Linear Regression for Quantum Technology Adoption Rates

A statistical approach to modeling the growth of quantum technologies from 2015–2024.

Linear Model

We model the adoption index \(Y\) (composite measure of companies, patents, and systems) as a linear function of year \(X\):

\[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma^2) \]

  • \(X_i\): year

  • \(Y_i\): adoption index

  • \(\beta_1\): annual adoption rate (units per year)

This gives a direct estimate of how fast quantum tech is being adopted. How fast the world can change via quantum technology.

Point Estimation

Least-squares estimators:

\[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

\[ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} \]

\(\hat{\beta}_1\) is the estimated annual growth rate in adoption of quantum technology.

Data Visualization - ggplot2

## `geom_smooth()` using formula = 'y ~ x'

Interactive Plotly Visualization

Hover to inspect yearly values, zoom, and toggle the fitted line on/off.

Residual Diagnostics - ggplot2

Residuals show no systematic pattern. Therefore, this supports the appropriateness of the linear model for the observed adoption growth phase.

R Code for the Plots and Model

library(ggplot2)
library(plotly)
set.seed(123)
years <- 2015:2024
adoption <- 12 + 28*(years-2015) + rnorm(10, sd=18)
df <- data.frame(year = years, adoption = adoption)
mod <- lm(adoption ~ year, data = df)

# ggplot2 bar chart + linear trend
g1 <- ggplot(df, aes(x = year, y = adoption)) +
  geom_col(fill = "#3498DB") +
  geom_smooth(method = "lm", color = "#E74C3C") +
  theme_bw()

# Interactive Plotly (lecture style)
xax <- list(title = "Year", titlefont = list(family = "Modern Computer Roman"))
yax <- list(title = "Adoption Index", titlefont = list(family = "Modern Computer Roman"))
fig <- plot_ly(df, x = ~year, y = ~adoption, type = "scatter", mode = "markers") %>%
  add_lines(x = ~year, y = fitted(mod)) %>%
  layout(xaxis = xax, yaxis = yax) %>%
  config(displayLogo = FALSE)

Interpretation of Adoption Rate

Estimated slope \(\hat{\beta}_1 \approx 28\) indicates an average increase of roughly 28 adoption index units per year. This reflects the accelerating commercialization of quantum technologies.

Linear regression provides a transparent starting point for tracking technology adoption trends.