Detection of embeddings in binary Markov chains

3.1 Using the total number of runs statistics

We introduce two hypotheses concerning the fraction δ ∈ [0, 1] of embeddings:

The hypothesis H₀ means that there are no embeddings and the stego-sequence Y^T₁ agrees with the cover sequence x^T₁. The composite alternative H₁ means there exist embeddings with some unknown fraction δ > 0. If the parameter of the cover sequence ε is known, then the null hypothesis, which will be denoted by H_{0, ε}, is simple; otherwise, H₀ is also a composite hypothesis. If the hypothesis H₀ holds, then the probability measure P will be denoted by P₀, otherwise, by P_δ. One similarly denotes the moments of random variables. The distributions P₀ and P_δ were found in [5].

Lemma 1. Under the hypothesisH_{0, ε}, the probability distribution of the stego-sequenceY^T₁ is as follows

where

is the minimal sufficient statistics withH_{0, ε}.

The statistics B_T is called the ‘runs test’ in [9] (it means the total number of runs). By virtue of (1), under the hypothesis H₀ the sequence of indicators I{Y_t ⊗ Y_t+1 = 1} consists of independent random variables with Bernoulli distribution 𝔅(2^-1(1 - ε)). Using the exact binomial probability distribution of the statistics B_T with the known value of ε, one may construct a randomized statistical test for the embedding existence with the given probability of the hrst kind error α, . However, for practical purposes, it is more convenient to use its asymptotic variant as T → ∞, which is given by the critical region

where t_α is the α-quantile of the standard normal distribution: Φ(t_α) = α.

Theorem 1. Let the model of embedding (4) hold. Then asT → ∞ the asymptotic size of test (7) for the hypothesesH_{0, ε}, H₁based on the total number of runs statisticsB_Tcoincides with a preassigned significance levelα ∈ (0, 1). The asymptotic expression for the power of this test in the case of the(1, 1)-model of embedding and of the family of simple contiguous alternativesH_{1, δ} : {δ = ρ/T^β}, β > 0, is as follows:

Proof. Under the hypothesis H₀, the De Moivre-Laplace limit theorem implies that

Hence, using (9) we have, as T → ∞,

In the case q = r = 1, under the alternative H₁, it follows from (1), (2), (4), (5) that the initial hrst moment of the random variable B_T is equal to

Using similar arguments, we calculate the initial second moment under the alternative H

For the variance, we have

By the construction (4) the random sequence {Y_t} satishes the strong mixing property [10, 11], and hence the central limit theorem for weakly dependent random variables holds,

In the case ε > 0, we take into account (10) and substitute δ = ρ/T^β as T → ∞ into the expression for the power. As a result, we have

Analyzing various values of β in this expression, we arrive at (8). The case ε < 0 is dealt with similarly. □

3.2 Using the short runs statistics

Let us construct the sequence of indicators of sign changes in the sequence Y₁, ..., Y_T ∈ V_T:

Next, we define the set of patterns in sequence (11):

here 𝔟_τ is the chain of τ successive 0’s bounded from the left and right by 1’s. Such patterns specify series of 0’s and 1’s of length τ + 1 in the stego-sequence {Y_t}. Further, we consider the disjoint random events ℭ_τ, τ ∈ ℕ ∪ {0}:

Lemma 2. Let the model of embedding (4) hold, q = r = 1. Then under the alternativeH₁ the probability distribution of the random events ℭ_τ is given by

where 𝔞_τ(δ, ε) → 0 as δ → 0, |ε| < 1.

Proof. Using the law of total probability for the model of embedding under consideration we hnd that

Theorem 2. Under the hypotheses of Lemma 2, the function 𝔞_τ(δ, ε) has the asymptotic expansion

Proof. We partition the set 𝔘_{τ+2, 1} = {u^τ+2₁ = (u₁, ..., u_τ+2) ∈ V_τ+2: w(u^τ+2₁) = 1}, |𝔘| = τ + 2, of binary vectors of length τ + 2, τ ≤ 3, with unit Hamming weight into three disjoint subsets:

Arguing as in the proof of Lemma 2, we have

Let us consider the case τ ≥ 2. The subset 𝔘^(j)_{τ+2, 1}, j ∈ {0, 1, 2}, contains sequences u^τ+2₁ ∈ 𝔘_{τ+2, 1} such that the events ℭ_τ ∩ {γ^t+τ+l_t = u^τ+2₁} are equiprobable under the alternative H_l:

Now (13) with τ ≥ 2 follows by substitution of (15) into (14). The case τ < 2 in (13) is considered similarly. □

Theorem 3. Under the hypotheses of Lemma 2, the function 𝔞_τ(δ, ε) has the second-order asymptotic expansion

where

The proof is similar to that of Theorem 2, the set of stego-keys 𝔘_{τ+2, 2} being split into classes of equiprobable events.

From Theorems 2, 3 it follows that under the alternative H_l (existence of embeddings) the probability distribution of the total number of runs of a given length differs from that distribution under the hypothesis H₀. In particular, for ε > 0 the probabilities of the events ℭ₀, ℭ₁, ℭ₂ increase as δ increases from 0 to 1, whereas for τ > τ_ε = 2 + ε(3 + ε)(1 - ε)^-1 the probability P_δ{ℭτ} decreases with the increasing of δ. This being so, we consider the statistics

where the statistics 𝓑_{T, 2} is the total number of series of 0’s and of 1’s of length 1 in the sequence {y_t}, and the statistics 𝓑_{T, 1} is related to the total number of runs statistics B_T by the relation 𝓑_{T, 1} = B_T - z_T-l - 1.

Using Theorem 1 from [5] one may show that under the alternative H_l the initial ftrst-order moments of the bivariate statistics (𝓑_{T, 1}, 𝓑_{T, 2}), as given by (17), read as

From (18) it is seen that for ε > 0 the mean number of sign changes or of two neighbouring sign changes is larger when the embeddings exist than in the opposite case.

Theorem 4. Let the model of embedding (4) holds. Then, asT → ∞, the statistical test for the hypothesesH_{0, ε}, H₁of the asymptotic significance level α ∈ (0, 1) based on the bivariate statistics (17) is given by the critical region

where the region 𝓓_{1, 2}is as follows

that is,

Proof. Using the fact that under the hypothesis H_{0, ε} the random variables {z_t} are independent and have the Bernoulli distribution 𝔅(2^-1(1 - ε)), and since the random variables z_t z_t+1 and z_sz_s+1 are independent if |t - s| > 1, we hnd that

Next, since the sequence of pairs (Z_t, Z_tZ_t+1) ∈ V₂ is 1-dependent, it follows that as T → ∞ the random vector 1T(BT, 1−12T(1−ε), BT, 2−14T(1−ε)2)′ has an asymptotic normal distribution 𝓝₂((0, 0))′, Σ₀), where

In Fig. 1 the region 𝓓_{1, 2} for the case ε > 0 is marked by the ‘+’ sign. Such a form of the domain follows from the asymptotical normality of the bivariate statistics (𝓑_{T, 1}, 𝓑_{T, 2}) and from expressions (18). To calculate the probability of the hrst kind error, we use the linear transform of the region 𝓓_{1, 2}. We apply the matrix Σ^{- 1/2}₀ to the unit vectors (1, 0), (0, 1) ∈ 𝓡² and construct the Gram matrix:

Figure 1

The region 𝓓_{1, 2} for ε > 0 and the scattering ellipses for 5 e [0, 1].

The angle φ between the vectors u_l and u₂ is expressed in terms of the coefficient of correlation:

Because of the joint asymptotic normality of statistics (17), the random variable

has an asymptotically exponential distribution with the parameter 1/2 as T → ∞. Hence, from the equation

we find c=26(1−ε2)2c1, 2 (the ellipse equation in Fig. 1: Q12 = c). The case ε < 0 is considered similarly with u1=Σ0−12(−1, 0)′, u2=Σ0−12(0, −1)′.

Lemma 3. Under the (1, 1)-model of embedding and the alternativeH_lthe random variablesz_t, z_sare independent if |t - s| ≥ 2, the random variablesz_t, z_sz_s+lare independent if |t - s| ≥ 2, and the random variables z_tz_{t + 1}, z_sz_{s + 1} are independent if |t - s| ≥ 3.

Proof. Let us consider the random variables z_t, z_t+k, k ≥ 2, and find the expectation E_δ{z_tz_t+k}, k ≥ 2:

Since the random variables z_t, z_t+k are binary and since cov, {z_t, z_t+k} = 0 for _k ≥ 2, then such variables are independent. A similar argument shows that the random variables z_t, z_sz_s+l are independent if |t - s| ≥ 2 and that the random variables z_tz_t+l, z_sz_s+l are independent if |t - s| ≥ 3. □

Now we will employ Lemma 3 to hnd asymptotic expressions for the hrst and second moments of the bivariate statistics (𝓑_{T, 1}, 𝓑_{T, 2}) under the alternative H_{1, δ}. The hrst-order moments were found in (18). In the course of the proof of Theorem 1 it was shown that

In view of Lemma 3 we have, as T → ∞,

Using the law of total probability, we hnd, for the model of embedding (1, 1),

We thus have

Using Lemma 3 as T → ∞ we hnd the variance DÒ{Bf2}: