Persistence Curves: A canonical framework for summarizing persistence diagrams

Abstract

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the Persistence Curves (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications—texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.

Appendices

Appendix A: Explicit calculations of bounds using Theorem 1

In this section, we will assume $C,D\in \mathcal {D}$ and utilize the definitions of the curves defined in Table 1. To ease notation we will write κ₁ in place of κ₁(ψ, C, D) and analogously for $\kappa _{\infty },\delta _{1},\delta _{\infty }$. We recall the following definitions.

$$ \begin{array}{@{}rcl@{}} \kappa_{1}(\psi, C, D) & =&\sum\limits_{(b,d)\in C\setminus{{\varDelta}}}\underset{ t\in [b, d]}{\max} |\psi(b, d, t)| + \sum\limits_{(b^{\prime}d^{\prime})\in D\setminus{{\varDelta}}}\underset{t\in [b^{\prime}, d^{\prime}]}{\max} |\psi(b^{\prime},d^{\prime}, t)| \end{array} $$

(A.1)

$$ \begin{array}{@{}rcl@{}} \kappa_{\infty}(\psi, C, D)\! &=&\underset{(b,d)\in C\setminus{{\varDelta}}}{\max} \underset{ t\in [b, d]}{\max} |\psi(b, d, t)| + \underset{(b^{\prime},d^{\prime})\in D\setminus{{\varDelta}}}{\max} \underset{t\in [b^{\prime}, d^{\prime}]}{\max} |\psi(b^{\prime}, d^{\prime}, t)|. \end{array} $$

(A.2)

$$ \begin{array}{@{}rcl@{}} \delta_{1}(\psi, C, D) &=&\underset{\eta:C\to D}{\inf} \sum\limits_{i=1}^{n_{\eta}}\underset{{ t\in [b_{i}, d_{i}]\cap [\eta_{b_{i}}, \eta_{d_{i}}]}} {\max} |\psi(b_{i}, d_{i}, t) - \psi(\eta_{b_{i}}, \eta_{d_{i}}, t)|. \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} \delta_{\infty}(\psi, C, D) &=&\underset{\eta:C\to D}{\inf} \underset{\substack{1\leq i \leq n_{\eta} \\ t\in [b_{i}, d_{i}]\cap [\eta_{b_{i}}, \eta_{d_{i}}]}}{\max} |\psi(b_{i}, d_{i}, t) - \psi(\eta_{b_{i}}, \eta_{d_{i}}, t)|. \end{array} $$

(A.4)

1.1 A.1 Betti-based curves

In this section we consider the curves based on the Betti number. As we will see the ψ function for each of the curves in this section is not continuous at the diagonal, which causes the δ bounds to be large.

1.1.1 A.1.1 Betti number curve (β)

Recall that the Betti curve is by taking ψ(D; b, d, t) := ψ^D(b, d) = 1 when (b, d) ∈ D, b ≠ d and 0 otherwise. The values for κ₁ and $\kappa _{\infty }$ are straightforward to calculate.

$$ \begin{array}{@{}rcl@{}} \kappa_{1} &=& \sum\limits_{i=1}^{n^{c}}1 + \sum\limits_{i=1}^{n^{D}} 1 =n^{C} + n^{D},\\ \kappa_{\infty} &=& 2. \end{array} $$

On the other hand, notice that if t ∈ [b, d] ∩ [η_b, η_d], then we have ψ^C(b, d) − ψ^D(η_b, η_d) = 1 when (b, d) ∈Δ⇒ (η_b, η_d)∉Δ and 0 otherwise. This means that the worst case scenario for the Betti curve is when all points are paired with the diagonal yielding the following δ values

$$ \begin{array}{@{}rcl@{}} \delta_{\infty} &\le& 1. \\ \delta_{1} &\le& \left( n^{C}+ n^{D}\right). \end{array} $$

Hence, we may conclude by Theorem 1 that

$$ \begin{array}{@{}rcl@{}} \|\beta(C)-\beta(D)\|_{1}&\le& (n^{C}+ n^{D})W_{\infty}(C,D) + L^{C}\land L^{D}, \end{array} $$

(A.5)

$$ \begin{array}{@{}rcl@{}} \|\beta(C)-\beta(D)\|_{1}&\le& 2W_{1}(C,D) + \left( L_{\infty}^{C}\land L_{\infty}^{D}\right)(n^{C}+ n^{D}). \end{array} $$

(A.6)

Neither of these bounds are desirable. Not only as the number of points increase both bounds tend to infinity, but also both bounds contain constants that are irrelevant to W₁ nor $W_{\infty }$. Moreover, the same occurs as the minimum lifespan grows.

1.1.2 A.1.2 Normalized Betti curve (s β)

According to Definition 2, we may define the normalized Betti curve by taking $\psi (D;b,d,t) :=\psi ^{D}(b,d)= \frac {1}{n^{D}}$ when (b, d) ∈ D, b ≠ d and 0 otherwise. Again we can get the κ values quickly

$$ \begin{array}{@{}rcl@{}} \kappa_{1} &=&\sum\limits_{i=1}^{n^{C}} \frac{1}{n^{C}} + \sum\limits_{i=1}^{n^{D}}\frac{1}{n^{D}}=2,\\ \kappa_{\infty} &=& \frac{1}{n^{C}}+\frac{1}{n^{D}}\le\frac{2}{n^{C}\land n^{D}}, \end{array} $$

On the other hand, for all η : C → D

$$ \begin{array}{@{}rcl@{}} \delta_{\infty} &\le& \underset{1\le i\le n_{\eta}}{\max} |\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})|\le \frac{1}{n^{C}\land n^{D}}.\\ \delta_{1} &\le& \sum\limits_{i=1}^{n_{\eta}}|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})| \le \sum\limits_{i=1}^{n^{C}} \frac{1}{n^{C}} + \sum\limits_{i=1}^{n^{D}}\frac{1}{n^{D}}=2. \end{array} $$

Hence, we may conclude by Theorem 1 that

$$ \begin{array}{@{}rcl@{}} \|\mathbf{s}\boldsymbol\beta(C)-\mathbf{s}\boldsymbol\beta(D)\|_{1}&\le &2W_{\infty}(C,D) + \frac{L^{C}\land L^{D}}{n^{C}\land n^{D}}, \end{array} $$

(A.7)

$$ \begin{array}{@{}rcl@{}} \|\mathbf{s}\boldsymbol{\beta}(C)-\mathbf{s}\boldsymbol{\beta}(D)\|_{1}&\le& \frac{2W_{1}(C,D)}{n^{C}\land n^{D}} + 2\left( L_{\infty}^{C}\land L_{\infty}^{D}\right). \end{array} $$

(A.8)

Even with the normalization, these two bounds still depend heavily on the number of diagram points and the maximum lifespan.

1.1.3 A.1.3 Betti entropy curve (β e)

By Definition 3, the Betti Entropy Curve is defined by taking $\psi (D;b,d,t)=\psi ^{D}(b,d) = -\frac {1}{n^{D}}\log \frac {1}{n^{D}}$ when (b, d) ∈ D, b ≠ d and 0 otherwise. If the points of C and D are indexed according to the optimal matching for $W_{\infty }(C,D)$ distance and if n^C ∧ n^D ≥ 3 (so that $\frac {1}{n^{C}\land n^{D}}\le \frac {1}{e}$), then by Lemma 3

$$ \begin{array}{@{}rcl@{}} \kappa_{1} &=& \sum\limits_{i=1}^{n} \psi^{C}(b_{i},d_{i}) + \sum\limits_{i=1}^{n} \psi^{D}(b_{i},d_{i}) =-\log \frac{1}{n^{C}}-\log \frac{1}{n^{D}} \le 2\log\left( n^{C}\lor n^{D}\right),\\ \kappa_{\infty} &=& \underset{1\le i \le n}{\max} \psi^{C}(b_{i},d_{i}) + \underset{1\le i \le n}{\max} \psi^{D}(b_{i},d_{i}) \le\frac{2}{e}, \end{array} $$

On the other hand, if the points of C and D are indexed according to the optimal matching for W₁(C, D), then by Lemma 3

$$ \begin{array}{@{}rcl@{}} \delta_{\infty} &\le& \underset{1\le i\le 1}{\max} |\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})|\le -\frac{1}{n^{C}\land n^{D}}\log \frac{1}{n^{C}\land n^{D}}.\\ \delta_{1} &\le& \sum\limits_{i=1}^{n}|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})|\le -\frac{n^{C}\lor n^{D}}{n^{C}\land n^{D}}\log \frac{1}{n^{C}\land n^{D}}. \end{array} $$

Hence, we may conclude by Theorem 1 that if n^C ∧ n^D ≥ 3, then

$$ \begin{array}{@{}rcl@{}} \|\boldsymbol\beta\mathbf{e}^{C}-\boldsymbol\beta\mathbf{e}^{D}\|_{1}&\le& 2\log\left( n^{C}\lor n^{D}\right)W_{\infty}(C,D) - \left( L^{C}\land L^{D}\right)\left( \frac{1}{n^{C}\land n^{D}}\log \frac{1}{n^{C}\land n^{D}}\right), \end{array} $$

(A.9)

$$ \begin{array}{@{}rcl@{}} \|\boldsymbol{\beta}\mathbf{e}^{C}-\boldsymbol{\beta}\mathbf{e}^{D}\|_{1}&\le& \frac{2W_{1}(C,D)}{e} + \left( L_{\infty}^{C}\land L_{\infty}^{D}\right)\frac{n^{C}\lor n^{D}}{n^{C}\land n^{D}}\log \frac{1}{n^{C}\land n^{D}}. \end{array} $$

(A.10)

1.2 A.2 Midlife-based curves

For this section, we will assume that all birth and death values are non-negative (so that b + d ≠ 0). The midlife-based curves, similar to the lifespan based curves, benefit from continuous ψ functions. Moreover, the bounds in this section take a form similar to those of the lifespan-based counterparts. For this section, we will use the notation ${u_{i}^{D}}:=\frac {\eta _{b_{i}}+\eta _{d_{i}}}{2}$, $U^{D}:={\sum }_{i=1}^{n^{D}}{u_{i}^{D}}$, and $U_{\infty }^{D} := \max \limits _{i}{u_{i}^{D}}$. Finally, because we are assuming non-negativity, we also have 2U^D ≥ L^D.

1.2.1 A.2.1 Midlife curve m l(D)

The midlife curve take $\psi (D;,b,d,t) := \psi ^{D}(b,d) = \frac {b+d}{2}$. The values κ₁ = U^C + U^D ≤ 2(U^C ∨ U^D) and $\kappa _{\infty } = U_{\infty }^{C} + U_{\infty }^{D}\le U_{\infty }^{C} \lor U_{\infty }^{D}$. We see then that for any η : C → D

$$ \begin{array}{@{}rcl@{}} \delta_{\infty} &\le& \underset{1\le i\le n}{\max} \left|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})\right|\\ &=&\underset{1\le i\le n}{\max} \left|\frac{(d_{i}+b_{i})}{2}-\frac{(\eta_{d_{i}}+\eta_{b_{i}})}{2}\right|\\ &\le&\frac{1}{2}\underset{1\le i\le n}{\max} |(d_{i}-\eta_{d_{i}})|+|(b_{i}-\eta_{b_{i}})|\\ &\le& W_{\infty}(C,D). \end{array} $$

If instead the indexing follows the optimal matching for W₁(C, D) we see

$$ \begin{array}{@{}rcl@{}} \delta_{1} &\le& \sum\limits_{i=1}^{n}\left|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})\right|\\ &=&\sum\limits_{i=1}^{n}\left|\frac{(d_{i}+b_{i})}{2}-\frac{(\eta_{d_{i}}+\eta_{b_{i}})}{2}\right|\\ &\le&\frac{1}{2}\sum\limits_{i=1}^{n}|(d_{i}-\eta_{d_{i}})|+|(b_{i}-\eta_{b_{i}})|\\ &\le& W_{1}(C,D). \end{array} $$

Therefore, by Theorem 1, we conclude

$$ \begin{array}{@{}rcl@{}} \|\mathbf{ml}(C)-\mathbf{ml}(D)\|_{1}&\le& 2\left( U^{C}\lor U^{D}\right)W_{\infty}(C,D) + \left( L^{C}\land L^{D}\right)W_{\infty}(C,D), \end{array} $$

(A.11)

$$ \begin{array}{@{}rcl@{}} \|\mathbf{ml}(C)-\mathbf{ml}(D)\|_{1}&\le& 2\left( U_{\infty}^{C}\lor U_{\infty}^{D}\right)W_{1}(C,D) + \left( L_{\infty}^{C}\land L_{\infty}^{D}\right)W_{1}(C,D). \end{array} $$

(A.12)

1.2.2 A.2.2 Normalized midlife curve s m l(D)

The normalized midlife curve take $\psi (D;,b,d,t) := \psi ^{D}(b,d) = \frac {b+d}{{2U^{D}}}$. The values κ₁ ≤ 2 and $\kappa _{\infty } \le 2$. Moreover, suppose L^C ≤ L^D. We note that $|U^{C}-U^{D}|\le {\sum }_{i=1}^{n}|{u_{i}^{C}}-{u_{i}^{D}}|\le nW_{\infty }(C,D)$.

$$ \begin{array}{@{}rcl@{}} \delta_{\infty} &\le& \underset{1\le i\le n}{\max} \left|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})\right|\\ &\le&\underset{1\le i\le n}{\max} \frac{|{u_{i}^{C}}-{u_{i}^{D}}|}{U^{D}} +\underset{1\le i\le n}{\max} {u_{i}^{C}}\frac{|U^{C}-U^{D}|}{U^{C}U^{D}} \\ &\le&\underset{1\le i\le n}{\max} \frac{|b_{i}+d_{i}-\eta_{b_{i}}-\eta_{d_{i}}|}{L^{D}} +\underset{1\le i\le n}{\max} \frac{4(U^{C}_{\infty}\lor U^{D}_{\infty})|U^{C}-U^{D}|}{L^{C}L^{D}}\\ &\le& \frac{2W_{\infty}(C,D)}{L^{C}\lor L^{D}} + \frac{4n(U^{C}_{\infty}\lor U^{D}_{\infty})W_{\infty}(C,D)}{\left( L^{C}\land L^{D}\right)(L^{C}\lor L^{D})}. \end{array} $$

Moreover,

$$ \begin{array}{@{}rcl@{}} \delta_{1} &\le& \sum\limits_{i=1}^{n}|\psi^{C}(b_{i},d_{i})-\psi^{D}(\eta_{b_{i}},\eta_{d_{i}})|\\ &\le&\sum\limits_{i=1}^{n}\frac{|{u_{i}^{C}}-{u_{i}^{D}}|}{U^{D}} +\sum\limits_{i=1}^{n}{u_{i}^{C}}\frac{|U^{C}-U^{D}|}{U^{C}U^{D}} \\ &\le&\sum\limits_{i=1}^{n}\frac{|b_{i}+d_{i}-\eta_{b_{i}}-\eta_{d_{i}}|}{L^{D}} +\sum\limits_{i=1}^{n}\frac{2|U^{C}-U^{D}|}{L^{D}}\\ &\le& \frac{4W_{1}(C,D)}{L^{C}\lor L^{D}}. \end{array} $$

Therefore, by Theorem 1, we conclude

$$ \begin{array}{@{}rcl@{}} \|\mathbf{sml}(C)-\mathbf{sml}(D)\|_{1}&\le& 2W_{\infty}(C,D) + \left( L^{C}\land L^{D}\right)\left( \frac{2W_{\infty}(C,D)}{L^{C}\lor L^{D}} + \frac{4n(U^{C}_{\infty}\lor U^{D}_{\infty}) W_{\infty}(C,D)}{\left( L^{C}\land L^{D}\right)(L^{C}\lor L^{D})}\right), \\ &\le&4W_{\infty}(C,D)\left( 1+\frac{U_{\infty}^{C}\lor U_{\infty}^{D}}{L^{C}\lor L^{D}}n\right) \end{array} $$

(A.13)

$$ \begin{array}{@{}rcl@{}} \|\mathbf{sml}(C)-\mathbf{sml}(D)\|_{1}&\le& 2W_{1}(C,D) + L_{\infty}^{C}\land L_{\infty}^{D}\frac{4W_{1}(C,D)}{L^{C}\lor L^{D}} \le 6W_{1}(C,D). \end{array} $$

(A.14)

1.2.3 A.2.3 Midlife entropy curve m l e(D)

In this final section we will discuss the bounds for the midlife entropy curve, which uses $\psi (D;b,d,t):=\psi ^{D}(b,d)=-\frac {b+d}{U^{D}}\log \frac {b+d}{U^{D}}$. Straightforward calculation reveals $\kappa _{1} = \log n^{C} + \log n^{D}$ and $\kappa _{\infty }\le \frac {2}{e}$. The bounds recovered for midlife entropy are the same as for life entropy. That is, if $r_{\infty }(C,D),r_{1}(C,D)\le \frac {1}{2e}$, Theorem 1 guarantees that

$$ \begin{array}{@{}rcl@{}} {}\|\mathbf{mle}(C)-\mathbf{mle}(D)\|_{1}&\le& 2\log\left( n^{C}\lor n^{D}\right)W_{\infty}(C,D) - \left( L^{C}\land L^{D}\right)2r_{\infty}(C,D)\log{2r_{\infty}(C,D)},{\kern12pt} \end{array} $$

(A.15)

$$ \begin{array}{@{}rcl@{}} \|\mathbf{mle}(C)-\mathbf{mle}(D)\|_{1}&\le& \frac{2}{e}W_{1}(C,D) - \left( L_{\infty}^{C}\land L_{\infty}^{D}\right)2\left( n^{C}\lor n^{D}\right)r_{1}(C,D)\log{2r_{1}(C,D)}. \end{array} $$

(A.16)

Appendix B: Proof of Lemma 4 in Section 5.4

For convenience, we will restate the lemma here:

Lemma 1

Let $f(t) = | \psi _{1}(t) \chi _{[b_{1},d_{1})}(t) -\psi _{2}(t) \chi _{[b_{2},d_{2})}(t) |$, where $\psi _{i}(t) = \min \limits \{ t-b_{i}, d_{i} - t \}$ for i = 1, 2. Then,

$$ \|f\|_{\infty} \leq |d_{2}-d_{1}| \lor |b_{2} - b_{1}|. $$

Proof

We need similar estimates to that in Lemma 1. Consider

$$f(t) = | \psi_{1}(t) \chi_{[b_{1},d_{1})}(t) -\psi_{2}(t) \chi_{[b_{2},d_{2})}(t) |,$$

where $\psi _{i}(t) = \min \limits \{ t-b_{i}, d_{i} - t \}$ for i = 1, 2. Recall that ψ_i(t) can also be expressed as

$$\psi_{i}(t) = \left\{\begin{array}{lll} 0 & \text{if } t\notin (b_{i},d_{i})\\ t-b_{i} & \text{if } t\in [b_{i},\frac{b_{i}+d_{i}}{2})\\ d_{i}-t & \text{if } t\in[\frac{b_{i}+d_{i}}{2}, d_{i}) \end{array}\right.. $$

Finally, define $m_{i} = \frac {b_{i}+d_{i}}{2}$.

Case 1: b₁ ≤ d₁ ≤ b₂ ≤ d₂ and [b₁, d₁) ∩ [b₂, d₂) = ∅. Then,

$$ \left\{\begin{array}{ll} \max_{t\in [b_{1},d_{1})} | \psi_{1}(t) | = (d_{1} - b_{1})/2 \leq (b_{2} - b_{1})/2 \\ \max_{t\in [b_{2},d_{2})} | \psi_{2}(t) | = (d_{2} - b_{2})/2 \leq (d_{2} - d_{1})/2 \end{array}\right.. $$

(A.17)

Thus, f(t) ≤ |b₂ − b₁|∨|d₂ − d₁|.

Case 2: b₁ ≤ b₂ ≤ d₁ ≤ d₂ and [b₁, d₁) ∩ [b₂, d₂) = [b₂, d₁). Then, we have three maximums to consider.

$$ \left\{\begin{array}{lll} \max_{t\in [b_{1},b_{2}]} | \psi_{1}(t) | \\ \max_{t\in [b_{2},d_{1})} | \psi_{1}(t) - \psi_{2}(t) | \\ \max_{t\in (d_{1},d_{2})} | \psi_{2}(t) | \end{array}\right.. $$

(A.18)

For the first, there are two subcases: b₁ ≤ m₁ ≤ b₂ and b₁ ≤ b₂ ≤ m₁. In either case we see

$$ \left\{\begin{array}{lll} \max_{t\in [b_{1},m_{1}]} t - b_{1} = m_{1} - b_{1} = \frac{(d_{1}-b_{1})}{2} \\ \max_{t\in [m_{1},b_{2})} d_{1} - t = d_{1} - m_{1} = \frac{(d_{1}-b_{1})}{2} \\ \max_{t\in [b_{1},b_{2})} t - b_{1} = b_{2} - b_{1} \end{array}\right. $$

(A.19)

Observe that for b₁ ≤ m₁ ≤ b₂, we have $\frac {(d_{1}-b_{1})}{2} - (b_{2} - b_{1}) = m_{1} - b_{2} \leq 0$. In each of the cases above, $\max \limits _{t\in [b_{1},b_{2}]} | \psi _{1}(t) |\le b_{2}-b_{1}$. Thus, $\max \limits _{t\in [b_{1},b_{2}]} | \psi _{1}(t) | \leq |b_{2} - b_{1}|$. Similarly, one may verify that when t ∈ (d₁, d₂), $\max \limits _{t\in (d_{1},d_{2})} | \psi _{1}(t) | \leq |d_{2} - d_{1}|$.

When t ∈ [b₂, d₁) = [b₁, d₁) ∩ [b₂, d₂), there are 5 subcases to consider.

• i) m₁ ≤ b₂ ≤ d₁ ≤ m₂;

• ii) b₂ ≤ m₁ ≤ d₁ ≤ m₂;

• iii)m₁ ≤ b₂ ≤ m₂ ≤ d₁;

• iv) b₂ ≤ m₁ ≤ m₂ ≤ d₁;

• v) b₂ ≤ m₂ ≤ m₁ ≤ d₁.

One may verify that each case is bounded by |d₂ − d₁|∨|b₂ − b₁|.

Case 3: b₁ ≤ b₂ ≤ d₂ ≤ d₁. We have three maximums to consider.

$$ \left\{\begin{array}{lll} \max_{t\in [b_{1},b_{2}]} | \psi_{1}(t) | \\ \max_{t\in [b_{2},d_{2})} | \psi_{1}(t) - \psi_{2}(t) |. \\ \max_{t\in (d_{2},d_{1})} | \psi_{1}(t) | \end{array}\right. $$

(A.20)

The first and third maximums follow the same arguments as in case 2. For the middle maximum, we see that we have four subcases

• i) m₁ ∈ (b₁, b₂];

• ii) m₁ ∈ (b₂, m₂];

• iii) m₁ ∈ (m₂, d₂);

• iv) m₁ ∈ (d₂, d₂).

For i), note that d₁ − b₂ < d₁ − m₁ = m₁ − b₁ thus, d₁ − b₂ < b₂ − b₁. We complete this case by noting that for all t ∈ [b₂, d₂), ψ₁(t) = d₁ − t ≥ ψ₂(t) ≥ 0. Moreover ψ₁ is decreasing over the interval [b₂, d₂). Therefore, the difference |ψ₁(t) − ψ₂(t)| is maximized at t = b₂ and the difference is precisely d₁ − b₂.

One may verify the remaining cases satisfy the conclusion. □