2.1 Piecewise Exponential Distribution

The hazard function, cumulative hazard function, PDF, CDF, quantile function of an exponential distribution r.v. \(t\) are: \[\begin{align*} h(t)&=\lambda\\ H(t)&=\lambda t\\ f(t)&=\lambda e^{-\lambda t}\\ F(t)&=1-e^{-\lambda t}\\ Q(p)&=\frac{-\log(1-p)}{\lambda} \end{align*}\] The hazard function, cumulative hazard function, PDF, survival function, quantile function of a piecewise exponential distribution r.v. \(t\) with breakpoints \(d_i\) are: \[\begin{align*} h(t)&=\begin{cases} \lambda_1, & t<d_1\\ \lambda_2, & d_1\le t<d_2\\ \cdots\\ \lambda_{r+1}, & t\ge d_r \end{cases}\\ H(t)&=\begin{cases} \lambda_1 t, & t<d_1\\ (\lambda_1-\lambda_2)d_1+\lambda_2t , & d_1\le t<d_2\\ \cdots\\ \left[\sum_{i=1}^r(\lambda_i-\lambda_{i+1})d_i\right]+\lambda_{r+1}t, & t\ge d_r \end{cases}\\ f(t)=h(t)e^{-H(t)}&=\begin{cases} \lambda_1 e^{-\lambda_1 t}, & t<d_1\\ \lambda_2 e^{(\lambda_2-\lambda_1)d_1-\lambda_2t} , & d_1\le t<d_2\\ \cdots\\ \lambda_{r+1}e^{\left[\sum_{i=1}^r(\lambda_{i+1}-\lambda_{i})d_i\right]-\lambda_{r+1}t}, & t\ge d_r \end{cases}\\ S(t)=e^{-H(t)}&=\begin{cases} e^{-\lambda_1 t}, & t<d_1\\ e^{(\lambda_2-\lambda_1)d_1-\lambda_2t} , & d_1\le t<d_2\\ \cdots\\ e^{\left[\sum_{i=1}^r(\lambda_{i+1}-\lambda_{i})d_i\right]-\lambda_{r+1}t}, & t\ge d_r \end{cases}\\\\ Q(p)&=\begin{cases} \frac{-\log(1-p)}{\lambda_1} & p< 1-e^{-\lambda d_1}\\ \frac{(\lambda_2-\lambda_1)d_1-\log(1-p)}{\lambda_2} & 1-e^{-\lambda d_1}\le p< 1-e^{(\lambda_2- \lambda_1)d_1-\lambda_2 d_2}\\ \cdots\\ \frac{\left[\sum_{i=1}^r (\lambda_{i+1}-\lambda_i)d_i\right]-\log(1-p)}{\lambda_{r+1}} & p\ge 1-e^{\left[\sum_{i=1}^r(\lambda_{i+1}-\lambda_{i})d_i\right]-\lambda_{r+1}d_r}\\ \end{cases} \end{align*}\]

The PwePred package provides dpwexpm(), ppwexpm(), qpwexpm(), rpwexpm() functions to the piecewise exponential distribution:

# Left Figure ------------------------------------------------------------
# use rpwexpm function to generate piecewise exp samples with rate 2, 1, 3
r_sample <- rpwexpm(50000, rate=c(2, 1, 3), breakpoint=c(0.3, 0.8))
hist(r_sample, freq=F, breaks=200, main="Density of Piecewsie Exp Dist", xlab='t', xlim=c(0, 1.2))

# piecewise exp density with rate 2, 1, 3 
t <- seq(0, 1.5, 0.01)
f2 <- dpwexpm(t, rate=c(2, 1, 3), breakpoint=c(0.3, 0.8))
points(t, f2, col='red', pch=16)

# exp distribution can be a special case of piecewise exp distribution
f1 <- dpwexpm(t, rate=2) 
lines(t, f1, lwd=2)
legend('topright', c('exp dist with rate 2','piecewise exp dist with rate 2, 1, 3',
                     'histogram of piecewise exp dist with rate 2, 1, 3'), 
       col=c('black','red'), fill=c(NA, NA, 'grey'), border=c('white', 'white', 'black'), 
       lty=c(1, NA, NA), pch=c(NA, 16, NA), lwd=2)

# Right Figure ------------------------------------------------------------
# CDF of piecewise exp with rate 2, 1, 3
F2 <- ppwexpm(t, rate=c(2, 1, 3), breakpoint=c(0.3, 0.8), lower.tail=T)
plot(t, F2, type='l', col='red', lwd=2, main="CDF and Quantile Function of Piecewsie Exp Dist", 
     xlim=c(0, 1.5), ylim=c(0, 1.5))

# CDF of exp dist is compatible with our package
F1 <- ppwexpm(t, rate=2, lower.tail=T)
lines(t, F1, lwd=2)

# plot quantile functions of both distributions
lines(F1, qpwexpm(F1, rate=2, lower.tail=T), lty=2, lwd=2)
lines(F2, qpwexpm(F2, rate=c(2, 1, 3), breakpoint=c(0.3,0.8), lower.tail=T), col='red', 
      lty=2, lwd=2)

abline(0, 1, col='grey')
legend('topleft', c('CDF of piecewise exp with rate 2, 1, 3', 'quantile function of 
                    piecewise exp with rate 2, 1, 3', 
                    'CDF of exp with rate 2', 'quantile function of exp with rate 2'), 
       col=c('red', 'red', 'black', 'black'), lty=c(1, 2, 1, 2), lwd=2)

2.2 Conditional Piecewise Exponential Distribution

The conditional survival function, CDF, PFD and quantile function of an exponential distribution \(t\) given \(t>T\) is \[\begin{align*} S(t|t>T)&=\frac{S(t)}{S(T)}=e^{\lambda T-\lambda t}\\ F(t|t>T)&=1-\frac{S(t)}{S(T)}=1-e^{\lambda T-\lambda t}\\ f(t|t>T)&=\lambda e^{\lambda T-\lambda t}\\ Q(p|t>T)&=\frac{\lambda T-\log(1-p)}{\lambda} \end{align*}\]

The conditional survival function and CDF of a piecewise exponential distribution \(t\) given \(t>T\) is \[\begin{align*} S(t|t>T)&=\frac{S(t)}{S(T)}, \text{ then plug in $S(t)$, $S(T)$}\\ F(t|t>T)&=1-\frac{S(t)}{S(T)}, \text{ then plug in $S(t)$, $S(T)$} \end{align*}\] The conditional quantile function of a piecewise exponential distribution \(t\) given \(t>T\) is \[\begin{align*} Q(p|t>T)=\begin{cases} \frac{\lambda_1 T-\log(1-p)}{\lambda_{1}}, & \hspace{-60pt} p <F(d_{1}|t>T), T <d_{1}\\ \frac{\left[\sum_{i=1}^{k-1} (\lambda_{i+1}-\lambda_i)d_i\right]+\lambda_1 T-\log(1-p)}{\lambda_{k}}, & \\ &\hspace{-60pt} F(d_{k-1}|t>T)\le p <F(d_{k}|t>T), T <d_{1}\\ \frac{\left[\sum_{i=m}^{k-1} (\lambda_{i+1}-\lambda_i)d_i\right]+\lambda_m T-\log(1-p)}{\lambda_{k}}, & \\ &\hspace{-60pt} F(d_{k-1}|t>T)\le p <F(d_{k}|t>T), d_{m-1}\le T <d_{m} \end{cases} \end{align*}\] The PwePred package provides ppwexpm_conditional(), qpwexpm_conditional(), rpwexpm_conditional() functions for conditional piecewise exponential distribution:

# Left Figure ------------------------------------------------------------
# CDF and qunatile function of conditional piecewise exp with rate 2, 1, 3 given t > 0.1
t <- seq(0.1, 1.2, 0.01)
F2_con <- ppwexpm_conditional(t, qT=0.1, rate=c(2, 1, 3), breakpoint=c(0.3, 0.8))
plot(t, F2_con, type='l', col='red', lwd=2, main="CDF and Quantile Function of 
     Conditional \nPiecewsie Exp Dist", xlim=c(0, 1.2), ylim=c(0, 1.2))
lines(F2_con, qpwexpm_conditional(F2_con, qT=0.1, rate=c(2, 1, 3), breakpoint=c(0.3,0.8)), 
      lty=2, lwd=2, col='red')

# compare with CDF and quantile function of unconditional piecewise exp with rate 2, 1, 3
t <- seq(0, 1.2, 0.01)
F2 <- ppwexpm(t, rate=c(2, 1, 3), breakpoint=c(0.3,0.8))
lines(t, F2, lwd=2)
lines(F2, qpwexpm(F2, rate=c(2, 1, 3), breakpoint=c(0.3,0.8)), lty=2, lwd=2)
abline(v=0.1, col='grey')
abline(h=0.1, col='grey')
legend('topleft', c('CDF of piecewise exp dist given t > 0.1', 'quantile function of 
                    piecewise exp dist given t > 0.1', 'CDF of piecewise exp dist', 
                    'quantile function of piecewise exp dist'), col=c('red', 'red', 'black', 'black'),
       lty=c(1, 2, 1, 2), lwd=2)

# Right Figure ------------------------------------------------------------
# use rpwexpm_conditional function to generate piecewise exp samples with rate 2, 1, 3 given t > 0.1
r_sample_con <- rpwexpm_conditional(3000, qT=0.1, rate=c(2, 1, 3), breakpoint=c(0.3,0.8))
plot(ecdf(r_sample_con), col='red', lwd=2,  main="Empirical CDF of Conditional 
     Piecewsie Exp Dist", xlim=c(0, 1.2), ylim=c(0, 1))

# compare with its CDF
lines(seq(0.1, 1.2, 0.01), F2_con, lwd=2)
legend('topleft', c('empirial CDF of piecewise exp dist given t > 0.1', 'true CDF of 
                    piecewise exp dist given t > 0.1'), col=c('red', 'black'), lty=c(1,2), lwd=2)

3.1 Hazard Rate Estimation with Known Breakpoints

The log likelihood function can be constructed as following: \[\begin{align} \log(L)=&\sum_{j\in D_1}\left[\log(\lambda_1)-\lambda_1 t_j\right]+\sum_{j\in C_1}\left[-\lambda_1 t_j\right]\\ &\quad +\sum_{j\in D_2}\left[\log(\lambda_2)+(\lambda_2-\lambda_1)d_1-\lambda_2 t_j\right]+\sum_{j\in C_2}\left[(\lambda_2-\lambda_1)d_1-\lambda_2 t_j\right]\\ &\quad +\cdots \\ &\quad +\sum_{j\in D_{r+1}}\left\{\log(\lambda_{r+1}) +\sum_{i=1}^r \left[(\lambda_{i+1}-\lambda_i)d_i\right]-\lambda_{r+1} t_j\right\}+\sum_{j\in C_{r+1}}\left\{\sum_{i=1}^r \left[(\lambda_{i+1}-\lambda_i)d_i\right]-\lambda_{r+1} t_j\right\} \end{align}\] where \(D_r\) or \(C_r\) are the index set of event time or censoring time that fall into \(r\)th interval of the piecewise exponential distribution.

When breakpoints \(d_1\) to \(d_r\) are known, we can take a derivative of \(\log(L)\) wrt \(\lambda\): \[\begin{align} \frac{\partial \log(L)}{\partial \lambda_1}=&\sum_{j\in D_1}\left[1/\lambda_1- t_j\right]+\sum_{j\in C_1}\left[-t_j\right] +\sum_{j\in D_2}\left[-d_1\right]+\sum_{j\in C_2}\left[-d_1\right] +\cdots +\sum_{j\in D_{r+1}}\left[-d_1\right]+\sum_{j\in C_{r+1}}\left[-d_1\right]\\ =&\frac{n_{D_1}}{\lambda_1}-\sum_{j\in C_1}t_j- n_{2^+}d_1\\ \frac{\partial \log(L)}{\partial \lambda_2}=&\sum_{j\in D_2}\left[1/\lambda_2+d_1-t_j\right]+\sum_{j\in C_2}\left[d_1- t_j\right] +\cdots +\sum_{j\in D_{r+1}}\left[-d_2+d_1\right]+\sum_{j\in C_{r+1}}\left[-d_2+d_1\right]\\ =&\frac{n_{D_2}}{\lambda_2}-\sum_{j\in D_2, C_2}(t_j-d_1)- n_{3^+}(d_2-d_1)\\ &\cdots\\ \frac{\partial \log(L)}{\partial \lambda_r}=&\frac{n_{D_r}}{\lambda_r}-\sum_{j\in D_r, C_r}(t_j-d_{r-1})- n_{r+1^+}(d_r-d_{r-1})\\ \frac{\partial \log(L)}{\partial \lambda_{r+1}}=&\frac{n_{D_{r+1}}}{\lambda_{r+1}}-\sum_{j\in D_{r+1}, C_{r+1}}(t_j-d_{r}) \end{align}\] where \(n_{D_r}\) is the number of events that fall into the \(r\)th interval; \(n_{r^+}\) is the number of events and censoring that fall into the \(r\)th to the end intervals.

Let all derivatives equal to 0, we obtain the MLE estimator of hazard rates: \[\begin{align} {\hat \lambda}_1=&\frac{n_{D_1}}{\sum_{j\in C_1}t_j+ n_{2^+}d_1}\\ {\hat \lambda}_2=&\frac{n_{D_2}}{\sum_{j\in D_2, C_2}(t_j-d_1)+ n_{3^+}(d_2-d_1)}\\ &\cdots \\ {\hat \lambda}_r=&\frac{n_{D_r}}{\sum_{j\in D_r, C_r}(t_j-d_{r-1})+ n_{r+1^+}(d_r-d_{r-1})}\\ {\hat \lambda}_{r+1}=&\frac{n_{D_{r+1}}}{\sum_{j\in D_{r+1}, C_{r+1}}(t_j-d_r)} \end{align}\]

3.2 Estimation of Breakpoints

3.3 Summary

The procedure of parameter estimation is summarized in the diagram below: