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$$Q(l_t) = \\sum_{k=t}^{t+N-1}(u_k\\bar{R_t}(l_t) - \\alpha_t\\hat{C_k})^2 + \\eta_t(r(l_t) - r(l_{t-1}))^2$$

$N$ = Window size
$u_k$ = is the buffer filling rate at some index between $t$ (the current chunk) -> $t + N-1$
$\\bar{R_t}(l_t)$ = is the average bandwidth of the current bitrate profile. We use average bandwdith of the curent window rather than the bitrate of the the chunk at $t$ becuase it'll consider what the near future of fragments will look like in size
$(u_k\\bar{R_t}(l_t))$ = is the required bandwidth
$\\hat{C_k}$ = is the estimated bandwidth
$\\alpha_t$ = is a weight used to inflate or deflate the estimated bandwidth (differential treatment)

$\\alpha_t$ > 1 to inflate bandwidth for complex scenes (1.1 - 1.5)
$\\alpha_t$ < 1 to deflate bandwidth for simple scenes (0.6 - 0.9)
Set $\\alpha_t$ to 1.1 for Q4 chunks and αt to .8 for non-Q4 chunks

The first term

$(u_k\\bar{R_t}(l_t) - \\alpha_t\\hat{C_k})^2$

This term is a penalty on how deviated the required bandwidth is to the estimated bandwidth. Minimizing this penalty term will aim to select the largest bitrate profile.

The second term

$\\eta_t(r(l_t) - r(l_{t-1}))^2$

is a penalty on bitrate profile change. $r(l_t)$ is the average bitrate of bitrate profile $l$. We don't use $R_t(l_t)$ because it represents the bitrate per chunk level. \\eta_t is the weight of the second term. The value can be either 1 or 0.The value is set to 0 when the chunk(segment) at bitrate profile at $t$ ($l_t$) and the previous chunk(segment) at $t-1$ are different in category of chunk size (Q1-Q3 vs Q4). For example, if a chunk at time $t$ is considered to be a Q4 chunk, the largest category of chunk sizes, and the previous chunk at time $t-1$ is considered to be a Q2 category chunk size, even though the bitrate may have changed from 720 at $t-1$ to 1080 at $t$, we don't penalize that because there was a change from a low quadrant chunk size (Q1-Q3) to the largest quadrant (Q4) so that whole term will be set to 0. However if the size difference went from Q2 at t-1 to Q3 at time t and the bitrate changed from 720 to 1080, then we do apply the penalty by setting the weight to 1.

Purpose-built for displays of data