Smart Inventory Management

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Model Evaluation

Assess ARIMA(1,0,1) forecast accuracy using MAE, RMSE, and MAPE metrics.

24.7402
MAE - Mean Absolute Error
MAE = (1/n) Σ|Yₜ − Ŷₜ|

On average, the ARIMA forecast deviates from actual demand by 24.7402 units. MAE treats all errors equally - no penalty for large errors.

29.9596
RMSE - Root Mean Squared Error
RMSE = √[(1/n) Σ(Yₜ − Ŷₜ)²]

RMSE = 29.9596 units. Penalises large errors more than MAE. Useful for detecting outlier forecasts. Always RMSE ≥ MAE.

4.8292%
MAPE - Mean Abs. Percentage Error
MAPE = (100/n) Σ|Yₜ−Ŷₜ|/|Yₜ|

Scale-free accuracy metric. MAPE = 4.8292%. Interpretation: Accurate Forecast (MAPE < 10%)

Actual vs Fitted - In-Sample ARIMA(1,0,1) Performance
Actual vs Fitted
Mathematical Derivations
MAE
$$\text{MAE} = \frac{1}{n}\sum_{t=1}^n |Y_t - \hat{Y}_t|$$
Unit: same as demand. Linear penalty - each error unit contributes equally. Robust to outliers.
RMSE
$$\text{RMSE} = \sqrt{\frac{1}{n}\sum_{t=1}^n (Y_t - \hat{Y}_t)^2}$$
Unit: same as demand. Quadratic penalty - larger errors are weighted more. Sensitive to outliers.
MAPE
$$\text{MAPE} = \frac{100}{n}\sum_{t=1}^n \left|\frac{Y_t - \hat{Y}_t}{Y_t}\right|$$
Unit: percentage (%). Scale-independent. Undefined when Yₜ = 0. Useful for cross-series comparison.
Accuracy Summary
MetricValueUnit
MAE24.7402Demand Units
RMSE29.9596Demand Units
MAPE4.8292%Percentage
Observations19Periods
Overall Assessment:
Accurate Forecast (MAPE < 10%)
MAPE guide: <10% = Highly Accurate | 10-20% = Good | 20-50% = Reasonable | >50% = Poor
Relationship Between Metrics
Always: MAE ≤ RMSE (equality only when all errors are equal)
When errors are normally distributed: $\text{RMSE} \approx \sigma \cdot \sqrt{2/\pi} \cdot \text{MAE}/\text{MAE}$
MAPE is undefined if any $Y_t = 0$; use RMSE in such cases.
Out-of-Sample Accuracy (Held-Out Test)

The metrics above are in-sample: they compare the fitted values against the very observations used to estimate the coefficients, so they flatter the model. Here the model is re-fitted on only the first 14 periods and then asked to forecast the 5 periods it has never seen. This is the honest measure of forecasting performance.

MAE (held-out)
17.68
Average absolute error on unseen periods.
RMSE (held-out)
20.04
Penalises large misses more heavily.
MAPE (held-out)
3.53%
Accurate Forecast (MAPE < 10%)
Period Actual Demand Forecast Error
2024-06 498.0 526.6 -28.6
2024-07 519.0 524.03 -5.03
2024-08 512.0 522.3 -10.3
2024-09 493.0 521.14 -28.14
2024-10 504.0 520.35 -16.35
Inventory Optimization