Experimental Data Summary

Cheatsheet Content

Experimental Data Overview (DSP context) This table summarizes experimental measurements from DSP analysis of .wav files, including impact characteristics, duration, peak time, mean decibel level (Mel_Mean_dB), and impact frequency (Impact_Freq_Hz). It also includes paths to generated spectrogram images (STFT, Mel, Laplace). Data Table from DSP Analysis File_No File_Name Impact Duration Peak_Time Mel_Mean_dB Impact_Freq_Hz STFT_PNG Mel_PNG Laplace_PNG 1 1.wav 0.0653 0.564 0.3246 -41.06 2250.0 /content/drive/MyDrive/laplacev2t1_results/1_stft.png /content/drive/MyDrive/laplacev2t1_results/1_mel.png /content/drive/MyDrive/laplacev2t1_results/1_laplace.png 2 10.wav 0.096 0.588 0.3251 -42.59 2250.0 /content/drive/MyDrive/laplacev2t1_results/10_stft.png /content/drive/MyDrive/laplacev2t1_results/10_mel.png /content/drive/MyDrive/laplacev2t1_results/10_laplace.png 3 11.wav 0.048 0.5707 0.2764 -41.61 4125.0 /content/drive/MyDrive/laplacev2t1_results/11_stft.png /content/drive/MyDrive/laplacev2t1_results/11_mel.png /content/drive/MyDrive/laplacev2t1_results/11_laplace.png 4 12.wav 0.0107 0.816 0.4623 -42.87 1312.5 /content/drive/MyDrive/laplacev2t1_results/12_stft.png /content/drive/MyDrive/laplacev2t1_results/12_mel.png /content/drive/MyDrive/laplacev2t1_results/12_laplace.png 5 13.wav 0.0533 0.5067 0.3107 -41.17 4687.5 /content/drive/MyDrive/laplacev2t1_results/13_stft.png /content/drive/MyDrive/laplacev2t1_results/13_mel.png /content/drive/MyDrive/laplacev2t1_results/13_laplace.png 6 14.wav 0.028 0.6373 0.3123 -44.35 2062.5 /content/drive/MyDrive/laplacev2t1_results/14_stft.png /content/drive/MyDrive/laplacev2t1_results/14_mel.png /content/drive/MyDrive/laplacev2t1_results/14_laplace.png 7 15.wav 0.0587 0.468 0.2409 -41.31 4875.0 /content/drive/MyDrive/laplacev2t1_results/15_stft.png /content/drive/MyDrive/laplacev2t1_results/15_mel.png /content/drive/MyDrive/laplacev2t1_results/15_laplace.png 8 16.wav 0.0587 0.5053 0.2455 -42.51 2250.0 /content/drive/MyDrive/laplacev2t1_results/16_stft.png /content/drive/MyDrive/laplacev2t1_results/16_mel.png /content/drive/MyDrive/laplacev2t1_results/16_laplace.png 9 17.wav 0.084 0.528 0.291 -42.81 4125.0 /content/drive/MyDrive/laplacev2t1_results/17_stft.png /content/drive/MyDrive/laplacev2t1_results/17_mel.png /content/drive/MyDrive/laplacev2t1_results/17_laplace.png 10 18.wav 0.036 0.8747 0.6286 -44.02 750.0 /content/drive/MyDrive/laplacev2t1_results/18_stft.png /content/drive/MyDrive/laplacev2t1_results/18_mel.png /content/drive/MyDrive/laplacev2t1_results/18_laplace.png 11 19.wav 0.052 1.076 0.8351 -44.53 1312.5 /content/drive/MyDrive/laplacev2t1_results/19_stft.png /content/drive/MyDrive/laplacev2t1_results/19_mel.png /content/drive/MyDrive/laplacev2t1_results/19_laplace.png 12 2.wav 0.048 0.528 0.2311 -41.38 4125.0 /content/drive/MyDrive/laplacev2t1_results/2_stft.png /content/drive/MyDrive/laplacev2t1_results/2_mel.png /content/drive/MyDrive/laplacev2t1_results/2_laplace.png 13 20.wav 0.0693 0.8547 0.4324 -42.61 2250.0 /content/drive/MyDrive/laplacev2t1_results/20_stft.png /content/drive/MyDrive/laplacev2t1_results/20_mel.png /content/drive/MyDrive/laplacev2t1_results/20_laplace.png 14 21.wav 0.0213 0.532 0.3064 -42.92 4125.0 /content/drive/MyDrive/laplacev2t1_results/21_stft.png /content/drive/MyDrive/laplacev2t1_results/21_mel.png /content/drive/MyDrive/laplacev2t1_results/21_laplace.png 15 22.wav 0.04 0.5 0.2313 -42.28 4687.5 /content/drive/MyDrive/laplacev2t1_results/22_stft.png /content/drive/MyDrive/laplacev2t1_results/22_mel.png /content/drive/MyDrive/laplacev2t1_results/22_laplace.png 16 23.wav 0.04 0.6027 0.228 -39.96 4875.0 /content/drive/MyDrive/laplacev2t1_results/23_stft.png /content/drive/MyDrive/laplacev2t1_results/23_mel.png /content/drive/MyDrive/laplacev2t1_results/23_laplace.png 17 24.wav 0.0413 0.8733 0.3726 -44.6 4125.0 /content/drive/MyDrive/laplacev2t1_results/24_stft.png /content/drive/MyDrive/laplacev2t1_results/24_mel.png /content/drive/MyDrive/laplacev2t1_results/24_laplace.png 18 25.wav 0.06 0.5653 0.2878 -39.8 4875.0 /content/drive/MyDrive/laplacev2t1_results/25_stft.png /content/drive/MyDrive/laplacev2t1_results/25_mel.png /content/drive/MyDrive/laplacev2t1_results/25_laplace.png 19 26.wav 0.0827 0.6933 0.4312 -44.82 1687.5 /content/drive/MyDrive/laplacev2t1_results/26_stft.png /content/drive/MyDrive/laplacev2t1_results/26_mel.png /content/drive/MyDrive/laplacev2t1_results/26_laplace.png 20 27.wav 0.0707 0.5707 0.2592 -42.84 2250.0 /content/drive/MyDrive/laplacev2t1_results/27_stft.png /content/drive/MyDrive/laplacev2t1_results/27_mel.png /content/drive/MyDrive/laplacev2t1_results/27_laplace.png 21 28.wav 0.052 0.572 0.3081 -41.51 4875.0 /content/drive/MyDrive/laplacev2t1_results/28_stft.png /content/drive/MyDrive/laplacev2t1_results/28_mel.png /content/drive/MyDrive/laplacev2t1_results/28_laplace.png 22 29.wav 0.116 0.4227 0.3455 -42.85 2250.0 /content/drive/MyDrive/laplacev2t1_results/29_stft.png /content/drive/MyDrive/laplacev2t1_results/29_mel.png /content/drive/MyDrive/laplacev2t1_results/29_laplace.png 23 3.wav 0.06 0.584 0.3013 -46.95 4875.0 /content/drive/MyDrive/laplacev2t1_results/3_stft.png /content/drive/MyDrive/laplacev2t1_results/3_mel.png /content/drive/MyDrive/laplacev2t1_results/3_laplace.png 24 30.wav 0.0493 0.5573 0.3258 -40.04 4875.0 /content/drive/MyDrive/laplacev2t1_results/30_stft.png /content/drive/MyDrive/laplacev2t1_results/30_mel.png /content/drive/MyDrive/laplacev2t1_results/30_laplace.png 25 31.wav 0.0307 0.52 0.2761 -42.65 4312.5 /content/drive/MyDrive/laplacev2t1_results/31_stft.png /content/drive/MyDrive/laplacev2t1_results/31_mel.png /content/drive/MyDrive/laplacev2t1_results/31_laplace.png 26 32.wav 0.0653 0.6227 0.3188 -43.76 2250.0 /content/drive/MyDrive/laplacev2t1_results/32_stft.png /content/drive/MyDrive/laplacev2t1_results/32_mel.png /content/drive/MyDrive/laplacev2t1_results/32_laplace.png 27 33.wav 0.0507 0.5413 0.2705 -42.98 2250.0 /content/drive/MyDrive/laplacev2t1_results/33_stft.png /content/drive/MyDrive/laplacev2t1_results/33_mel.png /content/drive/MyDrive/laplacev2t1_results/33_laplace.png 28 34.wav 0.084 0.6173 0.353 -39.7 5062.5 /content/drive/MyDrive/laplacev2t1_results/34_stft.png /content/drive/MyDrive/laplacev2t1_results/34_mel.png /content/drive/MyDrive/laplacev2t1_results/34_laplace.png 29 35.wav 0.0733 0.5893 0.3374 -42.91 4875.0 /content/drive/MyDrive/laplacev2t1_results/35_stft.png /content/drive/MyDrive/laplacev2t1_results/35_mel.png /content/drive/MyDrive/laplacev2t1_results/35_laplace.png 30 36.wav 0.116 0.4987 0.3395 -40.13 4125.0 /content/drive/MyDrive/laplacev2t1_results/36_stft.png /content/drive/MyDrive/laplacev2t1_results/36_mel.png /content/drive/MyDrive/laplacev2t1_results/36_laplace.png 31 37.wav 0.0573 0.7653 0.5591 -41.78 4875.0 /content/drive/MyDrive/laplacev2t1_results/37_stft.png /content/drive/MyDrive/laplacev2t1_results/37_mel.png /content/drive/MyDrive/laplacev2t1_results/37_laplace.png 32 4.wav 0.0373 0.5933 0.3 -43.29 4125.0 /content/drive/MyDrive/laplacev2t1_results/4_stft.png /content/drive/MyDrive/laplacev2t1_results/4_mel.png /content/drive/MyDrive/laplacev2t1_results/4_laplace.png 33 5.wav 0.052 0.6573 0.4232 -40.32 3937.5 /content/drive/MyDrive/laplacev2t1_results/5_stft.png /content/drive/MyDrive/laplacev2t1_results/5_mel.png /content/drive/MyDrive/laplacev2t1_results/5_laplace.png 34 6.wav 0.04 0.6133 0.2519 -40.45 750.0 /content/drive/MyDrive/laplacev2t1_results/6_stft.png /content/drive/MyDrive/laplacev2t1_results/6_mel.png /content/drive/MyDrive/laplacev2t1_results/6_laplace.png 35 7.wav 0.0547 0.624 0.4509 -40.86 375.0 /content/drive/MyDrive/laplacev2t1_results/7_stft.png /content/drive/MyDrive/laplacev2t1_results/7_mel.png /content/drive/MyDrive/laplacev2t1_results/7_laplace.png 36 8.wav 0.06 0.816 0.5276 -43.46 2250.0 /content/drive/MyDrive/laplacev2t1_results/8_stft.png /content/drive/MyDrive/laplacev2t1_results/8_mel.png /content/drive/MyDrive/laplacev2t1_results/8_laplace.png 37 9.wav 0.0453 0.4947 0.2802 -42.4 4125.0 /content/drive/MyDrive/laplacev2t1_results/9_stft.png /content/drive/MyDrive/laplacev2t1_results/9_mel.png /content/drive/MyDrive/laplacev2t1_results/9_laplace.png Ordinal Logistic Regression Model in DSP Context In Digital Signal Processing (DSP), like analyzing .wav files, we often extract various features. If these features (e.g., 'Impact', 'Duration', 'Impact_Freq_Hz') are used to predict an outcome that falls into ordered categories (e.g., "low severity," "medium severity," "high severity" of an event detected in the audio), an Ordinal Logistic Regression model is highly suitable. The formula generates the relationship between these DSP-derived features and our ordered outcome: $$ \text{logit}(P(Y \le j)) = \alpha_j - (\beta_1 \cdot \text{Impact} + \beta_2 \cdot \text{Duration} + \beta_3 \cdot \text{Impact\_Freq\_Hz}) $$ $Y$: The ordinal outcome variable, which could be a subjective rating or a classified severity level based on the DSP analysis. $P(Y \le j)$: The cumulative probability of the outcome being less than or equal to category $j$. $\alpha_j$: The intercept for each category $j$. $\beta_1, \beta_2, \beta_3$: Regression coefficients for the DSP features: Impact, Duration, and Impact_Freq_Hz, respectively. Interpretation of Coefficients The coefficients ($\beta$) indicate how changes in the DSP features affect the log-odds of moving to a higher ordinal category, assuming other features remain constant. Positive $\beta$: An increase in that DSP feature makes a higher ordinal outcome more likely. Negative $\beta$: An increase in that DSP feature makes a higher ordinal outcome less likely. This model helps us understand how the specific characteristics extracted from audio signals (like impact magnitude, event duration, or how often impacts occur) influence a categorized outcome. Understanding the Ordinal Logistic Regression Formula - Step by Step (DSP Focus) Let's break down this formula used to link your DSP features to an ordered outcome: $$ \text{logit}(P(Y \le j)) = \alpha_j - (\beta_1 \cdot \text{Impact} + \beta_2 \cdot \text{Duration} + \beta_3 \cdot \text{Impact\_Freq\_Hz}) $$ Step 1: The Left Side - $\text{logit}(P(Y \le j))$ $Y$: This is your ordinal outcome variable . In your DSP context, this might be a classification like "Low," "Medium," or "High" severity of an event detected in a .wav file. These categories have a natural order. $j$: This represents a specific category threshold. For example, if $Y$ has categories 1, 2, 3, then $j$ could be 1 or 2. $P(Y \le j)$: This is the cumulative probability that the event's severity (or whatever your $Y$ is) is in category $j$ or any category *below* $j$. So, for $j=\text{Medium}$, it's the probability of being "Low" or "Medium." $\text{logit}(\text{probability})$: This is a mathematical transformation. It takes a probability (which is between 0 and 1) and converts it into a value that can be any real number (from $-\infty$ to $+\infty$). This "log-odds" transformation is crucial because it allows us to model the relationship linearly. In simple terms: The left side translates the chance of an audio event falling into a certain severity level (or less) into a scale that the regression model can work with linearly. Step 2: The Right Side - The Predictor Part (DSP Features) This part calculates a score based on the numerical features extracted from your .wav files via DSP. $\text{Impact}$: This is a numerical feature you've extracted from the audio, representing the magnitude or intensity of an impact event. $\text{Duration}$: This is another numerical feature from the audio, representing how long an event lasts. $\text{Impact\_Freq\_Hz}$: This is a numerical feature, representing the dominant frequency or frequency range of the impact sound. (Note: using 'Impact_Freq_Hz' as per your data, instead of generic 'Frequency'). $\beta_1, \beta_2, \beta_3$: These are the regression coefficients (or weights) for each DSP feature. $\beta_1 \cdot \text{Impact}$: This multiplies the 'Impact' value by its weight. A larger $\beta_1$ means 'Impact' has a stronger influence. $\beta_2 \cdot \text{Duration}$: Similarly, 'Duration' is weighted by $\beta_2$. $\beta_3 \cdot \text{Impact\_Freq\_Hz}$: 'Impact_Freq_Hz' is weighted by $\beta_3$. $(\beta_1 \cdot \text{Impact} + \beta_2 \cdot \text{Duration} + \beta_3 \cdot \text{Impact\_Freq\_Hz})$: These weighted features are summed up. This sum represents the combined "score" or influence of all your DSP features on the outcome. In simple terms: This is a weighted sum of your DSP measurements. Each measurement (Impact, Duration, Impact_Freq_Hz) contributes to a total score, with its contribution determined by its $\beta$ coefficient. Step 3: The Intercept - $\alpha_j$ $\alpha_j$ (alpha-jay): This is the intercept . In ordinal regression, there's a unique intercept for each threshold between your ordinal categories. It acts as a baseline log-odds for being in category $j$ or lower, assuming all your DSP features (Impact, Duration, Impact_Freq_Hz) are zero. In simple terms: It's an adjustment for each cutoff point between your ordered categories, giving a baseline probability without considering the specific DSP features. Step 4: Putting it all Together - The Minus Sign The minus sign before the sum of predictors: The formula is structured so that as the combined "score" from your DSP features ($\beta_1 \cdot \text{Impact} + \dots$) increases, the value on the right side of the equation *decreases* (because of the minus sign). This means the $\text{logit}(P(Y \le j))$ decreases. A lower $\text{logit}$ value corresponds to a *lower* probability of $P(Y \le j)$, which in turn means a *higher* probability of being in a category *above* $j$. In simple terms: This model helps us predict the ordered severity category of an audio event. If your DSP features (Impact, Duration, Impact_Freq_Hz) result in a higher combined score, the model predicts a higher severity category for that event. The model learns the specific $\alpha_j$ and $\beta$ values from your data to make these predictions. This model allows you to quantify how the characteristics of audio signals, processed through DSP, relate to and predict an ordered outcome.