values to be an independent sample? Or do some of them depend on the prior year’s sample?
Note
In part C, we can test for autocorrelation at different lags, but not for any lag longer than a quarter of the length of the data series. Therefore, test for lags from 1 to N/4, where N is the length of the data series.
D. Fit a multiple linear regression model to the data, using only two of the three meteorological variables (precipitation and maximum temperature) to predict April 1 SWE. Report the trend in each meteorological variable. Estimate the overall trend in SWE, and the trend due to each meteorological variable alone. How much of the overall trend is due to the effect of a trend in the maximum temperature? (Note that if we were doing this for research, it would be good to also explore using only minimum temperature, or using the mean daily temperature calculated as (Tmax + Tmin)/2 – however, for simplicity, we will only look a maximum temperature in this problem.)
Note
In part D we want to quantify how much a change in each variable accounts for a change in SWE. We start with making a multiple linear regression model, such as one which looks like:
SWE = B0 + B1*(precip) + B2*(t_max)
We then have values for all regression parameters (each B value). Take the derivative of both sides with respect to time. Our regression parameter values are coefficients in this new equation.
dSWE/dt = B1d(precip)/dt + B2d(t_max)/dt
Then to find how much the trend in SWE is accounted for by the trend in precipitation we compute B1*d(precip)/dt, where d(precip)/dt in the slope of the trend in precipitation.
Problem 3: Peer Reviews (CEWA 565)#
Complete the peer review(s) that you were assigned on Canvas. Your review should be thoughtful, but it does not need to be long, nor does it need to focus on minor spelling or grammar mistakes. Write a few sentences describing the most important change that can be made to improve the draft (e.g. “reorganize the paragraphs to better explain X”, “add more supporting evidence for the topic statement”). Refer to the prior homework assignment to see guidelines for the draft project reports. (the rubric for the final report is available for your reference here)