set.seed(1)n_points <-50# Number of points to use in both partssigma <-2# Standard deviation for rnormaver <-0# Generate the tibble using n_points and sigmawd_gaussian <-tibble(index =1:n_points,y =rnorm(n_points, mean = aver, sd = sigma)) |>as_tsibble(index = index) |>mutate(density =dnorm(y, mean = aver, sd = sigma))# Plot the first tibblewd_gaussian |>ggplot(aes(x = index, y = y)) +geom_line() +theme_bw() +ggtitle("Generated White Noise Series")
# 2. Calculate and print the mean and variancesample_mean <-mean(wd_gaussian$y)sample_variance <-var(wd_gaussian$y)cat("Estimated Mean:", sample_mean, "\n")
Estimated Mean: 0.2008966
cat("Estimated Variance:", sample_variance, "\n")
Estimated Variance: 2.764863
# 3. Plot the Autocorrelation Function (ACF)acf_plot <- wd_gaussian |>ACF(y, type ="correlation") |>autoplot() +ggtitle("Autocorrelation Function of White Noise") +theme_bw()# Plot the histogram using the data from wd_gaussianhist_plot <- wd_gaussian |>ggplot(aes(x = y)) +geom_histogram(aes(y =after_stat(density)),color ="white", fill ="darkgrey", bins =10) +geom_line(aes(x = y, y = density)) +theme_bw() +ggtitle("Histogram with Theoretical Normal Density Curve")acf_plot
hist_plot
resource coe 4.2.3 “Simulation in R” Modern look
# Set seed for reproducibility of random numbersset.seed(1)# Generate a tibble with 100 random values from N(0, 1)wd_gaussian <-tibble(x =1:100, # Sequence from 1 to 100y =rnorm(100) # Generate 100 random values with mean 0, sd 1)# Plot the generated time series data as a line plotwd_gaussian |>ggplot(aes(x = x, y = y)) +geom_line()
# Create another tibble with data for histogramxd <-tibble(x =seq(-3, 3, length =1000), # 1,000 evenly spaced numbers between -3 and 3norm =rnorm(1000), # Generate 1,000 random values from N(0, 1)density =dnorm(x) # Compute normal distribution density for x)# Plot histogram of random values with density curvexd |>ggplot(aes(x = norm)) +geom_histogram(aes(y =after_stat(density)), # Histogram with density scalingcolor ="white", fill ="darkgrey") +geom_line(aes(x = x, y = density)) +# Overlay normal density curvetheme_bw() # Apply clean theme to plot
homework 4 question 3.a
this is sample code given by ai that uses an rexp function which we dont use but it does appear once in the book for one of the practice problems. I decided not to use it and show more of the math being done on the code
#| include: falseset.seed(42)# Parameterslambda <-1n <-500# Simulate white noise using the exponential distributions_t <-rexp(n, rate = lambda)w_t <- s_t - lambda# Plot the simulationlibrary(ggplot2)df <-data.frame(Time =1:n, White_Noise = w_t)ggplot(df, aes(x = Time, y = White_Noise)) +geom_line(color ="blue") +theme_minimal() +labs(title ="Simulated White Noise Process Using Exponential Distribution",x ="Time",y =expression(w[t]) )
hw 3.b
this was the code that was giving me the high density error for exponential white noise, I swap the dexp with dnorm and it produce a plot similar to that of the class png sample. but I dont think it is right because the class pdf expo line expands out like a normal distribution and the one that produce a similar one has the expo line start near -0.8.
swap this one mutate(density = dexp(w_t + lambda, rate = lambda))
with this one mutate(density = dnorm(w_t, mean = aver, sd = sigma))
# this was the code that was giving me the high density error for exponential white noise# Set up the environmentset.seed(1)# Step 1: Parameters for Gaussian White Noise (from Question 1)n_points <-50# Number of points used previously for Gaussian white noisesigma <-2# Standard deviationaver <-0# Mean for Gaussian white noise# Generate Gaussian white noise datawd_gaussian <-tibble(index =1:n_points,y =rnorm(n_points, mean = aver, sd = sigma)) |>mutate(density =dnorm(y, mean = aver, sd = sigma))# Step 2: Parameters for Exponential White Noise (from Question 3a)set.seed(1)n_exp_points <-500# Number of points for Exponential white noiselambda <-1# The rate parameter# Generate Exponential white noise datas_t <-rexp(n_exp_points, rate = lambda)wd_exp <-tibble(index =1:n_exp_points,w_t = s_t - lambda) |>mutate(density =dexp(w_t + lambda, rate = lambda))# Step 3: Convert to regular tibbles and prepare for merginggaussian_df <- wd_gaussian |>mutate(type ="Gaussian White Noise", value = y) |>select(type, value, density)exp_df <- wd_exp |>mutate(type ="Exponential White Noise", value = w_t) |>select(type, value, density)# Step 4: Combine both datasets for plottingcombined_data <-bind_rows(gaussian_df, exp_df)# Step 5: Plot superimposed histogramsggplot(combined_data, aes(x = value, fill = type)) +# Plot histograms for both distributionsgeom_histogram(aes(y =after_stat(density)), alpha =0.5, position ="identity", bins =20, color ="black") +# Add density lines for each distributiongeom_line(data = gaussian_df, aes(x = value, y = density, color = type), size =1) +geom_line(data = exp_df, aes(x = value, y = density, color = type), size =1) +theme_minimal() +labs(title ="Comparison of Gaussian vs Exponential White Noise",x ="Value",y ="Density" ) +scale_fill_manual(values =c("darkgray", "lightblue")) +scale_color_manual(values =c("Gaussian White Noise"="blue", "Exponential White Noise"="red")) +theme(legend.title =element_blank(),legend.position ="top" )
hw 3.c
both of this code get the same done
# Step 1: Generate the Exponential DWN simulation (from part a)set.seed(1)n_exp_points <-500# Length of the serieslambda <-1# Rate parameter# Generate Exponential white noises_t <-rexp(n_exp_points, rate = lambda)w_t <- s_t - lambda # White noise series# Step 2: Create the Random Walk Seriesrandom_walk <-tibble(index =1:n_exp_points,x_t =cumsum(w_t) # Cumulative sum to create the random walk) |>as_tsibble(index = index)# Step 3: Plot the Random Walk Seriesrandom_walk |>ggplot(aes(x = index, y = x_t)) +geom_line(color ="darkgreen") +theme_minimal() +ggtitle("Random Walk Series Using Exponential DWN Simulation") +labs(x ="Time", y =expression(x[t]))
# Using the Exponential White Noise series from part a)# Create the random walk serieswd_exp <- wd_exp |>mutate(x_t =cumsum(w_t) # Cumulative sum to create the random walk series )# Plot the random walk serieswd_exp |>ggplot(aes(x = index, y = x_t)) +geom_line(color ="blue") +theme_minimal() +ggtitle("Random Walk Series from Exponential DWN") +labs(x ="Time", y =expression(x[t]))
