Week 02 Practice Problems (Optional)

Published

Monday, August 31, 2026

Code 
using Printf

Attempt these before Wednesday. These problems are optional; keep your work for yourself. Open the worked answer after you have tried the problem.

The module 1 written test draws from this pool.

Persistence from a linear reservoir

A linear reservoir has release constant \(k = 0.3\). Its output under white-noise precipitation is an AR(1) process with coefficient \(a = 1/(k+1)\).

  1. Compute \(a\).
  2. Compute the autocorrelation at lags 1, 2, and 5 using \(\rho_h = a^{|h|}\).

\[ \begin{aligned} a &= \frac{1}{k+1} = \frac{1}{1.3} \\ \rho_1 &= a \\ \rho_2 &= a^2 \\ \rho_5 &= a^5 \end{aligned} \]

k = 0.3
a = 1 / (k + 1)
@printf("a = %.3g\n", a)
@printf("ρ₁ = %.3g, ρ₂ = %.3g, ρ₅ = %.3g", a, a^2, a^5)
a = 0.769
ρ₁ = 0.769, ρ₂ = 0.592, ρ₅ = 0.269

A single linear reservoir with one release constant turns white-noise precipitation into a persistent AR(1) output.

Estimation bias at short record length

The Kendall (1954) bias formula for the lag-1 autocorrelation estimated from a record of length \(n\) with unknown mean is

\[ E(\hat a) \approx a - \frac{1 + 3a}{n - 1} \]

where \(a\) is the true value and \(\hat a\) is the estimate (Mudelsee, 2010, pp. 56–57).

  1. At \(n = 50\) and \(a = 0.7\), what is the expected estimate, and what percentage of the true value does the bias represent?
  2. The bias-correction recipe inverts the formula: given an observed \(\hat a\), solve for the corrected \(\hat a'\). Show that \(\hat a' = [\hat a(n-1) + 1]/(n - 4)\), and verify that applying it to your answer from part a recovers \(a = 0.7\).

Part a.

\[ \begin{aligned} E(\hat a) &\approx 0.7 - \frac{1 + 3(0.7)}{50 - 1} \\ &= 0.7 - \frac{3.1}{49} \end{aligned} \]

a = 0.7
n = 50
bias = (1 + 3a) / (n - 1)
expected = a - bias
@printf("E(â) = %.3g, bias = %.3g, percentage = %.3g%%", expected, bias, 100 * bias / a)
E(â) = 0.637, bias = 0.0633, percentage = 9.04%

A 50-year record underestimates the persistence by about 9 percent.

Part b. In the bias formula, replace \(E(\hat a)\) with the observed \(\hat a\) and \(a\) with the unknown \(\hat a'\).

\[ \begin{aligned} \hat a &= \hat a' - \frac{1 + 3\hat a'}{n - 1} \\ \hat a(n-1) &= \hat a'(n-1) - 1 - 3\hat a' \\ \hat a(n-1) + 1 &= \hat a'(n - 4) \\ \hat a' &= \frac{\hat a(n-1) + 1}{n - 4} \end{aligned} \]

a_hat = expected
a_corrected = (a_hat * (n - 1) + 1) / (n - 4)
@printf("â = %.3g → â' = %.3g", a_hat, a_corrected)
â = 0.637 → â' = 0.7

The formula assumes a stationary AR(1) process with unknown mean and \(|a| < 1\). It breaks down for \(a\) above about 0.9, where higher-order terms matter.

Variance of a time average under persistence

A 50-year climate record has lag-1 autocorrelation \(a = 0.7\).

  1. Compute the variance inflation factor \(V = (1+a)/(1-a)\).
  2. Compute the effective number of independent years \(n' = n/V\).
  3. A colleague tests whether the 50-year mean differs significantly from the historical baseline, treating the years as independent, and gets a test statistic of 2.0. By what factor is the test statistic inflated, and does the result survive the correction?

Part a.

\[ V = \frac{1+a}{1-a} \]

a = 0.7
V = (1 + a) / (1 - a)
@printf("V = %.3g", V)
V = 5.67

Part b.

\[ n' = \frac{n}{V} \]

n = 50
n_eff = n / V
@printf("n' = %.3g effective independent years", n_eff)
n' = 8.82 effective independent years

Part c. The test uses \(\sigma_X^2/n\) for the variance of the mean, but the true variance is \(V\sigma_X^2/n\). The standard error is \(\sqrt{V}\) times larger than assumed, so the test statistic is \(\sqrt{V}\) times too large.

t_stat = 2.0
inflation = sqrt(V)
corrected = t_stat / inflation
@printf("inflation factor = %.3g, corrected t = %.3g", inflation, corrected)
inflation factor = 2.38, corrected t = 0.84

The corrected test statistic is well below any conventional threshold. The apparent trend comes from persistence alone.

The large-sample formula \(V = (1+a)/(1-a)\) assumes \(n\) is large enough that the finite-sample weights \((1 - h/n)\) are close to 1. At \(n = 50\) and \(a = 0.7\) this is a reasonable approximation.

Reproducing a published autoregressive series

Example 9.6 of Wilks (2011) generates AR(1) and AR(2) series from one printed innovation sequence, starting from \(x_0 = 0\). The first five innovations are \(\varepsilon_1 = 1.526\), \(\varepsilon_2 = 0.623\), \(\varepsilon_3 = -0.272\), \(\varepsilon_4 = 0.092\), \(\varepsilon_5 = 0.823\).

The textbook writes \(\phi_1\) for the AR coefficient we have been calling \(a\).

  1. For the AR(1) model with \(\phi_1 = 0.5\), compute \(x_1\) through \(x_5\) using \(x_t = \phi_1 x_{t-1} + \varepsilon_t\).
  2. The textbook also fits an AR(2) with \(\phi_1 = 0.9\) and \(\phi_2 = -0.6\), initializing \(x_0 = x_{-1} = 0\). Compute \(x_1\) for this model. The running text prints \(x_1 = 1.562\). Does your arithmetic agree?

Part a.

\[ \begin{aligned} x_1 &= 0.5(0) + 1.526 = 1.526 \\ x_2 &= 0.5(1.526) + 0.623 = 1.386 \\ x_3 &= 0.5(1.386) - 0.272 = 0.421 \\ x_4 &= 0.5(0.421) + 0.092 = 0.303 \\ x_5 &= 0.5(0.303) + 0.823 = 0.974 \end{aligned} \]

ϕ = 0.5
ε = [1.526, 0.623, -0.272, 0.092, 0.823]
x = zeros(5)
x[1] = ε[1]
for t in 2:5
    x[t] = ϕ * x[t-1] + ε[t]
end
for t in 1:5
    @printf("x_%d = %.3g\n", t, x[t])
end
x_1 = 1.53
x_2 = 1.39
x_3 = 0.421
x_4 = 0.302
x_5 = 0.974

These match Table 9.4 of the textbook to three figures.

Part b. The AR(2) recursion is \(x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + \varepsilon_t\), with \(x_0 = x_{-1} = 0\).

\[ x_1 = 0.9(0) + (-0.6)(0) + 1.526 = 1.526 \]

@printf("x₁ = 0.9(0) + (-0.6)(0) + 1.526 = %.3g", 0.9 * 0 + (-0.6) * 0 + 1.526)
x₁ = 0.9(0) + (-0.6)(0) + 1.526 = 1.53

The arithmetic gives 1.526, not 1.562. Table 9.4 also lists 1.526, and the next step in the textbook multiplies by 1.526, so the running text’s 1.562 is a digit transposition. The rest of the worked example is correct.

References

Mudelsee, M. (2010). Climate Time Series Analysis: Classical Statistical and Bootstrap Methods (1 ed. 2010.). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-90-481-9482-7
Wilks, D. S. (2011). Statistical methods in the atmospheric sciences (3rd ed.). Amsterdam, The Netherlands ; Elsevier.