# F25 Q1b

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/11/tee-c11-s1

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c11-s1/

## Question

**Question (5 marks).** For automated leaf-disease image classification, discuss factors affecting loss selection. For classes Normal, Mild, Severe with target [0,1,0] and predicted probabilities [0.1,0.8,0.1], compute and interpret categorical cross-entropy.

## Answer — step by step

### Given

Classes = Normal,Mild,Severe; target = [0,1,0].

Predicted probabilities = [0.1,0.8,0.1]; use natural logarithms.

### To find

Calculate categorical cross-entropy and interpret the result.

### Formula — which one and why

**Categorical cross-entropy.** For mutually exclusive classes with a one-hot target, all zero-target terms disappear. The loss is the negative log of the probability assigned to the true class.

yⱼ = target entry; pⱼ = predicted probability; c = index of the true class. Use natural logarithms unless another base is specified.



$$
L=-\sum_j y_j\ln p_j
$$





$$
L=-\ln p_c
$$



**Yaad rakhne ka cue:** One-hot mein sirf correct class ki probability ka minus ln.

### Solution — step by step

Select a loss according to the task and output meaning: real-valued regression often uses MSE/MAE, binary classification uses binary cross-entropy, and one-of-K classification uses categorical cross-entropy. Consider class imbalance, noisy labels/outliers, whether outputs are calibrated probabilities, and differentiability for optimization.

**Step 1: expand the categorical cross-entropy sum.**



$$
L=-[0\ln(0.1)+1\ln(0.8)+0\ln(0.1)]
$$



**Step 2: keep the term for the true class.**



$$
L=-\ln(0.8)
$$



**Step 3: calculate its natural logarithm.**



$$
\ln(0.8)\approx-0.223143551
$$



**Step 4: negate the logarithm.**



$$
L\approx0.223143551
$$



**Final answer:** loss ≈ 0.223144 nats. The true class received probability 0.8, so the loss is relatively small; probability 1 would give loss zero.