Piecewise Linear Modeling of EMA‑CUSUM Changepoints in Building Floor Sensor Data

Analysis Process

Question

A deterministic time series has 60 points: for t in [0,59], y = 2t+5+sin(t*0.5)2 when t<20, y = -t+65+cos(t0.3)3 when 20<=t<40, y = 0.5t+10+sin(t0.7)*1.5 when t>=40. The sensor was installed on floor 23 of a 45-story building. Compute the EMA with span=5 (alpha=2/6). Apply CUSUM changepoint detection on the EMA (cumulative sum of deviations from the mean). The equipment serial number is AX-7842-B. Find changepoints where the second difference of CUSUM changes sign, filtering to minimum spacing of 10. If more than 2 significant changepoints are found, fit piecewise linear regression using the top 3 changepoints as segment boundaries; otherwise fit a single linear regression on the full EMA. The building was constructed in 2018 at a cost of $89 million. Compute RMSE between the EMA and the fitted predictions. Return RMSE + number_of_significant_changepoints, rounded to 4 decimal places.

Iteration 1

Complexity Analysis

Complexity

moderate

Key Challenges

Ensuring correct implementation of piecewise time series generation
Accurately computing EMA with given span
Detecting changepoints with correct spacing
Choosing appropriate regression model based on changepoint count

Auto-Scrolling

Problem Dimensions

1. Data Generation and Preparation
Description: Create the deterministic time series, compute EMA, and prepare data for changepoint analysis
Strategy: Generate data first, then compute EMA, then CUSUM
Components:

• Generate y(t) for t=0..59 using piecewise formulas• Compute EMA with span=5 (alpha=2/6)• Calculate mean of EMA for CUSUM• Compute cumulative sum of deviations from mean

2. Change Point Detection
Description: Identify significant changepoints in the CUSUM series
Strategy: Analyze second differences, then apply spacing filter
Components:

• Compute second differences of CUSUM• Detect sign changes• Filter by minimum spacing of 10• Count significant changepoints

3. Model Fitting
Description: Fit regression models based on number of changepoints
Strategy: Choose model type based on changepoint count, then fit accordingly
Components:

• If <=2 changepoints, fit single linear regression to EMA• If >2, select top 3 changepoints as boundaries and fit piecewise linear regression• Generate predictions for all t

4. Error Metric Calculation
Description: Compute RMSE between EMA and predictions and combine with changepoint count
Strategy: Compute RMSE first, then add count
Components:

• Calculate squared errors for each t• Compute mean squared error and take sqrt• Add number of significant changepoints

Strategy

Establish foundational data structures and perform initial computations to enable subsequent analysis steps

Candidate Plans (2 Generated)

Plan 1

Tasks

knowledge

Confirm EMA formula and alpha calculation for span=5 (alpha=2/(5+1)) and verify piecewise time series definitions for t<20, 20<=t<40, t>=40

python

Generate deterministic time series y(t) for t=0..59 using the given piecewise formulas, compute EMA with span=5, calculate mean of EMA, compute CUSUM as cumulative sum of deviations from mean, and output arrays of t, EMA, and CUSUM for subsequent tasks

python

Compute second differences of the CUSUM series, identify sign changes, filter changepoints to enforce a minimum spacing of 10, count significant changepoints, and output the list of changepoint indices and the count

python

Based on the number of significant changepoints, fit either a single linear regression to EMA or a piecewise linear regression using the top 3 changepoint indices as segment boundaries, generate predictions for all t, and compute RMSE between EMA and predictions, finally output RMSE, number of changepoints, and the sum of RMSE and count rounded to 4 decimal places

Performance Metrics

Overall

0.85

Evaluation: This plan received an overall quality score of 0.85 based on effectiveness, task independence, and completeness.

Plan 2

Tasks

knowledge

Lookup numpy and scipy functions suitable for EMA, CUSUM, regression, and RMSE calculations to guide primary tasks

reasoning

Analyze the problem structure, identify key variables (t, y, EMA, CUSUM, changepoints, regression model), and outline the step-by-step computational workflow

python

Generate the time series, compute EMA, mean, and CUSUM as in plan A, ensuring correct indexing and data types

python

Detect changepoints by computing second differences of CUSUM, applying sign-change detection, enforcing a 10-point minimum spacing, and output the changepoint indices and count

python

Fit the appropriate regression model (single or piecewise) based on changepoint count, produce predictions, calculate RMSE, and return RMSE + changepoint count rounded to 4 decimal places

Performance Metrics

Overall

0.78

Evaluation: This plan received an overall quality score of 0.78 based on effectiveness, task independence, and completeness.

Selected Plan

Task	Tool	Query
1a	knowledge	Confirm EMA formula and alpha calculation for span=5 (alpha=2/(5+1)) and verify piecewise time series definitions for t<20, 20<=t<40, t>=40
1b	python	Generate deterministic time series y(t) for t=0..59 using the given piecewise formulas, compute EMA with span=5, calculate mean of EMA, compute CUSUM as cumulative sum of deviations from mean, and output arrays of t, EMA, and CUSUM for subsequent tasks
1c	python	Compute second differences of the CUSUM series, identify sign changes, filter changepoints to enforce a minimum spacing of 10, count significant changepoints, and output the list of changepoint indices and the count
1d	python	Based on the number of significant changepoints, fit either a single linear regression to EMA or a piecewise linear regression using the top 3 changepoint indices as segment boundaries, generate predictions for all t, and compute RMSE between EMA and predictions, finally output RMSE, number of changepoints, and the sum of RMSE and count rounded to 4 decimal places

Task 1a

Tool

knowledge

Query

Confirm EMA formula and alpha calculation for span=5 (alpha=2/(5+1)) and verify piecewise time series definitions for t<20, 20<=t<40, t>=40

Result

The exponential moving average (EMA) is defined recursively as EMA_t = alpha * y_t + (1 - alpha) * EMA_{t-1}. For a span (S) of 5, the smoothing factor alpha is calculated as alpha = 2 / (S + 1) = 2 / (5 + 1) = 2/6 ≈ 0.3333. This matches the value given in the problem statement.

The piecewise deterministic time‑series definitions are correctly transcribed:
• For t < 20: y = 2t + 5 + 2sin(0.5t)
• For 20 ≤ t < 40: y = -t + 65 + 3cos(0.3t)
• For t ≥ 40: y = 0.5t + 10 + 1.5*sin(0.7t)

These expressions preserve the original coefficients and argument scaling (e.g., sin(t*0.5) becomes sin(0.5t) and the amplitude 2 is explicitly shown). The formulas are ready for direct implementation in any numerical language.

Key practical notes:

When initializing the EMA, the first value EMA_0 is often set to the first observation y_0 to avoid bias.
For span‑based alpha, the effective window length is approximately 2*span – 1, so a span of 5 gives a responsive EMA that reacts within about 9 data points.
The piecewise functions are continuous at the boundaries (t=20 and t=40) if evaluated numerically, but the derivative is discontinuous, which is why a changepoint detection algorithm is appropriate.

These details directly support the subsequent EMA calculation, CUSUM analysis, and regression fitting required in the overall problem.

Auto-Scrolling

Task 1b

Tool

python

Query

Methodology

Extracted Data

Summary:

FORM B - Series data generated programmatically. All required arrays (t_series, y_series, ema_series, cusum_series) are created via list comprehensions and loops using the piecewise definitions provided in the query. No external data files are needed. The code snippets in relevant_values include the exact formulas and the EMA recursion with alpha=2/6. The mean of the EMA (mean_ema) is computed to serve as the baseline for the CUSUM calculation. The access method for each variable is direct assignment; no parsing is required.

Auto-Scrolling

Values:

t series = list(range(60)) # time indices 0-59
import math
y series = [2t + 5 + 2math.sin(0.5t) if t < 20 else -t + 65 + 3math.cos(0.3t) if t < 40 else 0.5t + 10 + 1.5math.sin(0.7t) for t in t series] # deterministic time‑series values
alpha = 2/6 # EMA smoothing factor for span=5
ema series = []
for i, y in enumerate(y series):
if i == 0:
ema series.append(y) # initialize EMA with first observation
else:
ema series.append(alpha * y + (1 - alpha) * ema series[-1])
mean ema = sum(ema series) / len(ema series) # mean of EMA values
cusum series = []
cum = 0
for ema in ema series:
cum += ema - mean ema
cusum series.append(cum)

Auto-Scrolling

Suggested Approach

Approach: Generate deterministic time series, compute EMA, CUSUM, detect changepoints, fit regression, compute RMSE

Methods:

piecewise linear regression
EMA via exponential weighted mean
CUSUM changepoint detection
RMSE calculation

Formulas: y_t = 2*t+5+2*np.sin(0.5*t) if t<20, y_t = -t+65+3*np.cos(0.3*t) if 20<=t<40, y_t = 0.5*t+10+1.5*np.sin(0.7*t) if t>=40 (+3 more)

Process: 7 steps — 1. Create array t from 0 to 59 and compute y_t using piecewise formulas

Libraries: numpy, pandas, statsmodels

Recommended Functions: pandas.Series, pandas.Series.ewm, numpy.cumsum, numpy.mean, statsmodels.regression.linear_model.OLS (+2 more)

Code

import math
import numpy as np

# Time indices
t_series = list(range(60))

# Deterministic time‑series values
y_series = [
    2*t + 5 + 2*math.sin(0.5*t) if t < 20 else
    -t + 65 + 3*math.cos(0.3*t) if t < 40 else
    0.5*t + 10 + 1.5*math.sin(0.7*t)
    for t in t_series
]

# EMA smoothing factor for span=5
alpha = 2/6

# EMA series construction
ema_series = []
for i, y in enumerate(y_series):
    if i == 0:
        ema_series.append(y)
    else:
        ema_series.append(alpha * y + (1 - alpha) * ema_series[-1])

# Mean of EMA
mean_ema = sum(ema_series) / len(ema_series)

# CUSUM series
cusum_series = []
cum = 0
for ema in ema_series:
    cum += ema - mean_ema
    cusum_series.append(cum)

# Second differences of CUSUM
second_diff = []
for t in range(2, len(cusum_series)):
    val = cusum_series[t] - 2 * cusum_series[t-1] + cusum_series[t-2]
    second_diff.append(val)

# Sign change detection for candidate changepoints
candidate_changepoints = []
for t in range(1, len(second_diff)):
    if second_diff[t] * second_diff[t-1] < 0:
        candidate_changepoints.append(t + 2)  # adjust index to original time

# Minimum spacing filter (>=10)
changepoints = []
last = -10
for idx in candidate_changepoints:
    if idx - last >= 10:
        changepoints.append(idx)
        last = idx

n_changepoints = len(changepoints)

# Regression fitting
if n_changepoints > 2:
    # Use first three changepoints as boundaries
    boundaries = changepoints[:3]
    segments = []
    start = 0
    for b in boundaries:
        segments.append((start, b))
        start = b + 1
    segments.append((start, len(t_series)-1))
    y_pred = []
    for (s, e) in segments:
        t_seg = np.array(t_series[s:e+1])
        ema_seg = np.array(ema_series[s:e+1])
        slope, intercept = np.polyfit(t_seg, ema_seg, 1)
        y_pred.extend(slope * t_seg + intercept)
else:
    slope, intercept = np.polyfit(t_series, ema_series, 1)
    y_pred = slope * np.array(t_series) + intercept

# RMSE calculation
y_pred = np.array(y_pred)
rmse = np.sqrt(np.sum((y_pred - np.array(ema_series))**2) / len(ema_series))

# Final result
final_result = round(rmse + n_changepoints, 4)
print(final_result)

Auto-Scrolling

Result

4.0788

Task 1c

Tool

python

Query

Methodology

Extracted Data

Summary:

FORM B - Series data generated programmatically. The list y_series contains 60 float values computed from the piecewise definition provided in the query and extracted from the analysis history (Task 0b). No external parsing is required; the expression can be evaluated directly in Python. The series is indexed by t from 0 to 59.

Missing data: None.

Access method: Direct assignment as shown in relevant_values.

No additional data structures are needed for the current task.

Values:

y series = [2t + 5 + 2math.sin(0.5t) if t < 20 else -t + 65 + 3math.cos(0.3t) if t < 40 else 0.5t + 10 + 1.5math.sin(0.7t) for t in range(60)] # deterministic 60‑point series

Suggested Approach

Approach: CUSUM changepoint detection via second difference sign changes

Methods:

compute EMA with span 5
calculate CUSUM of EMA deviations from mean
compute second differences of CUSUM
identify sign changes as candidate changepoints
filter changepoints to enforce minimum spacing of 10
count significant changepoints

Auto-Scrolling

Formulas: ema_t = (2/(5+1))*y_t + (1-2/(5+1))*ema_{t-1}, cusum_t = cusum_{t-1} + (ema_t - mean(ema)), second_diff_t = cusum_t - 2*cusum_{t-1} + cusum_{t-2} (+1 more)

Process: 7 steps — 1. Generate raw y series of 60 points from the piecewise definition.

Data Transform: Requirements: 1 item

Libraries: numpy

Recommended Functions: numpy.cumsum, numpy.diff, numpy.sign, numpy.where

Code

Auto-Scrolling

Result

[23, 40, 51] 3

Task 1d

Tool

python

Query

Methodology

Extracted Data

Summary:

FORM B - Series data generated programmatically. The EMA series (ema_series) and time indices (t_series) are created via list comprehensions and loops using the piecewise definitions provided in the query and extracted from analysis history (Task 0b). The list of significant changepoints after minimum spacing filtering is stored in filtered_changepoints, and its count is n_changepoints. All variables are directly assignable in Python; no parsing or file I/O is required. The EMA values are already computed and ready for regression fitting. The data_summary includes the exact code snippets for t_series and ema_series, and notes that the changepoint list and count are available as filtered_changepoints and n_changepoints.

Auto-Scrolling

Values:

t series = list(range(60)) # time indices 0-59
ema series = [ # EMA values computed from the deterministic series using alpha=2/6
7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5,
7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5,
7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5]

Auto-Scrolling

Suggested Approach

Approach: Piecewise linear regression on EMA with CUSUM changepoint detection

Methods:

Exponential moving average (span=5)
CUSUM cumulative sum of deviations from mean
Second difference sign change detection with minimum spacing
Piecewise linear regression using top 3 changepoints
RMSE calculation

Formulas: ema_t = (1-alpha)*ema_{t-1} + alpha*y_t, alpha = 2/(span+1), cusum_t = cusum_{t-1} + (ema_t - mean(ema)) (+3 more)

Process: 8 steps — 1. Generate the deterministic time series y_t for t=0..59 using the piecewise definition.

Libraries: numpy, pandas, scipy, statsmodels

Recommended Functions: numpy.arange, numpy.sin, numpy.cos, pandas.Series, pandas.Series.ewm (+7 more)

Code

import math
import numpy as np

# Time indices
t_series = list(range(60))

# Deterministic time‑series values
y_series = [2*t + 5 + 2*math.sin(0.5*t) if t < 20 else -t + 65 + 3*math.cos(0.3*t) if t < 40 else 0.5*t + 10 + 1.5*math.sin(0.7*t) for t in t_series]

# EMA smoothing factor for span=5
alpha = 2/6

# EMA series construction
ema_series = []
for i, y in enumerate(y_series):
    if i == 0:
        ema_series.append(y)
    else:
        ema_series.append(alpha * y + (1 - alpha) * ema_series[-1])

# Mean of EMA
mean_ema = sum(ema_series) / len(ema_series)

# CUSUM series
cusum_series = []
cum = 0
for ema in ema_series:
    cum += ema - mean_ema
    cusum_series.append(cum)

# Second differences of CUSUM
second_diff = []
for t in range(2, len(cusum_series)):
    val = cusum_series[t] - 2 * cusum_series[t-1] + cusum_series[t-2]
    second_diff.append(val)

# Sign change detection
candidate_changepoints = []
for idx in range(1, len(second_diff)):
    if second_diff[idx] * second_diff[idx-1] < 0:
        candidate_changepoints.append(idx + 2)  # map back to original index

# Minimum spacing filter (10)
filtered_changepoints = []
last = -10
for cp in candidate_changepoints:
    if cp - last >= 10:
        filtered_changepoints.append(cp)
        last = cp

# Number of significant changepoints
n_changepoints = len(filtered_changepoints)

# Regression fitting
if n_changepoints > 2:
    # Use first three changepoints as boundaries
    b1, b2, b3 = filtered_changepoints[:3]
    segments = [
        (0, b1),
        (b1+1, b2),
        (b2+1, b3),
        (b3+1, 59)
    ]
    y_pred = []
    for start, end in segments:
        idx = np.arange(start, end+1)
        coeffs = np.polyfit(idx, np.array(ema_series)[start:end+1], 1)
        y_pred.extend(coeffs[0] * idx + coeffs[1])
    y_pred = np.array(y_pred)
else:
    coeffs = np.polyfit(np.array(t_series), np.array(ema_series), 1)
    y_pred = coeffs[0] * np.array(t_series) + coeffs[1]

# RMSE calculation
rmse = np.sqrt(np.mean((y_pred - np.array(ema_series))**2))

# Final result
final_result = round(rmse + n_changepoints, 4)
print(final_result)

Auto-Scrolling

Result

4.0788

Iteration Evaluation

Decision

Analysis Complete

Decision Status

Analysis Complete

Quality Assessment

The iteration produced a complete, consistent result with all required calculations performed and verified.

Decision Rationale

All components of the query have been computed: EMA, CUSUM, changepoint detection, regression fitting, RMSE, and final sum. No missing steps or unresolved issues remain.

Synthesis

Result

The required value is 4.0788.

Auto-Scrolling

Key Findings

The EMA was computed with span 5 (α = 2/6) and the CUSUM of its deviations from the mean was used to locate changepoints via second‑difference sign changes. After enforcing a minimum spacing of 10 points, three significant changepoints were identified at indices 23, 40 and 51. Because more than two changepoints were found, a piecewise linear regression was fitted using the first three changepoints as segment boundaries. The resulting root‑mean‑square error between the EMA and the fitted predictions is 1.0788, and adding the three changepoints gives the final rounded result of 4.0788.

Auto-Scrolling

Final Answer

Result

4.0788

Answer Type

float

Focused Answer

Answer Type

Numeric Value

Selected Answer

4.08

Token Usage Summary
Model	openai/gpt-oss-20b
API Calls Made	27
Token Breakdown
Input Tokens	193,553
Cached Tokens	37,376
Output Tokens	13,562
Reasoning Tokens	1,765
Total Tokens	207,115

Cost Breakdown
Token Costs
Input Cost	$0.0117
Cached Cost	$0.0014
Output Cost	$0.0041
Reasoning Cost	$0.0005
Total Estimated Cost	$0.0172

Piecewise Linear Modeling of EMA‑CUSUM Changepoints in Building Floor Sensor Data

EMA Trends Reveal Three Shift Points, Guiding Piecewise Fit That Drives Final Accuracy Assessment

Question

Context

Summary Output

Final Answer

Focused Answer

Plan 1

Tasks

Performance Metrics

Plan 2

Tasks

Performance Metrics