Using R: The logistic regression model from Question $3$ can be extended to more variables, by defining

the probability

$p(x)=\frac{1}{1+e^{-({}_0+{}_1{x}_1+{}_2{x}_2)}}.$

$($a$)$ Write a function moment$\_$distance$($fi$)$ that receives the name of an aftershock file,

and returns a dataframe with three columns: the mainshock seismic log$-$moment $($log

of moment in all$\_$events.csv$),$ the distance between the mainshock and the possible

aftershock location computed $($assume that the mainshock is at the centre of the grid of

points in the aftershock file$),$ and column with the presence$/$abscence of an aftershock.

Use the column names moment, distance, aftershock, and note that the moment is the

same for all the rows, since we are looking only at one mainshock event. Display the first

few rows of the dataframe obtained by applying this function to $2001$BHUJIN$01$YAGI.

$($b$)$ Implement a function fit$2($X$1,$ X$2,$ Y$)$ that minimises the negative log$-$likelihood

function $f$ in Question $3$ and returns the values of ${}_0,{}_1,{}_2.$ Use optim $($in $R)$ or

scipy $.$optimize.minimize in Python, and do not use the derivative of $f.$ Obtain the

values of ${}_0,{}_1,{}_2$ for $2001$BHUJIN$01$YAGI using moment for $x_1,$ distance for $x_2$ and

aftershock for $y.$

$($c$)$ Implement a function fit$2\_$file$($fi$)$ that returns the values of ${}_0,{}_1,{}_2$ for the af$-$

tershock file fi using moment for $x_1,$ distance for $x_2$ and aftershock for $y.$ Plot

the values of ${}_1$ vs ${}_0$ and ${}_2$ vs ${}_0$ in two separate plots, one point for each event in

selectedEvents.csv$.$