---
title: "Reliability Growth Analysis"
output:
  rmarkdown::html_vignette:
    fig_width: 7
    fig_height: 5
    self_contained: false
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{rga}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

To run a Reliability Growth Analysis, start by loading `ReliaGrowR` and `ReliaPlotR`

```{r setup, warning=FALSE}
library(ReliaGrowR)
library(ReliaPlotR)
```

## Statistical Background

The **Crow-AMSAA** model (Non-Homogeneous Poisson Process, Power Law) describes the expected cumulative failures as

$$E[N(t)] = \lambda t^{\beta},$$

where $\lambda > 0$ is a scale parameter and $\beta > 0$ is the shape parameter. A value $\beta < 1$ indicates **reliability growth** (the failure rate $\lambda\beta t^{\beta-1}$ is decreasing over time); $\beta > 1$ indicates degradation; $\beta = 1$ gives a homogeneous Poisson process (constant rate). Parameters are estimated by fitting a log-log linear regression to cumulative failures versus cumulative test time [@Crow1974].

The **Duane model** predates the NHPP formulation and postulates that the cumulative mean time between failures follows a power law in cumulative test time: $\text{CMTBF}(T) = K T^\alpha$, where $\alpha$ is the growth rate. Empirically, managed reliability growth programs tend to achieve $\alpha \approx 0.3$–$0.5$ [@Duane1964].

**Piecewise NHPP.** When a design change or corrective action divides the test program into distinct phases, separate Power Law parameters are fit within each interval defined by the specified `breaks`. A vertical dotted line marks each change point.

## Crow-AMSAA Model

To run a Crow-AMSAA model, first set up some cumulative time and failure data:
```{r}
times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)
```

Then run the rga and plot the results:
```{r}
result <- rga(times, failures)
plotly_rga(result)
```

## Piecewise NHPP Model

To run a Piecewise NHPP, first set up some cumulative time/failure data and specify the breakpoint:
```{r}
times <- c(25, 55, 97, 146, 201, 268, 341, 423, 513, 609, 710, 820, 940, 1072, 1217)
failures <- c(1, 1, 2, 4, 4, 1, 1, 2, 1, 4, 1, 1, 3, 3, 4)
breaks <- 500
```

Then run the rga and plot the results:
```{r}
result <- rga(times, failures, model_type = "Piecewise NHPP", breaks = breaks)
plotly_rga(result, fitCol = "blue", confCol = "blue", breakCol = "red")
```

## Duane Model

To run a Duane Model, first set up some cumulative time and failure data:
```{r}
times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)
```

Then plot the results:
```{r}
fit <- duane(times, failures)
plotly_duane(fit, fitCol = "darkgreen", confCol = "darkgreen")
```

## Overlay Models

Multiple `rga` models can be overlaid on the same plot by passing a list of objects. Each model is rendered in a distinct color, and clicking a legend entry toggles all traces for that model.

```{r}
times1 <- c(100, 200, 300, 400, 500)
failures1 <- c(1, 2, 1, 3, 2)
result1 <- rga(times1, failures1)

times2 <- c(50, 150, 250, 350, 450)
failures2 <- c(2, 1, 3, 1, 2)
result2 <- rga(times2, failures2)

plotly_rga(list(result1, result2), cols = c("steelblue", "tomato"))
```

## References

## See Also

- [Life Data Analysis](weibull.html) — fit Weibull/lognormal models to failure-time data
- [Repairable Systems Analysis](repairable.html) — analyze recurrent events with MCF and NHPP models
- [Accelerated Life Testing](alt.html) — extrapolate life from accelerated stress conditions